Control Theory of Distributed Parameter Systems and Applications: Proceedings of the IFIP WG 7.2 Working Conference Shanghai, China, May 6â€“9, 1990

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner

IVlPl 159

X. Li, J. Yong (Eds.)

IJ

Control Theory of Distributed Parameter Systems and Applications Proceedings of the IFIP WG 7.2 Working Conference Shanghai, China, May 6-9, 1990

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Series Editors M. Thoma • A. Wyner

Advisory Board L. D. Davisson • A. G..I. MacFarlane. H. Kwakernaak .1. L Massey • Ya Z. Tsypkin • A. J. Viterbi

Editors Xunjing Li Jiongmin Yong

Dept. of Mathematics Fudan University Shanghai 200433 China

ISBN 3-540-53894-1 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-53894-1 Spdnger-Vedag NewYork Berlin Heidelberg

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FORWORD

The IFIP-TC7 Conference on Control Theory of Distributed Parameter Systems and Applications was held at Fudan University, Shanghai, China on May 6-9, 1990. More than thirty scholars from seven countries attended the meeting. There were five invited talks and about thirty contributed talks. This proceeding gethers most papers presented at the conference. The topics of this conference involve~ the following areas of distributed parameter systems: optimal control, identification, stability, numerical optimization t stochastic control, etc.

We would like to express our thanks to the following organizations which sponsored this conference:

State Education Commission of the People's Republic of China National Science Fundation of China International Federation for Information Processing (IFIP) Fudan University Institute of System Sciences, Chinese Academy of Sciences

We also would like to extend our gratitude to all the authors for their real intercsts in the conference and all members of the local organizing committee for their suggestions and supports. Our thanks also go to Professor I. Lasiccka, the Chairman of the IFIP-TC7, for her consistent helps in organizing the meeting and our colleagues at Fudan University for their cooperation which made the meeting really happen.

Xunjing Li and Jiongmin Yong

Department of Mathematics bSadan University Shanghai China

The IFIP-TC7 International Program Committee:

A. Bermudez, Fac. De Cienciaa, Santiago de Compostelo, Spain A. Butkovski, Control Institute, Moscow g. Curtain, Univ. of Groningen, Netherlands G. Da Prato, Scoula Normale, Piss, Italy R. Glowinski, INRIA, Paris, France K. Hoffman, Univ. of Augsburg, Germany G. Krabs, Teclmiache Hochschule, Darmstadt, Germany A. Kurzhanskij, IIASA, Laxenburg, Austria I. Lasiecka (Chairman), Univ. of Virginia, USA J. L. Lions, College de France and CNES, Paris, France U. Mo~co, Univ. of Rome, Rome, Italy O. Pironne~u, INRIA, Paris, Fra~lce P. Yvon, INttIA, Paris, France J. P. Zoleaio, Univ. de Nice, Nice, I~ance

The Local Organizing Committee:

Dexing Feng, Institute of System Science, Chinese Academy of Sciences Guangyuan Huang, Shandong Univ. Xunjing Li (Chairman), Fud~n Univ. Yongzai Lu, Zhejiang Univ. Laixiang Sun, Phdma Univ. Jingyuan Yu, Beijing Institute of Information and Control

LIST OF PARTICIPANTS

Banks, H.T. Center for Applied Mathematical Sciences, DRB-306, University of South California, Los Angeles, CA 90089-1113, USA

Butkovskiy, A.G. --Imtitutc of Control Sciences, Moscow, USSR Caffarelli, G.V.

., Dipartimento di Matematica, Univerita Degli Studi di Trento, 38050 Povo (Trento), Italy

Chavent, G --INI~IA, Domaine de Voluceau, Rocquencourt, B.P.105, 78153 Le Chesnay Cedex,

France Chen, Shuping

Department of Mathematics, Zhejiang University, Hangzhou, China Deng, Shaomei

Nanjing Insititute of Hydrology Ministry of Water Conservancy, Nanjing, Jiangsu 210024, China

Le Direct, F.-X. Department of Applied Mathematics, University Blaise Pascal Clermont-Ferm~d, B.P. 45-63170 Aubiere, France

Gao, Hang ---Department of Mathematics, Northeastern Normal University, Changchun, Jilin

130024, China Gao, Lin ---Inst i tute of Population Research, Chinese People's University, Beijing 100872, China Huang, Shaoyun ..... Department of Mathematics, Beijing University, Bcijing 100871, China Huang, Yu

Department of Mathematics, Zhongshan University, Guangzhou 510275, China Kappel, F.

Institut fur Mathematik, Karl-Franzens-Universitat Graz, A-8010 Graz, Elisabeth- strasse 16, Austria

Li, Chengzhi --Kiamcn University, Xiamen, Ft~jian 361005, China Li, Ping

Beijing College of Technology, Beijing, China Li, Xunjing - - D e p a r t m e n t of Mathematics, Fuadn University, Shanghai 200433, China Lu, Yongzai

Reesearch Institute of Industrial Process Control, Zhejiang University, Hangzhou, China

V!

Luce, R ......... Department of Applied Mathematics, University of Technology of Compiegne, 60206

Compiegne, Fiance Nakagiri, Shin-ichi ----Department of Applied Mathematics, Faculty of Engineering, Kobe University, Kobe,

Nada 657, Japan Pan, Liping

,, Institute of Mathematics, FUdan University, Shanghai 200433, China Peng, Shige

" Del~rtment of Mathematics, Shandong Univemity, Jinan, Shandong 250100, China Sakawa, Yoshyuki ~ D e p a r t m e n t of Control Engineering, Faculty of Engineering Sciences, Osaka Univ.,

Toyonaka, Osaka, Japan Simon, Jacques -----Department of Applied Mathematics, Univeristy Blaise Pascal Clermont-Ferrand, B.P.

45-63170 Aubiere, France Situ, Rong ...... Department of Mathematics, Zhongshan University, Guangzhou 510275, China Song, Wen ~ D e p a r t m e n t of Mathematics, Harbin Normal University, Harbin 150080, China Sun, Haiwei

.... Department of Mathematics, Zhongshan University, Guangzhou 510275, China Wang, Miansen

-Department of Mathematics, Xi'an Jiaotong University, Xi'an, Sanxi 710049, China Wang, Yun

Center for Applied Mathematical Sciences, DP~-306, University of South CMifornia, Los Angeles, CA 90089-1113, USA

Wang, Yuwen Department of Mathematics, Harbin Normal University, Harbin 150080, Clfina

Wu, Jingbo Department of Computer & System Sciences, Nankai University, Tianjin 300071, China

Xu, Yanqing Department of Computer & System Sciences, Nankai University, Tianjin 300071, China

Yong, 3iongmin ......... Department of Mathematics, F'adan University, Shanghai 200433, China Zhang, Weitao

Institute of System Sciences, Academies Sinica, Beijing 100080, China Zhao, Yahweh

,, Department of Mathematics, Shandong University, Jinan, Shandong 2,50100, China Zhou, Hongxin

..... Department of Mathematics, Shandong University, Jinan, Shandong 250100, China

CONTENTS

Methods and models to design mobile controls on surface A. G. Butkovskiy, V. A. Kubyshikin and V. I. Finyagina .......................... 1

A Geometrical theory for nonlineax least squares problems G. Chavent ...................................................................... 14

Dome.in variation for drag in Stokes flow J. Simon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

The existence of solutions to the infinite dimensional algebraic Riccati equations with indefinite coefficients

Shuping Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Optimal control for data assimilation in meteorology

F.-X. Le Dimet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 On the stability of open population large scale system

Hang Gao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Temperature control system of heat exchangers ~ a n application of DPS theory

Guangyuan Huang et al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Robust stabilization and finite dimensional controler design about a class of distributed parameter systems

Shun-ju Hu and Yian-Qin Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7

The asymptotic regulator design for nonlinear flexible structures with arbitrary constant disturbances

Chengzi Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Optimal control for infinite dimensional systems

Xunjing Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Numerical resolution of ill posed problems

R. Lucc and J. P. Kern~vcz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 6

Controllability and indentifiability for linear time-delay systems in Hilbert space

S. Nakagiri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 A generalized Hamflton-Jacobi-Bellman equation

Shi~e Peng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Dynamics and control of bending and torsional vibrations of flexible beams

Yoshiyuki Sakawa and Zheng Hun Luo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Strong solutions and optimal control for stochastic differential eautions in duals of nuclear spaces

B.ong Situ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Some new results on approximate controllability for semilinear systems

H. W. Sun and Y. Zhao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Vl l l

Optimal control for a class of systems a.ud its applications in the power factor optimization of the nuclear reactor

Miansen Wang, Zhifeng Kuang and Guangtian Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Single input controllability for spectral systems in Banach spaces

Jingbo Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Distributed parameter systems with measure controls

Jiongmin Yong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 The existence and the uniqueness of optimal control of population evolution systems

Jingyuan Yu, Ling Gao v~nd Guangtian Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Reachability for a class of nonlinear distributed systerns governed by parabolic variational inequalities

Y. Zhao~ Y. Huang and W. L. Chan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Analysis of the boundary singularity of a singular optimal control problem

Wei-Tao Zhang and De-Xing Feng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Analysis of the parabolic control system with a pulse-width modulated sampler

Hong Xing Zhou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

~TODS AND MODELS TO DESIGN MOBILE CONTROLS ON SURFACE

A.G.Butkovskiy, V.A.I{ubyshkin. V.I.Finyagina

Institute of Control Sciences

ProfsoJt~naJa 65, 1173~5 Moscow, USSR

The obJectes with a mobile heat source which periodically varies its position along an assigned traJecto~- on an object stu-face ~re considered. The object state is described by two-dimentional heat transfer equation. The problem to obta~ and maintain an object state closed to the assigned one is stated. Two t2~es of models are used to solve the problem. These are stationary models with distributed control and nonstatlonar~, ones in which heat solace movement is taken into account. The calculation method of controls making use of above two types of models has been developed. The paper contains the calculation examples of the source movement laws along the linewise object st~face trajectories, power of the mobile solace, 4b~amics of temperature field and grafical result representations.

~. INTRODUCTION

The systems with mobile sot~-ce, such as electronic, ion or laser

beams, possess some feattu~es complicating their modelling, desi~ and

analysis [i ]. The main of them are nonlinearity of controls and fast

movement of a so[uTce with respect to an object.

At present some publications highlight the developments on the choice

of models and designing of mobile controls [I-3]. They present in

sufficient details the investigations of the cases with

one-dimensional approximation of real objects. However in practice a

mobile power so~oe is most frequently surfacing the object along a

cLmvilinear trajectory. In this case one-dimensional models are too

warse approximation to be applied in practice.

2

The paper is concerned with the choice and validation of models of

objects with mobile action as well as the design of methods and

algorithms for calculating the source movement laws along a trajectory

on the object surface.

2. PROBLEM STATEMENT

Object whose state Q{x I .x2.t) is described by heat-transfer equ~tlons

(nonlinear in general case) with mobile heat source are considered

fa2Q ~Tt = a (Q)4~x2 a-~.2J - q(Q) + F(Xl'X2't)' (Xl'X2)ED' t>O. ( 2 . 1 )

c~x 2 1

Q(x I,x2,0) = Qo{Xl,X2), (x 1.x 2} E D,

Q + '~cT~nJ ix I "~2 )~i"

( 2 . 2 )

( 2 . 3 )

Here t is time. x=Cxl,x 2) is a spatial coordinate, D is a bo~uuded

domain of object determining, r is a domaln bo~u~dary, a(Q) is a

coefficient of thermal conductivity, G, ~. are constant coefficients,

Qo[Xl,X2) is an assigned function, ~Q/c~n is a derivative of the

external normal direction to r, Qp is an assi&nqed n~nuber, q(Q) is a

nonnegative ftuqction, determining heat removal from the object

surface, FCx I ,x2) is a mobile heat source having the form of

-

Here u(t) is power of heat source, ~5[xl,x 2) Is an assigned source

power distribution on the object relative to Its centre, ~(x I ,x2)~O, 00

II ~(Xl,X2)dxldx2 = I, -CO

(usually ~-'[x 1.x 2) has the form of the Gat~sian distribltion).

3

s(t)=(s1(t), s2(t)) is a position of the source centre in the assigned

domain G(D.

Assume that in the domah~ G th~ trajectory ~ with the length S is

assigned by parametzical equations x1=X1(Sl), x2=X2(Sl) and as the

parameter s I we have chosen the traJector~j arc length measured from

the arc beginning - the point 1 ° to its end - in the point 1 I, The

sotu-ce centre moves along the trajectory from 1 ° to 11 and inversely

periodically with the period T. Then the law of source movement is

fully determined by the trajectory equations X1(s l) and X2(s l) and by

the position of the source centre Sl(t) on the trajectory in each

moment of time

x (sl t l' x2(Sl(tll] '

In genegal form the control problem is stated as follows. A desired

object state Q*(x I 0x 2), (x 1,x 2)4G is assigned. It is required to find

the trajectory- ~cG, the law of a periodical source movement along the

traJecto~D~ s l(t) and the source power u(t) with the constraints

O~u~),~Um~ ~. which provide the object state Q(xS,x2,t) in the steady

mode (with t~00), whose deviation from an assigned state is minimal or

it does not exeed an acceptable value 8. The me~sure of such a

deviation can be assumed, for example, as the functional

~ = = ~ IQ'CXl,X2) Q(Xl,X2,t)] 2 - dx I dx 2 • t([tl +T] G

t1-~0

3. STATIONARY MODEL OF AN OBJECT

The problem solution in a general form is complicated by the fact that

over and above the object equation nonlinearity the controls X1(Sl),

X2(Sl), sl(t) entering in the equation are nonlinear too. Therefore

one has to t~e simple models for the problem solution.

4

It is ~uuown that ~u~der the condition of periodical heat source

movement along the traJecto~-~ an object state also becomes a

periodical function of time with t*00 i.e. the condition Q(x I 0x2,t+T] :

Q(X1,x2,t) is fulfilled. Then the object state can be represented in

the form of a sum of two componentes: an averaged one Q(xl,x 2) and m~

ocsillating one Qk[Xl,x2,t]. It can be sho,~au [3] that .~m ocsillating

component tends to zero with T~O under the norm L2(D). Then, with t*00

the object equation (2.1)-[2.3) can be approximatel~ ~ replaced b~:

a corresponding stationar V equation for the averaged component

Q[x I , x 2 )

2-

with botuuda~¢ conditions (2.2] and [2.3).

In equation [3.1 ) Q[x I ,x 2] is the averaged object state:

t+T

Q(Xl,X 2) : 1'I Q(Xl,X2,~)d%

t When t*~ it does not depend on time due to periodicity of Q(x I ,x2,t).

The averaged control F(x I ,x 2) has the form of

t+T

t

S

0

[3 .2 )

F ] It is implemented in (3.2) the substitution ~ = s,rC|Sl(t) | t(sl],

&. J

t[s I) is a function inverse to sl(t). Thus in a stead~- mode an

5

averaged object state can be dete~ined by means of a statlonarymodel

(3.1) , (2 .2) , [2 ,3 ) .

For a stationaz'y model the control problem is transformed as

follows. Find the source power O<~u[t)~<Umz~x , the trajectory IX1(Sl),

X2[Sl) } = = X(Sl) c G and the source centre movement law sl(t) alon~

the trajectory- ~ on the object surface. All of them are the controls

which b77 virtue of equation (3.1) provide a minimal value of the

func t ional

= O,*(Xl,X 2) - Q(Xl,X 21 dXld.X 2 [3.3) G

4. METHODS FOR CALCULATING THE TWO-DI~[ENSIONAL MOBILE CONTROLS

To solve the problems it is stk~ested to decompose it into two simpler

ones. I. To find the distributed control F*(xl,x P)_ which being

substituted into equation (3.1) for F(xl,x 2) of the form (3.2)

provides a minimal value or the one not exceeding the value 6 of the

functional (3.3). 2. To find the controls X~(sl). X2[Sl), sl(t], u(t)

which provide a minimal value or the one not exceeding an acceptable

value 6 of the functional

: ~F*[Xl ,X 2) - F[Xl,X2~L2(G ) (4,1)

Solution of the first problem by traditional methods (the principle of

maximum, dynamical programming, etc.) proves to be difficult due to

two-dimentionallty of the problem and nonlinearity of the object

equation. Therefore to solve the problem we propose to use the

substitution method.

The method is to substitude the chosen in a special way function

QA(Xl.X2), (x1°x2)~D which approximates an assigned state and the

following, requirements must be satisfied: a) it has to be a second

piecewise-continuot~ derivative; b) it has to satisfy botmdary

conditions (2.3); c) being substituted in equation (3.1) for Q(Xl,X 2)

it has to determine the function F*(xl,x 2) satisfying the constraints

F*(xl,x2)~0, I F*(xl,x2)dXldX2~Um~ z in domain G and it has to be equal G

to zero beyond G.

This method is rather vistmlized, it makes possibility to find

acceptable solutions for practice by simple and ~mvieldy computations,

and moreover the nonlinearity of equations slightly affects the

solution process.

The ftmction QA(Xl,X2) is conveniently assigned in the form of three

3

f~ctions "sewn" together QA(Xl,X 2) = ~ Qi(xl,x2),~i(xl,x2) where

I=I ~i(Xl,X2), i=1,203 are characteristic functions of the domains

respectively ~I=\G, ~2=FIE\~I (rlis a G-domain boundary, tie is the

E-nighbourhood of rl) and ~3=G\02 .

The function Q1(x1,x2) is assigned so that F*CXl,X 2) will be equal

to zero in the domain ~I' It is determined as the solution of equation

(3.1) with boundar~ r condltio~ (2.2) on the boLmdary r and ~1(Xl,X2),

(xl,x2)Er I on the boundary/ r I . The function ~1(x1.x 2) is assigned

following from physical considerations and updated in the course oZ

calculations.

In the domain ~3 the function Q3(xl,x2) either coincides with

Q*(Xl.X 2) or approximates it. Approximation becomes necessary when the

conditions a) and c) are not observed.

7

The domain ;]2 is ~ e d for transition from the function QI to the

function Q3' The kno~n~ methods of two-dimentional interpretation [~]

can be t~ed to assign Q2(Xl,X2). The value of the domain ~3 is chosen

so that in matching the constraints on F*(Xl,X 2) are satisfied.

Now consider in more details the problem of implementing the

distributed control by a mobile one. Let be assigned the distributed

control F*(x10x2). It is required to find the trajectory XI {Sl),

X2(Sl), the source movement law s~(t) along the trajectory and source

power u(t) all of which provide a minimal value of the functional

( 3 , 3 ) ,

In the sta~ed problem there are tl~ree controls (with the fixed source

form ~(Xl,X2)): the trajectory X1(Sl), X2(Sl), the sot~rce movement law

sl(t) along the ~raJectory and sotu~ce power u(t). It is known [3] that

with the fixed trajectory ~ the wider class of averaged controls

F(xl,x 2) can be obtained by the movement control sl(t) with the fixed

power u. That is why hereinafter the power u is to be referred to as a

constant value which is to be determined.

In some cases it can be additionally assigned that a source movement

occurs periodically at a constant speed in each period. Then, the

problem is stated to determln the trajectory ~ of a minimal lenght S

with which the (3.3) functional value does not exceed an assigned

admissible value O.

In other cases it is more reasonable to specify the source trajectory.

Then the problem consists in finding the power u and the source

movement law sl(t) which provide a minimal value or not exceeding in a

a@missable value of functional (4.1).

Consider the problem solution for the widely used in practice case,

namely for the l~lewise so~ce movement trajectory in a rectangular

domain.

X 2 , ~

b 2

l~I ~

0

' * ' \ I " , I " , l,,

,,,, - - - - ~

~ 2 . . . . . . . . . . ~i . . . . . . .

~, x I

Fig. I

Let in the rectangular object surface domain G=[O,b~]x[O,b 2] a mobile

source moves along the trajectory ~ shown in Fig, l. In this case the

source movement law in each llne can be calculated by the approximated

formulae which are derived from the assumption that a heat source is

located on a sufficiently smaller area compared with the total area of

the domain G [2] (a heat source is closer to a point one) and the

number of lines is sufficiently large. These formulae are

S

~i(s) = I F*(l~i'x2)dx2' 0~s~b 2, i=1,2 . . . . . N, 0

(4 .2) N

ti(s) = %i(s),T/ ~ %i(b2) ÷ ti(b2), i=I,2 ..... N, to(b2)=O,

i=I

Here I]i is a position of each line along the coordinate x I , N is the

total nL~mber of lines, T is the time of source movement along the

whole traJector;: from point 10 to point 11 (sotu~ce movement period),

Ti(s) is an intermediate variable, ti(s), O~s~b 2 is the function

being inverse to the source movement law si(t) in each line, O~<si~<b 2 ,

ti_1(b2)~<t~<ti(b2). It is assumed that the source centre moves from

line to line (from the point ~i-I to the point I~i, Fig.1 ) in the

infinitesimal time.

9

Implementation accuracy of distributed control by mobile one with

application of the above formulae improves if the line number is

increased and if the effective source dimension is decreased (the

source dimension is being approached to the point one).

5. CALCULATION FXANPLES OF TWO-DI~[ENSIONALMOBILE CONTROLS

Example,, I I . C ~ I c ~ a t { o r ~ of the d ( s ~ r { b ~ t e d c o n t r o l F ~ ( ¢ 1 . ¢ 2 ) .

The object is a rectangular plate D=(O,bl)X[O,b2), Fig. 1. The source

is moving along linewise trajectory. The assigned state is

Q*(Xl,X 2) : C = co~st., (Xl,X2)(D, q(Q) = O(Q 4- Q~).

Here G is a constant coefficient. Domains D and G coincide. Thus the

doma~t ~I is not available while domain ~3 is a rectangle

[ 8 , b l - S ] x [ S , b 2 - S ] .

Find the distributed control by the substitution method. The function

QA(Xl,X2) consists of two parts QA(Xl,X2) = Q2~2(Xl,X2)÷Q3~3(x1,x2).

Here Q3(Xl,X2)=C=co~s~., (Xl,X2)(0 3 and Q2(Xl,X2) serves for matching

with the boundary- conditions and it is represented as a product

Q2(Xl,X2)=~1(Xl),~2(x 2) where

k,(x I E) 2 + I, if O ~< x I ~< 8, j2

~I 2(xi 2 ) = CI I, if E ~< x I 2 ~< bi,2 - 8, ' ' I 2 '

[-k,[Xl,2- (bi,2 - S)] + I, if bl~2- S ~< xi,2 ~ b 1,2.

(x I , .x2)(~ 2 (5. I }

The coefficient k is chosen from the matching conditions between the

ftmction QA(XI'X2) and its derivative on the botuudary r I. In

particular for boundar~£ conditions (2.3) this coefficient is

10

k= ;t(S) 2 ¥ 2(I£

Fig.2 presents the approximating function QA(Xl,X2) and Fig.3 shows

the distributed control corresponding with this ftmction which has

been calculated by the substitution method.

Example 2. The object is the same as in F~:ample I. The assigned state

Q*(xl,x 2) has the form shown in Fig.4. The approximating function

shown in Fig.5 is obtained by smoothing the discontinuities of the

ftmction Q*(xl,x 2) in the domain ~3 and matching with the botundar-j

conditions of the function Q2(xl,x2) in the domain %" The

corresponding with this problem statement distributed control

calculated by substituting the approximating f~mction in the object

equation is sho~ In Fig.6.

Example 3, Imp~err,,e~tat{o~ o f ca lcu la te& &$st#{bate~ con t ro l s .

The sot~ce movement laws calculated by the approximated formulae (4.2)

for the ~inewi~e trajectory- are sho~vn in Figs,7 and 8. These laws

provide the distributed controls (Figs.3 and 6) described in Examples

I and 2.

6. TWO-Di~TIONAL NONSTATIONARY MODEL WITH MOBILE CONTROL

Calculations by approximated formulae require to be checked on digital

models taking into accotmt the sotn~ce movement. That is why the

two-dimentional model of a rectangular plate being heated by a mobile

heat source has been designed. The method of finite differences is

used to calculate the temperatture field making tu~e of equations

(2.1)-(2.3). The finite difference equations were formed by means of a

mesh with co,~$tant steps along spatial coordinates and a variable step

in time matching with the source movement law along the trajectory.

11

The trajectory and the movement law along the trajectory were

specified separetely.

The model takes much computer time due to a fast heat sotLrce movement

and. consequently, smaller mesh step size along the time coordinate.

As a result a computi~Ig time becomes unacceptable when is needed to

attain the stea<v mode. Therefore the temperature field is calculated

in two stage. In the first stage it is calculated the temerature field

of an object with averaged action (3.2). In approaching the steady

mode action (3.2) is replaced by mobile action (2.4) and a temperature

field pattern is finally determined.

The designed temperatL~e field is used for determining field

pulsations either in time or along statial coordinates. Fig.9 shows

the temperatute field in a steady mode. It has been calculated by the

model tunder the source movement law shown in Fig.7.

7. CONCLUSION

Two ripe of models are proposed to compute mobile controls on st~faoe:

I) stationary models with control which represents an action averaged

for the period of source movement or a distributed control neither

connected with the source movement; 2) nonstationary (full) models

taking into account the soL~ce movement.

The mobile control calculation consists of the following stages:

a) calculation of a distributed control making use of the stationary

model; b) calculation of the source power and the sot~ce movement law

along the trajectory which provide an averaged mobile sot~ce action.

i.e. the action which is close to the distributed control: c) checking

on a nonstationar~- model deviation of the obtained state from the

calculated one. the temperature variations in space and time. updating

the trajectory and the choice of a sotwse movement period.

1 2

r°(x)

. 4

',. ~ . 6 "' i X / I

\ • ",. P i g . B

13

The distributed control is calculated by substituting a specially

chosen function approximating an speslfied object state in the object

equation. An approximating function is performed by means of "sewing"

the ftu~ctions on the domains Joint where an action equals zero and an

obtained object state is determined.

This method allows us to find the controls for an object described by

nonlinear equations and it is visualized and easy to follow.

REFERJ~CES

I. Butkovskiy, A.G, and PL~stil'nikov, L.M. Mobile Control Distributed

Parameter Systems. (Ellis Hor~'ood Limited Publishers, Chichester,

1 987 ) .

2. Butkovskiy, A.G., Kubyshkln, V.A., Smirnov, A.G.,

Tverdochlebov, E.S., Chubar~v, E.P. A Method for Computing and

Implementing Distributed Control Signals. (Preprlnts IFAC 9-th

World Confess, vol. IX, 1984).

3. Breger, A.M., Butkovskiy, A.G., Kubyshkin, V.A. and Utkin, V.I.

Sliding ~{odes in Control of Distributed Plants Subjected to a

Mobile Multicycle Sigr~l. Automation and Remote Control, vol.41,

No.3. Part I, (1980).

4. Foux, I.D. and Prett, ~{.J. Computational Geomet~ for Design and

Manufacture. (Halsted Press: A Division of John Wiley & Sons, New

York - Chichester - Brisbane - Toronto). Ellis Horwood Ltd., 1979.

A GEOMETRICAL THEORY FOR NONLINEAR LEAST SQUARES PROBLEMS

Guy Chavent

CEREMADE, University of Paris-Dauphine,

75775 Paris C&lex 16, FRANCE

and

INRIA, Domaine de Voluceau-Rocquencourt

BP 105, 78153 Le Chesnay C6dex, FRANCE

Sllrrlmar'v

1 - Introduction

2 - Strictly quasieonvex sets

3 - Size ,, curvature conditions

4 - Application to Q-wellposedness of non-linear least squares

1 - Introduction

The motivation for tile geometricM tools described in this paper is the study of the non linear least

squares problem :

find x~ C s.t J(x) = |kp(x)-zli2 F = Min over C (1.1)

where

C is a convex subset of some linear vector space (set of admissible parameters)

F is an Hilbert space (data space)

q~ : C ~ F is a given mapping (parameter ~ outpu0 (1.2)

zE F is a given point (experimentld data).

We have given in (1.2) the interpretation of problem (1.1) in the context of parameter estimation in

PDEs or ODEs, but problems like (1.I) arise in various other areas, as approximation theory, control

theory, optimum design, inverse problems.

Problem (1. I) has been extensively studied when tp is linear (see for example reference [ 11). When

q~ is non linear, which is most generally the case in the above mentionned contexts, very few results

,are available : of course, compactness techniques allow in certain cases to proove the existence of a

solution x to problem (1.1), but give no indication on uniqueness of x, its continuous dependance on

the data z, or the absence of parasitic local minima. This last property is of great practical importance

when it comes to the numerical resolution of (1.1) on a computer, as it will guarantee that gradient

technique will be able to find the global minimunl ~,, and will not stop in some local minimum of J.

That is why we include this property (absence of local minimum) in our definition of well-posedness :

15

Definition 1,1. Let

d(x,y) = distance on C

be given. The non-linear least squares problem (1.1) is said to be Quadratically-well-posed

(Q-well-posed in short) on some open neighborhood ~ of q~(C) for the distance d on C iff:

i) for any ze ~ , (1.1) has a unique solutiort

ii) for any z~ ~ , J has no parasitic local minimum

iii) for zc 'It, any minimizing sequence xn converges to ~ for the distance d

iv) the mapping z-.ox is locally Lipschitz continuous from (~0", 11 II F) to (C,d).

(1.3)

We describe in tl~is paper sufficient conditions for problem (1.1) to be Q-well-posed on some

cylindrical neighborhood of cp(C), with explicit formula for the size of the neighborhood and for the

Lipschitz constant of z-.ox mapping.

Of course, such result requires that one is able to project z onto cp(C) in a unique and stable way.

That is why we shall devote paragraph 2 to the definition of strictly quasiconvex sets D of F, which

have this nice property for all z of some neighborhood 'Lt of D. Then we shall give in paragraph 3 two

sufficient conditions (size ,, curvature conditions) to recognize strictly quasiconvex sets, based on

geometric properties of some family of paths defined on D. Finally we shall apply in paragraph 4 the

above size ,, curvature conditions to D=cp(C) and obtain sufficient conditions for the Q-wellposedness

of problem (1.I).

The Q-wellposedness result has been used in reference [2] for the study of a plane wave detection

problem, in reference [3] for the estimation of a space-varying diffusion coefficient in a

one-dimensional elliptic equation and in reference [6] for the determination of the regularization

parameter when problem (1.1) is ill posed.

We present ill this paper only the main steps of tile construction, and refer the reader to references

[4] and [5] for more detailed discussion and all proofs.

2 - Strictly qnasicor~ve~.~e~s

We are givcn in this paragraph

F = Hilbert space (2.1) D C F

and we want to find conditions on D so that the projection on D is nicely behaved on some

neighborhood 'lY of D. The idea is to mimic the theory of projection on convex sets by replacing the

convex set by D and the segments of the convex by paths of D :

Definition 2.1. (path of D)

A mapping P:[0,L]~D is a path ofD iff:

16

v---~P(v) is in W2.*'([0,L])

IIP'(v)[[I:=I for a.e. ve [0,L]

(2.2)

(2.3)

Definition 2.2. (attributes of a path)

Let a path P be given. Then :

v~ [0,1.] is the arc-length along P

~5(P)~ L is the length of P

v ( v ~ P ' ( v ) is the unit tangent vector to P at P(v)

a(v~P"(v) is the accelleration vector to P at P(v)

p(v)- -a lla(v)lf 4 ~ I R + u {+~,) is the radius of curvature of P at P(v)

R(P)_ -a Inf Ess p(v) = Ilall~ > 0 is smallest radius of curvature along P 3'e 10./l(P)]

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

We have now to chose, among all possible paths of D, the ones which are going to play for D the

same role segments play for a convex ; we need for that :

Definition 2.3. (collection of paths)

A set !l ~ of paths is a collection of paths for 12. iff :

iP is made of paths of D

is complete, i.e. VX,Ye D, X~Y, 3Pe P such that P(0)=X, P(8(P))=Y

is stable with respect to restriction, i.e. VPe P , Vv',v"e [0,5(P)], v'<v",

the path P : vE [0,v"-v']---)P(v'+v) belongs to P .

(2.1o) (2A1) (2.12)

Of course, if D is convex, the set made of all segments of D is a collection of paths for D I So we

shall suppose in lh~ sequel thatone collection of paths ~ for the set D ba~ been chosen. This will be

emphasized by writing (D,~) instead of D (for example, when D=(p(C), one can take for ~P, provided

q) satisfies some conditions, the set of the images by (p of the segments of the convex set C as we shall

see in paragraph 4).

As we are going to study the projection of a point z~F onto D, the paths P joining two points of D

located at a distance of z slightly larger than d(z,D) will play a crucial role. Hence we define, for any

z~ F and rl>0 :

• '(z,'q) = {Pa ~'1 llP(j)-zll _<d(z,D)+rl, j=0,(5(P) }. (2.13)

The key number for the definition of quasiconvex sets is then defined by ;

k(z,rl) = Sup Sup Ess < z-P(v),a(v) >F (2.I4) l'~(z,rt) ve [0,~5(P)l

We can now give :

Definition 2.4, (quasiconvex and strictly quasiconvex sets).

• A set (D,IP) is tluasiconvex iff:

i) P is a collection of paths forD

17

ii) There exists a neighborhood ~ of D in F and a ~.s.c. function e:~/'-.--)]0,+~] such that :

ze 'U" k(z,al) < 1

0<r1<e(z)

• The set (D,P) is strictly quasiconycx if moreover :

z~ q J" the "distm'tce to z" function is

PG "P ::::, strictly quasiconvex

d(z,P) < d(z,D)+c(z) along the path P

(2.15)

(2.16)

The above definitions are quite technical, and we refer to reference [5] for a geon'tetrical

interpretation of k(z,rl). But the sufficient conditions to come in paragraph 3 will have a more natural

geometrical interpretation.

Proposition 2.1.

Let (D,F) be strictly quasiconvex. Then them exists a largest open neighborhood M of D, and a

largest t~.s.c, function ~:~---)]0,+ oo] satisfying (2.15), (2.t6),

The motivation for the introduction of strictly quasiconvex sets is the fact that on q./', the projection

onto D behaves as if D were convex :

Theorem 2.1. (projection on strictly quasiconvex sets)

Suppose that :

(D,~) is strictly quasiconvex (2.17)

Then :

i) uniqueness : for any ze 'If, there exists at most one projection ~( of z oll D.

ii) local minima. : if z~ 'If admits a (necessarily unique) projection ~( on D, the "distance to z"

function has no parasitic local minima on D distinct from ;k.

iii) continuity : if z0,zle ",3" admit projections ik0,~ t on D, and are close enough so that there exists

d>0 satisfying :

Ilzo-z ill+Max d(zj, n) < d < Min { d(zj,D)+e(zj) } (2.18) j=0,l j=0,1

then, for any path P going from X0 to Xt one has :

tl Xo-~21 II < ~(P) _< (l-k) -! tlz0-zlllr:. (2.19)

where k<l is defined by :

k = (k(zo, rl o)+k(z 1,'q i))/2 (2.20)

0 <rlj = d-d(zj,D) < ¢(zj) j=O,l

iv) gxistcnc.-e : if we suppose moreover that

D is closed in F (2.21)

then any ze ~ has a (unique) projection ~ on D, and any minimizing sequence X n satisfies

IIX~-XIIF--~0 and 5(Pn)--~0, where P,~ is any path of ~P going from X n to ~.

In the very simple case where D is an arc of circle of radius R and length L, one checks easily that

D is strictly quasiconvcx if and only if L/R < ~.

3 - Size .L.curvature conditions

18

We present in this paragraph two sufficient conditions for the strict quasiconvexity of a set D,

which yield cylindrical neighborhoods ~Y. We call them "size ,+ curvature conditions" because they

express the fact that in some sense the size of the set D is not too large with respect to its curvature,

i.e. the size ~ curvatme product of the set is not too large. The first condition is based on the notion of

global radius of curvature of a path, the second on the deflection of a path.

We begin the definition of the global radius of curvature, which was first introduced in [5].

Dcfinition 3. !

Let a t3ath P be given. Then, for any v,v'~ [0,~(P)], v+~v', we define the affine normal half space

N(v,v') to P at v seen from v' by :

N(v,v') = {z~ FI < z-P(v),~.v(v) >F -< 0 Vk~ IR, v+~.~ [Min(v,v'),Max(v,v')]} (3.1)

¢.r

N(v,v ') , - / \

~ . The normal half space at v to P seen from v', and the center of curvature C of P at v.

Of course, the center of curvature C of P at P(v), which is defined (cf.(2.8)) by :

C = P(v)+a(v) / Ita(v)ll

belongs to N(v,v') as < v(v),a(v) > = 0.

(3.2)

Definition 3.2 (Global radius of curvature)

Let a path P be given. Then, for any v,v 'e [0,5(P)], v~v', we define the global radigs of curvature

of P at v sgen from y' by :

19

PG (v,v') = d(P(v),N(v,v') n N(v',v)) E [0,+oo]

with the natural convention that PG (v,v')-- +,,,, if N(v,v') n N(v',v) = ~.

(3.3)

The global radius of curvature can be easily calculated :

Proposition 3.1.

Let Pe IP and v,v'e [0,8(P)], v~v', be given, and define :

N = Sgn(v'-v) < P(v')-P(v),v(v') >F (3.4)

D = (1- < v(v),v(v') >2)1/2 (3.5)

Then the global radius of curvature pG(vv') is given by :

p d v , v ' ) = if N > 0 and ~:v(v) ,v(v ' )>->0 (3.6)

N/D i f N > 0 and <v (v ) ) , v (v ' )>>0

The global radius of curvature is related, when v'--~v, to the usual radius of curvature p(v) :

Proposition 3.2

Let Pe ~ be given. Then, for almost all ve [0,5(P)] one has :

pc(v,v') --) p(v) when v'--4v

p~3(v',v) ~ p(v) when v'-.--)v

(3.7) (3.8>

We consider now tile worst possible case along a path :

Definition 3.3

Let Pe P be given. We define :

R d P ) = Inf p d(v,v'). vov'E lO.~i(p)l

(3.9)

Of course we see from proposition 3.2 that

RG(P ) ~ R(P) VPE ~P (3.11)

The motivatioa for the definition of the global radius of curvature comes from the following result :

Proposition 3.3

Let PE ~ and z~F be given. Then :

0 < d(z,P) < RG(P)

implies that the v -4 ItP(v)-zUt: function is strictly quasiconvex.

(3.13)

Coming back to ttle definition of strictly quasicoavex sets, we see that an immediate con~qucnce

of proposition 3.3 is the

Theorem 3,1 (R~-size~curvature condition)

Let (D,~P) be given.

If there exists R G > 0 such that

20

RG(P) > R o > 0 VPe P

Then :

(D,~) is strictly quasiconvex, with a cylindrical neighborhood off given by :

off = {z~Fld(z,D) < RG}

and an E(z) function defined, for any z6 off, by :

e(z) = RG-d(z,D) > 0

(3.]4)

(3.15)

(3.16)

This sufficient condition is constructive, as it gives simple explicit formul for off and e(z), an can

possibly be used as it is when a numerical estimate of R G is calculated on the the computer using

fomaulas of proposition 3.1. But this approach will often be unpracticable because of the huge amount

of computation required, and an analytical dctcmdnation of R G using these formula does not seem

very easy. Hence we develop now another (less precise) sufficient condition, which will be better

suited for analytical calculations, by searching for a lower bound Ro to Ro(P) in term of the deflection

of the path P :

Definition 3.4 (deflection along a path)

Let Pc P be given. For v,v'e [0,6(P)] the deflection of P between v and v' is :

0(v,v') = Arg cos < v(v),v(v') > ~ [0,rr],

and the largest deflection along P is :

0(P) = Max 0(v,v') . v.v'~ [0,6(P)I

(3.17)

(3.18)

The deflection can be estimated from the radius of curvature p(v) and the arc-length v using the

following result :

Theorem 3,2

Let Pc ~o be given. Then, for any v'~ [0,6(P)], ~0/~v(,,v')~ L~([0,~(P)], and :

so that : fa(e)

dv (9

(p) _< - / o p(v) Hence :

®(P) < ~(P)/R(P)

where lhe equality holds if and only if P is an at'c of circle.

(3.19)

(3.20)

(3.21)

Notice that the integrand in the right hand side of (3.20) is tile product of a size (arc length dr) by a

curvature (Up(v)), and so for (3.21). Hence any limitation on the deflection of a path implemented

using (3.20) or (3.21) is an actual size ,, curvature condition ! But paths with small enough deflection

should conceivably behave like segments (which ~tre zero deflection paths), or in other terms should

exhibit a strictly positive global radius of curvature. This is made precise by :

21

Theorem 3.3

Let P~ ~P be given. Then the following lower bounds for Ra(P) hold, depending on the deflection

O(P) of the path :

i) low deflection pa{h8

RG(P) = R(P) (3.22)

a s s o o n a s ."

0 < O(P) < x/2 (3.23)

ii) large deflection, p~IIls

Ra(P) _> R(P) sinO(P) + {~5(P)-R(P)O(P)} cosO(P)

a s S o o n as :

x/2 _< O(P) -< rr

(3.24)

(3.25)

We use now tile above result to obtain uniform lower bound on R G for all paths P of ~ .

Suppose we have found R,O,A such that :

R(P) ~ R > 0

O(P) < 0 < ~ for all Pc ~ .

A(P) _< A < + oo

(3.26)

Of course tile upper bound ® on tile deflection is supposed to be at least as precise as the always

valid upper bound resulting from (3.21), i.e. :

O -< zX/R. (3.27)

If 13 _< n/2 we find immediately from (3.22), (3.23) that Ro(P ) 2 R VP~ ~ , so that one can take :

R a = R (3.28)

in (3.14).

If x/2 < ® < n, we see from (3.26), (3.24) that :

RG(P) >_ R sin®(P) + {,~-RO(P)} cosO(P),

whose right-hand side is a decreasing function of O(P) on the [0,A/R] interval, so that :

RG(P) > R sin13 + {A.R13} cos@ "VPE T'.

Hence one can take :

R G = R sinO + [&-RO} cosO (3.29)

In order to reformulate (3.29) in a more convenient way, we notice that (3.27) can be rewritten in

an equivalent way :

13 = x A / R O~'c_< 1, (3.30)

where "t can be interpreted as follows :

, z=O means that all paths of ~o have a zero deflection, i.e. are segments, which implie~ that D is

convex t

• x=l means that the deflection of all paths have been estimated very roughly using the second

inequality of (3.20), i.e. as if all paths of P were arcs of circles.

Hence we shall call ~ the shaoe coefficient of the estimates R.13.A : x close to 1 means that either

the paths of ~ ,are close to arcs of circles, or tile estimates R,O,A are very loose ; x close to 0 means

22

that the paths of P are "close" to segments, i.e. that D is "close" to a convex (think of a piece of

granitated wall paper).

We can now rewrite (3.28), (3.29) using a: as :

I R 0_< ® < n / 2 (3.31)

We use now (3.31) to find conditions that will enure R G > 0. We have illustrated for that purpose

on figure 3.2 R G as a function of O for various values of'~. Clearly, if we dcfine:

e M = unique solution, in ]~t]2,rO, of the equation : tan0+{x -1-1 } 0---0 (3.32)

then :

R G > 0 as soon as O < @M. (3.33)

R G

x = 0

rr/2 OM n

deflection

®

FiKure 3.2 : The lower bound R G to global radius of curavaturc as function of the upperbound O to

deflection, for various values of the shape parameter x.

Hence @M aol~ears to be the..upper limit of deflections which yield a strictly positive R G. It is

noticeable that O M depends only on the shape coefficient "r of the estimates R,®,A : for x=l one has

OM=n, then ®M decreases quickly towards x/2 when 'r decreases towards zero (see figure 3.3).

Notice also that the function x---~ ®M is more easily expressed through its reciprocal :

'r = (1 - taa O M / O M )-l. (3.34)

23

1 . 0

0 . 7

{ ) . 0

10, 0

Th e t o MQ = /P l

J

........ ii

i t I t I '"'t i I " i I J 8 t I

0.0 0.3 0.7 1.0

l To,, ]

Figure 3,3 : The maximal deviat ion (gr~ as funct ion x.

[Oel t a/R~ ms,( ]

6, 7

3 . 3

0 . 0 I

0 . 0

i i " i j " ' i ~' t " i I l t I i

0.3 0 . 7 1..0

I -r~o ...)

Figure 3.4 : Tile maximal value (~v'R) M of/k/P, as funct ion of the shape parameter "~.

24

Aa equivalent way of writing (3.33) is obtained by imposing an ad-boc upper bound (,5/R) M on

A/R using the relation (3.30) between O and A/R. If we define :

(A/R)M = ®M/X = ®M - tan ®M (3.35)

then (3.33) is equivalent to :

R~ > 0 as soon as A ~ < (A/R) M, (3.36)

which is clearly a size ,, curvature condition. As shown on figure 3.4 (A/R)ta increases from rc to

infinity when I; decreases from 1 to 0.

Suppose that, without changing the estimates R and A, we refine the estimate ® on the deflection,

i.e. we diminish ®, and hence x : as we have seen, we expect a less constraining condition for R G >0,

which is clearly the case for (3.36) (as (,5/R) M increases), but seems apparently not true for (3.33) (as

®M decreases !). There is in fact no contradiction with the fact that (3.33) and (3.36) are equivalent :

when you refine your estimation O of deflection, its not surprising that even a smaller upper bound

~9 M may correspond to a less constraining situation !

We summarize these results in the

".Fheorem 3.4

Let (D,~) be given, let R,A,® and 1: be real numbers satisfying (3.26), (3.30), and define OM,

(A/R) M by (3.32), (3.35). If the 0-size ,, curvature condition :

O < O M (3.37)

or equiwdently :

,SiR < (~]R)M , (3.38)

holds, then R G defined by (3.3 t) satisfies :

R C > 0 (3.39)

and (D ,~ ) is strictly quasiconvex with ~ and ~ given by (3.15), (3.16).

One checks easily that theorems 3.1 and 3.4 recognize exactly all strictly quasiconvex arcs of

circles. More precisely, the Re-size ,, curvature condition of theorem 3.1 recognizes exactly all

quasiconvex curves, and the ~}-size ,, curvature condition of theorem 3.4 recognizes exactly all

strictly quasiconvex curves made of one arc of circle and one segment.

4 - Application to O-well posedness 9f non-linear least squares

We use in this paragraph the O-size ,, curvature condition of theorem 3.4 to study the

Q-well-posedness of the non-line.'lr least squares problem (I. 1), (1.2). We equip for that the set

D=cp(C) with the family of paths ~P made of the images by ¢p of the segments of the convex set C :

~to any x0,x le C we associate the path (4.1) \

~ P : t~ [0,1] --4 P(t) = ~(l-t)x0+tx 1)

and we suppose the q~ is regular enough so that :

P ~ W2.~(I0,11) VPe P (4.2)

2 5

Notice that (4.1), (4.2) alone does not imply that ~o is a collection of paths in the sense of definition

2.3, as t is not the arc-length v ! But we may still associate to any path P :

V(t) = P'(t) e F (velocity along the path) (4.3)

A(t) = P"(t) ~ F (accelleration "along the path)

which of course are different from v(v) and a(v) defined in proposition 2.2 (llV(t)ll F is not necessarily

equal to 1, and < V(t),A(t) > not necessarily zero !). The relation between V(t),A(t) and v(v),a(v) is

given, at any point where V(t) ~: O, by :

_ V( t ) v(v) - ITV(t)II~

(4.4) a(v)= A(t) _ v ( v ) < v ( v ) , A(t) >

IIV(t)ll~ IIV(t)ll~

which implies that :

IIv(v)IIF= 1 (not surprising !) (4.5)

IIA(t)IIF { V(t) A(t) _ 2~ 1/2 IIA(t)II._..~F

Ila(v)llv= IIV(t)ll~ 1- < ~ , IIA(t)IIF ~ ~ < IIV(t)ll~ "

We state now the main hypothesis on C and q~ : we suppose that one has been able to find a distance

d(x,y) on C such that :

(C,d) is complete (4.6)

there exists 0 < oc m < ~M such that :

Vx0,x 1 ~ C, for a.e. te]0A[ : (4.7)

c~ m d(xo,x t) < IIV(t)ll F < cz M d(xo,xl),

and :

there exists ® > 0 and R > 0 such that :

'v'x0,x 1 c C, for a.e. tel0,1[ : (4.8)

IIA(t)IIF / llV(t)ll1: < O, IIA(t)IIF / llV(t)ll2F < I/R.

Hypothesis (4.7) on V(t) implies that IIV(t)ll F ~: 0, Vt~ ]0,1[ for any path P of ~ , so that we see

from (4.4) that any path P can be reparametrized as a W2,'([0,8(P)]) function of the arc length v ;

hence ~ is now a collection of oaths in the sen.~e of definition 2,3.

By analogy with the case where the mapping q~ is twice differentiable, we shall say that ~,~ and cc M

,are lower and upper bounds to the singular values of q~'(x), x~ C, and that ~XM/Ct m is an upper bound

to the condition number of the linearized problems.

It is then easy to see using (2.8), (3.20) and (4.5) that O and R defined in (4.8) satisfy :

O(P) < O, R(P) > R > 0 VPe ~P. (4.9)

We can also define :

.4 = cc M diam C, (4.10)

which obviously satisfies :

6(P) -< A VPe ~o. (4.11)

In sight of (4.9), (4.11) and (3.21), we can suppose, without loss of generality, that (3.30) holds,

i.e. :

® = '~ A/R 0 < x -< 1. (4.12)

2 6

Then, if ® < @M defined in (3.32), all hypothesis of theorem 3.4 are satisfied, so that (gJ(C),~P) is

strictly quasiconvex. On the other hand, hypothesis (4.7), part iii) of theorem 2.1 and the

completeness of (C,d) imply the closedness of q0(C). Hence theorem 2.1 rewrites as :

Theorem 4.1

Let C,d,F and q0 be given satisfying (1.2), (4.2), (4.6), (4.7), (4.8), and A,z,R~,OM,(A/R)M be

defined by (4.10), (4.12), (3.31), (3.32), (3.35).

If the deflection size ,, curvature condition :

0 < ®M (4.13)

or equivalently :

Z~R < (A/R)M = O M - tan O M (4.14)

is satisfied, the non-linear least squares problem (1.1) is Q-well posed on the neighborhood :

= { z~ Fld(z,tp(C)) < R a } (4.15)

for the d(x,y) distance on C, and the following stability estimate holds :

°;md(Xl'X0) < f l UV(t)llFdt < (1-d/R)'l IIzI'Z01IF (4.16)

(where V(t) is the velocity along the path image by tp of [x0,xl])

as soon as z0,z t satisfy :

Itz i-zoll F + Max d(zj,q)(C)) < d < R o . (4.17) j=0A

A special case is given in :

Corollary 4.1.

Theorem 4.1 holds with hypotheis (4.8) replaced by :

there exists ~ _> 0 such that, Vxo,x t E C, for a.e. t a ]0,I[ : IIA(t)IIF < It d(xo,xl) 2

provided O and R and x are defined by :

O = (ll/~ln)diam C R = c~. 2/[~

"1; = am[ OtM,

and the size ~ curvature condition (4.13) or (4.14) rewrites :

(13]~m)diam C < (9 M

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

Notice that, when corollary 4.1 applies, "c is tile reciprocal of the (upper bound to) condition

number ct M/~,~ of linearized problems. Hcnce poorly conditioned linearized problems (o~fl ct m large)

will yictd a small x and hence (see figure 3.3) a O M very close (from above) to n/2. Hence, the

practical augmentation of the diam C caused by considering the case n/2 <- @ < r~ will be negligible as

soon as tile condition number aM/Oqn is larger than a few units.

Notice also that condition (4.22) can always be satisfied by reducing the size of C : theorem (4.1)

and corollary 4,1 yield a "local" non-linear inversion result, but with explicit expressions for the

localization.

27

References :

[1] Campbell S. and Meyer C., "Generalized Inverses of Linear Transformations", London : Pitman, 1979.

[2] Symes W.W., "The Plane Wave Detection Problei'n", Technical Report 90-1, Department of Mathematical Sciences, Rice Univerzity, Houston TX 7725 I, 1990.

[3] Chavent G., Kuniseh K., "L2-S lability of the Inverse ProbIem in a 1-D Elliptic Equation from an Ht-Observ:ltion '', (in preparation).

[4] Chavent G., "Quasiconvex Sets and Size ,, Curvature Condition, Application to Non-Linear Inversion", to appear in Journ,'tl of Applied Mathematics and Optimization, 1991.

[5] Chavent G., "New Size ,, Curvature Conditions for Strict Quasiconvexity of Sets", to appear in Siam Journ:d on Optimization and Control.

[6] Chavent G. "A New Sufficient Condition for tile Well-Posedness of Non-Linear Least Square Problems Arising in Identification and Control", in "Lectures Notes in Control and Information Sciences", Vol. 144, Analysis and Optimization of Systems, A. Bensoussan and J-L. Lions Ed., Springer, pp.452-463, 1990.

D O M A I N V A R I A T I O N F O R D R A G IN S T O K E S F L O W

Jacques S I M O N

C.N.R..S. a n d Univers i t~ Blaise Pasca l ( C l e r m o n t - F e r r a n d )

Dfipartement de MathSmatiques, Universitd Blaise Pascal, 63177 Anbiere Ccdex, France

I n t r o d u c t i o n .

The drag variations of a~ uniformly moving body in a Stokes flow, with respect to body variations, or more generally with respect to fluid domain variations, are investi- gated.

The variations of the initial fluid domain ~2 being represented by a vector field u, we are interested in an expansion of J (~ t+ u) with respect to u. We prove the existence of an indefinite expansion, aud we calculate the first and second order terms.

First variation value was calculated in [P], and for Navier-Stokes flow first variatioa was cMculated in [MS1] aa~d IS1] mad second one in [$2], using general rules for domain variations. However, full proofs were not given.

Proofs for similar problems were given given in [MS2] and [$1], based on usual implicit functions differentiation theorem. For fluid equations, assumptions of this theorem are not satisfied (see comment after (40) in section 6). Here, we give ma extension1 of this implicit function theorem, which allows us to carry on proofs.

Ou t l i ne s

1. Drag of a body in a Stokes flow 2. Drag variations 3. Velocity and pressure variations 4. Rough calculus of the derivatives 5. Differcntiabilty of an implicit equation solution 6. Proof of domain variation results 7. Application to drag minimization for given volume 8. Second variations 9. Rough calculus of second vmiations 10. hffinitc order diffcrcntiabilty, and second vm'iatlons proofs.

29

1. D R A G O F A B O D Y I N A S T O K E S F L O W .

We are interested in a motionless body/3 in a viscous incompressible fluid moving at a uniform velocity h. The flow is considered in a boundcd region A containing /3. Thus the fluid occupies the annulus shaped domain f~ = A \/3, whose boundary has an outside part 0A mad a.n inside part 0/3.

The velocity y = (yx, y2, ya) and the pressure p satisfy

- u A y + Vp = 0 and V.y = 0, in f~, (1)

0 on 0/3, y = h on cgA, ~ p d x = O , (2) Y

where the viscosity v is a positive real number, and h E IRa. The energy dissipated by the fluid is

J = ½ f~ ILy[ 2 dx (3)

where (Ly)ij = OiYi + OiYl. This energy is proportional to the drag, which is J/[h[. The velocity and then the

drag are uniquely defined by the following result.

T h e o r e m 1. Vv'e assume that

= A \-B, A is open and bounded, -B c A, OA and OB are Lip 1. (4)

Then, there exis~ a unique pair y E Hl(f~) 3, p E L2(I2), satisfying (I) and (2). Therefore (3) defines a unique J E JR.

Let us recall that tIk(~2) = {v l O W E L2(fl),O < ]¢x[ < k}. We say that c9f2 is Lip k if it is locally the graph of a Lipk(IR 2) function, where

Lipk(]R "~) = {v [D% is Lipschitz continuous on IR ~, 0 _< [o,[ < k - 1}.

We will use the following properties of a gencralized Stokes problem, which may bc found in [L] and [GR I.

T h e o r e m 2. (i) Assume that ~ sa, tistles (4), and let f E H - I ( ~ ) a, g E L2(~), k E HI (~ ) 3, r E IR, satisfy

~ gdx = f o , k.nd.s. (5)

Then, there exists a unique pair y E Hl(fl)3,p C L2(f2) such that

- t ~ A y + V p = f, V . y = g , y = k on Of~, ~ p d x = O . (6)

Conversely, if (6) has a solution, then (5) is satisfied. (ii) Assume ha addition that Of 2 is Lip 2 and f E L2(fl) 3, g E II~(f~), k e H'2(f~) 3.

tha~ Then y E H2(£/) 3, p E HI(~), and there exists a real e depending only on f~ such

1Mira + Ilvlln, _< 411fltL, + IMI>~, + Ilkll.~ +~).

30

l%emark. Theorem 1 is a particular case of theorem 2 since (5) yields foa h. nds = O. This is sat;isfied since foan ds = fa V1 dx = O.

2. D R A G V A R I A T I O N S .

We are interested in the variations of J(['/) with respcct to small variations of ~/. Let u be a vector field defined in all of 11-t.3 representing variations of ~. A varying domain is dcfined as

a + u = {x + u ( x ) I x e a} . (7)

We assume that u E L/pI(IP~3) 3, [lullLip, < 1. Then, u is a contraction thus f / + u satisfies (4). Therefore, by theorem 1, there exists a unique pair Yn+u E H I ( ~ / + u) a, pa+~ E LZ(f~ + u) such that

--uAya+u + Vpa+u = 0 and V . ya+u = 0 in f~ + u, (8)

y a + ~ = 0 on 0 B + u , y n + u = h on OA+u, [ p~+,,d:r=O. (9) a l l + u

Thcrcfore there exists a unique energy

J(f~ + u) = ½ [ d, . (10) J f l + u

We are looking for an expansion of J ( f / + u) with respect to small u. We will get an expansion for u in Lip ~, that is,

e(~ + u) = J(~) + J'(a; u) + (1~)

where o(t) denotes aa~y real number such that o(t)/t --+ 0 as t --* O. More precisely, the following result will be proved in section 6.

T h e o r e m 3. We assume Hint f~ satisfies (4) and Ofl is Lip 2. TlJen, the expansion (11) holds for all u E Lip 2 sud~ tlmt IlullLip < 1, with

fo Oya 2 = (12) -½

We denote by n = na a unitary vector field on 0f~, directed outside ft. And ~AIL = 11 • ?1.

IKemark . The expansion yields,

~ + , ]Lyn+,,l~ dx = ~ [Lynl2 d x - foau , , l~n l2 ds. (13)

r t e m a r k . In the case of a body in a fixcd experiment domain, the outside boundary is not supposed to vary. This is modelized by choosing u = 0 on A.

Thc varying body ft + u depends only on thc restriction u[aa. Howcvcr, wc usc a variation u(x) which is defined for all x E IR a since it provides a map, I + u, from f/ onto f / + u, which will be used for proofs.

31

R e m a r k . The intcgral in (12) is defined, since y 6 H2(~2) 3 thus ~ E L2(Oft) 3. In thc case where c9~2 is only Lip 1, it is not defined.

R e m a r k . As for any optimization problem, the first variation yields a necessary optimality condition. An example will be given in scction 7.

Moreover, based on j¢, we can use a gradient method to construct a locally optimal domain ~0.

R e m a r k . In particular, the Frechet differentiability, that is (11), yields variations depending on a small rcM parameter t. For a fixed u and for all 0 <_ t <_ t , , J ( f t + tu) is defined, and

J(f~ + tu) = J ( a ) + tJ ' (a ; u) + o(t). (15)

This is Gateaux differen~iability. However, as it will be scen in scction 7, it is not cnough for applications. This is the reason why we use Frcchct derivative.

3. V E L O C I T Y A N D P R E S S U R E V A R I A T I O N S .

3.1. Loca l v a r i a t i o n s y' ,p' . To get the variation J', wc will use thc varhttion y' of thc vclocity. Tha t is an expansion of the following typc,

y ~ + . = y~ + u ' (a; ~) + o(II~IIL,,,2)-

However y~+, lives in the domain ~2 + u which depends on u, but y9 ~md y'(ft; u) live on the fixcd domain ~. Thcrcforc this cxpansion cmmot be satisfied in "all of £/. It can1 be satisfied only in the intersection, for all smM1 cnough u, of the domains fl + u.

In the following result, such an expansion is given locally, that is in any strict subdomain w of f'/.

T h e o r e m 4. ~Vc assume that ~ satislies (4) and O~ is Lip 2.

(i) Given u E Lipl(]R3) 3, there exists a unique pair y' E H l ( ~ ) 3, p' E L2(~) satisfying

- - rAy ' + Vp' = 0 and V . y' = 0, in £~, (1G)

y' = - u , On on Oft, p' d~: = - u .pa d,s. (17)

Moreovc~, y' = y'(ft; u) and p' = p'(f~; u) are tiaear m~d continuous with respect to u, ~i-o~ Lip~(a~) ~ o n t o H~(a) ~ ~ d L~(a).

(ii) For any open set w such that "~ C a, and/'or all u E Lip 2 such that IlullL~., < 1,

y . + . = y~ + y ' (~; u) + o(ll,,llL~,,)

(ill) The variation J ' defined in thcorem 3 satisfies

in ~,(,,,)3, (Is)

in L2(w) 3. (19)

J/R fo u"ILYnl2 d~. J'(~2; u) = Lye. Ly'(~2; u) dx + ½ n (20)

32

(iv) For M1 u E Lip I,

~ Ly~ Ly'(~t; u)dx - jfon U,~]Lyet2 ds = - 2 foa Oyn ,2 • = u . I - N # j as . (21)

P r o o f o f p a r t s i and iv o f t h e o r e m 4. Here, we denote y -- y~, y' = y~(g/; u).

(i) To get the existence of a solution y',p' of (16)aand (17), it suffices to check the condition (5) of theorem 2, which here is foe u,~ n . -~, ds = O.

On each part of Of/, y is constant. Thus Vy = n~-~n. That is Oiyj -- n, 0Uion " Whence

the required condition is satisfied, since n . ~ = 2 i r t i • °o~ = ~ i O i ! l i = O. (iv) By definition, Ly . Ly' = ~ij(OiYj + Ojyi)(Oiy} + Ojyi) = 2 ~']ij(Oiyj + Ojyi)Oiy}. Thus, intcgrating by parts,

/ e Ly" Ly' dx = - 2 f~ ~iJ(OiOiyj + OiOjyi)y~ dx + 2 fon ~'2~iJniY)(OiYJ + OJYi) ds" (22)

By (1), ~ i Oiyi = 0 mad ~ i OiOiyj = ~Ojp. Thus the integral on ~2 in the right hand side is equal to

2 2 ~t 2 - - ; f , E i Ojp y) d~: = -; fa E l pOiy ~ ~l~ - -; f on E , PW Y;" dx = O,

since the integral on [2 is null by V . yt = 0, and since the integral on 0fl is null by = - u,,ni 0, = - ~ i = 0.

The integral on 0~2 in (22) is, since r~iy~ = --Unni~o, , = -u,,Oiyj, equal to

-2 ~ e un~ijOiYj( OiYj + OjYi) = - jfon unlLylZ ds,

which proves the first equality in (21).

Using again Oiyj = n i ~ and 2..,j ni o,, "= ~ j OjYJ = 0 , we get on OF/,

Oyj Oyi . 2

which proves the second equality in (21).

The parts ii and iii of theorem 4 are proved in section 6.3.

3.2. To t a l v a r i a t i o n s ~, i i. To get the boundary condition (17) on y', and to get the value (20) of J', we need a uniform dependence of ya+u on u up to the boundary O~2+u. This cannot be given by the expa~mion (18) which is necessary local. It is obtained by mapping ye+= on the fixed domain f2.

Denoting I the identity in IR 3, I + u maps f~ onto f / + u. Thus, tlm function ya+~ o ( I + u) lives in fL The uniform dependence is given by it's expaaxsion in all of f/. Thc following result will be proved in section 6.1.

33

T h e o r e m 5. We assume that ft satisfies (4) and 0~2 is L{p 2. Tl~en, for all u ~ L{p2(~3) 3, there exists ~(ft; u) e H~(f~) ~ a,,d ~;(n; u) ~ H i ( n ) ,

which are linear and continuous with respect to u, and which satisfy, for all u such that

fill[Lip, < 1,

v.+~ o (z + ~) = u~ + ~(ft; ~) + o(llultL~,,=) in H2(f~) 3,

p.+.o(I+~,)=p.+W4f~;,')+O(HL~p~) in H'(ft)

R.emark . The existence of a total variation ~ is necessary for our proof of thc existence of J~. However, knowing the exact value of 7) is not required.

To calculate the value of J ' , we will only use the value of thc local derivativc y~. In addition, a rough calculus of y~, pr and J~, carried on in section 4, is casy.

This is thc reason why we use at the same time the two different objects ~ and y~. In fact, they as-e related by

')(/3; u) = Y'(/3; u) + u ' V,Ju,

thus equations on ~, l) comc of cquations on y~,pq

4. R O U G H C A L C U L U S O F T H E D E I ~ I V A T I V E S .

Wc check here that J' has the value announced in thcorcm 3, when yf~+,,(x) and pa+u(x) smoothly depend on x and u. This assumption is not satisfied, and in section 6.1 we will give a proof bascd on rather differcnt ideas.

Wc differcntiatc with rcspcct to a r c a l parameter t, for a fixed u, since y~(~2; u) = ~ , J ( ~ + tu)l,=0.

4.1. D i f f e r e n t i a t i o n o f equa t ions . From (8) we get an equation satisfied in the domain ft + tu whidx vm'ics with t. Any point x fixed in ft, lies in ft + tu for small cnough t. Tlms,

-~Ay,+,~(z) + Vp~+,~(z) = 0 Vz e a,Vt < t~... (23)

By differcntiation with respect to t, at t = 0, wc gct the first equation in (16) :

- - v i , j ' ( x ) + V~,'(x) = 0 W e a .

Similarly, V-yf~+a,(x) = 0 yields the sccond equation in (16) :

V.y'(z) = 0 Vz ~ ft.

4.2. D i f f e r e n t i a t i o n of b o u n d a r y cond i t i ons . From (9) we get a boundary condition satisfied on the boundary O f / + tu which varies with t. For a givcn x in 0/3, the moving point x + tu lies in Ol3 + tu. Thus,

Va+t,,(x + tu) = 0 Vz e OB, Vt < t,,. (24)

By differcntiation with rcspcct to t, at t = 0, we get

u'(~) + u ( ~ ) . v , j ~ ( ~ ) = 0 w e a/3.

34

Since yf~ is null on 013, it's gradient at any point is proportional to the normal vector n. Thus Vyn(x) = nn(x) oa (x). Whence we get the boundary condition on 013 in (17) :

,j'(~) + u(~). ~ ( ~ ) ~ ( x ) = 0 vz E a~.

Similarly, the second boundary condition in (9) yields yn+,~(x) = h,Vx E OA, whence we get, on 0A and therefore on all of Off, the same condition.

4.3. D i f f e r en t i a t i on o f t he d rag . From (10), by the change of variable I + tu, wc get, Vt ~ t~,

J ( n + tu) = ½ L [Lyn+t,12 o (I + tu) l det[Oi( I + tu)i]l dx. (25)

Differentiating with respect to t at t = 0, and using ] det[...]] -- det[...] for small t and (det[...])'(0) = V.~,, we get

J'(a; u) = -~ ./o ((ILv[2)' dx + u. V(ILy~I 2) + [Lv~I2V • u) dx

f (Lv~.LV + ½V. (~tLvnl:)) g~.

This yields (20). By the formula (21), wc get the value of J ' stated in theorem 3. The integral condition on p in (17) is obtained by a similar calculation.

5. D I F F E R E N T I A B I L I T Y OF A N I M P L I C I T E Q U A T I O N S O L U T I O N .

To provc the cxistcncc of a total vaa'iation ~, and of higher ordcr total vari~tions, wc will use the following extension of the usual differcntiabilty result for implicit cqu~tdons solutions.

T h e o r e m 6. We give us - an open set 51 in a BanacJl space U, uo E U, two reflexive Bm~ach spaces A and B, - a map F :U x A e-~ B, such that F(u; .) E £ ( A ; B ) for a/I u E U, -a function ra : 51 ~-+ A, and a funcLion f : 5t ~ B , such ~hat

s (u , m(~)) = f(u) w E U.

(i) Assume that

u ~-~ F(u; .) is differentiable at uo into £(A; B), f is differentiable at uo, (26)

IIF(u0;x)ll. _> ~llxllA Vx e A, fox-some a > 0. (27)

Then, the map u ~-~ re(u) is diffcrentiable at uo. It's deriv~tive mr(u0; .) is the unique solution of

F(uo; m'(uo; v)) = f ' (uo; v) - 0~F(uo; m(uo); v) Vv E U. (28)

(ii) /n addition, assume ~ha~ for some integer k >_ 1,

u ~ F(u; .) and f are k times differen~iable a~ uo. (29)

35

T h e n , the m a p u ~ re (u ) is k times dif ferentiable at uo.

The new point here is that F(0, .) is not necessary a one to one map, since its range is not supposcd to be all of B. Whcncc, wc have to provc thc cxistcncc of a solution m'(uo; v) of (28).

P r o o f o f p a r t i.

B o u n d e d n e s s o f m. By (26),

IIF(u; x) - F(uo; z ) - O=F(uo; z; , , - uo)lln < II~tIA o(11~ - uollu), (3o)

thus

IIF(u; x) - F(uo; x)llB --< flllxllAII u - uollu. (31)

By cquation,

F(uo; re(u)) = (F(uo; re (u ) ) - F (u ; re(u))) + ( f (u) - f ( u o ) ) + F(uo; too), (32)

wherc ,no = m(uo). Therefore, ~llm(u)l lA _< ~llm(u)ll ,~llu - uol lv + "11~ - ~'ollu + I IZ(~o;mo) l l~ ,

In all the sequel, we assume that 11- - uollu _< @. Then,

llm(=)lt < 7 = ' ~ llF(=o; mo)tt~ + ~. (33)

C o n t i n u i t y o f m. By (32),

F ( u o ; m ( u ) - too) = ( F ( u o ; m ( u ) ) - F(u; re(u))) + ( f ( u ) - f ( u o ) ) , (34)

thus

~llm(u) - mollA _< (flllm(u)llA + r)ll ~ -- uolIu --< ( ~ + r)ll ~ -- uollu" (35)

W e a k d i f f e r en t i ab i l i t y o f m. Given a fixed v E U, wc denote u, = uo + tv, m , = m ( u t ) , and we assume t < to = ~/(2~ll ' l lv) . By (35), t m - 7( t - ' n o ) is bounded in A. Then, there exists li E A mid t , -~ 0 such that ~ ( m t , - too) --* i ~ weakly in A. By (32),

F(u0; re (u) - too) = r ( u ) - Ov, F(uo; m0; u - u0) + f ' (u0; u - uo), (36)

whcrc Y(II)

~ F ( . o ; . ~ 0 ; . - -0)) + ( S ( - ) - S ( - o ) - S'(-0;- - -o)). By (31) a n d (30), llr(,,)llzJ _< ~l lm(") - mollAllU - uollu + o(II~ - ,,ollu) = o(lt,, -

~ollu). In (36), we choose u = u t . , we divide by 6 , aad we let t,, --- 0. Then ,-(t,,) --~ 0,

aald thercfore at the limit, by weak continuity of F(u0; .), we get

F(~o; ~(~)) = - O . P ( ~ o ; too; o) + f ' (~o; o). (37)

36

D i f f e r en t i ab i l i t y o f m. By (27), (37) defines a unique #(v), which is linear and ¢ o . t i . . o . . ~ i t h respeCt to ~ By (36) and ( 3 % V(~0; m ( ~ ) - ~ 0 - , ( ~ - u0)) = ~(~). Thus,

llm(,~) - -~o - t~(u - uo)lla < ~ o(ll~ - ~ol)u).

P r o o f o f p a r t i i . Di f f e r en t l ab i l i t y o f m in a n e i g h b o u r h o o d o f u0. Now, wc assume that u ~-~ F(u; .) and f are differentiable in an1 open subset/g0 of /g contailfing u0. And we restrict u to Y =/g0 N {u I IIu - uo[Iu < ~-~}- By (27) and (31),

tlF(u;x)HD >- IIF(uo;x)tIB --liE(u; x) - F(uo;x)l[B >-- ~lI.~lia w e A.

Assumptions of part (1) are satisficd at any point u e Y. Thus, u ~-~ re(u) is diffcrentiablc in Y, aaad

F(u; m'(u; v)) = -O ,F(u ; re(u); v) + 1 % ; v), Vv e U, W E V. (38)

k -d i f f e r en t i ab i l l t y in Y. Now, wc assume that F(u; .) and f are h times diiferentiable in Ho. We define F1 : ]) x £(U; A) --* £(U; B) aa~d A : V ~-~ £(U; B) by

Fl (u ;y ) ( v )= F(u;y(v)) , f l (u)(v) = - O , F ( u ; m ( u ) i v ) + f f ( u ; v ) , Vv G U.

Then, (38) yields Fl(u;m'(u)) = f l (u) ,Vu E V. The map Fl satisfies (27), and F1 and f l azc k - 1 timcs diffcrcntiablc in 19. Thus, by prcvious step, m ' (u) is differentiablc in V.

Using k times the present step, wc get the k times differcntiability in ]).

k -d i f f e r en t i ab i l i t y a t u0. By (29), F aud f are k - 1 times diffcrcntiable in a neighbourhood L/0 of u0.

Then, by the previous step, u ~ re(u) is k - 1 times differentiable in a neighbourhood ]) of u0, and it's derivative satisfies Fk-l(u; m U'-I)) = f k - l (u ) in 1).

In addition, Fk-1 and fk-a are diiferentiable at u0. Then, by part (i), m (k-~) is diffcrcntiable at u0. That is the k-differentiability of m at u0.

6. P R O O F O F D O M A I N V A R I A T I O N R E S U L T S .

Here we prove theorems 3, 4 and 5 in three steps : -existcnce of eal unknown total variation ~), that is theorem 5. - existcnee of an unknown variation J* satisfying the expansion (11). - J ' has the announecd value, and yt is tile local variation

6.1 E x i s t e n c e o f a t o t a l va r i a t i on .

M a p p e d e q u a t i o n s . By thc map I T u , equations (8) and (9) satisfied by y~+,,, pfz+u yield equations satisfied by Y(u) = y~+,, o ( t + u), P(u) = pf~+u o ( I + u).

We denote [O,(I+u)y] the derivative matrix of the map I+u, M(u) = t"[Oi(I+u)j]-~ the transposed invcrse matrix, and Di(u) = ~ i j Adij(u)Oj. Using

(V f ) o ( I + u) = D(u) ( f o ( I + u)), (39)

we get

- , o ( u ) . (D(u)Y(u)) + D(u)Q(u) = 0, D(~). V(~) = 0, in ~,

37

Y'(u) = 0 on OB, Y ( u ) --- h on aA, Jn P(u) det[0~(_r + u)j l dx = 0.

By theorem 2, ya+~ e H:(f~ + u) a, pa+,, e H~(ft + u). Whence,

Y ( u ) e H2(12)3,P(u) E Hl ( f l ) .

To get homogeneous bound~y condition, wc consider 7t E C~_(1R3) 3 such that V . h = O, 7~ = 0 on 8B, h = h on 0A. Setting Z(u) = Y ( u ) - h , we get Z(u) E H~(a) ~ n HJ(a) ~.

Impl ic i t equa t ion . Wc define,

F : Lip:(It~a) ~ x H2(f~) 3 f~H01(a) a x H i ( a ) ~ L2(f~) a x Hl(~2) x ll~

F(~; z , p ) = (- ,DO,) . (D(~)Z) + D(~,)P ; D(~). Z ; [ P de@,(Z + ~J)l a~).

Then, the above equations yield

F(u; Z(u) , P(u) ) = - F ( u ; i~, 0). (40)

If F(0; .) could be a one to one map, then, by usu~ diffcrentiability properties of implicit equations solutions, we would get diffcrcntiabilty of u ~-4 (Z(u) , P(u)) . However, by condition (5) in theorem 2, the range of F(0; ") is not all the space. (Moreover, the rmlge of F(u; .) depends on u, then we cannot restrict the range at u to the range at 0). The differentiability will come of theorem 6.

Dif ferent iab i l i ty . Now, we check assumptions of theorem 6, for equation (40). The matrix inversion is differentiable into Lipl(lR3) ° = WI'~°(IR3) °, at any point

which is an invertible matrix. Then, tt ~ M ( u ) is differcntiable from Lip2(IFt3) a into l,Vl'°°(IK3) 9, at any u such that IlUllLip, < 1. Since F is linear with respect to (Z ,P) , quadratic with respect to M, and three-linear with respect to u by the determinant function, then u ~ F(u , .) is differcntiablc, at any such u, into £(Lip'z; L ~ x H L x IR). That is the first part of assumption (26).

In addition, this yields the differentiability of u ~ F(u; h; 0), which is the second l)art of(26).

At last, F(0; Z, P) = (f ; k; r) is the generalized Stokcs problem stated in theorem 2, with g = 0. Thus, assumption (27) follows from part ii of thcorcm 2.

All assumptions of theorem 6 being satisfied, u ~-* (Z(u), P(u) ) and therefore u H (Y (u ) , P (u ) ) m'e differcntiable from Lip2(lRa) 3 into H2(a) 3 x//~ (a), at any u ~uch that llullci~,, < 1. This proves the existence of ~ m~d 15, that is theorem 5.

6.2 Ex i s t ence o f a t o t a l va r i a t ion J ' . By ~heorem 3.3 in [$1], the existence of 7) implies the existence of J~ satisfying expansion ( l l ) . For sake of completness, we give the following proof.

In the right hand side of (10), wc use the change of variable i t+ u. Using (39), m~d det[...] > 0 ~inCe tI'ItL~' < X, we get

J(a + u) = / 2;ilD;(u)ig(u ) + D2(u)Y~(u)] 2 det[0;(I + u)a] dx. (41) da

38

By the differentiability of M, u ~ Di(u)Yj(u) + Dj(u)Yi(u) is differentiable from Lip2(]l-~3) a into HI(~/). Since the determinant map is nmltilinear, u ~-~ det[Oi(I + u)i ] is diffcrentiable into nipl(1R 3) = WI'°°(IRa). Thus, the intcgratcd function is diffcrcntiable with respect to u into W~'~(Q) (L ~ would bc enough here), at any poiut such that ]lu]}L~,, < 1. Therefore it's integral is differentiable, that is there exists J' satisfying expansion (11).

6.3 Value o f the der iva t ives . By Icmma 1.2 of IS1], the existence of total variations E H2(~) a and li E H~(Q), implies the existcnce of local variations y' E Hl( f l ) 3

and p' e L2(~) satisfying expansions (18) and (19). By theorems 3.1 and 3.2 of [$1], cquations (8) and (9) imply that y' and p' satisfy (16) ~md (17). This proves the part ii of thcorcm 4.

By theorem 3.3 of [$1], the vMue of J ' is givcn by (20). This proves the part iii of thcorem 4.

Duc to (21), the valuc of J ' is given by (12). This provcs theorem 3.

7. A P P L I C A T I O N TO DR.AG M I N I M I Z A T I O N FOR. G I V E N V O L U M E .

"VVc arc interested in the minimization of J for bodies/3 which have z~ given volume v, in a fixed cxpcrimcnt domain A. Wc ~sumc that 0A is Lip "~, and wc denote

7)aa = {B I ~ C A, az~ is Lip ~, fon ,t~ = ,,).

T h e o r e m 7. Assumc that ~here exisgs I3o such that

/30 e z)o., J(A \/30) <_ J(h \-9) V/3 e Z),,~.

Then~

cOy t¢ o is ¢onstmJt on 0/30. (42)

I~emark. This condition is satisficd by any possible/3o. Howcvcr, the existencc of such a/30 is not provcd, although it is cxpectcd.

P r o o f of t h e o r e m 7. Lct u satisfy

E Lip2(~3) 3, / u,,ds=O, u = 0 on 0A. (43) U

Jo 13o

As it will be proved in a moment, there exists a path U E C1(]R+; LipZ(I[~3) a) such that,

dU(t) =u, Jt~ dx=v, U(t)--O on c0A. (44) I,=o dt 0+u(t)

Then, for small t, 13o + U(t) E :Da~, thus J(B0 + U(t)) > J(13o). By the chain rule tlmorem,

a J(/30 + U(t))[,=0 = Jt(/3o;u) > O. dt

39

By (12) ~his yields, since u = 0 on A,

f0 u , l ~ t 2ds 0 for all such that c")rl,l_ < U

13o f0 u~, = ds O.

13o

Tlfis implies (42). It remains to check (44). The map u ~-* ft~+, dx = ft~ dct[0i(I + u)j] dx is multi-

linear, and thcrefore differentiable. It 's derivative a~ 0 is u ~-~ f~ V " u dx = fo. ~,, ds. Now, let u satisfy (43), let w e Lip2(]I~3) ~ satisfy fo6ow, ,ds ¢ O,w = 0 on OA,

and denote V(t , c) = fg+(t,+~o) dx. Then the implicit equation

v ( t , = >__ o,

has a solution t ~ c(t) which is C 1 in a neighborhood of O. Indeed V is C 1, mad ~--~Y(0,0) = foGw, ,ds ¢ O. The solution satisfies, ov °V (o O~c'(O~ -b-i-(O,O)+-ff/'~ , j t ~ = O. Since ~;,(0, O) = fob u,, = O, this yield, c'(O) = O.

Thus, U(t) = tu + c(t)w satisfies U'(0) = u and therefore (44).

ILemark. Should we proved only directional diffcrentiability, that is (15), wc would get J~(A \/30) k 0, only for v satisfying Bo + iv E 7?,a,Y/ < t , . This condition yields ft~o+,, dx = f130 dx ,V t < t , , which is satisfied only for constant v. Therefore we woukl obtain much less iifformation on/3o.

8. S E C O N D V A R I A T I O N S .

Wc arc looking for a second order expansion of J, that is

J(~2 -t- u) -- J(f/) + J '(ft ; u) + ½J"(12; v, u) + o((HuIILip2)~). (45)

The following result will be proved in section 10.2.

T h e o r e m 8. (i) ~¥e assume that f~ satisl~es (4) and Of~ i~ Lip "~. Then, there ex i~s a bilincar contimmus mai~ ,1"(~2;-,-) on Lit;2(IRff ) 3, sucfi tlmt

expmasio, J (45) holds for all u E Lip 2, II~IIL~,,, < L

(ii) Wc ~lSSUlne hi addition ~hat 0~2 is Lip 3. ThcJa,

where m~iquc y" E H~(f~) z m~d l/ ' E L2(~2) m-c deigned by

--uA~j" ÷ Vp" = 0 a.nd V . y" = O, hi f2, (4G)

on 0a, (47) y" + 2(u , V)y ' + (u . (u . V ) V ) y = O

r being a real depending linearly on a11 its arguments.

40

Moreover, y" = y ' ( f / ; u, u) and p" = p"(f~; u, u) are billnear and continuous with respect to u, froln Lip2(IRa) a into H1(ft) a and L2(~2). For any w such tl~at ~ 6 .f~,

ya+. = ya + y'(a; u) + y"(a; u) + o((ll llL, ,) in

pa+~ = pa + p ' (a ; u) +p"(f~;u,u) + o((llul[Li,,) 2) in H-l(w).

R e m a r k . As for first variations, proofs for second variations are based on uniform dependence on ya+u up to the varying boundary Oft + u. This comes of second order differentiability of ya+~ o ( I + u). In fact, in section 10 we will get infinite diffcrentiability order.

R e m a r k . The second variation J" yields a sufficient condition for local optimality. Let a domain f~0 in some class :D,d satisfy,

J ' ( a 0 ; u ) = 0 and g " ( a 0 ; u , u ) > 0 , VuClZ~,

where/A~ is the tangent cone at 0 in Lip 2 to {u [ f~0 d- u e :Dad}. Then, f~0 is locally minimal in Z),d.

Moreover, based on J" , the velocity of gradient methods may bc improved.

9. R O U G H C A L C U L U S O F S E C O N D V A R I A T I O N S .

E q u a t i o n s d i f f e ren t i a t ion . Twice differentiating equation (23) with respect to t, ~t t = 0, we get the first equation in (46) :

- v ( A y " + VV")(x) = O, W ~ ft.

Similarly, we get (V. y")(x) = O, Vx e a .

B o u n d a r y cond i t i on d i f fe ren t i a t ion . Denoting " the second order derivativc with respect to t at t = O, we have,

(yn+~, o ([ + tv))" = y" + 2(u. V ) J + (u. (~. V)V)y. (48)

Thus, twice differentiating boundary condition (24) with respect to t, at t = 0, we got, Y"(x) + 2(u(x)-V)y'(.~) + (u - (u . V)V)(yn)(x) = 0 for an x e 08. Similarly we get the same equation on A, and therefore on all of gt, that is boundary condition in (46).

D r a g d i f fe ren t i a t ion . Twice differentiating (25), we get

J"(f2; u, u) = I j f ([Lya+,, [2 o ( I + tv))" + 2([Lya+,, 12 o ( I + tv)) ' (V • U)

+ [Ly[ e ( (V. u) 2 - V u . '~Vu) dx,

since V-u and (V. u) 2 - Vu. t~Vu) are the first and second derivatives of the determinant function, and since [det[...][ = det[...1 for small t.

41

By (48), the right hand side is

((IL jI ) '' + v(ILylb' + v)v)(IL,JI

+ ' + u . V(ILyI ))(V- + In,A2 ((V. _ w - ' r w ) ) dx,

where R = (u. (u-V)V)(ILyl 2) + u . ~7(ILy]2)(V • u) + tLyl2((V, u) 2 - V u . t~Tu) . To get the announced value of J " , it remains to ctmck, and this is left to the reader,

that R = V. (u(u. V)(IL,jRI~)) + V. ((u(V- u) - (u- V)u)ILynl~).

10. I N F I N I T E O R D E R D I F F E R E N T I A B I L I T Y , A N D S E C O N D V A R I A - T I O N S P R O O F S ,

10.1. In f in i t e o r d e r d i f fe rent iab i l ty . Here, wc prove tha.t u ~ J ( f /+u) is indefinitely differcntiable. However we do not calculate high order derivatives since, ~m seen for second derivatives, length of formulas increases with differcntiability order.

T h e o r e m 9. We assume that 12 satisfies (4) and 0 ~ is Lip 2. Then, at any point u such that Hu]]L{r, < 1,

u ~ J(l~ + u) is indcmdte ly diffcrentiable from Lip2(IR~) a h2to IF~,

u ~-* Yu+, o ( I + u) is indctlnitcly diffcrcntiable f~'om Lip2(IRa) 3 into /:i2(~)3,

u ~-* pa+,, o ( I + u) is indefinitely differentia, hie from Lip2(IR.a) 3 into H~(f l) .

Proof . In section G.1, we got the differentiability of the map u ~-~ F(u; .) defining tile implicit equation (40). In fact, by a similar proof, this map is k times diffcrentiabIe for any given k.

It stdtlccs to remark that matrix inversion is k times diffcrcntiablc into W l'°°(f~)9, at any point which is an inertible matrix. Therefore, the map u H M ( u ) = t r [Oi ( I+u) j ] - i is k times diffcrcntiable from Lip2(]R3) 3 iuto W1'°°(~) 9, at any u such that ]]UllLi~,, < 1. In addition the determinant function being multilinear, u ~-~ de t [0 i ( /+ u)j] is k times differcntiable into IR.

In addition this yields the k diffcrentiability of u ~-~ F(u; 74 0). Now, we can use part ii of theorem 6, for implicit equation (40). Whcucc, u

(Z(u) , P (u ) ) and therefore u ~ (Y(u), P ( u ) ) = (yfl+,, o ( I + U),l,,~+,, o ( I + u ) )a rc k times ditfercntiablc from Lip2(IR3) a into H2(fl) 3 × HI(~) .

The k differentiability of u ~ J ( f / + u) follows by (41).

10.2. P r o o f o f t h e o r e m 8. By theorem 9, u ~ J( f t + it) is two thncs diffcrentiable, which is paz't (i) of theorem 8.

By theorem 3.3 of [$2], the second vm-iation f " of any two times diffcrcntiablc flmction u ~ f ( ~ + u) is related to the first va.riation by

f " ( a ; u , u ) = ( f ' ) ' ( a ; u, u) - f ' ( f l ; (u . ~7)u), (49)

42

Expansions of y, p. The existence of y" and p" satisfying second order expansions follows from the sccond order differen~iabitity of u ~-~ (yn+u o (I + u),pn+u 0 (I + u)), by using twice tcmma 1.2 in [St].

Equa t ion differentiat ion. By theorem 3.1 of [$1], (16) yields -u~(y ' ) ' + V(pt) ' = 0. Thus, by (49), --rAy" + Vp" = 0 in f~, which is the ilL'st equation in (46).

Similarly, we get thc second equation in (46), V- y" = 0 ill gL

B o u n d a r y condi t ion differentiat ion. The boundaxy condition in (17) may be written as y' + (u- V)y = 0 on a l l Thcrcforc, by thcorcm 3.2 of [$2], (y')'(~2; u, u)+ 2(u. V)y'(12; u) + (u. V)2y = 0 on 012. Thus, by (49), we get the boundary condition on y" in (47).

Drag differentiat ion. The equality (20) may bc written as

fn ' (uJLy[2)) dx. J'(~};u) = (Ly. Ly' + 2V .

By theorem 3.3 in [Sl], we get

Therefore, using (49) and integration, wc gct the announced vMue for J"(fl; u, u). To get (49), the present ~pplication of theorem 3.3 in [$2] requires u E Lip't(IR~) 3.

Therefore, the formulas axe proved for u E Lip 4. All the terms of these formubus being continuous with rcspcct to u in Lil ;'t, and Lip "I bcing dense into Lip 2, they hold for u E Lip ~.

References [GR] V. Girault & P.A. Raviart: Finite clement methods for Navicr-Stokes equations. Springer-Verlag, I986. [L] O.A. Ladyzhenskaya: The mathematical theory of viscous incompressible fluids. Gordon & Breach, 1963. [MS1] F. Murat & J. Simon: Quclqucs rSsul~a~s sur le controle p~u: un domaine gSomStri- quc. Research report, P~u'is 6 Univcrsi~y, 1974. [MS2] F. Murat J~ J. Simon: Stir le conf, role par un domainc g(}omdtriquc. II.esc~,rch report, Paris 6 Univcrsity, 1976. [P] O. Pironneau: Optimal shape design for elliptic systems. Springcr, 1983. [$1] J. Simon: Diffcrcntiation with respect to ~hc domain in boundary value problems. Numerical Functional Analysis &: Optimization, 2, pp. 649-687, 1980. [$2] J. Simon: Second variations for domain optimization problems. In: Control cstimation of distributed parameter systems. F. Kappcl, K. Kunish & W. Sc~t)l)achcr cds. Internationaa Series of NumericM Mathematics, 91, 1)I). 361-378, Birkhauscr, 1989.

The Existence of Solutions to the Infinite Dimensional Algebraic Riccatl Equations with Indefinite Coefficients *

Shuping Chen

Zhejiang University, Hangzhou, China

Abstract. Necessary and sufficient conditions arc established for the existence of selfadjoint solutions and positive definite solutions to the algebraic Riccati equations in Hilbert spaces with indefinite coefficients.

~1. Definitions, Notations and Formulation o f Main Results.

Let X, U, V and Y be separable Hilbert spaces. Let A be the infinitesimal generator of a Co-semigroup e at, t >_ 0, on X with dense domain ~(A). Let B E £(U,X), G E £(V,X) and C E £(X,Y) . Here, we use £(H1,H2) to denote the Banaeh space of linear bounded operators mapping from Hj into H2, endowed with usual operator norm, and denote £(1t) = £(H, H). The inner product in a Hilbert space H is denoted by (., ")u and the induced norm is denoted by I1" Ibr. The subscript H will be suppressed if it can be understood from the context.

In this paper, we consider the following algebraic Riccati equation

( A ~ ) s : (S~I, a ~ ) + (a~, Sz~)-(SQSz,, ~ ) + (msx~, ~ ) = O, (1.i) zl, z2 ~ ~(A),

where Q = C*C and Ms = BB* - 62GG *, and 6 > 0 is a parameter. Here, T* denotes the adjoint of the linear operator T.

It is well-known that the Riecati equations play an important role in the control theory for linear systems. In particular, (AR.E)o arises from the standard linear quadratic cost control problems (L-Q) and the filtering problems of infinite dimensions [1,3], and has been studied in some details by a number of authors (see [7] for example and the references cited therein). For L-Q problems with conflicting objectives one will encounter (ARF~)6 with 6 > 0. Moreover, there is a close connection between (At7~)6, 6 > 0, and the dual Reccati equation

(AR.E)~ : (Px,,axa) + (Ax , ,Px~)+(PMsPx, ,xa) - (Qxl,x2) = o, (1.2)

zl,z2 E I)(A).

Whereas (Altgb3)~ has applications to the two-person zero-sum differential games [5] and has recently been found crucial to the study of H~-optimal control problem via state-space approach ([2]).

The aim of the present paper is to establish conditions under which (aRE)s, 6 > O, have selfadjoint solutions and/or positove definite solutions.

* Project supproted by the NSF of China.

44

We shall use the following definitions and notational conventions throughout the paper. The generator A is called to be stable if the semigroup cAf is exponentially stable, i.e., there are positive constants m and oJ, such that

Ilea'll _< vt >_ o. (1.3)

The pair (A, B) (resp. (C, A)) is said to be stabilizable (resp. detectable) if there exists a K E £(X, U) (reap. g e L:(Y, X)), such that A + B K (resp. A + K C ) is stable. Unless otherwise stated, we shall always assume the stabilizability of (A,B) and the detectability of ((7, A). A selfadjoint operator T E f..(H) is said to be nonnegative, denoted by T > 0, if (Th, h) >_ O, Vh E H; T is said to positive definite, denoted by T > 0, if (Th, h) >_ ~lllhll 2, Vh E H, for some r/> 0 independent of h. In the latter case, T is boundedty invertible and T -1 > 0. We shall also denote by r , (resp. F~) the set of bounded selfadjoint solutions to (ARB)6 (resp. (ARE)~).

Before going further, let us review a result of Zabczyk [7].

T h c r e o m 0. Suppose (A, B ) is stabilizable and (C, A) is detectable. Then, there exists a unique nonnegative solution So E Fo, and moreover,

F = A - QSo is stable. (1.4)

Since the semigroup e Ft and hence e e'r is stable, we can define operators L E £(X) and Lo E £(X, X) by

and

where X £(X, X) can be readily calculated:

(L'/)(~) = f0' eF( ' - ' ) f (s)ds '

and

jf$ o o

(L])( t) = er( ' -O](~)ds , ] E X, (1.5)

Lo/= (L/)(0), ! e X, (1.6)

- L~(O, oo; X). Their respective adjoint operators L* E £(X) and L; E

! e x, (1.;)

Our main results are now stated as follows.

T h e o r e m 1.1 (a) There exists an S$ E F$ with the property

F$ = A - QS~ stable

i f and only if I - - 6~G*L*QLG > O.

Such S$ ia unique oa Ion K as it exists, and has a representation

& = So - $ " L o G ( I - ~2G'L*qLG)-IG*L~.

(i.8)

(1.9)

(1.1o)

(1.11)

45

(h) There exists a positive definite silution S~ E P~ if a~d only if So > 0 and

I - 62G*L'QLG - 6~G*L*oSo1LoG > 0. (1.12)

In this case, (I.11) gives the unique positive definite solution to (ARB)s and (1.9) is also satisfied.

§2. Preliminary Results.

In this section, we shall establish a number of laminas which will be used to prove our main results.

L c m m a 2.1. Let So be the unique nonnegative solution to (ARE)o and let F = A - QSo. Then,

O) Ss E £ ( X ) is a selfadjoint solution to (ARE)s if and only if A - S~ - So is selfadjoint and satisfies

( A z 1 , F x z ) + ( F x I , A x 2 ) - ( Q A x , , A a r ~ ) = 6 z ( G * z 1 , G * x 2 ) , x, ,x2 e ~)(A). (2.1)

(ii) l f P s ~ $, then, for all y E Y and w f. IR, we have

Ily[I ~ - ~2llG'(iwl + F ) - ' e * y l l z = i1[I - C A ( i w l + F)- 'c*ull ~. (2.2)

Proof. (i) can be derived through a straightforward calculation. We proceed to prove (ii). Suppose there exists san S$ E rs . Then, we obtain (2.1) with A = S$ - So. Note that (2.1) is equivalent to

( a ~ , (ions + F ) ~ ) + ((i~x + F)~, axe)- (q,Xxl, ax2) = ~(G* ~ , G* x~), (~.3)

and (io~l + F ) -~ E £ ( X ) for all ft E/R. Hence, by takin~ x~ = z2 = (i~vI+ F) -~C*y in (2.3), we immediately obtain (2.2). The proof is thus completed. []

L e m m a 2.2. Let S~ E r~ and A = S~ - So. H F6 = A - QSs is also stable, then

inf IIX- ca(ion1 + F)-~C*II~(y) > 0. (2.4)

Proof. If FS is stable, then iwI + Fs is boundedly invertible for all w E k2. Note that Fs = F - C ' C A , one can easily verify

I - CZx(i~Z + F ) C = [t + CZ~(io~I + F ~ ) - ' C ' ] - ' . (2.5)

Suppose that (2.4) is violated. Then, there exist sequences {w,} and {y .} , ~a, E IR, y , 6 Y with Ily, ll = 1, such that

yn = [ I - C A ( i w I + f ) - J C * l y , ~ 0, as n ~ co. (2.6)

The stability of F~ implies

a = suP l lCA( iwI+ F~)-1C*IIc(r) < co. (2.7)

46

Hence, by virtue of (2.5), we obtain

1 = Ily. !! = I1[ I + CA(iwI + Yt)-a C.lg. II < (1 + ~ ) I I E l l -'-' 0, ~ ,-, ~ co.

This is a contradiction and therefore (2.4) must be true.

In what follows, we shall introduce

T(i~) = G ' ( i ~ I + F ) - ~ C ' ,

and then define IITtI~ = tmP[a(T* ( iw )T( iw ) )] 112,

t d

where a(.) stands for the spectral radius of bounded linear operators. consequence of Lemma 2.1 and/. ,emma 2.2 then is

Coro l l a ry 2.3. / f there exists an $6 6 F~ satisfying (1.9), then

1 IlYlloo < ~. (2.11)

The following lemma is an alternative version of the result by Yakubovich [6].

L e m m a 2.4. Let L* e £(X) be given by (I.7). Then

I[G*L'C'tlc(y,v) = I[TII--, (2.I2)

whe~ y = L~(0, oo; Y) and v = L2(o, co; V). From Lemma 2.4 and the fact that

IIG'L'QLG]I = ]IG'L'C'I[ 2, (2.13)

we obtain

Coro l l a ry 2.5.

(2.8)

[ ]

(2.0)

(~10)

An immediate

The conditions (i.10) and (2.10) are equivalent.

We conclude this section with the following result.

L c m m a 2.6. Let So E Fo be the unique nonnegative solution. Then,/'or any St E F t , we have

ss - so < 0. (2.14)

Proo[. By Lemma 2.1, we see that A = St -- So satisfies (2.1). For x E :D(A), let ~(~) = e~tx. It is seen that l~(t) E D(A) = D(F) for all t _ 0 and limt--.o ~(t) = 0 since F is 8table. By (2.1), we can compute

d(A¢( t ) , ~(t)) = (QA~Ct), A~(~)) + 62(G'¢(t), G'~(0). (2.15)

Itence, it follows that

g ( ~ , x) = - [(QA¢(t), a~(O) + 6~(a'~(O, a'~(O)ldL (2.16)

47

The right-hand side of (2.16) converges because ert is exponentially stable. Further- more, A is bounded and D(A) is dense in X, so (2.16) is valid for all x E X. Then, (2.14) follows from (2.16). I-I

§3. Proof of Theorem i . I .

Proof of (a). The "only if" direction follows from Corollaries 2.3 and 2.5. We proceed to prove the "if" direction. Suppose the condition (1.10) is satisfies. Then, (1.11) makes sense and defines a bounded linear operator S# E £(X). By Lemma 2.1, to show S6 E re, it suffices to show

A = -~2LoG[I _ 62G*L*QLG]-lG*L~ (3.1)

satisfies (2.1). To this end, let us introduce the first order differential operator

l(t) 1(0. (3.2) D :

Clearly, for any T E £(f11,t12) and f E L2(0,oo;Hl) with D f E L2(O, oo; It]), we have

D T f = TDf . (3.3)

Moreover, one can easily verify ttle following identities:

G*L~Fx = DG*L~x, Vx E V(A) = V(F). (3.4)

LoDg = -g(0) - F'Log, Vg 6 L~(O, ooi V(A)) with DO 6 X, (3.5)

DL'o = L~g(O) + L'Dg, Vg 6 X with Dg 6 X, (3.6)

DLg = LDg, VO E X with Dg E X. (3.7)

For notational convenience, we shall set E, = I - 6UG°L*QLG and henceforth. For g E V with DO E V, let f = E~Ig or g = Esf . Then, by means of (3.3)-(3.7), we can deduce

DO = O f - 62D{G*L'QLG]}

= D] - ~UG'L*QLGDI - 5UG'LoQLoGI. (3.8)

From (3.8), it follows that

D E [ I g = E[1Dg + 6UE~IG*L~QLoGE[l g. (3.9)

Let f = E~IG*L~x, x E D(A). Then, f E V since Ee i~ an isomorphism on V. With a standar argument, we can claim that G'L*QLG] E L°°(O, oo; V). Furthermore, from

I(t) - ,~a" eF(~-')dsQ er'(~- ')G](r)dr = a ' e r t z , (3.10)

we can also conclude that ] E L~(O, co; V) and, in particular,

l i~ IIf(t)ll < co, lim f ( t ) = lim G'ee':r = G'x. (3.11) 1--*oo 1-40 1--~0

48

With the above preparation, we can do the following verification. Let

A = --62LoGETIG*L~

and let x,x~ e :D(A). Set f = E~'IG*L~xl, we have

( A Fx l, x ~ ) = -~f~ (LoG ET ' G* L~Fxl, xa)

= -62(LoGE~lDG*L~zl, x2)

= -62(LoGDET'G*L~z,, z2) + 64(LoGE'i'G'L;QLoGE'[IG*LO'xl, x~) (3.12)

= 62(Gf(O), z2) + 62(F*LoGE~'lG'L~zl, x2) + (AQAxl , x2)

= 6~(G'=,, G'=~) - (a=,, F=~) + (AQA=1, =~).

Tiffs shows that the operator given by (3.1) does satisfy (2.1). It remains to show that the solution S, given by (1.11) satisfies (1.9). To prove this, it suffices to show (see [4])

~ t l : "= l l~d t < v,~ e x, (~.13)

where F~ = a - QS~ = F - c ' c a ( , ~ ) , (a.14)

with A = A(6) being the operator given by (3.1). BY the well-known Parseval equality, we see that

II(,:,,,z + .P)-~=ll2d~ = I I : ' = l l ~d t < oo, x. ~ X (3.15) c¢.

since F is satble. Let us assume for the moment that I - (iwI + F)-1C*CA(6) is houndedly invertible for all w E IR and

= sup i l [ 1 - ( i~I + F ) -~ c'c,',(6)1-~11 < co. (3.16)

Then, from

it follows that

( i~I + F ) - ' = [ I - ( i~I + F ) - ' c " a a ( ~ ) ] - ' ( i . : I + F) -1, (3.17)

/o : = ileF,xll2dt = II(iwI + F6)-~xll2dw o o

k C II(iw/+ P)-~xll~d~ < _< ¢0,

as desired. We now proceed to show (3.16). If it is not true, then there exists sequences w, E/R and x . E X, with Ilx, II = 1, such that

[ I - (i~,, 1 + F ) - I C * CA(~)]=. --, 0, as n --* co. (3.18)

Let E, = CA(6)x, . Then, from (3.18), we obtain

[ I - C A ( ~ ) ( i w . l + F ) - l C * I ~ . --, 0, as n ~ co. (3.19)

49

By (2.2), we see that [ I I - C A ( 6 ) ( i o J . I + F ) - ' C * I [ ~ >__ 1-6211T11~o > 0, and hence (3.19) yields CA(6)Xn = ~, --* O, as n --* oo. This fact together with (3.18) lead to x , ~ 0, as n --* oo, a contradiction! Thus, we see that (32.16) must be true and (3.13) is proved.

Finally, let Ss be defined by (1.11) and let St E I s . Then, with a simple manipulation, we obtain

Note that F6 is stable, the same argument az employed in proving Lemma 2.6 enables us to derive

ss > Yr. (3.22)

This shows that the selfadjoint solution S~ with the property (1.9) is a maximal element in Pt and hence is unique.

Proof of (b). With the conclusion (a) in hand, the "if" direction would be a trivial matter.Let us now turn to the "only if" direction. If there exists an Ss0 E I'6o which is positive definite. Then, from (2.14), it follows that

So _> s60 > 0. (3.23)

Moreover, by (2.2), we see that 1 - 6gllTlloo > o, which implies that (1.10) holds for all 5 E [0, 6o). Hence, Ft i~ 0 for all 6 E [0, 5o) and (1.11) gives the solution $6 that satisfies (1.9). With a similar argument as used to establish (3.23), it can be shown that

s~ > Sto > o, v6 e [0, 60). (3.24)

Since So > 0, we may write (1.11) into

$6 = S~ol~ { l + 62 5o1D LoG[I - 62G'L* QLG - ~2n* r * ~ - l , ,c~-I~. r* c - J l ~ - J el l2 "-" ~ - ' O ' 0 xJOXlJ v X,tO,., 0 I 'JO "

(3.25) The above is valid at least for 6 sufficiently small. Then, from (3.24) and (3.25), we can derive the necessity of the condition (1.12) without much difficulty. The proof is thus completed. [ ]

Remark. The same technique can be used to establish conditions for (ARE)~ to have positive definite solutions. These results together with their applications to H ~°- optimal control problem of infinite dimension6 will be given in [2].

R e f e r e n c e s

[1] A. V. BMakrishnan, Applied Functional AnMyBis, Springer-Verlag, New York, 1976.

[2] S. Chen, X. Li, S. Peng and J. Yong, work in preparation.

[3] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, Vol.8, Springer-Verlag, New York, 1981.

[4] It. Datko, Brtending a theorem of A. M. Liapunov to Hilbcrt syace, $. Math. Anal. Appl., 32 (1970), 610-616.

50

[5] h. Ichilmwa, Linear quadratic differential games in a llilbert space, SIAM J. Control &. Optim., 14 (1976), 120-136.

[6] V. A. Yakubovich, The frequency theorem for the ease in which the state space and the control ~pac¢ are lIilbert spaces, and its application in certain problems in the ~yn$hesis of optimal control II, Sibrisk Mat. Z., 16 (1975), 1081-1102.

[7] J. Zabczyk, Remarks on the algebraic Reccati equations in Hilbert space, Appl. Math. Optim., 2 (1975), 251-258.

OPTIMAL CONTROL FOR DATA ASSIMILATION IN METEOROLOGY

Frang;ois-Xavier Le Dimet Universit6 Blaise-Pascal

63177 Aubi~re Cedex, France

INTRODUCTION

To carry out a numerical forecast in meteorology 2 items are needed:

- a numerical model modelizing the evolution of the atmosphere. The equations used are the general aquation of fluid dynamics plus a thermodynamic equation and an equation for water under several phases ( liquid, vapor, solid). Therefore we obtain a system of 6 nonlinear partial differential equations ( 3 components of the wind, temperature, atmospheric pressure, humidity). A very simplified model (2-D, without thermodynamic and humidity) is given by the Shallow Water Equations:

OU+uOU+v Ou - f v + g O h 0t O--x- 0y - ~ x = 0

~t + u OV + v OV + fu + g O h = o Ox Oy Oy

Oh u Oh Oh . ~ [Ou.Ov~ ,, - - + ~ + V ' t n / ~ t - - I = U

Ot Ox Oy ~Ox Oy]

In these equations u and v are the components of the wind, h is the geopotentiat, proportional to the height opf the atmosphere, f is the Coriolis parameter. After discretization in space we obtain a system of ordinary differential equations for which an initial condition has to be provided before a numerical integration giving the weather forecast.

-Data: the data used in meteorology are provided by the synoptic network each 12 hours, additional data are given by satellites, aircrafts and ships. From a general point of view these data arte heterogeneous in quality and density ( in space and in time).

The problem is to insert these data into the model in such a way that the resulting solution of the differential equation

52

i) is not too far from the observation ii) is in agreement with the general properties of the

atmosphere. The dynamic system representing the atmosphere has attractors, the forecast has to be located on this attractor.

To fulfill this requirements we have proposed, see e.g. Le Dimet and Talagrand (1986), to use optimal control methods which have the advantage to transform this problem into a problem of unconstrained optimization for which standard algorithms are available.

OPTIMALITY CONDITIONS

For sake of simplicity, we will consider the problem after discretization in space. In the following X denotes the state of the atmosphere, X belongs to a space Z ,

d X = F ( X ) + B.U (1~ dt X( 0 )= V (2)

U is some control parameter, in a space ILl, for instance representing the boundary conditions. V is the initial condition in 'g, B is a linear operator from U to I~. We suppose that if U and V being given the differential system has a unique solution between 0 and T. The difference between a solution of the model and the observation is measured by the cost-function J defined by:

T

J ( u, v ) = 1 L 11 c . x - Xob 1t 2dt

C is a linear mapping from 5~ to O. Therefore the problem is to determine U* and V* minimizing J. From the numerical point of view, we need to determine the gradient of J with respect to U and V before carrying out a method of unconstrained optimization.

First order op t imal i ty condi t ion

Let H = ( Hu , Hv) be a perturbation applied on the control variables U and V. For any variable Z depending on Y we define its Gateaux derivative Z. in the direction H as given by:

53

-- z(Y+,~H O- z (v) Z ( Y,H ) = lima.-, o (x

The directional derivatives of X and J are deduced from (1) and (2):

=[oF].~ + B.° (3) dt LOXJ

.]( U, V, H ) = ; ( C . X - Xobs,C.Xt) dt (4:

[d~] is the Hessian matrix of H with rcspect to X.

The gradient of J is obtained by exhibiting the linear ^

dependance of J with respect to the perturbation H. Let us introduce the adjoint variable P, take the inner product of (3), (4) with P and integrate from 0 to T. We get •

L(I I ) dX p})dt= OF .~+BHu P dt -~- , (5;

After an integration by parts we see that if the adjoint system is defined as the solution of the differential equation •

dP + f O F ] t -~- kaxj "P = Ct'(cX'Xobs)

with the condition at time T: P(T)=O. (7)

(6)

Then the components of the gradient of J with respect to U and V are:

Gradu(J(U,V))=-B t .P Gradv(J(U,V)) =-P(O)

54 Second Order Opt imal i ty Condi t ions

The matrix C represents a linear application from the space :~ into the space of observations O. If only few observations are given then the problem may have several solutions. For practical purposes it is important to link the uniqueness of the solution to the number of observations, this can be done by computing the hessian matrix of J with respect to the initial condition ( we will suppose that B=0).

-dX-=F(X) (l 0[ dt

x(o)--u ( 1 1 )

The second derivative of the cost-function with respect to the initial condition is obtained by deriving the mapping:

V J : U ~ - P (0) Let us consider K a perturbation on the initial

condition for X. The directional derivatives of X and P are obtained from (IO)-(1 I) and (6)-(7):

2 _[oF(x)1 (12[

x(ot-- K (131

It d__EP+[02F ~ p+[0F] t.P=c,.c.,X (14) dt [OX 2 LoxJ

A

V(TO = o (15)

Let us introduce the bidual variables Q and R, we take the inner product of (12) and (14) by Q and R respectively and integrate between 0 and T, it comes:

T

d i - ' ~ ~d[ kLOXJ ' k[ox z J [~--~j. ,R 0 (16)

BB

Eq. (16) is integrated by parts, therefore if we set Q and R as solutions of the differential equations:

It aQ + PF]'.Q-[0 F .P .Q--C'.C.R dt l_OX.l lax 2

dt LOXJ" R= 0 (18)

with the initial condition:

Q ( T ) =0 (19) Then from (16) it remains:

-(K, Q (0)1)= ( P'(0), R (0)) (20~ From (11) we have:

P(0)---(VJ)'. ( U , K ) ( 2 1 ) Let H be the Hessian matrix of J, then it verifies:

H . K = V 2 J . K =- ( V J ) ' . ( U , K ) (22] Therefore the columns of the matrix H are equals to the N

different values of the vector Q at time 0 obtained by N integrations of the differential system (17), (18), (19) with initial conditions on R:

R (0)=Ei, 1 < i < N (23)

where the Ei are the vectors of the canonical base. Eigenvectors and eigenvalues can be computed by carrying

out an iterated power method. Numerical results .

The domain which has been considered is a rectangle of size L = 6000kin by D=4400km. The discretization has been done in finite difference with a centered scheme, the parameters of the discretization in space are Ax = 300kin, ,Sy = 220kin. Inn time we have used a leap-frog scheme with a timestep of At = 600s.

Figure 1 shows the original initial fields o f geopotential (Fig. lA)and wind (Fig.IB). The "true" fields are computed by integrating the shallowwater equations from this initial condition. "Observed" fields (Fig.3) are computed from the

56

. . . . . . . . . . . . . . . . . . .

% N

Figure I.A Figure I.B Figl. A. Optimal Initial Condition for the geopotential Figl. B. Optimal Initial Condition for the wind

~'=' -~-~A~.< ~ , , ' q ~ " ~ . . . . . . . . . . . . . . . . . ° .

. . . . . . . . . ; . . . . . . . . S ' "

z • / • • • . . . . •

. . . . . . . . . . o . . . . . . .

Figure 2.A Figure 2.B Fig 2. A. Observed Initial Condition for the geopotential Fig 2. B. Observed Initial Condition for the wind

lO "1

l 0 "l

10 "s

lO . r

i "\ 10 "j

o 'zo : ~ 2,;1 .;0 5~ r,,1

,",,'umber o f i l e r a t i o n s

l O "1

19 .3

IO,~"

;0 "~

Figure 3. Evolution of the normalized cost-function

fi7

true fields by adding a random noise. The retrieved initial condition, after having performed the method, does not show any visual difference with the "true" one.

Figure 3 is the evolution of the cost-function with the number of iterations for 4 different algorithms of unconstrained optimization: QN1 and QN2 are quasi-Newton algorithms while TNI and TN2 are truncated-Newton methods. These results show the importance of the choice of the method of unconstrained optimization on the total cost of the method. A systematic comparison of several methods of optimization applied to this problem can be found in Zou, Le Dimet, Navon, Nouaitler (1990).

PHYSICAL CONSTRAINTS

From meteorological experiments it is well known that the atmosphere carries only few gravity waves in comparison of what could be expected from the governing equations. This is due to the fact that the system has an attractor and the natural evolution of the atmosphere lies on this attractor, if the initial condition ,which is provided to the model, does not belong to the attractor then the numerical solution countains gravity waves progressively damped but nevertheless preventing a short term forecast. The mathematical stucture of an attractor is very complex even for simple and low dimensional differential systems. Therefore these additional constraints may be handled into two differcnt ways:

- by filtering gravity waves - by using an approximation of the attractor.

Fi l ter ing of gravi ty waves It is possible to filter the gravity waves by

adding a constraint on the time derivative of X. In practice this can realized by using penalization-type method.

With pena l iza t ion methods an additional term is added to the cost function which take the form:

J (X) J(X) +

Thc effect of the penalty term is to damp the fast waves in the solution and to change the conditionning by an appropriate choicc of

58

the parameter e. The crucial problem is the choice of e, when e goes to 0, then the solution will converge toward a steady state solution, there is no physical evidence on the choice of e.

An alternative method is to use a p e n a l i z a t i o n - r e g u l a r i z a t i o n method. If H is a given parameter we define the functional q) by:

• (X) = 0 if X < H and ~(X) = (X-H) 2 , if X~H The penalized functional will be:

The adjoint system is derived as the solution of:

d . + JOEl'.F__ +

dt LOXJ E

1 crp, dX

0t

with the condition at time T:

L I

Then the gradient is given by:

The advantage of such an approach is to apply the penalization only if it is necessary, furthermore the parameter H can be chosen on physical considerations such as an estimation of the velocity of gravity waves.

Let us define Uad by:

By using standard methods it is possible to get: Theorem I

If F is lipschitzian and Uad is nonempty then the problem of optimization has a solution U*.

5 9

Theorem 2 The penalized-regularized problem has a solution U~ and

U~ ~ U* when e --* 0. Another advantage obtained in adding some penalization

term in the definition of the cost function is to control the regularity of the solution.

Approximat ions of the a t t rac tor After dicretization in space, the shallow water equation

can be linearized around a standard state. It is shown ( see e.g. Daley (1982)) than the spectrum of the linearized operator can be splitted into two parts corresponding to two differrents waves:

- Rossby waves which the slow part of the motion - Gravity wave are the fast components of the motion. In a basis of eigenvectors the evolution of W=(Y,Z), the

state of the atmosphere, between 0 and T verifies: dY = Ay.Y + Ny ( Y, Z ) dt

dZ =Az.Z +Nz (V, Z) dt

In this expressions Y is the Rossby component of the motion belonging to '~ (the Rossby manifold), Z is the gravity component in 72: (the gravity manifold). Ay and Az are diagonal matrices, Ny and Nz are the nonIinear parts of the equations.

If a noisy observation Wobs = (Yobs, Z~bs) is given at time 0, it is not enough to use W(0) = (Yobs, 0) because the nonlinear term in the equation will bring back gravity waves in the solution. To prevent the development of gravity waves we can consider the problem of determining the optimal initial condition U* in Y minimizing J with:

j(U)=_I_ GIY(~)-You~il +llz(~)- Zob~ll)dt 2

with the constraints: dY = Ay.Y + Ny( Y, Z) dt

Y(0) = U Az.Z + N z ( Y , Z ) = 0

The optimality condition is obtained by introducing the adjoint system. Let Q and R be the adjoint variable to Y and Z, the adjoint system is defined as the solution of the differential system:

60

dE + Ay.P +[ONY]t.P [0NY] t dt 0Y ] "[ -0--Z-] "Q = v ( tO- Yob,(t)

Q ' / P = Z ( t)-Zobs(t)

Then the gradient with repect to the initial condition on Y is given by:

V J = - P ( O ) It is important to point out that the dimension of the

differential system is be integrated can be lower than before because the integration is carried out only on the Rossby component.

CONCLUSION A crucial point for improving numerical weather

forecast his to get efficient methods for data assimilation and which are able to be implemented on operational models of very large dimension ( of the order of 106). Optimal control methods seems well-suited for this purpose, nevertheless a lot of studies remain to be done from the dtudies on dynamical systems modelizing the atmosphere to the numerical analysis of these problems.

ACKNOWLEDGEMENT The numerical results presented in this paper have been

made on the CRAY2 belonging to the Centre de Calcul Vectoriel pour la Recherche du CNRS.

REFERENCES Le Direct F.-X. and O. Talagrand (1986): Variational Algorithms for Analysis and Assimilation of Meteorological Observations: Theoretical Aspects. Tellus, 38,4, 97-110.

Zou X, Le Dimet F.-X., Navon I.M., Nouaitler A. (1990): A Comparison of Efficient Large-Scale Minimization Algorithms for Optimal Control Applications in Meteorology. Submitted to S IAM Journal on Optimization.

ON THE S T A B I L I T Y OF OPEN

POPULATION LARGE SCALE SYSTEM*

Gee Hang

( D e p a r t m e n t o f M a t h e m a t i c s , N o r t h e a s t

N o r m a l U n i v e r s i t M , C h a n g c h u n , j i [ i n 130024 , C h i n a )

Abstract

I n t h i s P a P e r , hie w i l l d i s c u s s t h e L ~ a p u n o v t s s t a b i l i t y o f open

p o p u l a t i o n l a r g e s c a l e s y s t e m , o b t a i n t h e t i m e - v a r ~ i ~ g I o ~ e r c r i t i c a l

v a l u e and t h e t i me-va t , , , i ng uPPer c r i t i c a l v a l u e o f a b s o l u t e

b i r t h - r a t e , and g i v e t h e s u f F ~ t z i e n t c o n d i t i o n s For t h e l a r g e . ' ; t a l e

s~-'stem t o be s t a b l e and a s ~ , m p t o t i c a l l y s t a b l e , mear,bJhi l e g i v e t h e

necessar's., c o n d i t i o n f o r t h e l a r g e s c a l e s,#stem t o be s t a b l e . These

r e s u l t s me?, p r o v i d e a s t r i c t m a t h e m a t i c s t h e o r y f o r p o p u l a t i o n

cot-it i"O I •

I n t h i s p a p e r , t h e f o l l o w i n g oPen P o P u l a t i o n l a r g e s c a l e s y s t e m is

d Jscussed [1 = :

L P (r, t) =A (r, t) P (r, t)

P (r.O) =F (r)

P ( 0 , t ) = m ( t )

i n Q=g2 X ( 0 , o o ) , ( I )

i n S-2 = ( 0 . R ) , ( 2 )

i n ( 0 , ~ ) • ( 3 )

inhere L = ..°~., + ~ - ÷ M ( r , t ) M ( r , t ) = d i a B ; ( ~ z 1 ( r . t ) ~ ( r . t ) ) . C~r ~t " .....

P ( r ' L ; ) : ( P l ( r , t ) ) i = n i " F ( r ) = ( f i ( r ) ) i = n l ' ~I' ( t ) : ( ~ P i ( t ) ) i ~ 1 are

n rows | c o l u m n v e c t o r f u n c t i o n , A ( r , L ) = ( 8 ( r , t ) ) . n . is a n re&is ij z,j=i

r, c o l u m n s m a t r i x , and t h e f u n c t i o n s P i ( r , t ) , 6t i ( r , t ) , t" i ( r ) and

"-Pi ( t ) d e n o t e r e . ~ p e c t i v l y ~ge d e n s i t v f u n c t i o n , m o r t a l i t ~ v . l ~unc t ion ,

~: The Pr'oj~'ct SuPPOrted by N a t i o n a l N a t u r a l Sc ience FoulJdat ion ot: Oh ina

62

i n i t i a l age p a t t e r n , a b s o l u t e b i r t h - r a t e f u n c t i o n o f t h e i t h

P o P u l a t i o n grouP a t a g e r and t i m e t , a i l ( r , t ) d e n o t e s t h e m i g r a t i o n

r a t e f H n c t i o n ~h i ch i.~ m i g r a t e d f r o m t h e j t h Pop~, I ~ t i o n g r o u p

i n t o t h e i t h p o p u l a t i o n g r o u p , R is

r a c e •

hie suPPose t h a t

a). Vi(r,t):l~i (r,t)-ail (r,t)

v t ( r , t ) d r - . - ÷ ~ , a n d o

t h e h i g h e s t a g e o f tt~e hui l ian

v a r i m i n q

is s u b j e c t e d t o t h e c o n d i t i o n :

j~ { v ( x , t ) d x < + o o i f r < l ~ , o i

3 / 2 , 3 / 2 ( r , t ) = ~ o >O, v ( r , t ) 611 (CI~) - j ~

where 6~:=Q' >< (0,T), Q' =(0,R'),0<R f <I~ and R' may be arbitrarily

3 / 2 , 3 / 2 c l o s e t o R, T maM be a r b i t r a r i l 2 l a r g e t~l i th T < + ~ , a n d H

is t h e S o b o l e v s P a c e : 9 a,

3/2, .~12 b> a (r,t) ~H (G), Q =Q >< (0,'r),

%-~ ( r , h ) = O , j = ] , 2 . . . . . n , a n d ] r i : ~ : . ] , 8 ( r , t ) . ~ 8 . i~ aij lJ

c> £ ( r ) 6 H I ( ~ - a ) , f ( r ' ) > O , i = l , 2 . . . . . n , $ ° ( I ; ) ~. 111 ( O , T ) , £ i i

. ~ ° i ( t ) > O , i : I , 2 . . . . . n , T < + o o .

F i r s t o f a l [ , ~e g i v e a [emma.

l~*mma I clwa~.- We assume t h a t t h e a s s u m p t i o n s a ) , b ) , ar id c ) a r e

f l i l f i l t e d f o r p r o b l e m ( I ) - - < 3 ) , and { r I ( r ) " ~ i ( t ) } , i = l , 2 . . . . . n ,

s a t i s f y t h e c o m p a t i b i l i t ~ r e l a t i o n s { ~ . ~ . } ~ 3 Then t h e r e e x i s t s

a u n i q u e f u n c t i o n P ( r , t ) b e l o n g i n g t o I~I H 3 1 2 " 3 1 2 ( Q ) s a t i s f . ~ i n g i=I T

t h e p r o b l e m ( I ) - - < 3 ) - F u r t h e r m o r e P ( r , t ) = ( p ( r , t ) ) s a t i s f i e s t h e i

re I a t i on :

r n

÷q (r-t)+~P i (t-r) } exp [-~v (×,x+t-r)dx ] . • o f

(4>

~" I[- [.~_-)] h a v e d i s c u s s e d t h e s t a b i l i t y of c l o s e d P o P u l a t i o n s y s t e m . Tn

t h i s P a p e r , ~le d i s c u s s t h e L~ ,aPunov ' s s t a b i l i t y o f t h e open P o P u l a t i o n

s:.,~<tem < 1 } - - { 3 ) .

L e t us

~e s e t

63

i n t r o d u c e a f u n c t i o n and a d e f i n i t i o n .

K~ (r,s,t)=14 i (r,t)exp [ - ~ v i (x,x+t-r)dx ] . 8

D e f i n i t i o n 1. Me d e f i n e

- fo . ~PiO (t) : (r, O, t) ~°]C ( t - r ) dr, (5)

~:[~O ( t ) i s c a l l e d t h e t i m e - v a r y i n g l o w e r c r i t i c a l v a l u e o f a b s o l u t e

b i r t h - r a t e S ° (1~) f o r t h e i t h s u b s y s t e m s o f p o p u l a t i o n l a r g e s c a l e t

s y s t e m ( 1 ) - - ( 3 ) .

T h e o r e m 1. f~ssume t h a t t h e c o n d i t i o n s o f Lemma I a r e s a t i s f i e d ,

i f S ° i ( t ) ~ P l e ( t ) , i = l , 2 . . . . . n , t h e n , t h e l a r g e s c a l e s y s t e m

( 1 ) - - ( 3 ) i s s t a b l e .

P r o o p . F rom Lemma 1 and [ 9 ] , i~e know t h a t l i P ( . , t ) I I L I ( ~ ) E O [ O , T ] .

b l i t h t h e a i m o f c o n v e n i e n c e , we w r i t e l i p ( . , t ) l l = l l P ( . , t ) l l 1 ( ~ 2 ) "

I n e ~ { L t a t i o n ( 1 ) , ~te h a v e t h a t

ar P l ( r ' t ) + ~ P±(r't)*~(r't)Pi(r't)='~-~'j=l a l jp j ( r , t ) ,

we i n t e g r a t e b o t h s i d e o f t i l l s e c l u a t i o n w i t h r e s P e c t t o r i n ~;~,

a c c o r d i n g t o c o n d i t i o n a ) and p l ( r , t ) ~ O , ~Je o b t a i n t h a t

dt-Ct II P i ( r , t ) II =S° i ( t ) - Rl P l ( r ' t ) dr+'~'j--1 aiJPJ ( r , t )dr -

From condition b), ae have

° So" -~d II P ( r , t ) II = .cL S-], II P t ( , t ) II [ S ° t ( t ) - ta t p : t ( r , t ) d r ] ( 6 ) d t d t t = l " - ~ = 1

From l..em.la 1 and c o n d i t i o n s b ) and c > t t h e r e e x i s t s

i n e q u a I i t y :

P i ( r , t ) :~ S ° i ( t - r ) exp [ - ~ v i (x. x+ t - r ) dx ] ,

substituting the above inequality into (6>, we have

f o l l o ~ l i n g

64

n d lJ (. t)n <~ [5o (t)-%0- (t) ]. (7)

a V % ' 1:I i ~ o F r o m t h e c o n d i t i o n ~° i ( t ) < ~ ° [ c ( t ) . o b t a i n ~]fll P ( . , t ) I I < 0 , h e n c e .

II P ( . . t ) l I ~C I1 I ? t f .

Th i :~ .~hoJu~ t h e .%".,'stem ( I ) - - < 3 ) i:< s t a b l e . T h e o r e m ~ i s p r o v e d .

Thf,¢~r~m 9 . Nssume that the conditions of lomma I a r o r s a t i s f i e d ,

i f %01 ( 1 : ) ' - ~ % 0 . ( t ) , i = 1 , 2 . . . . . n , O < o ~ < l , t h e n , t h e I ~ r g e .':~¢ale i c a;5,:qtem < 1 > - - ( 3 ) i s a ~ y m p t o t i c a l I~.' . ~ t a b l e .

P r o o F . hh-~ f i r : < t ppo~ /o t h a t

I Tm ~ . ( t - r - ) d r ' = O . i = ) , 2 . . . . . n ( 8 ) t --~ .>5 0 1

Tn f ~ c t , i f t h o r e e x i s t s l o ' e q i l a l i t ~ (8) d o e s n ' t h o l d f o r i 0 . t h e n

hip ~ an P r o v e e a s i |5, t h a t 0 < % 0 i ( L ) <~-:H f o r L E ( 0 , ¢ - : ' ) . S o t I ~ e

[ l . k -R , l.k ] so t h a t ~ o ( t ) >O_2R f o r t #- E k . : ;et [ t k - R , L k ] -

1 ~ = II k , ou~ir;~ t o

C~<:-: ~0 R k~_ tk

C

f r o m fl~i.~:¢ i | ' l oq l l81 i t ' . , , , hlo o b t a i n tl-~#]t

C. mesl : . :-'~: 2]~1 " k = 1 , 2 .....

~ k 5b-~l. SN= I ] I] wo h a v e t h a t

k:1 k" ~Plo c ( t )

~Pi0 (tl-~Pi:o(L)=~P i (t) [ 1 - ( t ) ] o ~Oio

C 1 <<-- [I-- ]=-~ (~>O)

2R ,]

f o r t ~. SI t " I~le i n t e g r a t e b o t h s i d e o f i n e q u a l i t y ( 7 ) u~ i th

t ~>u [ O . T ] a r i d s u P p o s e S N c [ O , T ] , t h e n , f o l l o l l i n g

h o | d.~ : T" n

u F u + ~ ~ r ~ (o-%0fe (t0 ]dr. I1 P II o i=1 i

_ ~T - ,,~ II P'll + 0 [ ~ i o ( t ) - ~ ±o c ( t ) ] d t

< II F II - ~" A d t " ~ II I" II - A H ( L - SIf 2M

r e s p e c t t o

i n e q u a f i t s ,

65

IF T is s u f f i c i e n t l y l a r g e so t h a t N ma~ be s u f f i c i e n t l w l a r g e , t o o ,

t h e n , ~e o b t a i n t h a t tl P ( . , T ) II < 0 . T i l l s n o t t u r e , b e c a u s e II P ( - , t ) II

> 0 r o t t C ( 0 , o o ) , h e n c e e q u a l i t ~ ( 8 ) h o l d s f o r i = 1 . 2 . . . . . n .

Now, me wan t t o P r o v e [ [ m l l P ( . , t ) II = 0 . t -~ oo

From t h e e q u a l i t y ( 4 ) , me h a v e t h e i n e q u a l i t y

O-.~pi(r,t)~2fi(r-t)÷~:t (t-r)* ~ p.(s ,s*t-r)ds. o J=1

We can p r o v e t h e i n e q u a l i t ~

n n

Pj (s,s÷t-r) <~%-~ p~CO, t-r) j :1 j:1

f o r s E g2 arid pj ( O , t . - r ) = l ~ ( r - t . ) + c P j ( t - r ) . I n F . a r t i c u t a r ~

we h a v e £ ( r - t ) = O , h e n c e J

O<p i (r,t),;;so (t-r)-R ~ 5O ( t - r ) . i j = l J

From t h i s i n e q u a l i t ~ and ( 8 ) , o b t a i n t h a t l i r a fl P ( . , L ) li : 0 , t - ~ o o

l d t ' ge t ~ca i e :~:,~stem ( 1 ) - - ( 3 ) i s a s y m p t o t i c a l l y , s t a b l e .

F' I'Oved. Nou,, ~e introduce following functions:

i e t > k ,

i . e . t h e

T h e o r e m 2 i s

a i ( r , t ) = m a x { a l j ( r , I ; ) [ j ~ i , ] < j ' - ~ , n } ,

+ n f r Ki(r 's ' l ' )=K~(r 'O't)~ ~ s R~(r'x'L)aj(x'x+L-r)dx"

D(-Finition ?. Me d e f i n e

~O R 5 ° + ( t ) = K+ ( r , O, t } SO ( t - r ) d r " (.q) i c z i '

5 ° + ( L ) i':~ c a l l ed t h e t i J l l e - v a r y i r l ~ u p p e r c r i t i c t l l v a l u e o f a b s o l u t e i c

b i r t h - r . ~ t ~ ( P i ( t ) f o r t h e i t h subs>"J;tems o f p o p u l a t i o n I ~ r g e s c , h l o

St#steul ( 1 ) - - - ( 3 ) .

Th~:,,",r,,, i ",~. (~Ssunto t t~d t t h e c o r l d i t i o n c ; oF Lamina 1 a r e s a t i s f i e d , i f

5O i (t) ~,- ~ 1 5O+ io (t;), i--|,2 ..... n, ~i >|, then, the Large scale

66

Proof. For t>R, we have that

?~ Ii P ( . , t ) II = (S ° ( t ) - Ri ( r , t ) pi ( r , t ) dr) i=1 i

n R

=}-] [ ~P i ( t ) - I o K~ ( r , O, t ) S°i ( t - r ) d r - i : I

- / ;rKi(r,s,t) ~-~, a p. (s,s+t-r)dsdr ] . o o 3~i i~ 0

01~ i i l g t 0

n n

p ( s , s + t - r ) < __~=I ~P j ( t -r) ;)=I j f o r s (~ Q a n d 1,;>!~., ~le l~ave t h a t

n E J .

dt z A c c o r d i ng t o t h e c o r l d i t i o n o f t h i s t h e o r e m , i~le k n o w t h a t

d -([~ Ii p ( . , I s ) Ii > 0 , hence f o r t > R , !1 I ' ( . , t ) II ) II P ( . , R ) tl . From

D e F i n i t i o n ~, we hove t h a t

io ( t ) > ~ i ( r , t ) P i ( r , t ) d r > ~5t l P ( . , t ) 11 > O 1 >O,

From t h i s i n e q u a l i t ~ ~e get

n

d I1 P ( . , t ) I1 ~-~-J= + dt = 1 [ S ° t ( t ) -S° ic ( t ) 3 . n

=1 '~ie ( t ) I n t e g r a t i n g t h e b o t h s i d e o 1 ' t h i s i n e q u a l i t y o n [ R , L ] , a,e d e d u c e

i l P ( . , t ) i l ; ~ H I ' ( . ,R) 11 +C 1 ( a l - I ) ( t -R) .

I f t i s s u f f i c i e n t l t ~ l a r g e , t h e n , II p ( . , t ) II .roam t a k e ar l ) i t r a r i 1~

l a r g e v a l u e . T h i s s h o w s : t h e l a r g e s c a l e s y s t e m ( 1 ) - - ( 3 ) i s n o t

s t a b l e . Theorem 3 i s p r o v e d .

No~., we d i s c u s s t w o s p e c i a l c a s e .

I n t h e f i r s t p l a c e , i f n = l , s?s tem ( 1 ) ~ ( 3 ) d e g e n e r a t e i n t o

fol t o m i n g s y s t e m :

a , t ' + ~) Q r p ( r ") - ~ T P ( r ' t ) + R ( r ' t ) p ( r ' t ) = O "

p ( r , O) =f (r) ,

p (0 , t )=$° ( t ) ,

(10)

(11)

(12)

67

D e f i n i t i o n 3- tde d e f i n e

$° o ( t )= fR .F o ~L ( r , t ) 9~ ( t - r ) e x p [ - _oI~ ( x , x + t - r ) d x ] d r , (13)

9 P c ( t ) i s c a l l e d t i m e - v a r y i n g c r a t i c a l v a l u e o f a b s o l u t e b i r t h - r a t e

5 ° ( t ) f o r t h e s y s t e m ( 1 0 ) - - ( 1 2 ) .

O o r o l l e y 1. I~ssume t h a t !~ ( r , t ) , f ( r ) ,SO ( t ) s a t i a f y t h e

c o n d i t i o n s a ) and c ) , Then, i f SO ( t ) < S O ( t ) t h e s y s t e m ( 1 0 ) - - ( 1 2 ) o is s t a b l e ; i f SO ( t ) < 8 . S O c ( t ) , 0 < 8 4 | , t h e s y s t e m ( 1 0 ) - - ( 1 2 ) is

a s y m P t o t i c a l l ' ~ s t a b l e ; i f SO ( t ) > 8 ~J° e ( t ) , a > 1 , t h e s y s t e m 4 1 0 ) - -

( 1 2 ) is n o t s t a b l e .

P r o o f . Owing t o R ( r , t ) = O , hence ÷

$° e ( t ) =So~ ( t ) =s ° e (t) +

(9P: ( t ) a n d S° e ( t ) a r e g i v e n by ( B ) and ( 9 ) i n , . ,h ich t h e i n d e x i i s

removed ) , f r o m Theorem I - - Theorem 3, we can o b t a i n t h i s r e s u l t o f

Corolle~ I.

I n t h e n e x t P l a c e , Per t h e s y s t e m ( 1 ) - - ( 3 ) , i f

IJ. 1 ( r , t ) : . . . = l ~ n ( r , t ) = ~ ( r , t ) , (14)

me have f o l l o w i n g r e s u i t s -

C o r o l l e y 2- Assume t h a t t h e c o n d i t i o n s o f Lemma 1 a r e s a t i s f i e d

and ( 1 4 ) h o l d s - Then, t h e r e e x i s t s ¢~ic ( t ) , w h i c h is c a l l e d t h e

t i m e - v a r y i n g c r i t i c a l v a l u e o f a b s o l u t e b i r t h - r a t e 9P i ( t ) f o r t h e i t h

s u b s y s t e m s o f p o p u l a t i o n l a r g e s c a l e s y s t e m ( 1 ) - - ( 3 ) and g i v e n :

FR 9°ic ( t ) = ~0 ~ ( r " t ) s o i ( t - r ) e × P [ - ~o ~ ( x , x + t - r ) d x ] d r . (15)

I f 9 P l ( t ) ~ 9 ~ i c ( t ) , i = 1 , 2 . . . . . n , t he l a rge s c a l e sys tem ( I ) - - ( 3 ) is

s t a b l e ; i f 5 ° t ( t ) < a s o i ¢ ( t ) , O < a K 1 , i = 1 , 2 . . . . . n , t h e l a r g e s c a l e

sys tem 4 t ) - - ( 3 ) is a s y m p t o t i c a l l y s t a b l e ; i f ~ J P i ( t ) > a l ~ i e ( t ) , a l > l ,

i = 1 , 2 . . . . . n , t h e l a r g e s c a l e s t ' s t e m ( 1 ) - - ( 3 ) i s n o t s t a b l e .

n n

p r o o f . Suppose p ( r , t ) : Pi ( r , t ) , f(r)~----~ E I ( r ) a n d ~P ( t ) =

n =<~_] 9P I ( L ) , tl~en, the fuctions ( p ( r , t ) , f ( r ) , S ° ( t ) } sa t i s f y Problem i=I

( 1 0 ) - - ( 1 2 ) . [Je n o t e t h a t

tl P ( . , t ) II =11 p ( . . t ) II a n d I! F i l =11 E l l

From C o r o l l e y 1, we may o b t a i n t h e r e s u l t s o f O o r o l t e y 2,

68

C o r o t l e y 2 s h o w s : i f ( 1 4 ) h o l d s , t h e n , P o p u l a t i o n m i g r a t i o n f r o m

one s u b s y s t e m s t o o t h e r s u b s v s t e m s d o e s n ' t a f f e c t t h e s t a b i l i t v o f

l a r g e s c a l e s y s t e ~ l l (I)--(3)-

REFERENCES

[I] Song Jian, Yu Jingvuan & Li Guangs, uan, S c i e n t i a Sini(a, 24(1981),

3 : 431 - - q ~ 1 4 .

[ 2 ] Song .:Tian ~ Yu 5 r i n g v u a n , M a t h . P l o d e l l i n ~ , 2 ( 1 9 8 1 } , 1.

I 3 ] Feng D e x i n g F~ Zhu G u a n g t i a n , 3 o u m l a l oF P l a t h e m a t i c a t R e s e a r c h

and E ~ ( p o s i t i o n , 3 ( 1 9 8 3 ) , 3 : 1 - - - 6 .

[ ~ ] Song 3 i a n ~ Chert R e n z h a o : $ c i e n t i a S i n i c a , 2 6 ( 1 9 8 3 ) , I 2 : 1 3 1 4 - - t 3 2 5 .

[ 5 ] Chen Renzhao I~ Gao Hang, S c i e n c e i n c h i n a . . 3 3 ( 1 9 9 0 ) , 8 : 9 0 9 - - ~ J 1 9 .

1:6] Chen Ren:-:hao, Ke×ue T o n g b a o , 3 0 ( 1 9 8 5 ) . - 6 : 7 1 I - - - - 7 1 6 .

[ 7 ] Chen R e n z h a o E Gao Hang, 3 o u r n a l oF N o r t h e a s t Norr~lal U n i v e r s i t y , .

1985, 3 : 1 - - - 6 .

[ 8 ] Gao Hang; 3 o u r n a l o f N o r t h e a s t No rma l U n i v e r s i t y , , 1989, 3 : 1 3 - - - 2 0 .

19] L iong . . ~.L.IPx Magenes , E . , Non-Ho.nogeneoHs Bound,~r'.,. V a l u e P r o b l e m s

and f : t P P l i c a t i o n s , I , l I , $ p r i n g e r - ~ ; e r l a g , 1972.

'rEMPEI~ATURE CONTROL SYSTEM OF IIEAT EXCIIANGERS ---AN APPLICATION OF DPS T|IBORY

I t u a n g G u a n g y u a n , Hie Lei , Zhao Yaowen ~cpattlent af ffathewati¢, Shangdang 0niversit~, $i~an,~h~adaag 150t00, P. L Ch(ns

Wu Qibin~ Yang Weiming, Liu Jian Dcparti|eat of Autolatics,Qiagda0 Chemical Institute of Technology, qlngd~0, Shangd0ng 2~0tZ, P,R.C~i~a

Abstract

To t h e f e a t u r e of l o n g d e l a y t i m e . v a r y p a r ¢ ~ m e t e r a n d n o n l i n e a r eLs .

in Lhe l o n g t i m e - d e l a y s y s t e m , u s i n g t, he d i s L r i b u t e d p a r a m e t e r t h e o r y , w e

g i v e a n e w k i n d of m i x e d c o n t r o l p l a n ~ t h e f e e d f o r w a r d c o n t r o l p l u s

i n d e n t ' i f y i n g m o d i f y i n g f e e d b a c k . Via t h e r e a l t i m e e x p e r i m e n t o n a

e x c h a n g e r , i t i s s h o w n t h a t i t i s m u c h m o r e s u p e r i o r Lo t h e p o p u l a r o n e s

b a s e d on t ,hc h l m p s p a r a m e t e r t h e o r y .

[. I n t r o d u c t i o n

W e a L h c r i n t h e c o n t r o l t h e o r y o r t h e e n g i n e e r i n g p r a c t i c e ~ L h c

p r o b l e m abol~L Lhe c o n t r o l to t h e s y s L e m w i t h l o n g L i m e - d e | a y ( m o r e

exac t , l y , i . e . Lhe s y s t e m w h o s e p u r e l y d e l a y t i m e i s m u c h m o r e g r e ~ t t e r

l .han it:¢ t r a n s i t i o n Lime) i s n o t c o m p l e t e l y s o l v e d b y now. One o f t h e

ll~aiJl r e a s o n f o r t i l e p u r e l y L i m e - d e l a y i s t i m d i f f e r e n c e b e t w e e n Lhe

s i L u r t t i o n s of t i l e s y s t e m ) ~ i n p u t a n d o u t p u t . [ n t h e lul~)t)~ p a r a m e t e r

t h e o r y , t h e d e l a y t i m e if. i s r e g a r d e d a s a p a r a m e t e r o f s y s L e m i t s e l f , a n d

w h e n in m u c h l o n g , t h e r e i s no a l l - r o u n d L h e o r e L i c a l r e s u l t s y e t . l~ut

t h e o u t s t a n d i n g o f t h e d l s L r i b u t e d p a r a t a e L e r t h e o r y i s j u s t c o n s i d e r i n g

t.he f e ; t L h e r o f t h e s y s L e m t s d i s t r i b u t i o n , t h u s we c a n c o n c l u d e t h e

s y s L e m ' s t i m e - d e l a y p h e n o m e n o u b y i t s p h y s i c a l p a r a m e t e r a n d s p e c i a l

l ~ | ' o l ~ r y ill s o h ~ L i o n , t h e n t h e e f f e c t i v e m e t h o d s to o v e r c o m e t h e L ime-

d e l a y c a l l b~r f o t t n d Oll t .

As i s w e l l k n o w n , f e e d f o r w a r d i s t h e e f f e c t i v e m e t h o d s to o v e r c o m e

Lhe t i m e - d e l a y . M a k i n g f u l l u s e of t h e d i s t r i b u t e d p a r a m e t e r t h e o r y , w e

c a . e a s i l y a c h i e v e t h e ~ o m p o u n d f e e d f o r w a r d c o n t r o l p l a n , w h i c h i s

ob viol~.,~ in p h l s i e a t s e n s e a n d e f f e c t i v e t o , , , :pc w i t h d i s t u r b a n c e

Int~;tl lwhiic, O[ c o u r s e it, '~ soL e n o u g h to e n s t l r ~ t i l e a c c u r a t e t o h a v e t h e

f e c d f o r w a r d O l ' l l y , c ~ o n s i d e r i n g t h e e n s s e t , i a l m e a s u r e a n d m o d e l i n g e r r o r ,

So i t ' s n e c e s s e r y t o i n d e t i f y t h e s y s L e a l p a r a m e L e r f r o m t h e s i n g n a l s o f

inl~llL a n d o u t p u t a n d m o d i f y t h e f e e d f o r w a r d c o n t r o l l e r . O b v i o u s l y , t i l e

c o n s e q a e n c c * o w e to t h i s i s m u c h m o r e r e a s o n a b l e t h a n t h a t o w e to s i m p l e

d i r e c t s u p e r p o s i t i o n f e e d b a c k . T h e r e f o r e we g i v e t l ae g e n e r a l c o n t r o l

p a t t e r n Lo t h e | o t t g t i m e - d e l a y s y s t e m a s f o l l o w s :

f e e d b a c k = f e e d f o r w a r d ÷ i d e n t i f i c a t i o n

T h e c o n s t r u c t i o n of t h e s y s t e m c o n t r o l i s s h o w n a s F i g . 1 .

70

T h e d i f f e r e n c e b e t w e e n t h e p l a n a b o v e a n d t h e u s u a l a d a p t i v e c o n t r o l

b a s e d o n t h e l u m p s p a r a m e t e r t h e o r y is thaL in Lhe p l a n a b o v e t h e

f e c d f o r w a r d c o n t r o l l e r it+. f o r m e d b y t h e s p e c i a l d i s i ; r i b u t e d p h y s i c a l

f e a t u r e of t h e s y s t e m a n d t h e p a r a m e t e r s w h i c h n e e d to be i n d e n t i f i e d

l a t e r a r e al l p h y s i c a l o n e s . So t h e c a l c u l a t i o n n e e d e d is m u c h l e s s .

- '~,Ji~Ittth+~jI,:~ iap,.~l- ~i " sy.~tcm "]--u,Hl'nl I "+'a'"°" t r ° t i"I'tlt"~l I +,,.! +..- .... -+ } I ++,++.+,:+.,+o,:o...,+,..,,+,+,5'

,. mn rc, t re+ + P t+t~;. |

2. HODEL AND MATHEMATICAL ANALYSIS

T h e s k e t c h of t h e h e a l e x c h a n g e r i s s h o w n a s F i g . 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c o l d w a t e r --~-

~-- hot water

0 ......... L ~... x FIG.2

and i ts mathematical model can be ob ta ined by p h y s i c s as ( l ) ( h e a t

conduc t ion omit ted):

' +v ' - d rPtP- ud r P~'=a{ Q- P } epQ¢ c p Q ~ - a ( P - Q } ,

B o u n d . Cond . Q{O,L)=Q,( t ) , P{L,L)=Pt (t}

l n i t i . Cond . Q(x ,O}=Q, (x ) , P (x ,O)=P I{x},

(1)

w h e r e P a n d Q a r e t h e t e m p e r a t u r e s o f t h e h o t a n d co ld w a t e r s r e s p e c t i v e -

ly, a n d c ,d - - - s p e c i f i c h e a t s , p , r - - - d e n s i t i e s , v , u - - - v e l o a i t i e s , a - - - t h e

h e a t - t r a n s f e r c o e f f i c i e n t , x - - - p l a c e , a n d f - - t i m e .

In p rac t i ce , v or u is the con t ro l v a r i a b l e or the main d i s t u r b a n c e

:,,i(! Q(L,t} is the conLrolled var iab le . Q. and P~ can be also d i s t u r b a n c e s .

Obviously Q(L,t} and the t ime -de | ay a re non l inea r ly d e p e n d e n t on v and u.

3. COMPOUND CONTROL SYSTEM

Since all the d i s t u r b a n c e s are measurable ,we ob ta in a scheme of feedforward con t ro l from solu t ion to ( l ) . t h e measu r ing s i g n a l s

71

Q(L,L) ,P(L,L) ,Q(0,L) a n d P ( 0 , t ) c a n be u s e d f o r t h e i d e n t i f i c a t i o n of

t h e s e p h y s i o a l p a r a m e t e r s . I n e x p e r i m e n t s we f i n d that . t h e h e a l t r a n s f e r

c o e f f i c i e n t a d e p e n d s on v a n d u . To m a k e t h e c o n t r o l m e t h o d s m o r e

e f f i c i e n t , leL~s suppose

a = f ( p v , r u } {2}

w h e r e t h e f u n c t i o n f n e e d to be d e f i n e d a n d n o t e g = c p , g l = d r , g { o r g f )

mean~ t h e r m a l c a p a c i t a n c e . T h e n al l t o g e t h e r t h e r e a r e f o u r i n d e p e n d e n t

p a r a m e L e r s , i .e . g , g j ,a~and L. WiLhout l o s s i n g g e n e r a l i t y we a s s u m e u i s

c o n t r o l v a r i a b l e . T h e s c h e m e o f c o n t r o l s y s t e m is a s F ig .3 .

, d i~ t~ , 'bance Q°,P~,v'-" I I [ [. t ' |

. . . . . - ............

f e e d f o r w a r d c o n t r o l l e r ] - a - - . - . ~ ] ~ _ ~ i d e n t i f i c a t i o n ] FIG.3

4. DEFFERENCE BETWEEN DPS AND LPS IqETIIOD

T h e r e a r e t h r e e f o l d s :

I ) . E a c h p a r a m e t e r in ~he m o d e l g i v e n b y D P S m e t h o d h a s c l e a r p h y s i c a l

s e n s e , b u t in i t s d i s c r e t e f o r m (LPS m o d e l ) al l t h e p a r a m e t e r s a r e d e p e n -

d e n t on t h e l e n g t h o f x a n d t , a n d w i t h e a c h oLhe r ;

2). T h e c o n t r o l s c h e m e o f f e e d f o r w a r d a s a b o v e i s b a s e d o n t h e p h y s i c a l

p r o c e s s , e s p e c i a l l y t h e r e l a t i o n b e t w e e n v , u , a n d QiL,L) i s n o n l i n e a r ;

3), The c o m p u t a t i o n t ime in t h e i d e n t i f i c a t i o n o f p a r a m e t e r s in F i~ .3 . is

much l e s s t h a n t h a t in LPS a todel , b e c a u s e all t o g e t h e r t h e r e a r e ,1

independenL parameters to be identified h e r e .

5. EXPERIMENS AND CObIPARESON

T h e o r i g i n a l c o n t r o l s y s L e m ( b y LPS) :

. . ~ " ........... ~ h o t waLer

cold waCer~ ~ ~ f low t r a n s .

f low t r a n s . --~" f e e d f o r w a r d t e m p . t r a n s .

co, ,t ,-o~ler i 1' ~ t a n k o f

"- ~ t a n k o f w a ~ e r ~ . . . . . . . . . . . . . . . . . . . ~ ' ( ~ ) - - ~ h e a L e r - * " w a t e r

. . . . . . . . . . . . . . . . . . t FIG,,I

72

Only t h e s t e a d y s i t u a t i o n c o n t r o l i s c o n s i d e r e d h e r e , a n d t h e d y n a m i c

p r o g r e s s c o n t r o l i s a l s o f e a s i b l e b y t h e p r i n c i p l e m e n t i o n e d a b o v e . In t h e

s L e a d y s i t u a t i o n , Q ) , P ~ , u , v a r e al l c o n s t a n t , a n d t h e m o d e l i s s i m p l i f i e d a s

f o l l o w s :

V ! --- cp Q:{ a(P-Q) (3) druP~ = a(P-Q) B.C." Q(0)=Qe; p(L)=pI {,l)

S o l v i n g t h e d i f f e r e n t e q u a t i o n a b o v e we h a v e t h e sl.ead~" s t a t e s o l u t i o n .

Q{ = Q~+A(P(-Q~) (5) P, = P&-B(Pt-Q e) (6)

where A={1-exp[(b~ -b) l ] } / {1-b, b't exp[(b, -b) l l } B = { l - e x p [ (b~ - b ) l ] } / { b b ~ - e x p [ ( b ~ - b ) l ] }

b = a / c p v ; b = a / d r u

So t h e r e l a t i o n s h i p b e t w e e n t h e s y s t e m ' s i n p u t a n d o t l t p u t call be o b -

t a i n e d .

(Q(-Qo)/(PI-Q#) = A (7)

1) S c h e m e ]. T h e aim is to e n ~ u r e t h e t e m p e r a t u r e a t t h e cool w a t e r e x i t

Lo a c h i e v e t h e v a l u e g i v e ~ l ) u s i n g 1~ ~ co))Lrol , l t s p r i n c i p l e is s h o w n a s

F ig .5 .

I pvl l IP(O, I ) IQ(0, l )

I I ] . . . . . . . x" I T ! ) ....... ~'I I I . . . . . . . . . . . . ~'J

. . . . . . . . . . . . . . . . . . . . Jl

~r(L, )) e : ~ c l i ~ l t ( ] e r

%

i , C ) ..... - - -

ratine ~i~e~ ~i1 t|~: ~om~u~cr $ I

UR'Si FI8.5

T h e f e e d f o r w a r d c o n t r o l c a n be c o n s t r u c t e d f r o m the f o r m u l a r ( 5 ) , w h e r e

Qo i s t h e m e a s u r e d t e m p e r a t u r e in t h e cool w a t e r e n t r a n c e , r u a n d p v a r e

c o r r e s p o n d i n g l y t h e f l o w i n g a m o u n t o f w a r m w a t e r a n d coo l o n e , q i s t h e

t e m p e r a t u r e g i v e n .

73

As f o r t h e r e a l i z a t i o n of i d e n L i f i c a t i o n , w e c a n n o t i c e the.

r e J a L i o n s h i p of s t e a d y s t a t e (51 a n d <51. S u p p o s e

R = ( Q I - O , ) / ( I~ - 0 . ); w=(r ') -I~, ) / 1 0 0 - l ' l )

X={bb,b) L); B=iO,ln((R-l) /(tq-l l ))

we have

B=AX 18)

B a s e d on f o r m u l a r ( 8} ,we c a n i n d e n L i f y t h e p a r a m e t e r b l , a r id b | L , u s i n g t h e

L e a s t - S q u a r e m e t h o d o r o t h e r s , c o n s e q u e n L l y , t h e v a l u e s o f p a r a m e t e r ~ & / c * L

a n d a / d * L n e e d e d to be i n d e n L i f i e d in t h e m o d e l a b o v e a r e w o r k e d o u t , b y

Lhe f o l l o w s

b L = a / ( c p v ) * L ; b L=a/(dru)*L

T h e r e a l t i m e e x p e r i m e n L i s o p e r a L e d b y a n I B M - P C - X T c o m p u L e r w i t h 16

D/A an( t 2 A/D. T h e r e a r e s i x v a r i o u s w h i c h , :an be m e a s u r e d , ~ e h e r e t~ ,P l t ,%

'Qt a v e a l l l r a n s m i t l , c d i n t o 0 - 1 0 mA e l e c L r i c s i g n a l hy Lhe h o t - I * e s i a t : t n c e

I . e m p , , r o t u v e s e n s o r , p v b y a n e l e c t r i c l o n ~ , - r a n g e r o t o r f l o w m e L e r a n d r u by

a g a s l o n g - r a n g e r o L o r f l o w l r t e t e r p l r )8 l g a s e l e c L r i c l . r , t n s m i L i o n .

T h e f l o w i n g or p r o g r a m is s h o w n a s F i g . 6 .

21 S c h e m e 2. W h i c h h a s Lhe s a m e ain) a s S c h e m e l , buL us i r tK ru a s

, :onbl 'o l . T h e c o n L r o l p r i n c i p l e i s a l l nosL Lhe s a m e :rs uhowt ) it) l : ig ' .5 . B a s e d

on f o , - t n u l a r ( 5 ) ( t h e f o r m u l a r (6) c a n ' t , b e u s e d ) f o r I.h*~ c o r r e s p o n d i n g v a l u e

of PQ i s n ) L k n o w n y e t , a n d it. v a r i e s m u e h l , L h e o r e k i c a l l y we c a n g i v e t h e

v a i t l e of r u c o r r e s p o n d i n g Lo t h e g i v e n v a l u e q by s o l v i n g t h e e q n a t i o l r

d i t ' c c L l y , but; i t ' s i m p o s s i b l e in p r a c t i c e , f o r il.'~'-; a t r : t n s c c n d e n t i a l

i : q t l a L i o r l c o n c e r l l , i n g r u . [ l l L h i s case,we (2alrl g e t the n e , r d c d v a l u e of r u t )y

t h e n u m e r i c a l b i s e c t i o n s e a c h meLhod o r t i r e n u m e r i c a l i t e r a t e d

i n L e r p o l a t i o n meLhod .

"Fire r e a l i z a L i o n o f i d e n t . i f i c a t i o n ira S c h m e m 2 is j u s t t h e s a m c a s in

S c h e m e 1. T h e f l o w i n g of p r o g r e s s i s s h o w n in I , ' ig .7.

3) The experimenLial r e su l t s (FIG.8 is for scheme I and FIG.9 for scheme

2} a n d khe c o m p a r i s o n wiLh t h e c o m m o n LPS p l a n . {:q,>. ~he f o r m ni l P7I

[,'rom t i re c o m p a r i s o n a b o v e , i t ' s s h o w n I, h a t Lhe c o n t r o l p l a t l s b a a e d on

DPS are much super ior to tha t based on LPS in ovcrcoming time-delay and

deducing thc t r ans i t ion timc and so on. Besides,the accurate in s teady

s t a t e i s s a t i s f i e d a n d t h e q u l i t y o f t h e c o r r e s p o n d e n c e in t h e d y n a m i c

p e o g r e . s s i s h i g h , L e o . T h e s i m p l e a d a p t i v e c o n t r o l c a n s a t i a f y Lhe s y s L e m

w i t h ~LII I.he o p e v a t . i o n .~iLuaLion Lo s a t i s f y t h e i d e a l s e q u a n c e

au L o m a L i e a l l y .

7 4

I i n i t i a l pro,lram I

I drawin!j scheme I

I i n i t i a l deeidin,q I I of p ick iwmtime I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[decidin:} picl~iug-t.ime as 5 sec . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

I pickin.9 Q ( j ) . p v ( j ) , r u [ . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

Icahzulatin.q P from modell . . . . . . . . . . . . . . . . . . . . . . . . .

I i n i t i ; d pr~,'lrcm I

I dvawin,~l scheme I

I i n i t i a l dechth~,j I [ o f p ick in : l - l . in le I

Imal~in~) p ick ing- t ime as 5 sec.l . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

k . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ipickin9 I'(j),woi'kin.~l ~" in PID[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I outvut hitln a l i m i t O<V<,! I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II,ickilxtl I '( . i ) .cah:ulal . i luj V in I'[IJI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

lout.put with a l imi t O<V<,t I . . . . . . . . . . . . . . . . . . . . . . . . . .

[pickiml ~|(,J),['(,i) I

. . . . . . . . . . . . . . . . . . . . . .

Idra~in;i dots Q( j ) ,P ( . i ) l . . . . . . . . . . . . . . . . . . . . . .

~'--,~"--- Is {] s teady? ~Y

[t-Y.L-ts fl equal to tim value ,.jiven? I I ~N I . . . . . . . . . . . . . . . .

L I i d e n t i f i c a t i o n l I . . . . . . . . . . . . . . . .

I'L-N "- Is the picki,x9 number aclnieved? IY

end FIG,6

1 Idck in ! l I J ( , i ) . r ,v ( . i j , ru ( , i ) I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I,. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I ealeulat . in: l ru from model I

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Iconver t in9 ru in to U,outputl . . . . . . . . . . . . . . . . . . . . . . . . . . . .

'~--N S - - - Is Q steady?

~v Ly., I s Q equal l.o the value .~jiven?

l i d e n t i f i c a t i o n l

I Z - ~ - Is pickin~j numbcr achieved?

end F16.7

35i . . . . . . . . . . . . .

31 l" . . . . . . :.,~ ~,.

f " ~ ' ~ " " ~,"'~L

t 1 3 , 8 5 l i r i ,

I s 6

- l o

30

IB

Pl/l+

75

, 40

]

t 2g

lq-

",,./ I i ,

• - ' - - - - -....,,-_._.._, lq, l

30

. . . . . . . . . . . . . . QII

~eL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

FIG.8 lqG.,q

1 l d i s t u r b a n c e ] p u r e l y 1 t r a n s i t i o n I s u p e r l a c c u r a t e o f l I I Ide lay Lime I t ime I pcrCellL Is teady s taLe

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

{ 1 6 5 - 1 3 7 . S L / h l <20 see . t l 1 . 7 5 mi l l . I 5 . ! % l 0 . 3 ! % [ } Scheme I I . . . . . . . . . . . I . . . . . . . . . . . I . . . . . . . . . . . I . . . . . . . . . . . I . . . . . . . . . . . 1

I (LPS) I 137.5-100 I <20 1 8 . 1 8 I 3 . 3 I 1 .27 I I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ISchcme2(LPS)t I . IO-200 { 0 .18 rain. I 6 .25 I 1.1 { 0.{~3 I

f eedback [ 90-50 { some 1.0 I U5 1.8 I <1.5 1 ~ i t h P i l l t . . . . . . . . . . . ] . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . 1 (one l o o p { 1.15-100 I some 1.0 1 35 2.,'g I < l . f i I c o n l ; r o l ) I . . . . . . . . . . . I . . . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . { (OI)S) l I00-168 l some I.O { 3Z.5 1,2 I <i.5 I

76

In t h e o t h e r h a n d , w e c o m p a r e t h e S c h e m e 1 a n d 2 . S h o w n b y t h e e o m p a r l s o n

of t h e i r r a t i o s o f t h e t r a n s i t i o n t ime to t h e f l o w i n g a m o u n t v a r i e d a n d t h e

r aL ios of t h e s u p e r a m o u n t to t h e f l o w i n g a m o u n t v a r i e d , w e c a n d e f i n i t e l y

g e l a c o n c l u s i o n t h a t t h e p l a n u s i n g f l o w i n g a s c o n t r o l i s m u c h e f f i c i e t i o n

to o v e r c o m e t h e d i s t u r b a n c e o f coo l w a t e r ' s f ] o w i n g . T h i s c o n c l u s i o n g i v e a

p r o o f of t h e r e a s o n a b i l i t y of t h e m a t h m e t i c a l mode l a b o v e a n d on t h e m e a n -

w h i l e , i t p r o v i d e a r e a s o n a b l e c o n t r o l p l a n f o r t h e p r a c t i c a l e n g i n e e t ' i n g .

r e f e r e l ~ c e

1 . G g u a n g - Y u a n t i u a n g , "ApplicaLior~ of D i a t r i h u t e d P a r a m e t e r S y s t e m H e t h o d s

in E x c h a n g e r ' * A u t o m a t e d T e c h n o l o g y ~ 1 9 8 2 , N o . l

2 . G a n g - Y u a n l l u a n g , < < T h e r o y in D i s t r i b u t e d P a r a m e t e r SysLem>> P u b l i s h e d

b y Sha l~Dong U n i v e r s i t y

3 . < < h u t o , u a t i o n in C h e m i c a l I n d u s t r y > > P u b l i s h e d by Z h e j i a n g U n i v e r s i t y

~OEII~T ~TABILIZAT~ON AND F I N I T E DIHENSIONAL CONTROLLE~

DESIGN ABO~T A C L A ~ O F D~ST~IBUTED PA~A~EI'E~ SYSTEMS

Hu S h u n - J u Xu £ i a n - Q i n

b e p a r ~ m e n ~ or ' c o m p u t e r a n d ~ y s ~ e m

~ c i e n c e s , N a n K a i U n i v e r s i t y

A b n e g a t e : i n ~ h l s paper, we d i s c u s s e d ~ h e ~ o b u s t n e s ~ Or"

d i s c r e t e s p e c t r a l s y s L e m ~ u n d e r t h e p e r t u r b a t i o n s o i ' ~ h e o p e r a t o r A,

~ a v e t h e c o r r e s p o n d i n ~ m a x i m a l r o b u ~ n e s B m a r ~ i n a n d ~ h e m e ~ h o d

o1" F i n i t e d i m e n s i o n a l r o b ~ c o n t r o l l e r d e s i g n . I n a d d i t i o n , we

e x t e n d m a j o r r e s u l t s t o a l a r c e r c l a s s oi" s y s t e m s , w h i c h ~ a ~ i ~ y t h e

~ p e c ~ a l d e c o m p o s i t i o n a s s u m p t i o n a n d t h e u n s t a b l e p a P ~ i s t ' I n i t e

d i m e n s i o n a l .

Key ~ o r d s : d i s c r e t e s p e c t r a l system~, ~ o b u s t n e s ~ mar¢in ,

r o b u s t c o n t r o l l e r , spectra l d e c o m p o s i t i o n a s s u m p t i o n

INTRODUCI"ION

0I" / a t e y e a r ~ , t h e r o b u s t n e s s ~ t u d y o i ' d i s t r i b u t e d p a r a m e t e r

~ y s t ~ m s i s p a i d a t t e n t i o n t o m o P e a n d m o p e F U r d i F 1 ' e r e n t

c l a s s e s o i ' s y s t e m s a n d d i t T e P e n t c l a s s e s o i ' p e r t u r b a t i o n s , p e o p l e ~ x v e

d i ~ ' ~ r e n t m e t h o d s o ~ r o b u ~ c o n t r o l l e r d e ~ i ~ n , 1 ' o r e x a m p l e

[ i J , [ ~ J , [ 3 J . ~ u t , t h e a u t h o r s o1" t h e s e p a p e r s a x e a l l t ' r o m f r e q u e n c y

d o m a i n t o d e s i g n t h e ~ o b u s t c o n t I ~ 3 l t e r s ot" ~ y ~ t e m s I ~ r t h e

p e r t u r b a t i o n s o i ' ~ h e i r t r a n s t ' e r f u n c t i o n s . So 1"at , n o o n e h a s s t u d i e d

t h e r o b u s t n e s s oi" d i s L r i h u L e d p a r a m e t e r s y s t e m s u n d e r p a r a m e t e r

p e P t u r b a t , i o n s , H o w e v e r , i n t h i s p a p e r , we a r ~ ~'rom L i m e d o m a i n t o

d i s c u ~ i n d e t a i l t h e I ~ o b u s t n e s s o f d i s c r e t e s p e c t r a l s y s t e m ~

u n d e r p e r t u r b a t i o n s o~' A, ~ i v e t h e m a x i m a l r o b u s t n e s s m a P ~ i n a n d

d e s i g n m e ~ h o d o~' ~ ' i n i t e d i m e n s i o n a l c o n t r o l l e r s . F u ~ h e r m o r e , ~ e e x t e n d a b o v e - m e n t i o n e d m a j o r r e B u l t ~ t o a l a r ~ e r c l a ~ o r E y ~ t e m ~ .

C o m p a r e d w l ~ h [ 1 J , ~ h e m e b h o d o~" c o n t r o l l e r d e s i g n i n t h i s p a p e r h a s

t ' o l l o w i n ~ c h a r a c t e r s : F i r s t , we h a v e n o ~ t o m a k e i ' i n i ~ e d i m e n s i o n a l

a p p r o x i m a t i o n o~' s y s t e m s , s o ~ h e a p p l ~ o x i m a ~ i o n d e ~ r e e h a s n o e ~ ' ~ c L o n

~ h e r o b H ~ n e s s ; ~ e c o n d , t h e ~ r a n s r e r t ' u n c ~ i o n G÷A o1" ~ h e p e r t u r b e d

~ y ~ t e m h a ~ n o t ~o h a v e ~ h e ~ a ~ e n u m b e r o i ' u n s t a b l e p o l e ~ a s t h e

t r a n s I ~ P ~ n c t i o n G o f t h e o r i $ i n a l s y s i ~ e m . ~ o , ~ e o v e ~ o m e m a j o r

s h o r L c o m i n C s oi" ~ h e ~ e ~ h o d i n [ 1 ] . F o ~ d i s c r e t e s p e c t r a l s y s t e m s ,

t h e s t a b i i i z i n ~ s p e e d a n d t h e m a x i m a l r o b u s L n e s ~ m a r ~ i n c a n b e c h o s e n s u r I " i c i e n b l y l a r & , e , L h e ~ e c a n n o t b e c o n s i d e r e d i n [ 1 J ,

T h e p a p e r i s d i v i d e d i n t o I ' i v e p a ~ e . ~ i , t h e d e ~ r i p t i o n ol" d i s c r e t e s p e c L ~ a l s y s t e m s a n d c o r r e ~ p o n d i n ~ p e r t u r b a t i o n s ; ~ a n d ~ 3 ,

r e s p e c t i v e l y ~ h e e r I ' e c ~ o ~ p e ~ u ~ b a L £ o n s Ro~ ~ o p a ~ t s oF ~ h e ~ y s ~ e m ;

~ 4 , t h e r o b u ~ n e s ~ a n d t h e c o n t r o l l e r d e s i g n ; ~ , t h e e x t e n s i o n o r

m a j o r r e s u l t s i n a l a r C e r c l a s s o r s y s t e m s .

78

~I THE D E ~ G ~ I P T I O ~ OF SYSTEMS A~D PErTURBATIOnS

First, we give the concept oF (D~ operator

Definition I: A llneaP operator F onBanach space X

(I}) operator, if" F has a Pe~ula~ point at least and Lhe~e

unconditional basis ( ~ n ) ~ = l O~" X , a c o m p l e x number sequence

and p o s i t i v e integer N O such LhaL

limllnl = " + ~ , A n ~ A m , 1 " 0 ~ V n ,m>N o

F [ ¢ i ' " ' " '¢NoJ~ [ ~ i * " ' " '~NOJ

and the spectrum or" F on space [~j , . . + , @No j are

N o { A I , 1 2 , . . . . ~Wo} , where [ ~ , , . . . ,~NO j i s t h e space LhaL ( ¢ ~ = 1

s p a n

is cal led

e x i s t an

"(An J'~'=i

'Fhe dlscret~e Sl~ect.ral system%haL we will dlscuss ts

~<t) = Ax(L) ÷ Bu(t) LeO (I~

where the state space is Hilber% space H , A is a (D) operator on H,

It~.q elgenvalues and generalized eigenvecLors are (X~: I and {~n)7=i ,

without lost of ~enePa~iLy, we assume RelnaReln+ I ,n:i,2, ...

Go~espondin g ei~envalues and ~enerailzed elgenvecLors of" A n are

{~}~=i and (~}~=j ,((~n '~n )} is a biortho~onal sFs~em. FuLhePmoPe,

we ~uppose LhaL Lher~ is a HiiberL space V such LhaL I(~--#V is a

cont.inuous dense embedding. I npuL space U is HilberL space, input

operaLo~ B satisfies the cond±tlon HI(B)S,O;V,H) , t ha t . iS, B~LCU,V),

L(H,V) is the set oi" boundedllneaP operators on H, and i'oP every

u(t)~L2(O.t;U).

l l JoS(L-s )Bu(s)ds i | H ~ ~I|Ul tL,(O. t ; O ~

fo~ some constant c>o, where S(L) is a C o semigPoup on H that A

generates (the existence or the C o seml~oup wlll be proved in the

second pa~L of the paper) .

In light of paper [4J, the system (I) is well-posed under

above-mentloned condltlons, that is, Lhe equaLion (I) on H has an

unique wild solution Top every u.~_L,(O.t;U9 and every initial values

Xoeil ,

S ( t - o ) B u ( o ) d o

Let T be the perturbations ~or operator A , the system under

p e r t u r b a t . i o n s T Is

~ ( L ) = ( A * T ) x ( ~ ) + B u ( L ) LaO ( ~ )

~uppose the stabilizlng speed or the closed loop system that

we w1~h is not less LhaD ~.~, where ~<O ~ then ~can choo~e a na.hure

79

numbeP N such that ~eAN~A>ReAN, i .Let Slmspan(~ }N= , ,H2--span(@ }~=N. ,

, A mA/H which is meant that A is the limitation of A o n H ,

I-I,2 . Similarly, V" can be divided into V ~ VleV 2 , where Vi=H l ,

H2~-~V 2 is a c o n t i n u o u s d e n s e embeddlnc~ We define B as B t u = B u / V L

,i=I,2, l ~ r every n~Lz(O,~;U). From p a p e r [4] , we know 81~(U,V i) ,

B ~ ( U , V 2 ) , a n d B 2 satisfies the condition HIfBz,S2,U;V2,H2),where

S2(t) is the C O semlCrou p on H 2 that A 2 cenerates Suppose

perturbations T satlsI~y T I | I E H l , T H 2 ~ 2 ,then ' rffi(~ t ~ 2 ) ,where T L e T / i t ~

,I=1,2 .in f a c t , T satisrylng the c o n d i t i o n exist {enerally, f o r

example, the perturbations or chances oi" the interior dampinc

coefficient and the bendin~ stiffness in elastic v i b r a t i n ~ systems

under no exterior dampin~s , the perturbations of every cofficient

in heat conductin~ systems ,and so on .

Thus, we divide the system ( 1 ) into the system-i and the

system-2 , divide the perturbations T for the system ( 1 ) Into the

perturbations T i f o r the system-I and the perturbations T 2 For the

symtem-I .

Later, we let p(A) denote the resolvent set of A, o(A) denote

the s p e c t r u m o f A .

< 2 THE EFFECTS OF PERTURBATIONS FO~ THE SYSTEM-2

To o b t a i n o u r m a j o r r e s u l t s i n t h e s e c t i o n , we h a v e t o

i n t r o d u c e f o l l o w i n C i e m m a s .

LEHHA i : ( s e e [ S i ) . S u p p o s e t h a t E is a l i n e a r o p e r a t e r o n B a n a c h

s p a c e X, F ~ ( X ) , ~ ( X ) i s t h e s p a c e o F b o u n d e d l i n e a r o p e r a t o r s o n X

, r i s a n a r b i t r a r y c o m p a c t s u b s e t o f p ( E ) . i f

IIFIl<minllk(<,E) l1-1 then F c p ( E + F ) .

LEMMA Z: ( s e e [ 6 ] ) . S u p p o s e t h a t L i s a ( D ) o p e r a t o r o n H i l b e r t

space X ,F is a linear operater on X , I])0 ,they satis/'y

itFxll~PllU6xlt 6 < J , f o r V x ~ O ( L )

I f the eicenvalues ( Z n ) ~ = £ o f L satisfy

7n=cnP[i+o(n--i)l , p(I--6)>3/2 , where n=1,2,3,...

then L+F is still a fD) operator .

Now,we can obtain our major results in this section .

THEOREM 1:If' sup(Rek: kc-~(A)}<+~, then A cenerates a C O

seml~roup S(L) on H, A 2 ~enerates a exponential stable semiCroup

S2(L) on H 2 . Futhermo1~e , for an arbitrary constant a satlsFyln~

keAN~)o>~ekN. I ,there Is some constant H>O such that

supIRe~, :~,~-op(A2÷Tz)} ~ c~

8O

o n l y i1" b o u n d e d l i n e a r p e r t u r b a t i o n s T 2 o n 1t 2 s a t i s f y t h e c o n d i t i o n

t i t 2 t l ~ ( a - o ~ t N

w h e r e o i s t h e e x p o n e n t i a l ~ r o w t h c o n s t a n C of" S 2 ( t ) . When AN÷ t and

A t a p e s i m p l y , we c a n t~ake o ~ e A N + t a n d A o = s u p ( ~ e X ' X ~ o ( A ) > = R e ~ t

P r o o f : B e c a u s e A i s a ( D ) o p e r a t o r , t h e r e a r e a n u n c o n d i t i o n a l

b a s i s ( ~ > ~ = t o r ' H, a c o m p l e x s e q u e n c e ( A n ~ = 1 a n d p o s i t i v e i n t e g e r

N O s u c h t h a b

~ T I X . I = + ~ , A ~ A , r o t n . m > N o , n ~

A~,~ "= A,~¢ n , n>N o

A [ ~ l ' " ' ' ' ~ N o] ~ [@t . . . . '~o J

~ n d t h e s p e c t r u m o f A l i m i t i n ~ o n l e t ' " ' ' ' ~No j i s (A t ~r t = l

we l ~ t x l = ~÷~ , % . . . . . ~N o J ' ~ " L ÷ s . ~ " ~ . . 2 . . . . J

O b v i o u s l y H = X l O X 2 , A X I ~ X £ , A X 2 ~ 2 . O p e r a t o r A.~X t i s a b o u n d e d l i n e a r

o p e r a t , o r s i n c e Jt I i s a l ~ l n i t ~ e d i m e n s i o n a l s p a c e , ~ o o p e r a L o r A/~( t

g e n e r a t o r s a C o - S e m i ~ r o u p . I~y H i l J e - Y o s i d a L h e o r e m , we k n o w t h e r e

e X i ~ i . s H ~ O , A o > S U p ( ~ e A : A ~ - O p ( A ) ~ s u c h ~ h a ~

il L C ~ I - A ) - t i ~ x s il ~ N s t l x t t l / ( ~ e k - A o ) ~ , ~ e k > £ o

F u C h e r m o r e . i . h r o u ~ h d i r e c t ~ l y c h e c k i n ~ we k n o w e a s i l y A o = ~ e k t w h e n k i i s s i m p l y .

N e x C , we d i s c u s ~ o n X 2 . F O r a n a r b i C r a r y A s a t J s ~ ' y i n ~ ~eA>A o

, w e k n o w A ~ p ( A ) b y ~ h e h y p o t h e s e s i n t h e o r e m l . F o P a n a r b i t r a r y x 2 ~ z • '~e have

,co x 2 =n~N÷ t ~,',~bn

( A I - A ) - S x 2 " ,~N+sc~, ", A - A ~

-- 0

cO O t ~- ~ ____~&:&__ - MoA-A o n=N-'t",~ ~ o A - A ~ ~n

FoP a n a r b i t r a r y { i v e n p o s i t i v e £ n t e ~ e r K , we m a y d e d u c e b y a n a l o g y :

(. l, Ce~. _.,ik o ) k n~N÷ i((n (~__k ~k

Obviously ,

81

~.~._)ko)k I ~ I , t ' o r a l l p o s i t i v e i n t e g e r k ~ l

Le t . H 2 d e n o t e t h e uncondi t . tor=al b a s i s c o n s t a n t w i t h r e s p e c t t o b a s i s

( ~ } ~ t h e n we h a v e

( ReA-~n ~ k I i [ ( h l _ A ) - t j k x 2 1 1 ~, 1 [ lr~.~N+te(r~ - . . . . . . . @~ ! [

( ReA_A o } k ( ~ _ j o ) ~¢

H ~ I I x II

( k e A - A o )

t 'Or k = = l , . 2 , . . .

A + a J . n , H=JEieX 2 , (~kJ--A)-i~(i~; X i , (X I -A ) - t } (% (_; X z , X i a n d ~(2

a p e c l o s e d l ~ n e a P s u b s p a c e ~ o f ' H, SO, ~ ' o r V x ~ H , V A S a t £ s ~ y i n ~ k t e A > £ o

• ~ e h a v e

X ~-~ X t *I. X 2 X1~6~)( i x X 2 ~ R 2

k ~ ( . ~ . ; A ) k x == i~()k;A)kXi + R ( . ~ . ; A ) k x 2

| | I ' ~ ( ~ ; A ) k X | | ~ | | ~ ( ~ ; A ) k X l | f -('- I| P~ (~ ;A)~X2~t

t l x / II + l l x 2 II ( R e ~ _ h o ) k ( R e A _ A o . ) ~ +

B e c a u s e k a n d 1

e x i s t ~ H a s u c h t h a t

m a x ( H j , H 2 } ( l l x t l l + llx211 )

(keA--Ao)k

~ 2 a r e C l o s e d l i n e a r subspaces o i " H , t h e r e

l l x t l l .e I I x 2 i l ~. tl:}llxli

l l k ( ~ ; A ) k x l l ~ t max(Hi,H2~H 3 l l x l l k=1 ,2 , ( k e ~ _ h o ) k . . . .

I I k f ~ k ; A ) E I I $ "1 m a x ( M I , M 2 ) H ~ K ' I , , 2 , ( R e A _ ~ o ) k ' " + "

By I i l [ J e - Y o s i d a t h e o r e m , o p e r a t o ] P A { e n e r a t e s a ( ] o s e m i ~ r O t l l D S ( t )

o n I ! . When e i C e n v a l u e . ) ' l i s s i m p l y , we c a n t a k e Ao=i t eA j b y p ~ e c e d i n ¢

d i s c u s s .

S i m i l a r l y , h ~ c e n e r a t e s a C o s e m i g r o u p S 2 ( t ) o n I | 2 , a n d I ' o e

V o s a t i s l ~ y i n ~ O ) o ) s u p ( • e X : ~ o ( A 2 ) } = R e A N + ~ , t h e r e e x i s t s a c o n s t a n t

n u m h e P H)O s u c h t h a t

I I S 2 ( t ) l l ~ He ° t t~ .O

~o S2(I . ) i s exponenC~ie l l y s t a b l e . By H i l l e - Y o s i d a theorem , we know

82

f O P V A s a t i s f y i n ~ ~el>a>o ,

H H

T h u s

lIT211 £ ( ~ - - o ) / N ( I I R ( A ; A 2 ) I I - I

F r o m l e m m a 1 , we k n o w t h e c o m p a c t s u b s e L ( A 2 o r p l A 2 ~ 1s

i n c l u d e d i n p ( A 2 + T 2 ) . F o r ~ s a t i s t y i n { ~ e X > o i s arbitrary ,

we c o m p l e t e the pPoo t " .

THEOREH 2: S u p p o s e T 2 i s a b o u n d e d l i n e a r o p e r a t o r o n H 2 , t h e

e v J ~ e n v a l u e s of" A 2 s a t l s f y i n ~

AnfcnP[i+O(n-l~] , n = N + l , N÷~ , . . .

where c is a positive constant numbe~ , p>~/2 . Then A2+T 2 is a (D9

operator .

Proof: Let ~0 ,by lemma 2 ,we know the result .

REMARK: Aitho~h p>3/2 is demanded here, yet most o f practical

distributed parameter systems all satisfy the condition . F O r example,

elastic vibratin~ systems, heat conductive systems, and so on. So the

c o n d i t i o n i s n o t s t P o n ~ .

~ [ E O ~ E H 3: Under the conditions of" theorem i and theorem

,the system (A2+T2,82) is well-po~ed on H 2 and A2÷T x lene~ates an

exponentially stable semi~roup S2(t) , whose convePcence speed can

approximate ~ s u f f i c i e n t l y

P r o o f : B y lemma 2.~ i n paper [4], we k n o w that B z satisfies

the condition Hs(B2,~2,U;Vm,M2) h~cause T 2 is a bounded operator and

82 satisgies the condition ||£(B2,S2,U;V2,]| 2) • SO the system

~A2+'|" 2 ,B 2) is well-posed o n H 2

A~aln, Az+T z is a (D) operator by theorem i and theorem 2 ,and

So f~om proof of theorem i we know fox ~ 02<02 , there exists H4>O

s u c h that

|JS2f t>J~ ~ M 4 e°2 t f o r all t~O

We take o2<0 , so A2eT x ~enerates exponentially stable semic~oup

.Futhe~more , because 02 c a n approximate ~ suITiciently,we call ~2(t)

s--exponentially stable .The proof is finished .

T o sum u p , w h i l e a ~ - e x p o n e n t i a l l y s t a b l e d i s c r e t e s p e c t r a l s y s t e m i s p e r t u r b e d b y o p e r a t o r T 2 , t h e p e r t u r b e d system k e e p s

~ - e x p o n e n t i a l ~ y s t a b , i e ' o n l y i f T 2 s a t i s f y I | T 2 t I , ~ ( ~ - c ~ / M . , w h e r e o

83

is exponentially g ~ o w t h constant o r S2(L)

determined b y A 2 , ~ satisfyin¢ Re~NZ~>o>Me~N~ i

stabilizin~ speed of the closed

Additionally, due to A n satisfyin~

,under many circumstances ,

~Mef~n-kn+i) I .... > +~

In this circumstances, t h e maximal n o r m (a-o)/H

perturbations T~ may be c h o s e n adequate ia~e .

l o o p system that

A n = = ( 3 n P [ l ÷ O ( n - l ) J ,

when n---)+~

constant M>o I s

can he chosen by the

y o u wish.

n=l,2, ...

o ~ the permissible

, then

< 3 THE EFF'~TS OF PERTURBATION FUR THE SYSTEH-2

~uppo~ T~ i~ the perturbations f'oP the sytem-i , TIf[Pj]NX N

N • r~ can be w~iLten T = Z . p w h e r e " , i ~ , J = l t j E i j E t j = E e ~ 3 ) N w N

= i l t ' o r k= i , l = j ekt 0 1"o~ otherwise i,j=i,2, ... , N

Thus , the system-I u n d e r t h e p e r t u r b a t i o n s T i is

N ~(t~ ~ ( A t + t ~ j = ~ p i j E i ) x ( t ) + B ~ u ( L )

FoP t'IniLe dlmensional systems with the

Fop Vx~H~

cJass or" parameter perturbations , their robustness have been discussed in many papers

, fop example [ V ' J , [ 8 ] , [ g J , [ 1 0 J , All Controller design methods can

be divided into two classes by parameter chan{es oi" systems: When

the parameter chan{es bein C ~nall , people desicn the robust.

controller by sensible f'unctions, for examples [BJ,[gJ ; When the

parameter chan~es bein CbiC ,people d i s c u s s the robustness o F systems

under the condi t ion that parameters chance in some prescribed convex

sets , in this circumstances , there are two kinds oF desJcn

methods with the difTerent objective in ~obust controile~ desicn

: the t'irsL is the cuaranted cost method that Euaranted the cost t'unction is up[~r bounded, for example method I and method ~ in ['(];

the second only ensure the close~ loop system is stable, for example

method a in [7]. Because we consider majoriy the stability or" closed

loop sys~ms, we yill use the design method 3 in [TJ that is chanced

into~uarantin C the stabilizin C speed oI" closed loop systems.

Uerore we do this , we have to make the followinC assumptions :

~sumptions: (i~ The permi~ible perturbations 'l't chan~e only

in a compact subset oT R NXN that is ,aLj~p~a%b j i,j=i 2, N • w ~ ° ' ' "

Furthermore , aij~O , b~jaO ;

( i i ) The s t a b i l i z i n C, s p e e d of" t h e c l o s e d l o o p

system i s n o t l e s s t h a n ~ l , w h e r e a<O . ~ o t h e c o s t F u n c t i o n I s

2~t T T ~=I~-- [X ( t )Qx(t )+u ( t ) R u ( t ) ] d t

w h e r e Q ~ ' ~ . QaO ,Re~ H~H . ~>0 ;

T t

84

(iii) All stale variables are a v a i l i a b l e ;

(iv* [AI+TI,BI] forms a controllable pair , where

is a arbitrary permis~able perturbation operator ;

i

(V) [ A I ÷ T t , Q 2 ] f ' o rms a c o m p l e t e o b s e r v a b l e p a i r ,

is a arbitrary permissible perturbation operator , T 1

( Q 2 ) T C Q ) Z . Q , ( Q 2 ) T is t r a n s p o s e o1" QZ.

Under these assumptions, we can design a robust controller by

the Tollowing method because these assumptions ensure that evePy

step o r the design method i s solvable .

The method i~ :

(I) ~olve the RJccati equation :

S. T ~ -iT o A I + A I S o - S o B t ~ BIS o +Qo=O , w h e r e Qo=Q ;

t T , ( 2 ) E v a l u a t e Co=-R- B t S o

(3) E v a l u a t e FO=AI+ BIC 0

(A) Evaluate T ( S o ) = 6 [ ~ - t ~ ~j =s ETtj S_Eu ~j +~N2So } . where

N N

~=,,O~lX' l lE , j l l , , ~ , j Z =.[max(~ la~j I , l b ~ j i ) j 2 ;

(~) Determine Po such that (Qd)o = Q0- PoT(So ) inde~'inite. Set j=l;

(O) Solve the equation :

S j F . I _ I +FTJ-tS - S ' B t R - t B T S j + Q J f O J J

- t w h e r e Q J = P ~ Q 4 - t + P ~ - I Y ( S J - I ) ' PjaJlqj [I

- t T ( 7 ) E v a l u a t e Cj =Cj _ t - ~ B t S J , Fj = A t + B I t J

( 8 ) E v a l u a t e T ( S ) ~ 5 [ ~ - t ~ E T . s . E + o N e S ] J t ~ = l t J J t J j

N N • 1 " I E ~ , , j j j , , j = , I I , ' ~ ~ = , [ m a x ( l a ~ I , J b j)j2 ;

K:m

( 9 ) D e t e r m i n e pj s u c h t h a t ( Q d ) j ~ //i _IQ¢ _ l + ( p j _i+pj ) T ( S j ) i S a

indefinite matrix . If pjal , evaluate u(t)=-e-atR-IB~Sjx(t)

Then u(L) iS the feedback controller rule. If not, set j f j + l , continue (6) ,

, w h e r e

~4 ROBUSTNESS AND CONTROLLER DESIGN OF SYSTEMS

S u p p o s e t h a t t h e s t a b i l i z i n g s p e e d o1" t i l e c l o s e d l o o p s y s t e m Is not. less than [al, where a<O satisfies ReANaa>o)ReXN+ i , then we

85

can design the r o b u s t controller o£ t h e system (1) by the Following

p r o c e d u r e :

(1 ) E v a l u a t e t h e e i ~ e n v a l u e s and t h e n o r m a l i z e d e i c e n v e c t o r s o f A ; (2 ) D i v i d e t h e s y s t e m ( 1 ) i n t o t h e ~ y s t e ~ - I and t h e s y s t e m - 3 by ~1 ; (3) Design a robust controllep K o f the system--J by ~3, where we Jet

the Maximal r o b u s t n e s s mar¢in b e (a--o)/H , t h a t i s .

m a x ( [ a , i [ , ~ b i # ] ~ = ( a - O ) / 2 N .

( 4 ) T h e n f o x a l l t h e l i n e a r p e r t u r b a t i o n o p e r a t o p T 2 s a t i s f y i n g

l i T | l o ~ ( a - O ) / H , w h e p e g T l l o = m a x ( l t T t l l t , | IT21}~ , tlT21t i s t h e o p e P a t o P

n o r m o f T z , l ITj l l t i s d e f i n e d a s I I T l l l i = m a x ( [ a j ~, ~ b t j ~} , K i s t h e

r o b u s t c o n t p o l l e p o f t h e s y s t e m ( 1 ) .

T h u s , we ~ i v e a r o b u s t c o n t r o l l e r d e s i g n ~ e t h o d w h e n a

d i s c r e t e s p e c t r a l s y s t e m i s p e r t u r b e d b y a b o u n d e d l i n e a r o p e r a t o r

T in the part of operaLop A, and point out the maximal robustness

stabilizing mar~in is (aw)/N .This controller make t h e c l o s e d loop

system ~ - e x p o n e n t l a l l y s t a b l e , where a<O i s chosen a r b i t r a p i l y . gnder many circumstances~ the maximal robustness m a r g i n can be chosen

sufficiently bi~ .

As compared with the contrOller design method in paper [lJ , we

overcome its two ma~or shortcomlngs. Moreover, in this method, we can

choose the stabiiizin6 speed and the robustness margin , these are not

consided in [ I ] .

~ 5 EXTENSION OF H A J O k RESULTS

L e t o ~ ( A ) = o C A ) ~ C~ where C~={AeC : ReA~O} i

o ~ ( A ) = . ( A ) n C~ , where C~={AeC : Re~<6~

D e f i n i t i o n 1: O p e r a t o r A s a t i s f i e s the . s p e c t r a l d e c o m p o s i t i o n 4 .

a s s u m p t i o n a t s o m e p o i n t ~ , w h i c h i s m e a n t t h a t o p ( A ) i s b o u n d e d a n d

s e p e p a t e d t'~om o p ( A ) such t h a t a r e c t i f i a b l e , s i m p l e , c l o s e d c u r v e can

be drawn so as t o enc lose an open se t c o n t a i n i n ~ GIn{A) i n i t s i n t e r i o r

and, o ~ ( A ) i n i t s e x t e r i o r .

T h e C l a S s o f s y s L e m ~ t h a t we c o n s i d e r ape t h e s y s t e m s t h a t o p e r a t o r A s a t i s f y t h e s p e c t r a l d e c o m p o s i t i o n a s s u m p t i o n i n some

po in t p<O and o~(A) i s the se t c o n t a i n i n g on l y f i n i t e po intS.

F u t h e r m o r e , we a s s u m e t h a t A g e n e r a t e s a C O s e m i ~ r o u p . I n t h i s

c i r c u m s t a n c e s , t h e s y s t e m ,can b e d i v i d e d i n t o t h e system--1 ( A I , B i )

and t h e system--2 ( A 2 , B 2) by t h e s p e c t r a l p r o j e c t i o n Pp , where

;lt=P~H ,H2m(I--PH)H ape ~ e s p e c t i v e l y t h e s t a t e s p a c e s o f t h e system--1

and t h e s y s t ~ m - 2 ~ t h e s p e c t p a . l s e t s o f A 1 and A2 ape r e s p e c t i v e l y

86

o ~ ( A ) a n d o~(A> ; B L u - B u / H . i - t , 2 , ~ o r e v e r y u ~ L z [ O , ~ ; , } . By t h e

deTlnltlon oF the spectral composition assumption, we know that

(AI,B i ) i s f i n i t e d imens iona l

For e v e r y one oT t h i s c l a s s o f system~ , i f Ax ~enera tes a exponentially stable semi~roup ,that Is, the~e exists some H>O

,o<O,such that IlS2(L)II~Ne °t , then there is the followin¢ theorem .

THEOREH 4; For V ~ sat}sfyln~ ~<~<0, if T 2 satisfy iIT211~(~-o)/H

, t h e n s u p ( E e ~ : k< -Op(A2+ T2) )< ~

Proof : Because R(~,A2)x ffi $ ~ e - ~ t S 2 ( t ) x ( t ) d t , t ~ r Vx~D(A2~

, V K s a t i s F y i n C ReK)o , we c a n d e r i v e

It E e ( ~ , A z ) II - I ~ ( ~ e ~ - - o ~ / H

~o - j

By lemma l , we know ( ~ ) c p ( A 2 + T 2 ) , T o t V~ s a t i s I " y l n ¢ g e ~ > o . So ,

s u p < R e ~ : ~ O p ( A 2 ÷ T 2 ) ~ < a

we f i n i s h t h e p r o o f .

F rom t h i s t h e o r e m , we know t h e s y s t e m ( A 2 B 2 ) u n d e r t h e

p e r t u r b a t i o n s T 2 s a t i s f y i n ¢ l I T 2 l i ~ ( o - o ) / H i s a i n p u t - o u t p u t s t a b l e

, w h e r e i n p u t - o u t p u t s t a b l e i s m e a n t t h a t t h e r e a l p a r t o f

e i ~ e n v a l u e s o f t h e s y s t e m i s n o t b i ¢ ¢ e r t h a n ~ . i f A 2 ~ e n a r a t e s a

a n a l y t i c s e m i ~ r o u p , A2+T ~ w e n e r a t e s s t i l l a a n a l y t i c s e m i w r o u p b e c a u s e

T2 i s b o u n d e d on H 2 . I f s o , t h e s y s t e m ( A 2 + T 2 , B 2) i s ~ - e x p o n e n t i a l l y

s t a b l e b e c a u s e a n a l y t i c s e m i ~ r o u p s s a t i s £ y t h e s p e c t r u m d e t e r m i n e d ~rowLh a s s u m p t i o n .

S u p p o s e t h e p e r t u r b a t i o n s T s a t i s T y TH~cH l . TH2cH 2 , T c a n be

divided i n t o p e r t u r b a t i o n s T i f o r t he system-I (AI,B ~ ) 8nd

p e r t u r b a t i o n s T 2 f o r t h e s y s t e m - - 2 ( A 2 , B 2 ) . F rom a f ' o P e s a i d d i s c u s s we

know the system (A~+T2,B 2) i s a i n p u t - o u t p u t s t a b l e when T 2 s a t i s f y

llT21l~(~-o)/H ,where ~>o can be chosen. So, i f we design the robust

c o n t r o l l e r of (AI,B I) with maximal robus tnes s ma~ff}n (o-o)/H by the

m e t h o d i n ~3 , t h e c o n t r o l l e r j u s t i s t h e r o b u s t c o n t r o l l e r o f t h e system (A,B) w i t h the same maximal robus tness m a r t i n . I f A 2 c e n e r a t e s a a n a l y t i c semi{~oup , t he c losed loop system i s m-exponentially stable. But i n this section, the stabilizin¢ speed J~| and the maximal ~obustness mar¢in have some limits and can not be

chosen arbitrarily.

87

KEFEKENCE

[ l J R u t h F C u r t a i n a n d K e l t h G l o v e ~ : R o b u s t s t a b i l i t i o n o f '

i n f i n i t e d i m e n s i o n a l s y s t e m s b y t ~ i a i L e c o n t r o l l e r s , S y s t e m s

a n d C o n t r o l L e t t e r s 7 , 1 ~ •

[2J Ruth F Curtain: Robust stabilizability of normalized coprime

t'acLors: the infinite dimensional case , Report TW 2~I ,

U n i v e r s i t y O f ~ r o n i n ~ e n , NL , 1 9 8 ~

[3] Hathukumalli Vidyasa~a~ : The ~raph metric t'o~ unstable

plants and ~obustness estimates ~r t'eedback stability ,

IEEE T~ansacLions on Automatic Control , v o J AC-~ NOB ,1~84 .

14J T. KaLo : ~ Perturbation Lheory fop linea~ operator ~ ,

p u b l i s h e d i n 1086 .

[~J Lo Y u - H u : P r o p e r t i e s o f e i ~ e n v a l u e s oF a c l a s s oF d i s c r e t e Spectral systems, Natbmatlc ~ournal 7 , i 9 8 7 . ( i n chinese)

(6J Euth F .Curtain: Equivalence o F input--output stability and

e x p o n e n t i a l 6 ~ L a b i l i L y f o r i n t ' i n i L e d i m e n s i o n a l s y s t e m s

a p p e a r e d .

|'/J O°l. Kosmidou and P. Bertrand: int.~.Control 3 , 19~3

[BJ K e r m l n ~hou and P~amond P. Kha~oneckar: S I A H j GonLrol and

Optimization 6 ,i~88 ,

tVJ Byte and Burke: IEEE Trans. Control Zl, 19Td .

| l O J K r e i n d i e E: I n t J C o n t r o l 8 , 1~08 .

THE ASYMPTOTIC REGULATOR DESIGN FOR

NONLINEAR FLEXIBLE STRUCTURES WITH

ARBITRARY CONSTANT DISTURBANCES

Li Chengzhi

Oelrrrtment of Comzmter ~ieace ~ System Science, X~w2n

Univerdty, Fujian 361005, P. R. C.

This paper investigates the finite-dimensional asymptotic regulator design for nonlinear

flexible structrues with arbitrary constant disturbances. The problem is to design a feedback

controller such that resulting closed-loop nonlinear system will be stable and the controlled

output will be regulated. With some assumptions, this paper presents explicit sufficient con-

ditions for the existence of the finite dimensional asymptotic regulator which stabilizes and

regulates nonlinear flexible structures.

1. INTRODUCTION.

The servomechanism problem for D. P .S . has recently been researched by some authors

(for example. S . A . Pohjolainen, T. Kobayashi, U. Hiroyuki and I. Tetsuo). But they

only considered the linear D. P .S . which baced on linearized models of the actual problems.

In actual operation, any controller operates on the actual structure and not the linearized

model. To author ~ s knowledge, there is no known systematic servomechanism theory for

nonlinear D. P. S . . The purpose of this paper is to generalize the robust regulator theory of

finite demensional nonlinear systems E3] to infinite dimensional nonlinear systems.

2. PRELIMINARIES.

The flexible structures considered here must satisfy the generalized wave equation:

t~(t) + 2@fi(t) + Aou(t) = FEu( t ) , f i ( t ) , f ( t ) ] (2. 1)

which relates the displacements u ( x , t ) of a structure Q from its equilibrium position due to

applied force distribution F ~ u ( t ) , t i ( t ) , f ( t ) - ] , The operator A0 is symmetric, time-invareant

differeential operator whose domain D(A0) is dence in the Hilbert Space H0=L2 (Q) with the

usual inner product (. ,. ) and associated norm t 1- [ ,the operator A0 is bounded bellow by

(Aou,u)>lallui[2 , a > 0 ( 2 . 2 )

and therefore has a square root A~/2 and a bounded reverses operator A~ I. The damping term

in (2. 1) is 2 @ t i ( t ) , where @ > 0 . The applled force distribution is seperated into the con-

89

trol force and the disturbance force.

FEu(t) ,u(t) , f ( t ) ] = ReEf(t)] + FdEu(t) ,,i(t)]. The control force are produced by m actuators with influence forces bi in Ho

m

e c E f ( t ) ] = Bo:(t) = ~ b , ( z ) f , ( t ) . (Z. 3)

The disturbance forces are given by

Fd[u(t.) , , ; ( t ) ] = Fo[u(t),u(t)] + Ew, ( 2 . 4 )

where Po is uniformly Lipschitz continuous in all its arguments, i .e. , there is a constant

k > 0 such that

II P0(,,, ,,;,)Fo(,,2,,;2) 11 ~< k[ 11 ,,,-~2 II 2 + II ~,-~, 11 =3,/2 and F0(0 ,0 ) = 0. The w is an unknown constant disturbance. The E is an unknown opera-

tor beling to L(Rq ,H0) . Measurements are made by p sensors with output:

y(t) = [y, ,. . . . . . , y , ] r = C0u(t) + Dou(t) (2. 5)

where y~(t) = (e~ ,u) + (all,u) ,i== 1, "-- ,p , and the influence functions cj, di are in H0.

When the state of (2. 1) is defined by v ( t ) = [ u ( t ) , t i ( t ) ] r , we have

v(t) = Av(t) + Bf(t) + h[v(t)] + Ew y(t) = cv(t) ( 2 . 6 )

where

[0] 0 ] B = Bo ' Fo(. , . ) , C = [Co, Do']

A = [ 0_Ao -2@I ] , D(A) =D(Ao)(~D(A~/2)

and D(A) dences in the Hilbert Space H~D(A~/2 )@Ho with the "energy norm" defined by

I1 v IIi = 1t ~ 112 + 11 A~/zu 112 ( 2 . 7 )

Lemma 2. 1 : The h (v ) is also uniformly Lipsehits continuous with same L-constant k.

Lemma 2 . 2 : There exists the unique continuous "mild" solution v ( t ) for (2. 2) given

as following •

v(t) + f' = v ~ ( O v ( o ) ou~(v,) (Bf(~) + h [ v ( ~ ) ] + gw)ds

Proof: See [4. p. 1833 I •

Now vce are able to pose the following control problem.

Problem 2 . 3 : Find a finite dimensional control system:

i~ ( t ) = y(t)-~ z(O) = 0 f ( t ) = Kz(t)

(2. 8)

where K E L ( R p , R m ) , y is an constant reference signal in Rp, such that the resultant sys-

tem

90

will behave in following way:

1) System (2. 9) will be stable.

2) The output y(t) will be regulated to an arbitrary constant reference .~, i.e. , y(t)--~

.~ as t--~oo, in spite of the disturbance w.

3, Stability of The Nonlinear D. P. S.

lemma 3.1 : When @ > 0 , A generates a group U^(t) and there exist positive constants

M and q such that

v~(O II. ~< Me'" t ~ 0 Proof: See El'] for detail

Lemma 3 .2 : When ( ~ > 0

then

we have

that

Proof: Let

]3ecauce of

the following equation is true

CA-I B = -CoA-ol Bo

J z ' = [ a , H ]

CA'IB = HBo

c = E~ ,nOA

= E-HAc,G-2@H~

=HAo = Co. According to ( 2 . 2 ) , we have

CA-I B =-CoA-olBo. | Theorem 3 .3 : When 8 > 0 , if rank rCoA~lB0]=p, then there is KEL(Rp ,Rm)such

generates an exponentially stable semigroup Ux(t) on H=H(~)Rp, i.e. , there are positive

constants M, and g such that

II ts~<t) II < Mie", t >~ O. Proof i See Appedix A for detail. |

Theorem 3. 4 : When 8 > 0, w = 0 and ~ = 0, if

1) rank~CoA~IBo~=p

2) k < g / M j

then the "mild" solution of system (2. 9) is exponentially stable.

Proof: Considering the "mild" solution of (2 .9 )

v ( t ) ] ~ ( o ) , z(t) j = Ua(t)[z(O) ]-t- foU2(C's)[h[v(S)OJ]ds'

91

we have

, r . ( , ) l to('>1 P(0)l J0M,~-.<"> II Lz(:)J I1 L:(OJ I1 < M,e,' II Lz(O)J I1 + il

According to the GronwalF s Inequali ty,

[~,:oI r,,(O)l 11 Lz(t)J 1t < M, I1 Lz(O)j 11 , ' ~ ' , ' - ' " - - - M, e-'~

where

p(o) 1 M2 = M~ II Lz(O)J [I

b~- g-M,k > O. !

d~

4, MAIN RESULT.

Considering following algebra equation:

gw [~ "qr,,l+ [',<;~']+ [_~ ]= O o ~, <"~ Difinition: The is called the equilibrium point if it is the solution of (4. 1).

Z

Lemma 4 .1 : There is one and only one equilibrium point if

, [~ ?]" , < 1/k (4. 2)

Proof: From ( 4 . 2 ) , we have

[;]=-[D %'<]'["<o>]-[D "<~'r.-~ 0 J L-yJ (4.3)

]3eeauce of

where

, [::]-[:1, < , [; o"1" l_Z~-Zz_l

(4.4)

[A o ] k,=kll c fl 0. 2) rank[CoA~iB0]=p.

3) k < g / M , 4) the equilibrium point of (4. 1) exists.

92

Proof: Considering the system ( 2 . 8 ) , there K is given in theorem 3.4. When w----'0

and ~j--= 0, according to theorem 3. 4, we know that the "mild" solution of resultant control

system (2. 9) is exponentially stable.

According to condition 4 ) , we have

where [S] is the equilibrium point o f ( 4 . 1 ) .

The "mild" solution of ( 2 . 9 ) is given as following:

,.,(o] r , , (o) l ;,tua(t_s ) h E v ( s ) J - h ( v ) _ . , r v ] ,(,)j= v~(')[z(0)J+ ([ o ]-,%j}ds

r v ( O ) ] v v , = v~(,)([~(o)j-[= ], + [~]+ ff~(~-,)["E"(S)3oJ'(")]a, We have

r,,(',] r,'l r,,':o,]_r ,,] r,,(..+,'] II Lz(t)J-Lzd I1 ~< M, II Lz(O)J Lz_l II e',' + II L=(s)-=4 II ds

Similar to the proof of theorem 3. 4, we have

["(")] r"] , , > / o , II Lz(t)J'LzJ II -<.< ~-" where

[,,(o)]_[,,] M~ = M;( 11 L~(O)J L~J II )

Since, as t --,-c~,

y( t ) = Cv(t) --,. Cv

According to ( 4 . 1 ) , we have

Li m y( t ) = Cv = y i g ~ e ; o

REFFERENCES

[ 1 ] M. J. Balas, Distributed parameter control of nonlinear flexible structures with linear

finite-dimensional controllers. JOURNAL OF MATHEMATICAL ANALYSIS AND

APPLICATIONS 108 p 5 2 8 ~ 5 4 5 (1985).

[ 2 ] S. Pohjolainen, Robust controller for systems with exponentially stable strongly contin-

uous semigroups, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICA-

TIONS 111 0622- -636 (1985).

[ 3 ] Li Chengzhi, Robust regulator design for a class nonlinear system with unmeasurable

constant disturbances. JOURNAL OF XIAMEN UNIVERSITY, to be published.

[ 4 ] A. Pazy, nSemigroups of Linear Operators and Applications to PDE 'p. University of

Maryland, College Park, 1974.

93

IS] R. F. Curtain and A. J. Pritehard, "Infinite Dimensional Linear Systems Theory".

Springer-Verlag, Berlin/Heidelberg/New York, 1978.

[6"] T. Kato, "perturbation Theory for Linear Operators". Springer-Verlag, Berlin /Hei-

delberg/ New York, 1976.

APPENDIX A: Proof of Theorem 3.3

Proof of Theorem 3. 3:According to the condition (3 .1) ,let K----eKt such that

cr(-CoAh~BoKt) ~ {~ E ~ IReL ~ 0}, where, e ~ O. Then the operator X may be written as

:. = 2 ( e ) = A, + ~Az,

where

°o], BK1 ]

Az=[O0 0 J"

Obviously o (A1)=o(A) 0 {0} and 0Ep(A) . So the spectrum of At satisfies the spec-

trum decomposition assumption [-6, P. 178]. This decomposition holds also for the perturbed

operator A(e) [6 , p379"] at least if

O < , < r o = m ' , ( I [ [0 0 Btt',] ~ r 0 j II • II R(Z;A,) II ( + 1)- ' ,

where F' is a circle of radius r* centered at the origin, running in U,,(0)-----{).E {~ [[~.1"~5}

Co(A). Let

P(e) = ~iIR(Z;7t(e) )dZ,

then the operator A(e) may be decomposed according to the decomposition of the original

space

~i = H Q R, = H + (~) e n ( ~ ) ,

where H + ( O = P ( e ) H , H ( e ) = (I-P(e))H, respectively. Let A+ (~)and X(e) be the restric-

tions of X(e) on H + (¢) and H-(0 . Then A + (e) is bounded and finite-dimensional,and

P(~)~.(e) = 2(e)P(e) = A + (e)P(e)

( I - P(e))"A(e) C - A ( e ) ( 1 - P(e)) = A- ( e ) ( I - P(t)) . The strongly continuous semigroup Ux(t) will also be decomposed according to the de-

composition of the space H. The parts U^÷¢,~ (t) and Ux¢,~(t) are strongly continous semi-

groups, with infinitesimal generators A+(e) and A- (e ) respectively [-6, p212]. Clearly,

94

the semigroup Ux(t ) will be exponentially stable if the semigroups U^+(o ( t ) and U^-(,)(t)

are exponentially stable.

Stability of U^÷(o ( t ) : Since A + (e) is finite-dimensional, stability of IDA*(,)(t) can be

seen from the spectrum.

According to the Perturbation Theory [ 6 ] , the eigenvalues ~ ( e ) , i = 1, "-- , p , of the

operator A + (e) near e = 0 are given as a converging Puiseux series

k ( e ) = ~,k + el+~aij -+- " " ,

where P i l l , and ~, i= l , ' . . ,p , are the eigenvalues of operator P(O)A2 on H+(O) . Be-

cause of

0

we have

R e k ( O , i = 1 , . " , p .

Then U^*(,)(t) will be exponentially stable, for sufficiently small positive values e.

Stability of Un-¢o(t) : Since Ux¢0)(t) is given as

U~(o)(t) = LCA_~U.(t)_CA. j

we may compute Ux(o)(t) as

XO r 1 LOA'~UA(Ox~J

An easy computation proves that U^-(~)(t) is exponentially stable on H - ( 0 ) .

Let

U(e) = P ( e ) P ( O ) --}- (]-e(e))(I-e(o)) ,

V(e) = FI - - [P(O)-P(e)]z-] ~' {P(O)P(e) + ( I - - P(O)) (I-P(e)) } Becauce of

R(,~,; A(e) ) = R(A; A(0) )-eR(~,; A (e ) ) AzR(,~; -A(O) ) ,

we have

where

P ( e ) = P(O) q- eP(e) ,

T~(e) = 2fl--~if B(Z;2(e) )A2R(X;-A(O) )dX. l'

After an easy computation ,we have

II P<e) 11 ~< ( i - r ° ) z ( 2 q- to) r" ar~

11A(0)P(e) U ~ [aro -I- r" ( l - t o ) ] ( 2 -I- ro) (1- ro) r* ar]

Jl ~ ( e ) ~ ( o ) tJ ~< M" , where

= M ~ ,

a = II A~ 11

95

. . . . . ro [2 (1 - ro )a (2+ro) r " ]-' Thus, when u<...e~..%mln/-~,[ ~ j } = c ' , w e have

tl e ( o ) - p ( , ) II < 1, v ( t ) 6 L ( T i )

V ( t ) E L(11)

U ( e ) V ( e ) = V(e )U(e ) = I , (5. 1)

U ( e ) : H - (0) --~ H - (e) ,

V ( t ) : H - (e) ~ H - ( 0 ) ,

U ( t ) = I + tU(e) ,

V (e ) = I + eF(t) ,

where II V(¢) II, II 0¢~) II, II V(~) • ~¢0) II, II X¢O) • U(O II are uniformly, bounded

for O < c < c " .

Becauce of

V(~)TA-( , ) ( t )U( t ) = T,.(,w(,)%, ( t ) ,

on H(0) ,we have.

V ( e ) A - (e)U(e) -~ A - ( 0 ) + e ( I - - P ( O ) ) G ( t ) ( I - - P(O))

where

G(e) = [A2 + V ( t ) A I "1" A,~7(t) q-- eV(t)A2

+ V(e)At 'U(e) + tA2U(r) + ezV(e)AU(e)]

Obviously there is a G > 0 , such that II ~(~) II ~<G for all 0 < e < e ' . Thus

II ( z -P (0 ) )a (~ ) (m, (0 ) )1 t < a , a t , o.

Since .T,(0) is an infinitesimal generator of an expontially stable semigroup and G(e) is

uniformly bounded, ¥ ( t ) ~ , ( 0 U (e) is an infinitesimal generator of an exponentially stable

semigroup on H(0 ) for sufficiently small opsitive values t.

According to (5. 1 ) , Tx(o(t) is expontially stable on H - ( t ) . |

OPTIMAL CONTROL FOR INFINITE DIMENSIONAL SYSTEMS

X u n J i n g Li

D e p a r t m e n t o f M a t h e m a t i c s , F u d a n U n i v e r s i t y

S h a n g h a i 200433 , C h i n a

Abstract This is a survey of some of the works on optimal control

theory for infinite dimensional systems carried out by the

research group of Fudan University in recent years.

Key words Maximum principle, dynamic programming, optimal control,

distributed parameter systems, stochastic systems.

AMS(MOS) classifications. 49B27, 93C25, 93E20.

I. Distributed Parameter Systems

Let X and Y be Banach spaces with the embedding X c Y being dense

and continuous. The dual spaces of X and Y are denoted by X" and

Y , respectively. Let U be a given metric space and T>O be a

constant. The admissible control set is U =Lm(O,T;U). Let A = ad

{(t,s) ~ [O,T]x[O,T] lO-~s<t-<T} and ~ be its closure. The optimal

control problem can be stated as follows:

Minimize T

J(v(-)) = f~ fO(t,x(t;v),v(t))dt

subject to the following

(l.i) t

x(t;v) = G(t,o)x(o;v) + fO G(t,s)f(s,x(s;v),v(s))ds ,

o-<t-<T,

( 1 . 2 ) ( x ( o ; v ) , x ( o ; v ) ) ~ S c X x X ,

( 1 . 3 ) v ( ' ) ~ U a d

We assume the f o l l o w i n g

This work was partially supported by the National Natural Science

Fundation of China and the Chinese State Education Commission

Science Fundation.

97

e [H0) X is strictly convex.

(HI) The evolution operator G: A -->L(Y,X) is strongly continuous

in A and there exist constants M>O and Os~<l, such that

M V (t,s) ~ A H ,,C(t,s),,~(v,x ) ~ (t-s) ~

Moreover, G: A -->~(X,X) is also stFongl¥ continuous and

Gfs,s) ~ I , ¥ s~[O,T] ,

where I is the Identity opermtom on X.

(HI) The mappings f: [O,T]xXxU --> Y, fo [O,T]xXxU --> ~ and their

Frechet derivatives f and fo are strongly continuous. X x

{H3) The set S is convex and closed in XxX.

LI and Yong [I0] proved the Following maximum principle.

THEOREM I.I Let (HO)-(H3) hold. Let u(-), x(-)=x(';u) De a solu-

tion of the optimal control problem. Let R-Q be of flnlte

codimensJon in X, where T

R = {~X{~ = ~0 Gl(T't){f(t'x(t)'v(t)) - f(t,x(t),u(t))}dt,v(.)~Uad }

Q = (x I- G (W,O)Xo{(Xo, Xl)~ S)

and GI(',-) satisfies

T (1.4) GICL,s)F = GCt,s)y + ~Gl(t,r)fxCr,x(r),uCr))Gl(r,s)y dP

s

V ( s , t ) ~ A , yEY .

Then, there exists a (~(-),~°)~O, such that

o ~O,

* ~iG'(s, t )f3(s, x(s), u(s) }~(s)ds (1.5) ~(t) : G (T,t)~(T) +

Te •

+ o ~G (s, tlr~(s,x(s),~(s))ds,

V t ( I O , T ] .

[ I . B ) <~(t),f(t,x(t),u(t))> + e°f°(t,x(t),u(t))

= max {<~(t),f(t,x(t),v)> + ~°f°(t,x(t),v)} V a.e. t~[O,T] vEU

<~(O),Xo-X(O)> - <~(T),x-x(T)> ~ 0 V (Xo, Xl)~ S.

98

Remark 1.1 S={(Xo, Xl) }. Thls is an optimal control problem wlth

fixed end points. Then Q={xI-G%(T,0)xo)}. Hence, if R is of finite

codimension in X, then the maximum principle holds. This result

contains that of Fattorinl [3].

Relm~rk 1.2 S={x0}xQ1, Q =Q%-GI(T,0)x O. Hence Q is of finite

codimension in X Is the same as QllS so. Hence, provided QI is of

finite codimension in X, the maximum principle holds, This is the

result of Li and Yao [9].

Optimal periodic control problem. Remark 1 . 3

Assume

(H4) G(t+T,s+T) = G(t,s),

f(t+T,x,v) = f(t,x,v),

f°(t+T,x,v) = f°(t,x,v),

s = ((x,x) tx ~ x}

THEOREM 1.2 Let (HO)-(H4) hold and let Range (I-GI(T,O)) be of

finite codimension in X. Then the maximum principle holds for the

periodic optimal control problem, i.e., there exist ~o~-0 and ~(.)

satisfying (I.5)-(I.S) with ~(O)=@(T).

Remark 1.4 We know that if X is reflexive, then, bw changinE the

norm to an equivalent one, we maM assume that X" is strictlM convex.

Also, if X is separable, then, by Day [2], ma M also do the above~

Thus, we see that (H0) is general enough to cover almost all cases

which interest us (e.g., X=C([-r,O];~n),Ll(~n),ete.).

Example I.I Let r>O, X=C([-r,O];~n). Then, X is separable. Thus,

we may endow a new norm to X so that X * is strictly convex.

Consider the following functional differential system

dx(t) (1.7) dt - f(t'xcv(t)) '

where f: ~xXx~m--~ n is a given map and x E X is defined by t

x (e) = x(t+e), V e e [-r,O] ,

whenever x(-) is continuous. Furthermore, le t f°:RxXxRm~ be given.

Assume that (H2) and (H4) hold for the maps f and fo. Now, let

G ( . , - ) be the solut lon operator of the var lat lonal equation 1

(i.8) d~x(t) d ~ - f ( t ' x t ' u ( t ) )~x t ' Y

where fy ( t , x t , v ) is the Frechet derivat ive of f ( t , x t , v) in x t. From

99

Hale [4 ] , we know t h a t G,(T,O) l s compact f o r T>r. Thus, i f t h e

pe r iod T>r, t h e n t h e Range o f I -G , (T ,O) be o f f i n i t e c o d i m e n s l o n i n

X. Hence by theorem 1.2, we get the maximum principle for the

optimal periodic control problem of functional dlfferentlal system

(1.7) without any additional condition. Here we eliminate the

conditions imposed by Colonius [I] or Li and Chow [8] in proving

the similar result.

2. S t o c h a s t i c Sys tems

Let (Q,Y,P) be a p r o b a b i l i t y s p a c e w i t h f i l t r a t i o n ~t . Le t B ( - ) be

an Rd-va lued s t a n d a r d Wiener p r o c e s s . Assume

yt = ¢ {B(s)i O~s~t} ,

Let U be a given metric space. An admissible control v(-) is a ~t_

adapted measurable process wlth values in U, such that

sup Elv(t)I m < , , Y m=l,2,... O~t~l

Denote the set of all admissible controls by Uad .

The optimal stochastic control problem can be stated as follows:

Minimize 1

J(v(-)) = E ]~ l(x(t;v),v(t))dt + Eh(x(1;v))

subject to the following stochastic system

(2,1) dx(t;v) = g(x(t;v),v(t))dt + c(x(t;v),v(t))dB(t) ,

x(O;v) = x ° ,

v(') ~ U ad

Assume the mappings

g: ~n X U--~ ~" , ~: R n x U--~ ~(~d ~n) ,

I: ~n X U ---) R , h: R" --> R .

and their derivatives gx' gxx' ~x' ~xx' 1 ,x ixx, hx' hxx are con-

tinuous. Assume gx, gxx,~x, Crxx, lxx, hxx are bounded, g, V, ix,

h are bounded by C(1+Ixl+[vl) The stochastic maximum principle x

was discussed by a lot of papers when the diffusion coefficient

independs on control. Dr.PenE [II] derived a maximum principle

when the diffusion coefficient depends on control. Assume (u(-),

x(.)=x(.;u)) is a solution of the optimal stochastic control

problem. Then there exists an unique solution (p('),K(')) of the

first order adjolnt equation

100

~ "(x(t), u(t } - dpCt ) = g~CxCt),u(t))p(t) + cr~ ) ) K j ( t .1=1.

(2.2) + l*(x(t),u(t))]x dt - K(tldB(t)

p(1) = h'(x{l)) . X

Let (P(-),Q(.}) be the unique solution of the second order adjolnt

equat ion

- dP(t) = [ g~(x{t),u(t))P(t) + P(t)gx(X(t),u(t))

d •

+ ~ o'J (x(t),u(t))P(t)crl(x(t),u(t)) X X

J = l I1 '~' •

(xct),u t %ct) + (xCt ,u(t),

42.3) + Ixx(x(t),u(t)) + gx'x(x(tl,u(t))p(t)

]=1

PC1) = h (x(1)) . X X

THEOREM 2 . 1 L e t u ( - } and x C - ) = x ( . , u } be an o p t i m a l s o l u t i o n of" t h e

o p t i m a l s t o c h a s t i c c o n t r o l p r o b l e m . L e t ( p ( . ) , K ( - ) } and ( P ( ' } , Q ( - ) )

satisfy 42.2) and (Z.3), respectively. Then the maximum principle

_ 1 ~'(x(t) u(t))P(t)~(x(t),u(t)) 2

- l(x(t),u(t}) - g'(x{t),u(t))p(t}

(2.4) - <K(t) - P(t)cr(x(t),u(t)),cr(x(t),u(t))>

1 max {- ~ ~'lXlt),v)Plt)0-tXlt),vJ" " " " " " " " " "

vEU

- iCx(t),v) - g'(x(t),v)p(t)

- <K(C) - P(t)cr(x(t),u(t)),cr(x(t),v)>} .

holds a.e.a.s.

Provided ~r independ~ on v, then 42.4) reduce to

( 2 . 5 } - l ( x ( t ) , u ( t } ) - 

: max {-l(x(t),u(t)) - <pCt),gCx(t),v)>) . vEU

This is proved by Haussmann [5].

Dr. Hu [6] discussed the following optimal stochastic control problem:

101

Let

where A J

Assume

then the maximum

Let X,Y,V be Hilbert spaces. A Wiener process with values in Y is

a ~t adapted stochastic process B(.), such that, for any eEY,

<B(.),e> is a real Wiener process in ((l,~,Ft,P), wlth the correlation funct i on

E<B(tl),e><B(t2),e'> = ( t r Q) min(t l , t2),

V e,e'~ Y, tl,tzzO ,

where Q is a positive selfadjoint nuclear operator defined on Y,

i.e., for any orthonormal basis {e}} of Y, we have

Z > < ~ . t r Q = <Q e l, ej j=1

gt = ¢ (B(s); Oss~t) .

and {e j} be the orthonormal basis of Y such that

Qe =Ae , J J J

is the eigenvalue of Q and

Aj > O, j = 1 , 2 , ' ' ' , --)- Aj < ~o . J=t

g: [0,I] x X x V --~ X ,

v: [0, I] x X-~ ~(Y,X) ,

I: [0, I] x X x V --) ~ ,

h: X ---> ~ ,

G: X --) ~m be all continuously Gateaux dlfferentiable, gx' ~rx' Gx be bounded

and g, v, 1 , h be bounded by c(l+Hxl). x x

Hu and Peng [7] considered the following optimal stochastic

control problem: minimize

minimize I

J(v[.)) = m J~ l[x(t),v(t))dt U

subject to (2.1) and

x(l;v) ~ Q

Assume Q is of finite eodimension in L2(~),

principle holds.

102

(2, S)

s u b j e c t t o

(2.6)

!

J (v ( . ) ) = E[o

t

x ( t ; v ) : etAx 0 * ~0

t

l ( t , x ( t ; v ) , v ( t ) ) d t + E h ( x ( l ; v ) )

e ( t - s ) ^ g ( s , x ( s ; v ) , v ( s ) ) d s

(t-s)^ e ~(s,x(s;v))dB(s) ,

E G ( x ( 1 ; v ) ) = 0 ,

v{ . ) a U d ,

tA where A is a given infinitesimal generator of a C -semigroup e

o and an admissible control v(.) ~ U is an adapted measurable

ad

process with values in UcV such that

s u p E l y ( t ) I z < +m • o 'c t ;Sl

Hu and Peng [7] proved

Theorem 2.2 Let u(') and x(.)=x(.;u) be an optimal solution of the

problem (2.5)-(2.8). Then there exist A~, u E ~ m such that

I~I = . I.I = : I

and (p(.),K(.)} satisfying m

p(t) = e (i-t)^ (h:(x(l))k + G'(x(1))v} x

I • ~i js-t,^ (g~(s,x(s),u(s))p(s)

d + ~ ~'(s x(s),u(sl)~j(s)}ds

J = l

~t (s-tlA" - e K(s)dB(s) ,

such t h a t

w h e r e

H(t,x(t),u(t),p(t)) = max H(t,x(t),v,p(t)) , vEu

H(t,x,v,p) = k l(t,x,v) + <p,g(t,x,v)> .

3. C o n n e c t i o n b e t w e e n Maximum P r i n c i p l e a n d D y n a m i c P r o g r a m i n g

Given (s,y) ~ [0, I] x ~n assume the mappings

1 0 3

g: [0, I] x R n x U--+ R n ,

I: [0, i] x R n x U -->

h: ~n __)~

and their derivatives gx' ix. hx are continuous.

control problem is the following:

minimize

subject to

J(s,y;v) = ~1

8

(3.1)

Denote

l(t,x(t;v),v(t))dt + h(x(l;v))

dx(t;v) _ g(t x(t;v) v(t)) dt ' ' '

x ( s ; v ) = y ,

v(') e U = L®([s,I];U) . ad

The optlmal

V(s,y) = inf{J(s,y;v) I subject to (3.1)} .

Let u(-) and x(.)=x(.;u) be a solution of the optimal control

problem. According to the maximum principle, there exists ~(-)

satisfying

{3.2)

such that

d~(t) dt - gx(t,x(t),u(t))~(t) - l'(t,x(t),u(t})x

~(I) : h'(x(1)) x

(3.3)

where

H(t,x(t),u(t),~(t)) = max HCt,x(t),v,~(t)) vEu

a . e . t~[s,l] ,

H(t,x,v,p) = -l(t,x,v) - <p,g(t,x,v)> .

The Bellman's dynamic programming says: provided V(',.) continuously

differential, then V satifies

v av(t,x)) (3.4) av(t,x) + sup H(t x, ' T = 0 at

vEU

v ( 1 , × ) = h(×)

It is very know that if V(-,-) is second order continuously differential

then

~(t) = aV(t,xft)) Ox

But,the value function V(-,-) may not be smooth. Dr.Zhou [12] proved

104

THEOREM 3.1 Let u(-) and x(.)=x(.;u) be a solution of optimal

control problem. Then

D- V(t,x(t)) c {~(t)} c D + V(t,x(t)) , x x

and

D:,xV(t,x(t))c{(H(t,x(t),uCt),~Ct)),¢(t))}cD[,xV(t,x(t)) ,

w h e r e D ÷ and D- I s t h e s u p e r d i f f e r e n t i a l a n d s u b d l f f e r e n t i a l ,

r e s p e c t i v e l y .

F o r o p t i m a l s t o c h a s t i c c o n t r o l p r o b l e m :

1

m i n i m i z e J(s,y;v) = E [ l ( t , x ( t ; v ) , v ( t ) ) d t + E h ( x ( 1 ; v ) ) B

s u b j e c t t o

( 3 . 5 } d x ( t ; v ) = g ( t , x ( t ; v ) , v ( t ) ) d t + c ( t , x ( t ; v ) , v ( t ) ) d B ( t )

x ( s ; v ) = y ,

v ( . ) ~ U a d '

I , h, B ( - ) , where g, ~,

following:

THEOREM 3.2

U a s § 2 . Dr. Zhou [ 1 3 1 , [ 1 4 ] p r o v e d t h e a d

Let u('), x(')=x(';u) be a solution of the above problem.

respectively, and

V(s,y) = inf (J(s,y;v)[ subject to (3. S)} .

D 2 P ÷ (p(t),P(t)) e V(t,x(t)) X

THEOREM 3.3 The value function V is a viscosity solution of the

HJB e q u a t i o n

a V ( t , x ) f . O v ( t , x ) + sup GL%x,v, Ox '

a t veu

V ( I , x ) = h(x) ,

where

aZv(t--~'x}J = o

Ox z J

1 = - ~ ' ( t , × , v ) S - " ~ ( t , x , v ) G ( t , x , v , p , S ) g

- - l ( t , x , v ) .

THEOREM 3 . 4 L e t u ( . ) and x ( . ) = x ( - ; u ) be an o p t i m a l s o l u t i o n , t h e n

t h e maximum p r i n c l p l e h o l d s , i . e .

G(t,x(t),u(t),p(t),P(t))-<K(t)-P(t)~(t,x(t),u(t)),~(t,x(t),u(t))>

T h e n

Let p('), and P(') satisfy (2.2) and (2.3),

105

= max {G(t,x(t),v,p(t),P(t))-<K(t)-P(t)~(t,x(t),u(t)),~{t,x(t),v)>} . vEu

References

[1] F. Co lon iu s , Optlmal P e r i o d i c C o n t r o l , S p r i n g e r - V e r l a g , 1988.

[2] M.M. Day, S t r i c t c o n v e x i t y and smoothness of normed spaces ,

Trans . Amer. H~th. Soc . , 78(1955) , 516-528.

[3] H.O. F a t t o r l n l , A u n i f i e d t h e o r y of n e c e s s a r y c o n d i t i o n s f o r

n o n l i n e a r nonconvex c o n t r o l sys tems , A p p t . ~ t h . & Opt im. ,

15(1987), 141-185.

[4] J.K. Hale, Theory o f Funct ional D l f f e r e n t l a l Equations, Spr lnger -

Verlag, 1977.

[5] U.G. Haussmann, General necessary conditions for optimal

control o£ stochastic systems, Math. Programming Study, 6[1976).

[6] Y. Hu, Maximum principle of optimal control for Marker processes,

Acta Math. Sinica, 33(1990), 43-56.

[7] Y. Hu and S.G. PenE, Maximum principle for semilinear stochastic

evolution control systems, to appear.

[8] X.J. Li and S.N. Chow, Maximum principle of optimal control for

functional differential systems, J.Optim. Theory & Appl.,

54(1987) , 33S-360.

[9] X.J. hi and Y.h. Yao, Maximum p r i n c i p l e of d i s t r i b u t e d pa r a me t e r

sys tems wi th t ime l ags , Lec tu re Notes i n Cont ro l and I n f o r m a t i o n

S c i e n c e s , Spr inger-Verla4~, No.7S(1986), 410-427.

[10] X.J. Lt and J.M. Yong, Necessary c o n d i t i o n s o f op t i ma l c o n t r o l

f o r d i s t r i b u t e d pa ramete r sys tems, to appear i n SIAM J . C o n t r .

& Optlm.

[11] S.G. Peng, A general stochastic maximum principle for optimal

control problem, to appear in SIAM J.Contr. & Optim.

[12] X.Y. Zhou, Maximum principle, dynamic programming and their

connection in deterministic controls, J.Optim. Theory & Appl.,

65(1990) , 363-373.

[13] X.Y. Zhou, The connection between the maximum principle and

dynamic programming in stochastic control to appear in

Stochastics and Stochastlcs Reports.

X.Y.Zhou, Maximum principle, dynamic programming and their

connection, Ph.D thesis, Fudan University, 188S.

[14]

NUMERICAL RESOLUTION OF ILL POSED PROBLEMS

R. Luce, J.P. Kerndvez

Universitd de Technologie de Compi~gne,B.P. 649 Compi~gne France

Abstract

The aim of this paper is to give some numerical methods and results for the resolution of ill posed

problems in linear P.D.E's.

Firstly we have studied a simple example namely the Cauchy Problem for the Laplacian Operator. To

solve this problem we have applied the" Hilbert Uniqueness Method "(H.U.M.) developed by J.L. Lions

[1] [2], and we have discuss the numerical problems encountered. We have then compared these results

with those obtained using methods of Optimal Control: Regularization Method and Duality Method.

Secondly we have applied the same methods to an ill posed problem for a linear parabolic equation.

Key Words: I11 Posed Problems, Exact controlabillity, Optimal Control, Regularization, Duality

Method.

I THE CAUCHY PROBLEM FOR THE LAPLACIAN OPERATOR

1.1 Problem formulation

Let (1) be the ill posed problem:

(i) f -Au = 0 inf2

~ = 0 onS

= f onS ~ t

Two conditions are imposed on the same part S of the boundary. The two boundaries need not

necessarily be disconnected.

The objective is to find a function u satisfying (1) on f~. This example is a standard model of an

ill posed problem in sense given by Tykonov in [3].

This problem can be posed as a problem of exact eontrolabillity:

To find u such that (1) is true is equivalent to finding a control v on S' such that u satisfies equations

(2) and (3)

107

t ' -Au = 0 inl2

(2) u = 0 onS (3) = f

~. U = V Oia$'

This problem of exact controlabillty does not always admit a solution, as it depends on the function f.

Also The H.U,M~ method pe.~its the determination of the space G which must contain the functio.n f

must belong in order that v belon~ to L2 S.(_~,

1.2 Application of the H.U.M method.

We consider the following equations (4) and (5) :

'-Aq~ = 0 in

(4) ~ ~ = g onS

l = 0 onS '

where (4) defines alinear operator K ........

K:~(S) ~ >L2(S9 ] g V=~ !

f -Ag =0 inf2

(5) ~ ~ ; 0 onS

l ¥ = V onS'

where (5) defines a linear operator L . 2 -I !

L : L (S') >H (S)

v ~v

Thtts (4) and (5) to~ether define a linear operator A, such that

-I A : ~(S) ....... >H (S)

Using the frequently cited theorem of the unicity of the Cauchy problem it can be easily demonstrated

that (6) defines a norm on ~t(S).

(6) I g I = U--~ IIL2(S,)

Let ~(S) be endowed with the norm defined by (6). We then consider the hilbert space G be the

completion of .~(S) with respect to this norm (6)

Proposition 1 : The application A can be extended to an isometry of G onto G'

A :G(S)g > G ' ~

Proof: The proof of this proposition is given in [4]. The demonstration is based on Hahn Banach

Theorems.

108

2 2 Proprosition 2: K define an isometry of G onto L (S') and L an isometry of L (S') onto G'.

Proof : The proof is given in [4].

Propostion 3: G is a dense subspace of HI(s) and G' is included with density in H- ](S).

Proof : The proof is given in [4].

Figure n ° 1

The ill posed problem is now well posed. Indeed if f belongs to G' it there exists only one g in G

satisfying A g = f, the control v is given by v =Kg, and the sotution u is V.

1.3 Numerical Resolution of A g = f.

Proposition 4: The resolution of the linear equation A g = f amounts to the minimization of 3(g) with

gEG.

I ~v 2 L - < f , g (7) J(g) = / < A g , g >G,,G- <f,g>G,,G =~H II 2(S, ) >G',G"

Proof : The proof is given in [4]

The results presented in §1.3 shows that (7) admits a minimum if f belongs to G'. But given a

function f, it is difficult to determine whether or not it belongs to G' or not. Suppose that f belongs to G'

so (7) admits a minimum; thus f belongs to H" 1. Since H 1 is included with density in G, a basis of It t

also is a basis of G. Under these conditions 7 can be minimized using the finite element method together

with a conjugate gradient method [5][4].

For a numerically attainable f the control v obtained is accurate, although the norm of the control g in

H|(S) may be very large. But a perturbation 5f of f, small in the norm of H- I(S), may be large in the norm

of G' so that the corresponding ~g may be large in the norms of both G(S) and HI(s). As a consequence,

the control v = ~ is very sensitive to slight variations of f. Numerical examples which confirna these

results are presented below.

The domain of the first example is a crown (fig n°2). The function f desired is represented on the

figure n°5. The figures n°3 and 4 show the control g and v obtained by solving the minimization problem.

The figure n°5 shows that the function numerically constructed f is correct, and agrees with the desired f.

figure n°2

109

Control 7

"7 ) . . . . . . . . . d . . . . . 2,~ " ' ~ "

- - g Controlwlth,xO--~ - - - g Control wi~a0.¢~ l I I I I I I I I I I I Ip l l l l l l l l IIIII

figure n°3 V On ~

9

0

-9 . . . . 0 4 211

Valae on S' of lhc exact v control x v nodes values on S" obtained for the g control

-4 . . . . . . . . . ,~ . . . . . . . . . Ofdesilte d 2n

" 2/t: x f node, s values obtained with g eonlrol ( 8 ~ - ~ )

figure n°4 figure n°5

Thus, for a crown the results are very good, and even when the function f is pertubed, tile results

always are accurate.

The domain of the second example is a disk (fig n°6), the boundary S being by the part of circle 3g beetween; and --~-, the boundary S' being by the complementary part of the circle. comprised

S

7.E67 ....

, . .,., .. m. • ~ 4-

5 . 1 ~ ~ gcontzol wlth ~0

- - - - gconl~,~l w i t h t i e ~


110

The figure n*7 shows the controls g obtained with different values of the step of the mesh. Notice that

the control g does not converge, the amplitude of g increases with decreasing step size.

On the other hand the controls v obtained are accurate (figure n°9), and the numerically reconstructed

function f is very good (figure nO8).

]955

~915

~.~5

x f nodes v slu¢~ obtained (A 0 ~'~ 2 )

1.0

-20

. . . . . . , , • [ . . . . . , ~ l l l l I l , l , , l l l l ~ l I I 1 1 l " ' ' ' ' l

2v desirod Ao :~2x and AO = 32~. ) 2 - - - v n o d e s v a l u e s o n S ' w i t h


However when the function f is slightly pertubed according to a uniform law (fig n°I0) the control v

obtained is not accurate (fig n°l 1).

0,1E-2

0.SE-~ t

OAE-5

Perturbation of f

i

-1

-6 d2~

~" - - f - fperturbed 2 •

figure n°10

So the H.U.M. is not directly numerically applicable.

Y i

: i i - l i

v N f " "! i \ ' , ,4 i

figure n ° 11

1.4 Regularization method.

The regularization method consists to add to the cost function the term ~ II g 112HI(s), we define JE(g) as

e 112HZ($) (8) J (g ) = J(g) + ~ U g

and we consider the minimization problem of J~(g) for a fixed e. For a detailed explanation see[4].

rain (J(g) + 211 g 1t2ti(S)) (9)

g e HI(S)

For a not too small e, we find a good approximations for f, v and g. As e tends to 0, the

approximation of f always is good, while those for g and v are not (fig n ° 12, 13 and 14).

412

-25

-4. '~.r

~ v e

20

-10

-40

-70

,,,,,,,,

v

CI) v dcsired (~ v e = l.e-5 ""'~. s l.e-10 ~)v £=1.e-12 (~v £=1.C-12

111

(10 ~°gtfl!llperturbed~ f obtalnedl) .

2

2.0

figure n°12

. . . . . . . . . . . ' . . . . . ~ . . - " S ...... g cpsilon = l.g-5 2

figure n°13

11" SEA

lEA

-6F_.A

t 3F..5 $

2 ..... g epsilon = I.E-12 2

6E5

g elp~;llon I E 13

figure n°14

The regularization method gives interesting results but the difficulty lies in the choice o f the parameter

e. The behaviour o f the results, according to ~ depends on several parameters : the geometry o f the

domain, the size o f the boundary S,ete. The method presented in the fol lowing section permit, in part, to

resolve this problem.

1.5 Dua l i ty me thod[6] [4 ] .

W e have

- an isometry L : L2(S') ~ G'(S)

- G'(S) dense in H-I (S)

t~ttJ,cb~ll B = { z ~H'~S) /Uz-r~J . . z (S ,) ~(1~,0)} Therefore V f in H" 1 (S) ] v e L2(S') such that Lv E B.

112

1 ¢) Consider the problem (I1) Inf 2 II v II~z

v E '/tad

By the l%nchel_ Rockafellar theorem: , l 2

1(12) m f ~llvlll . l l = I n f {p(v)+ ' l (v)}=Sup { - p * ( - s ) - y * ( s ) }

I v ~ %~ v~t.~ ~ L ~ . . . .

O si v E ~J'ad (Lv E B ) where p(v) = 1 II v 1122 and ?(v) = s i v ~ % ~ (Lv ~ B) and

1 ,,~ ~*(s) - - sup ((s,v)L~ ~,,)] -- sop ((s,v)L~] { (s,V)L2- 0(v) } = ~ J~ s v e 'Ead

tends to 0, the results get worse.

-3

- 7 . . . . . . . . i . . . . . . . i . . . . . . . . ...r.- ,3=_ S 2 2

13=0.1

500

-400

1300

,20oo ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . aL 2 2

13 = 0,001 .....

600

-200

-1000

figure n°15

. . . . . . . . . . ' . . . . . . . . . ° . . . . . . . ~ S 2 2

I~ = 0.005

~00o

00o

3000

2 13 = 0.00(~ 2

p*(s) -- Sup v~L 2 vcL2

,, , After some manipulation we find

I I Where s spans a dense subset o f L2(S ') as g spans HI ( s ) and

[ - A ~ = 0 dans ~ ~3fn

( t4) ~ ~p = g su rS (15) S = ~ v s u r S '

t = 0 sur S'

Finally t h e dual problem is (16) I Min p*(-s)+y*(s) ] .... [ subject to (2) (g ~ I-II(S)) I

For a not too small 13, we obtain an acceptable reconstruction for v and even g. However, as 13

113

V 1 . 0 - - -

( ~ ~ II~=O.1) . ~ : ) , (0=o.oo5) ~ v 0~=o.ooi) C)v(t~=o.ooo4)

figure n ° 16

In fact this method is more interesting than the others because it gives a result with an error on f less

dmn a given 13, and thus 13 provides a measure for the accuracy of the numerically constructed f.

The methods and the results, that we have presented in this section for the solution of the Cauehy

problem for the Laplacian operator, can be applied to all ill posed linear equations according to the

definition of Tikonov [ 1 ].Thus, in the next section we have considered a linear parabolic equation with a

control applied on a part of the domain fL

II A LINEAR PARABOLIC SYSTEM WITH CONTROL ON

ILl Problem formulation

(17) IYt-Ay =ZqV in Q B,C, y(x,t) = 0 on ~- .C. y(x ,O)= 0

Let

q = to x ]0,T[ Y~ = F x ]0,T[ with Q = ~ x ]0,T[ I"

Z o indicatory function of q

The objective is to find a control v such that y(T) is equal to a given function Yl" The problem, as in

file proceeding examples, is to determine the space G in which Yl must lie such that v belongs to L2(q).

This problem does not have a straightforward, however, the H.U.M. method allows its resolution.

Let II.2 Application of the H.U.M. method

~ -Pt-AP = 0 in Q ['Yt-AY = Xq0 in Q

(18) ~ p = 0 on E (19) / Y = 0 on

l p ( x , T ) = g , y ( x , 0 ) = 0

where (l 8) and (19) define a linear operator A

A : ,~ (~ ) > L2(~) I

I ),(x,'r) Using the Hill-Yosida Theorem, we can show that (20) defines a nom~ of D ( ~ )

114

1

(20) II g 11~(£2)'= ( ~Q (xqv)2 dx dt) 2

Then we consider the hilbert space Gi- ~ be the completion of ~ ( ~ ) with respect to this norm (20). The

situation is recapitulated in the figure n°17 where L2(~) is include with density in Gn, M is a closed

subspace of L2(q) and G~is include with density in L2(~).

K ......

figure n ° 17

This problem can not be solved by solving directly A g = Yt because the space Gf2 is very small

[2], however it is dense in L2(f~) so that it is perhaps amenable to solution by the duality method.

II.3 Duality method We consider the following ball

We define '/Lad as the set of admissible controls [%Lad = [ v ~ M / y(x,T)' ~ .....

(21) 2. . ) v a "~ad ~" C lql

2 This problem admits, at least, one solution i fy I belong to L (~)

We apply the duality method to this minimization problem and after manipulation we obtain:

.... m',',',l [ (21) ¢:~ IgnfL2 (J(g)=l l ls(g) l l2z~-(g, yl)L2+l~llgl lL 2)

f -pt-Ap = 0 inQ

with s defined such that (22) ~p = 0 on Z (23) s = ZqP

L n(x,T) = g II.4 Numerical results[4]

For numerical simulations we consider to a trucated basis of L2(~) with ~ = ]0,1[ and ~ = ]a,b[

( 0 ~ a < b ~ I ) N

The control g is decomposed on the eigenfunctions of the Laplace operator g(x) = i ~ g i sin i g x .

Figure n°18 shows the behaviour of g as a function of N, the number of terms in the series

description of the function g, and as a function of the parameter Ik This behaviour is similar to that

discribed in the preceeding paragraph. Figure #'19 shows the influence of the parameter a and b on the

control g. The control g tends to increase outside of the domain co.

115 l , , l ....

23 .4 r g I l l

13.9 / ~ ' k

. 5~ 0" . . . . . . . . d 5 . . . . . . . . d:i . . . . . . . . 1:0 O g control with 11 ~ 0,0l (~) g control with [~ : 0.001 N : 5

.... II"llll i ~ I I

i 113"649 i.~ . . . . . . . . ~.,333 . . . . . . . d.66~/ . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . fig uren°18

65.1 :l g ........ c / 15 """'" s 38,7 / e ~ ~;~.85

-14.~.;-_ ~. ~ . = , ~ o ~'"1o : i~ ~: ~ p,= o~!,,.~,,,,,~::2~ g ~ = o . o l ) ~ g (t~= o.,oooD N = , t 5

figure n ° 19

15

~g/~ a = 0 .333 / \ 1

- 159.9 6.0 . . . . . . . . 6,333 . . . . . . 6.667 . . . . . . . 1,0 ,, ~ g (1~= 0.0l) O g ( 13= O.0001 ) N = 1 5

We demonstrated in this paper that the H.U.M. method applied to the numerical resolution of ill posed

problems in PDE's does not always work, particular/ when the controtabillity space G' can not be

identified. In addition, with the regularization method one obtains acceptable numerical results. However,

these results depend on a parameter e, which is not easily controlled. The duality method yields a good

numerical soludon to these problemes depending upon a parameter 13. In the limit as 13 tends to zero the

optimality system of the H.U.M. method is recovered. In addition 1~ gives a measure for the error in the

function f.

References: [I] J.L LIONS

Contrrlabit6 Exacte, Perturbation et Stabilisation de Syst~mes DistribuEs (Tome 1) Collection Recherches en Mathrmatiques AppliquEes, Masson

[2] .J.k LIONS Cours de la Sorbonne (Paris) automne 89

[3] M. LAURENT~V,V.G. ROMANOV, S.P. SHISTSKO 111 Posed Problems of Mathematical Physics and Analysis American Mathemetical Monograph

[4] R. LUCE Study of ill posed problems Thesis, University of tecnology of Compiegne, (To appear in december 1990)

[5] R. GLOVINSKI, C. H LI, J .k LIONS A Numerical Approch to The Exact Boundary Controllability of The Wave Equation, Dirichlet Controls:Description of the Numerical Methods. Research Report UH/MD-22 January, 1980

[6I H. BREZIS Analyse Fonctionnelle, Throrie et Applications Collection Mathrmatiques Appliqures pour la Ma~tfise, Masson

CONTROLLABILITY A N D IND EN TIFIA BILITY FOR LINEAR TIME-DELAY SYSTEMS IN HILBE1LT SPACE

S.Nakagiri

Department of Applied Mathematics, Faculty of Engineering

Kobe University, Nada, Kobe, 657, 3APAN

1 I n t r o d u c t i o n

Let f2 C R" be ~ bounded domain with smooth boundary Of 2, and H = L~'(~2) be tile usual L~-sp~e. A model control system under consideration is described by the following parabolic partial differential equation with time delays

(1.1) Ou( t ' x ) - - .Au( t , x )+ +s , x )ds+b(x ) f ( t ) , t > O, xEg t , Ot

where A is an elliptic differential operator of the second order, b E L~(f~) denotes a system controller, f E L~o¢(IL+; C) denotes a control function, h > 0 is a delay time emd a(s) is a real scalar function on Ii, ~ [-h, 0]. The boundary condition attazhed with (1.1) is, for simplicity, given by the Diricldet boundary condition

(1.2) ulon=O , t >_ 0

and tile initial data is given by

(1,3) s e [ - h , 0 ) , ea,

where g ° e L2(12), gl E L2(lh; ll~o(ft)). This condition oi, the data g = (90,91) is needed for the regularity of solutions for (1.1).

We are interested in the controllability problem for the system (1.1)-(1.3) both in tile L2-space and the appropriate function space over L 2 and the identifiabitity problem of the unknown operator A and the initial data g = (gO, g l ) . The purpose here is to give conditions on the controller b or the initial data g such that the controllability or the identifiability holds.

Let H = Lz(~), V = Hl(f~) and g = (gO,g,) E M2 =-- II x L2(£; V) and A0 be the realization of A with the boundary condition (1.2) in H = L~(12). Then the system (1.t)-(1.3) can be included in the following generM class of abstract time-delay systems (S) in a Hilbert space H :

du(t) o N

(1.4) = Aou(t) + 7Aou(t - h) + f l a(s)Aou(t + s)ds + ~ blfi(t), t >. 0 dt ~' i= ,

(1.5) = o °, = e I-h, o).

117

This type of equations is studied by Blasio, Kunisch and Sinestrari in [2,3] and Blasio [1] and recently by Yong and Pan [14] for quazi-line~r equations. Itowever, the system theoretical concepts such as controllability and identifiability has not been studied until now. So we investigate the problems of both H-appro~mate and M2-function space controllability and spectral mode controllability for tlm time delay system (S), and the identifiability of an operator A0, a constant 7, a real scalar function a(s) and an initial data g = (9° ,9 ') e M~ in (S) by certai , observation.

2 L i n e a r t i m e - d e l a y s y s t e m s a n d s t r u c t u r a l o p e r a t o r F

In this section we give exact description of the time-delay system (S). Let H and V be complex Itilbert spaces such that V is dense in H and the inclusion map { : V --* H is continuous. The norms of H, V and the inner product of H are denoted by l" I, II-II and < .,. > , respectively. By identifying the antidual of H with H we may consider V C H C V*. Let a(u, v) be a bounded sesquilinear form defined in V x V satisiyiug G£rding's inequality (2A) Re ~(~, , ) _> coll~ll ~ - c~l~l ~, where Co > 0 and ca > 0 are real const~mts. Let Av be the operator azsocie.ted with this sesquilinear form

(2.2) < ~, A0~, > = -~ (~ , , ~), ~, ~ e V,

where < .,. > denotes also the duality pairing between V and V* . The operator A0 is e~ bounded linear form from V into V*. The realization of A0 in H is also denoted by Ao. It is proved in Tanabe [11; Chap.3] that Ao generates an analytic senfigroup etAo =T(L) both in H and V* and tha tT( t ) :V*- -~Vforeach t > 0 .

Let a(s) be H61der continuous in Ih. For the brevity of notations, we introduce Stiltjes measure 17 given by

(2.3) r/(s) = --TX(_~_hl(s) -- a Ao : V --~ V*, s e lh,

where X(-c~,-tq denotes the characteristic function of ( - c o , - h i . Under the above conditions Tanabe [12] has constructed the fundamental solution W(~) of (S) as the solution of the following integral equation with delay

{ I' /o (2A) W(t) = T( t ) + T( t - s) hdr t ( ( )W( ( + s)ds, l > 0

O t < 0 ,

where O denotes the null operator. This fundamental solution W(t) is strongly continuous both in H and V* and

W ( t ) : V* --. V for each t > 0. Then for each t > 0, the operator valued function Ut(.) given by

. n

(2.5) u,(s) = ]]h w ( t - ~ + 0d ,Tf f ) : v --. V, a.e. ~ e h

is welt defined.

1 1 8

Let controllers b~, controls fl and initial data g in the system (S) be assumed to satisfy (2.6) b, e H, f, e L~o¢(ll.+; C), (i = 1, ..., N),

(2.7) g = (go, gl) E t i x Z2(Ih; V) =-- M~.

Under the conditions (2.6), (2.7) the (nfild) solution u(t) of (S) exists uniquely mid is represented by

(2.8) u(t) = W(t)g ° + Ut(s)gl(s)ds + - s)bifi(s)ds, t >_ O. h i=l

Here we note that the function u(~) in (2.8) satisfies the integrated for,r, of (S) by T(t) (cf. Nakagiri [6]).

The state space M2 = H x LZ(Ia; V) of the system (S) is a IIilbert space and the adjoint space M~ of M2 is identified with the product space H x L2(lh; V*) via the duality pairing

(2.9) < g, f >M, = < gO, f0 > + < gl(s), f ' ( s ) > ds, h

g = (gO, gt) e M~, f -= (fo, f~) e M~.

Now we introduce the structural operator F studied in Nakagiri [7] and Tan,be [13] on the abstre.ct space setting.

Let /71 : L : (h ; V) --* L2(/h; Y') be give,, by

(2.10) [F,g'](~) = ( ¢ ) ~ l ( ¢ _ ~) a.e. s e I~.

Tllestruc,ura~ opertator £[~ : M~---)-a~{//~ isde~ned by F~--- ( I O ) 0 FI i.e.,

(2.11t [Fg]O = gO, [Fg ] 1 = ]719 1 for g = (gO gl) e M~.

3 S p e c t r u m o f t h e g e n e r a t o r a s s o c i a t e d w i t h ( S )

Let u(t; g) be tile solution of (S) with b, = 0 (i = 1, ..., N) and the segment ut be given by u~(s; g) = u(t + s; g), s E Is,. The solution sc~rfigroup S(t) associated with (S) is defined by (3.11 s(~)o = (~(~; g), ~,( .; g)), t _> o, g c M~.

S(t) is a C0-senfigroup on M2 and its infinitesimal generator is denoted by A. in what follows we investigate tile structure of tile spectrum a(A) of A under the following condition that (3.2) tlle inclusion map i : V --+ H is compact.

Tile condition (3.2) is assumed throughout this paper. Then the resotvent (~ -Ao) -1 is compact for ~ome )~ E p(Ao), so that according to the Riesz-Schauder theory the operator A0 has a discrete spectrum

(3.3) ~(Ao) = {~,,; ~ = t, 2, ...}.

Let rn(A)

(3.4)

be given by

119

re(A) = 1 + 7e -xa + / ~ , eX~a(s)ds

and define the characteristic operator A()~) : V -'~ V* by

(3.5) zx( ) =

Then the spectrum a(A) is completely determined by the entire function m() 0 and (A0) (see Jeong [4]).

T H E O R E M 1 . The spectrum a(A) is given by

(3.6) o(A) = oo(A) u

odA) = re(x) = o } , ap(A) = {A; re(A) # 0, __-Z7 ~ e a(Ao)}. rnt ~ l

Each nonzero point of g~(A) is not an eigenvalue of A and is a cluster point of a(A). Thc point spectrum a~,(A) consists only of discrete eigenvalues wilh finite multiplicities.

If A is in a resoivent set of A, then by Theorem 1 the inverse A(A) -1 : V" --~ V e.xists. Further for Re )~ sufficiently large, A(.k) -~ is given by the Laplace transform of W(Q.

L E M M A 1 . For A e ap(A),

(3.7) Ker (A - A) = {(~o °, c~'9°); A(A)~o ° = 0}.

For the characterization of Ker (A - A) *, t = 1, 2, .... in terms of A(A) we refer to [4],[7]. Since each A E ap(A) is an isolated eigenvalue, the order k~ of )~ as a pole of ( z - A ) -1 is finite. The spectral projection Px and the nilpotent operator Q~ for A E ap(A) are defined respectively by

1 (3.8) Px = ~ i fr~ (z - A)- idz

(3.9) Q~ = ~ (z -- A)(z -- A)- ldz ,

where F~ is a small circle with center ~ such that its interior and P~ contains no points of a(A). Let M~ = Im P~ be the generalized eigenspace corresponding to the eigenvalue ), of A. Then we have (3.10) Q~ = O, hn Q~ c . M ~

and the useful relation (3.1I) Ker (A - A) -- f14~ VI Ker q~

(cf. Suzuki and Yamamoto [113]). The following direct sum decomposition of the space M2 (see e.g. Sato [5]) is essential in our study.

L E M M A 2 . For )~ E ap(A),

(3.12) .h4a = Ker (A - A) ka, /'vI2 = A4x $ I m ()~ - A) k~.

120

4 ] : / - a p p r o x i m a t e a n d M 2 - f u n c t i o n s p a c e c o n t r o l l a b i l i t y

In the system (S) we define the controller/3o : C jv --* H by

N

(4.1) Boy = ~_,vibi, v = (v,,..., vt¢) e C t~. i----1

The ~ttainable subspaces ~o in / / and 7~ in tt//2 are defined by

= u f' w/ , - . / .0:¢ .1 . . , ,1, o">}, (42) 7~ o t > 0

{Z' } T¢ = U S ( t - s)(Bof(s),O)ds; f e L2([0, t]; C iv) , t > 0

(4.3)

respectively.

D E F I N I T I O N 1. The system (S) is suid to be H-appro~mately controll~tble (resp. M2-function space controllable) if Cl(7~ °) = tl (resp. if Cl(l~) = M2).

In view of (3.2), there e~sts a set of eigenvalues a~ld eigenvectors {#,, ¢,0; j = 1, 2, ..., d,,, n = t.2....} of the adoint A~ such that ~,, are distinct from each other and d,, = dim Ker (#,, - A;). It is well known (Kato [51) that a(A;) = {~,,; n = 1, 2, ...}.

First wc give a. result on tlm //-approxinaate controllability (cf. Na&agiri mid Ya- mamoto [9]).

T H E O 1 Z E M 2 . Assume that re(O) ¢ 0 and thc system of generalized eigcnvectors of Ao is complete m H. Let B,, (n=l,2,. . .) be N x d,, matrices given by

(4.4) B,, = (< b,, ¢,0 > ; i I i,. . . , N, j -~ i, 2,..., 4 , ) .

Then the following two stalemenls are equivalent: (i) the control system (S) is II-approximalely controllable; (it) r unkB , ,=d ,~ for each n >__ l.

For the .Mz -function space controllability we require the following Lemm~t.

L E M M A 3 . Assume that the system of generalized eigenvectors of Ao is complete in t I , 0 ~[ a(Ao) , 3' -~ 0 and re(O) 7t 0 , then the system of generalized cigenveclors of A is complete in M: , that is,

(4.5) Cl(Span {)vt~ ; A E op(A)}) : M~.

T H E O R E M 3 . Under the assumption in Lemma 3, the following two statements are equivalent: (i) the control system (S) is M2-function space controllable; (it) rankB, , = d,~ for each n > l.

121

5 S p e c t r a l m o d e c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y

(s.1)

(5.2)

where H with initial data

Let U be a complex Hilbert space and B0 : U ---* H be a controller which is bounded. We consider the following control system with the controller Bo :

du(O = Ao~(O + %4o,~(t - h) + ~(~)Ao~(t + ~)as + Bof(O, t > 0 dt h - -

u(0) = gO, u(s) = g ' ( s ) a.e. s e [ - h , 0),

f E L~o¢(R+; U) denotes a control function. The observed 'transposed' system in (~o °, ~o') E M2 is defined by

........ ; dr(t) = A;v(l) + 7A;v(t - h) + a(s)A;v(t + s)ds, dt .', v(0) = ~o , v ( s ) = ~'(s) ,,.e. s e [ - h , 0 )

v ( t ) = s;~(t ) t > o ,

(5.3) ~ _> o

(sA) (sz)

where y(t) denotes the observation of transposed system and B~ : H --~ U. Let A be the infinitesimal generator of S(t). Then we can imbed the system (5.1),

(5.2) into the state space M2 as the following control system (~) without delay :

(5.6) dh(t) = A~z( t )+By(t ) , t > 0 dt

(5.7) ~(0) = g e M~,

where B : U--~ Me is given by B f = (Bof, O), f E U. The nfild solution 5(t) of (E) is given by

~0 t (5.8) (u(/; g, f), u,(.; g, f ) ) = S(t)g + S(t - ~)Bf(s)da, t >_ O,

where u(t; g, f) is the nfild solution of (5.1),(5.2). Associated with the observed system (5.3)-(5.5) we introduce the operator AT() 0 by

(5.9) zx~,(,~) = :~ - ,~(:,)A;.

Let us denote by {ST(Q},_>o the C0-senfigroup on M2 corresponding to the observed system and by AT its infinitesimal generator. Then by Theorem 1 we see that the point spectrum %,(AT) of AT is given by

i.e., av(AT ) is the mirror image ap(A). The transposed system (~W) in M2 induced by the system (5.3)-(5.5) is given by

(5.10) df~(t)_ = ATe(t), t > 0 dt

(5.11) ~(0) = ~ = (~0 # ) e M~

(s.12) 9 (0 = B '~ (0 t >_ 0,

122

where B* : M2 "--* U is given by B*g ~- B~g °, g = (gO gZ) E M~. Then the observ;~tion ~(t) of (ET) is represented by

(5.13) 9(0 = B'Sz( t )~ t >_ o.

Tile attainable subspace ~ for (E) and the unobservable subspace AfT for (ET) are defined by

- - u {/o' ,1; .)}, t>o

(S.l~) H r = r"l I<e~ B*S~(t), t>0

respectively.

DEFINITION 2. (1) The system (X~) is sMd to be ,k-controllnble for A E crp(A) if Ct(7-£) D ./Via.

(2) The system (ET) is said to be A-observable for ,~ E ap(AT) if AfT Iq M~" = {0}.

Ill DefiIfition 2 the symbol Mx T denotes the generalized eigenspace corresponding to the isolated eigenvalue ~ of At. ]?or ~ E %(AT) the symbols p~r, Q~ denote the spectrM projection and the nilpotent operator corresponding to the eigenvalue ~ of AT, respectively. We set Q~ = Pa, (Q~')o = p [ for notational convenience.

THEOILEM 4 . Let )~ E ap(A) be given. Then the following statements (i)-(xiii) are equivalent:

(i) the system (E) is )~-controllable;

(it) the system (ET) is ~-observable;

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

N r c Im (~ - At ) k* C M2 ;

kx

( A ICe,. B ' ( 0 ~ Y ) C h~, (~ - At) ~ C M~ ; j=O

Cl(Span{Q~Bf ; 0 <_ j < k~ - 1, f E U}) = 2~4~ ;

(N I<e~ B*(QT) i) n M~ = {0} ; j=O

Cl(Im ()~- A) + I m B) = M2 ;

c t ( tm (;~/'~ - AP~) + hn PxB) = M~ ;

el(Ira ~x(~) + I m Bo) = It ;

Ker AT(~) f-) Ker B; - {0} ;

1(er ( i - A~) n K~r B* = {0} ;

123

(xii) F.Ma C CI(FT4);

(xiii) AfT fl M T C Ker F*.

Let ~ E ap(A) and da be the dimension of thc eigenspace Ker ()~ - A). Then by Lemma 1 we have

(5.16) da = dim K e r ( ~ - A ) = dim K e r ( ~ - A T ) = dim KerAT(~) < oo.

We denote the basis of Ker AT(~) by {~p°l, ..., ~oOa~}. For the practical case where the controller B0 is given by (4.1), we have by apply-

ing Theorem 4~ the following verifiable conditions for spectral mode controllability and observability.

T H E O R E M 5 . Let )~ E ap(A) be given and the controller Bo be given by (4.1). Then the following statements (i)-(vii) are equivalent:

(i) the system (~) is ~-controllable;

(ii) the system (EIT) is ~-observable;

(iii) rank (< hi, ~o° i > ; i ~ 1,.. . ,N, j ---* 1,2,...,da) = d~ ;

(iv) hn ()~- A) + Span {(bi,0); 1 < i < N} = M2 ;

(v) hn A(,k) + Span {bi; 1 < i < N} = H ;

(vi) Ker AT(~) n ({bi; 1 < i < N}) j- = {0} ;

(vii) Ker (), - AT) Cl ({(hi, 0); 1 _ 0

dt h -

(6.2) u(0) = gO, u(s) = t ics) a.e. s e [ -h , 0) (i = 1, ..., N).

Here A0, 7, a(s), g, = (gO, g~) E M~ (i = 1,..., N) are unknown quantities to be identified.

In this section we suppose the following conditions:

(I) the operator A0 associated with the seaquilinear form (2.2) enjoying the condition (2.1) is unknown;

(II) tile real constant 7 is unknown; (III) the real scalar function a(s) is unknown but is known to be HSlder continuous

in In; (IV) the initial conditions g, = (gO g!) E M~, i = 1, ..., N are unknown.

124

Under the above conditions there exists mild solutions u(t; gi), i = 1, ...,N of (i tS). By the model system (ITS) m we understand the system (6.1), (6.2) in which Ao, 7, a(s) and g~ = (gO, g~), i = 1,...,N are replaced by A~', 7", a"(s) and g~ = (g7 '°, g ~ ) , i = 1 , . . . ,N, respectively. The model states of (ITS)" are denoted by u(t; g'~), i = 1, ..., N. An 'm' superscript means that the quantity is known.

D E F I N I T I O N 3. The unknown quantities A0, 7, a(s) and gl = (gO, g~), i = 1, ..., N in (ITS) are said to be identifiable if

(6.3) Ao = A~, 7 = 3 'm, a(s) = a'~(s) s e lh,

(6.4) gi = g~" in M~ (i = 1, ..., N)

follows from the relations

(6.5) ~,(t; g,) = ~(t; gT') in ,r~, t _> t0, i = 1,..., ~v

for some io >_. 9.

Here we introduce the transposed model system (I/S)T , that is the system (RS)'" in which A~ is replaced by its adjoint (A~')*. Let S~(i) be the semigroup associated with the transposed model (RS)T. The infinitesimal generator of S'~(t) is denoted by A~. In view of (3.2) and Theorem 1, ap(A~.') consists of discrete eigenvalues. Let )~ E ap(A~) and Ker (~ - A N ) be the eigenspace corresponding to the eigenvalue of A~, which can be calculated by Lennna 1. We denote the basis of Ker ()~ - A ~ ) by {~p~, ..., ~'dx}, where d~' = dim Ker (~ - A~). The structural operator associated with (itS) ~ is denoted by F " .

THEOI- tEM 6 . Let A'~ satisfy the assumption in Lemma 3. If the set of initial conditions {g'~, ..., g~]} satisfies

(6.6) r~nk (< ~ , F"g 7 >M, ; i ~ 1, ..., N, j ~ 1, 2, ..., dx) =- d~x

for each )~ e ap(A'~),

the~ all m , 7, ~(~) ~ d g, = (g?, g~), i = I , . . . , N i , ( i t S ) are ide~tifiabl~.

The above theorem improves the results in Nakagiri and Yamamoto [8] and Jeong [4]. Complete proofs of all theorems in this paper will appear elsewhere.

R E F E R E N C E S

[1] G. Di Blasio, The linear-quadratic optimal control problem for delay differential equations, Rend. Accad. Naz, Lincei, 71(1981), 156-161.

[2] G. Di Blasio, K. Kunisch and E. Sinestrari, L2-regularity for parabolic partial inte- grodifferential equations with delay in the highest.order derivatives, J. Math. Anal. Appl., 102(1984), 38-57.

[3] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50(1985), 231-263.

125

[4] J-M. Jeong, Spectral properties of the operator associated with a retarded functional differential equations in Hitbert space~ Proc. Japan Acad., 65 A(1989), 98-101.

[5] T. K~to, Perturbation Theory for Linear Operators, Second edition, Springer, Berlin- Heidelberg-New York, 1976.

[6] S. Nakagiri, Optimal control of linear retarded systems in Banach spaces, J. Math. Anal. Appl., 120(1986), 169-210.

[7] S. Nakagiri, Structural properties of functional differential equations in Banach spaces, Osaka J. Math, 25(1988), 353-398.

[8] S. Nakagiri and M. Yamarnoto, ldentifiability of linear retarded systems in Banaeh spaces, Funkcial. Ekvac., 31(1988), 315-329.

[9] S. Nakagiri and M. Yamamoto, C~ntrollability and observability of linear retarded systems in Banach spaces~ Int. J. Control, 49(1989), t489-1504.

[10] T. Suzuki and M. Yamaanoto, Observability, controllability and feedback stabilizability for evolution equations, I, Japau J. Appl. Math., 2(1985), 211-228.

[11] H. Tanabe, Equations of Evolution, Pitman, London, 1979.

[12] tI. Tanabe, On fundamental solution of differential equation with time delay in Ba- nach space, Proc. Japan Acad., 64 A(1988), 131-134.

[13] H. Tanabe, Structural operators for linear delay-dlfferentiat equations in Hilbert space, Proc. Japan Acad., 64 A(1988), 263-266.

[t4] J. Yong and L. Pan, Quasi-linear parabolic partial differential equations with delays in the highest order spatial derivatives, to appear.

A Generalized Hamilton-Jacobi-Bel lman Equation *

Shige Pent Department of Mathematics, Shandong University

Jinan, Shandong 250100, China and

Institute of Mathematics, Fudan University Shanghai, 200433, China

Abstract. We interpret the following fully nonlinear second order partial differential equation

{ O~u+inf{£(x,a)u+ fCz, u,O~ua(z,a),a)} =0, (z,t) eO×(O,T), for (z,t) • D × [0,T]; u(x,T) = g(x).

as the value function of certMn optimal controlled diffusion problem. Where A E/R I: is control domain. £(z, a) is a second order elliptic partial differential operator parametrized by the control variable cr E A C/.R k. A particular case of this equation is when f = f(z, a). In this case, the equation is the well known ttamilton 3acobi Bellman equation.

The problem is formulated as follows: The state equation of the control problem is as classical one. The cost function is described by a solution of certain backward stochastic differential equation.

§I. Introduction

It was known that a solution of a second order linear parabolic (or elliptic) equation can be formulated as a functional of a solution of some stochastic differential equation. This kind of interpretation has been found it's important applications both in theory of partial differential equations and that of stochastic differential equations, such as large derivation, optimal control theory, martingale problem, variational and quasi variational inequality~ e.c.t.

A natural and interesting problem is to obtain a sinfilar interpretation for a system of parabolic (or elliptic) partial differential equation. A recent result in backward stochastic differential equation (see Pardoux and Peng [7]) produces new insights to this direction. In [11], we interpreted a systems of second order quasilinear parabolic partial differential equation as a solution of a backward stochastic differential equation. This backwr;d equation is associated with some classical Ito's forward stochastic differential equation. When the backward equation is linear and one dimensional, the corresponding system of equation becomes a linear one. So the classical probabilistic interpretation can be regarded as a special case of our formulation.

The same idea can be also applied to generalize Ha~filton-Jacobi-Beliman (HJB) equation.

* p~tially supported by the Chinese N~tional Natural Science Fund,~tion.

127

Let D be a domain in/R", Let A be a nonempty set in/R k. Let

b(~, a) : ~ " x ~ k ~ ~,,, a) : x . _ ,

Consider the following stochstic control problem parametrized by the initial data z E/R'*

f dyCs) = bCYCs),,,Cs))ds + aCYCs), aCs))~Cs), (1.1) ~ y(O) = x E ~" ,

where W(s);s > 0 is an d-dimensional standard Wiener process, a(s),s > 0 is an Jr,-adapted process taking vahes in A, called control process, y(.) is called trajectory corresponding to oh(.). If b and a satisfy some suitable condition, then this system is well defined. With this system, we can introduce the following backward stochastic differential equation: For any given t E [0, T), we look for an adapted pair (p(-), q(-)) that solves uniquely

{ /2' /2' (1.2) p(s) = g(y(T- t)) + f(p(r),q(r),y(r),a(r))dr- q(r)dW(r),

s e [o, T - tl.

where J'(z, u, q, c~) is a given real function defined on IR" x/R x/ /~ x A, and g is a given real function defined on/R n. The existence and uniqueness for the above equation is obtained by P~rdoux and Peng [7]. Then we can define the so called cost function as follows

&,Ca(.)) = Ep(0) = p(0).

The value function of this optimal control problem is defined by

u(z,O = inf J~.'("(.)). ,,(.)

We will show Lhat this value function can be charecterized by the following generalized HJB equation

a,u + inf{£Cz, v)u + fC~,u,a~ua(z,a),a)} =0, (x,t) ~ D x CO, T),

uCz, t) = ~(z), for (z,t) E dD × [0,TI; u(z,r) = ~(:c).

where /~(z, v) is a second order elliptic differential oprator parametrized by a E A. When p depends only on (x, a), the above equation becomes a cla.~ical HJB equation.

This kind of problems can be applied to financial problem, where the term f is called utility function (C.Ma, private communication.).

§2. Backward SDE and Systems of Parabolic PDE

2.1. Backward Stochastic Differential Equation

We begin with presenting a recent result of adapted solution of backward stochastic differential equation. (see Pardoux and Peng [7]).

128

Let (1], jr, p) be a probability space equipped with filtration 7t. Let {W(t), r > 0} be a d-dimensional standard Wiener prosess in this space. We assume

= o{w, ;o < s < t}.

We denote by ]d2(0, T;/R r~) or AI:(/R n) the set adapted proceescs such that

F Let the following functions be given.

f (p ,q , t ,o~) : /R m × £(/R'~;/R m) × [0,7'] × i2 ~ / R ' , Q(oJ) : o - , ~ " ,

W e a.$sume

(//2.1.1) for each (r,q) e ~ " × Z ( ~ ; ~'~), :(p,q,-) e ) t ~ ( ~ ' )

(//2.1.2) for each (t,w) e [0,T l × f~, f(p,q,t ,w) is continuously differentiable with respect to (p.q), their derivatives fu and fq are bounded.

(//2.1.3) Q(w) is 7T measurable, and EIQ[ 2 < oo

Consider the following backward stochastic differential equation

f" (2.1.1) p(t) = Q + f(p(s),q(s),s)ds - q(s)dW(s).

Our problem is to look for a pair of adapted N"*x /~(~;/R m) valued processes (p(s),q(~)) which ~olves equation (2.1.1). We have

Proposition 2.1.1. We assume (H2.1.1)-(tt2.1.3),. Then, there exists an unique pair (p(.),q(.)) in At2(0,T;/R '~) x )t2(0,T;Z(/Ra;/R")) which solvcs equation (2.1.1). We have

Esuplp(t)p < oo. t

If we assume further more

f K = sup{IO@)l 2 + If(o,o,t)12dt) < oo, t¢

then we have

(2.1.2) sup Ip(t,w)t 2 < Ke c(T-t).

129

The proof for the existence and uniqueness can be found in Pardoux and Peng 17]. The proof for the boundness can be found in Peng [9], or [10].

We can also consider the following type of backward equation

(2.1.3 3 p(t A r) = O + f(pCs),qCs),s)ds - qCs)dWCs). h r hr

Indeed, we have

Proposition 2.1.2. We assume (H2.1.1), (H2.1.2) and (H2.1.4). Then there exists a pair (p(.),q(-)) in ~2(0, T;/R m) x ,M2(O, T; Z(/Ra;/R"~)) which solves equation (2.1.3). Such solution is unique in the foUowing sence: if both (pl(.),ql(.)), (p2(.),q2(.)) solve (2.1.2), then,

(p'(~ A ~),q'C~ ^ ' ) ) = (fC~ A ~),q~C~ ^ ~)), ,v~ e [0,fl.

The proof can be found in Peng [11].

L2. Probabili~tic formulation for System of parabolic PDE

In this subsection, we will formulate a system of quasilinear parabolic p~tial differential equations as a solution of certain backward stochastic differential equation, of type (2.2), associated with some forward (classical) stochastic differential equation. We first introduce the forward equation.

Let

Let D a domain in ~m with boundary aD = S. We denote Q = D × (0,T). assume

(i) a(z) is of class C~(#),

b(~) is of cla~ c 'c0) ; (H2.2.1) (it) S is a manifold of Ca;

(iii) ~,~A~)6,~i >- ?1~1 ~, v~ e O.

where ,8 > 0 is a constant and aii = .~[aa*]ii. For any given (z,Q 6 Q, consider the following forward equation defined on [0, T l

I, v(o) ~:.

For any given t E [0, T], we define the follwing stopping time

= rx,t = inf{s e [0,T-t i ; (y(s) ,s ) GO x lo, T - t ) } .

From (H2.2.1), the diffusion process y(-) and related stopping time r are well defined.

130

(2.2.2)

Where

Then, we consider the associated backward stochastic equation: defined on s E [0, r]

f; // p(~^~) = ~ ( y ( 0 ) + /(pC~),qC~),y(0 ) ~ + qC~)~w(~). Ar Ar

f(p, q, ~) : m". × ~ (~d × m,~) × m " -+ m" , ~(~)Ia" - , /R".

We assume

(i) ]'(p, q, x) is continuously differentiable

(H2.2.2) in/R '~ × ~(/R ~ × IR") × Q, the derivatives arc bounded;

(ii) ~(z) is of C 3.

We can now define u : ,Q ~ IR '~ by

uCx, t) = ~'~p(o). We will show that the function u(z, t) solves the following system of parabolic PDE

{ u~ + ~(.)u(~,O +/CuC~,O,u~(~,O~(~),~) =o, (~,0 c Q, (2.2.3) u(x, T) = ¢1(x), Vx E D,

u ( ~ , 0 : ~(~), w e s , t e [O,T).

Eere we denote, for u(~) = (~'(~),..., ~"(~)),

l$t : O t l t , U t : : , U z : Ox t t~ U x - - " " . . "

where

S,2 $

We need the following lemma

£ 1 1 =

Proposi t ion 2.2.1. We assume (1t2.2.1) and (H2.2.2), we assume also the following compatibility condition

[~c,,~)~(~) + I(~(~), ~ ( ~ ) ~ ( ~ , r ) , ~ , r ) ] ~ : o.

If D is bounded, then, (2.2.3) has an unique solution in C 2,1 .

The proof of this lemma can be found in [4], (Th.7.1, Ch.VII).

With this lemma, we can assert

131

Theorem 2.2.2. We assume the same conditions in Iennna 2.2.1. Then, for any given (z, t), the solution of the system of parabolic equation (2.2.3) has the following interpretation

uCz, t) = ~,tpC 0) = p(0),

where p(t) is determined uniquely by (2.2.1), (2.2.2).

Remark. Particularly, when p(.) (of (2.2.2)) is valued in/R (n=l), (2.2.3) becomes a parabolic equation. Even in this case, it is still a nontrivial extension of the classical probabilistic interpretation. In fact, the classical case can be described by setting

f = fo( ) +

§3. Optimal Stochastic Control: Dynamic Progranuning

In this section, we introduce a generalized form of optimal control system where the cost function is determined by certain backward stochastic differential equation discussed in §2. We will show that the principle of dynamic programming, known as Bellman's principle, can also apply in this situation.

We introduce the set of admissible control in usual sence: Let A be a compact set of/R k. An admissible control is a collection of

(i) a probability space (fl, jr p) equipped with a filtrition 4;

(it) a d-dimensional standard Wiener process (W(t);t > 0), such that = < < t);

(iii) a progressively measurable process a( t ) ; t > 0 taking its value in A.

We denote the set of admissible controls by ..q. For given admissible control and initial data z E/R ~, we can consider the following stochastic control problcm

(3.1) { d (S)yC0) == + where b(s,a), a(s, or) are respectively/R '~- valued and /~(~,t;/R") valued functions defined on/R a x/R k. We assume that

(H3.1) / b and a are continuous in i s, a), and continuously differentiable in z, their derivatives bx, a~ are bounded.

' k

Obvioursly, the solution of y(.), called the trajectory corresponding to the control a(.), is well defined and

(3.2) Ely=,aC')(s)l ~ < Clzl ~,

where C is a constant independent of z, a(.). We can define a stopping time by

r = rx,t = inf{CYCs),s) ¢ D x [0,t)}.

132

We now introduce the foHowlng backward stochastic differential equation: let f(p,q,x,a) be a real function defined on/it;< (~a)*x /R"x /g k ((/Rd) * denotes the adjoint space of/Rd). For any given continuous function g(x) : / 2 r' - , /R satisfying iS(x){ ___ C(1 + {x{), we consider

{ z: (3.3) p(s A T) = gCYCT)) fCPCr),qCr),YCr),a(r))dr- qCr)dWCr),

8 e [% ~']. We o, s su lne

f is continuous in (p, q, x, a) and continuously differentiable (H3.2) in (p,q,z), its derivatives fp, ./'q, A are bounded.

According to proposition 2.3, (3.1) is well defined. Since p(s) is ~-adapted and W(0) = 0, thus p(O) = Ep(O). We can introduce the following generalized cost function

p(0) = g(z,t;g(.),a(.)); (x,t) 6 D × (O,T).

From proposition 2.1, for given g(.), J(z,t ;g(.) ,a(.)) is uniformly bounded. Thus we can define

vc~, t ;g( . ) )= inf YC~,t;g(-),~(-)). ,~(.)6.4

We have the following regularity result about V. To simplify, we set D = /R" . In this case, ~ = t, thus

{ // // (3.4) p(~) = 9(y(O)+ /(p(~),q(~),y(~),~(~))d~- q(~)ew(~), e 10,~].

Lemnia 3.1. We suppose (H3.1) and (II3.2). We have

(i) Let g(x) be uniformly Lipschitzian. Then we have

(3.5) tv(z,t;g(-))- vc~,t; 9(-))t < V~lz-~1 vz,~ e ~ - v t .

where Cg is a constant depending only on the Lipschitian constant of g(.).

(it) Let gl (x) be continuous and bounded, Then

(3.6) IV(z,t.; O'(-))- v(~,~;~(.) + 9,(-))1 <_ Csup 19,(~)1. z

(iii) If gl (z) is continuous, nonnegative and bounded, then

JCz, t;g(.) + gl(.),a(.)) >__ J(x,t;g(.),a(.)), v(~,t; ~(.) + siC.)) >_ v(~,t; ~(.)).

Now we can generalize the well known dynamic programming principle.

133

We assume

(H3.3) gC x) is uniformly Lipschitzian.

We have the so called ;Bellman's principle (in form of Nisio [6]). We still set D = LR r'.

Theorem 3.2. Let (Ha.1)-(H3.3) hold. Then w e have

(3.9) V(x, t+hig( . ) )=V(z , t ;V( . ,h ,g)); Vz, Vt+h<_T.

§4. Hamilton-Jacobi-Bellman equation

In this section, we fix g = gCz) and set

uCxJ)-vCx, T-t;g(.)), t e [O, TI.

Under certain conditions, this function is the solution of the following fully nonlinear partial differential equation

(4.:) a , . + i ~ { z ( ~ , ~). + f ( . , .~ . (~, . ) , ~ , . ) } = 0,

(4.2) ~C~,T) = ~C~),

where 71 II

i ,1"=1 =-----1

1 aij(~,~) = ~1o(~,~)~ (~,~)]i~,

u,: = (u= , , ..., u=° ) .

To simplify, we set D =/R n. We have

Theorem 4.1. Let D = IR" and let us assume Ctt2.1)-(H2.4). Let ~ E C~'~(IR ") be a solution of (4.1), (4.2). Then u = ~.

Theorem 4.2. Let D = / R ~ and let us assume (tt2.1)-(tt2.4). Let u E C~JCiR'*). Then u is a solotion of HJB equation (4.13, (4.2).

If we assume

(i) (not loss of generality) let b = 0; (ii) aly(x, a), i, j, = 1, .., n, f(p, q, x~ a) are continuous

in (x, ce), for eac~ a E A, they arc twice continuously (H4.1) differcntiable with respect to (p,q, z). All those derivatives

are bounded; (iii) A,]~l 2 _< Ei~" alY( z, v)~i~y < ~1~1 ~ ~ > A, > 0 a~o constants; (iv) s(~) is c~(~,,).

134

Then, the following proposition obtained by Krylov [3] provides the existence (and uniqueness) result of HJB equation (4.1) for non degenerate case. For the domain D C/R n, we assume

(H4.2) / andD =l~b=l{z; ¢(x):> 1 on > 0}0D.be a bounded domain such that ¢ is in C3(/R '~)

We have

Proposition 4.3. Let (H4.1), (It4.2) hold, Then The tIJB equation (4.1) with the boundary conditions (4.2), (4.3) has unique solution u e C 2'1 (D × (0,T)). The norm of u in C'a'x (D × (0, T)) is bounded.

Detail treatment of the above problems can be found in our forthcoming papers 111], [13].

References

[1] R.Bellm~n, Dynamic Programming, Princeton Univ. Press, N.J. 1957. [2] W.tt. Fleming and R.W. gishel, Deterministic and Stochastic Optimal Control,

Springer-Verlag, 1975. [3] Krylov, Nonlinear Elliptic and Parabolic Bquations of Second Ordcrj D. Reidel

Pubfishing Company, Dordrecht, 1987. [4] O. Ladyzenskaja, V. Slonnikov, N. Uralceva, Linear and Quasi-linear Equations of

Parabolic Type, Translations of Maths Monograghs, Vol.23 AMS, Providence Rhode Island, 1968.

15] EL.Lions, Optimal Control o] Diff~ion Processes and Hamilton Jaeobi Bquations, Part 1, 2, 3, Commu. in PDE, 1983.

[6] M.Nisio, Some Remarlcs or~ Stochastic Optimal Controls, Proc. 3rd U.S.S.R.-Japan Syrup. Prob. Theory, Lecture Notes in Maths 550, Springer, Berlin, 1976.

[7] E. Pardoux and S. Peng, Adapted Solution of Backward Stochastic Equation, to appear in Systems and Control Letters.

18] S. Peng, A General Stochastic Mazimum Principle for Optimal Control Problems, SIAM J. Cont. Vol.28 No.4, 1990.

[9] S.Peng, Backward Stochastic Differential Equation and lt's Application in Optimal Control, to appear.

[10] S. Peng, Stochastic ltamilton-Jacobi-Bellman Equations, to appear. [111 S. Peng, Probabilistic Interpretation for Systems of Parabolic Partial Differential

Equations, submitted. [12] S. Peng, Maximum Principle for Stochastic Optimal Control with Nonconvez Con-

trol Domain, to appear in Proceedings of 9th International Conference on Analysis and Optimization of Systems, Antibes, 1990.

[t3] S.Peng, A Generalized Dynamic Programming Principle, to appear.

DYNAMICS AND CONTROL OF BENDING AND TORSIONAL VIBRATIONS OF FLEXIBLE BEAMS

Yoshiyuki Sakawa and Zheng Hua Luo

Department of Control Engineering, Osaka University

Toyonaka, Osaka 560, Japan

1. INTRODUCTION

Modeling and control of flexible beams has received a great deal of attention in

recent years. This problem has arisen in the area of space and industri,'d robots with

lightweight and flexible arms as well as in the area of flexible space structures. In this

paper, we consider a flexible beam having a rigid tip body at the free end. It is assumed

that the cross section of the beam is geometrically symmetric and the shear center axis

coincides with the centroidM axis. We considered the coupled bending and torsionM

vibrations of the flexible beam, when the mass center of the tip body does not lie on the

shear center axis of the flexible beam [1]. Here, we consider the flexible beam having

a rigid tip body, of which the mass center lies on the centroida.1 axis of the beam. In

this case, it will be shown that the decoupled bending and torsional vibrations occur,

and the motion is governed by decoupled partial differentiM equations and boundary

conditions.

We first derive decoupled partial differentiM equations with decoupled boundary

conditions as a mathematicM model of the vibrations. Then each set of equations will

be rewritten together as an evolution equation in a properly defined Hilbert space. We

design a finite-dimensional dynamic compensator using sensor outputs to construct a

feedback control system on the basis of the previous results [2].

136

2. MATHEMATICAL MODELING OF A BEAM WITH A TIP BODY

We consider a slender flexible beam which is rotated in a horlzontal plane by a motor

M1 which may suppress the bending vibration. It is also rotated around a centroidal

axis of the beam by another motor Ms which may suppress the torsional vibration.

The beam of length L, having uniform mass density p per unit length, uniform flexural

rigidity E l , and uniform torsional rigidity GJ, is clamped on a vertical shaft of the

motor Mi at one end and has a tip body rigidly attached at the free end, as shown

in Fig. 1. Let Xo, Y0, Z0 designate an inertiM Cartesian coordinate axes, where );7o

and Y0 ,axes span a horizontal plane, and Z0 axis is taken so that it coincides with the

vertical rotation shaft of the motor M1. Let X1, ]I1, ZI(Z1 = g0) denote a coordinate

axes rotating with the motor 3//1, and let X~.,Ys, Z2(X2 = X1) denote a coordinate axes

rotating with the motor Ms. Let 0(t) be the angle of rotation of the motor 3'/1, and let

~,,~(t) be / l i e angle of rotation of the second motor Ms.

Let Q denote the mass center of the rigid tip body. It lies on the centroidal axis

of the beam. Let e be the distance between the beam's tip point and the point Q. It

is assumed that e is small. We take another coordinate axes Xa,Y3, Za attached to the

tip body, where Xa is the beam's tip tangent. Since the tip body is a rigid body, it is

characterized by mass ra, and two moments of inertia dE and J0 with respect to the

lines QXa and QZz, respectively.

Let y(t, z) and ~a(t, z) denote the transverse displacement and the angle of twist

of the beam, respectively, at position z (0 < ~ < / ) ) and at time t. Both y and ~a are

assumed to be small. For the transverse vibration we use the Euler-Bernoulli model

with internal viscous damping of the ¥oigt type [3]

O~y(t, ~) E10~y(t, ~) E± O'y(t, ~,) + =

p OtO~ 4 p tgz ~

where 61 > 0 is a small damping constant of the beam material. Since the beam is

clamped at z = O, we obtain

y( t , O) = O, v'Ct, O) = O. (9.)

The torsional vibration is governed by [4]

137

where ~2 > 0 is a small damping constant for the torsional vibration, pt¢ 2 is the mass

polar moment of inertia per unit length of the beam. Since the beam is clamped at

z = O, we obtain

, , ( t , 0) = 0. (4)

O ~ - - ~ ®z

zo

(b)

Fig. 1. Bending and torsion of a flexible beam with a rigid tip body.

In the same way as in [1], using tile Lagrangian method, we obtain the equations

of motion of the tip body as follows:

mid(t, L) + c~)'(t, L)] - El[y"(t, L) + 261~)'"(t, L)] = -m(L + c)b'(t), (5)

138

mc[~(t, z) + c~'(t, L)] + JoF(t, ~) + E±[y"(~, L) + 2~1y"(~, ~)]

= -[J0 + mc(L + c)]/~(~), (6)

JE(~(t, L) + OJ[p'(t, L) + 252(o'(t, L)] = -J~(~,~(t), (7)

where a dot denotes the time derivative, and a prime denotes the derivative with respect

to the spatial variable z. Equations (5)-(7) give the decoupled boundary conditions for

the bending and the torsional vibrations.

The equations of motion of the rotation motors can be respectively written as

JlO(t) + 7it~(t) = Ti(t) + EIy"(t , 0), (8)

J2~,,,(t) + 72~bm(t) = T2(t) + GJ~'(t , 0), (9)

where J t and J2 are the moments of inertia of each motor, 71 and 72 are the viscous

friction coefficients, and :/~(t) and T2(t) are the torques developed by each motor. It is

clear that the bending moment EIy"(t,O) works on the motor shaft of M1 as a reaction

of the bending motion, and the torsional moment GJ~'(t , O) works on the motor shaft

of Mz as a reaction of the torsional motion.

3. EVOLUTION EQUATIONS AND CONTKOL SYSTEM

We treat the partial differential equation (1) together with the boundary conditions

(2), (5), and (6) for the bending vibration in the form of an abstract second-order

evolution equation in an appropriate Hilbert space Hz = L2(0~ L) x R 2 with the inner

product < (,,,, ~ , ~) , (.,,-=, ~) > . ,

fo ~ [ ~ ,,,~ ] [~ ,1 00) = p , , , (~)~(~)d~ + [ ~ ~ l mc Jo + mc 2 v3

where the first component of an element of H1 corresponds to the bending distribution

y(t, .) and the remaining components correspond to y(t, L) and y'(t, L), respectively.

Define the operator Ai on / /1 in to/ /1 by

A ~ : (~(.), ,,,~ + . ~ , , ~ z + (Jo + m ~ ) ~ ) r, (11)

Furthermore, let us define the operator ~i by

Ziu = u(i4)( . ) , -EIe; ' (L) , EIu~(L) , (t2)

139

o ( ~ , ) = {, , = ( , ,~ ,~, , , , ,~)r lu, ( - ) e u~ (0 , L) , , ,~ = ~, , (L) , ( l a )

where D(~l) denotes domain of ~1. Also let us define an element fh E Hi by

~1 = - (~ , m(L + c), & + .~c(L + c)) ~". (14)

Then it is easily seen that, by using the operators A I and }]1, the differential equations

(I) together witll the boundary conditions (2), (5) and (6) can be written as

h1~(~) + 2 6 1 ~ ( ~ ) + ~.~(t) = U,~(t), (i5)

where u(t) = (ul( l) ,u2(t) ,ua(t)) T 6 D(EI) . If we define an operator At on Hi by

A1u = ATi)31u, D(Aj.) = D(}]I), (15) can be written as

where 1(0 = 0(0.

It can be proved in the same manner as in [1] that the operator A1 is selfadjoint

and positive de£nite on 'Hi, and the inverse operator A71 exists and is compact. Conse-

quently, the operator A1 has the eigenvalues {An} and the corresponding eigenfunctions

{¢,~(z)} satisfying

&¢,~(~) = ~ , ,¢ , . ( . ) (lr)

Let ¢ , . ( ) = (y,,(),y,,(L), y;,(L)) r Then (17)i,uplies

(ESle)~!?)(~) = ~,,,,,(~), ] - EIy:: '(L) = mJ,~[y,~(i) + cy~,(L)], j (18)

#t '1 2 i EIy , , (L) --- A,,lmcy,,(L) + (go + mc )y,,(L).

From (18) the dgeavalues A,, and the eigenfunctions tb,,(z) can be calculated easily.

The eigenvalues arc given by

A,, : (Etlp)(i3,~IL) ~,

where fl,~ are the positive solutions of

1 + cos fl ¢osl~ fl - (m/p)(fl/L)(sin fl cos*, fl - sinl~ fl cos fl)

- 2 ( ' } --~e) ( ~ ) = s i n f l s i n h f l - ( J° + inca ) ( ~ ) a ( s l n f l c o s h f l + s i n h f l c o s f l )

+ (mJolP~)( f l lL) ' (1 - cosfl coshf 0 = 0. (19)

It is easily seen that

140

,e,, - , ( ~ / 2 ) + ,,~ (,, ~ oo).

Similarly, the partial differential equation (3) together with the boundary conditions

(4) and (7) for the torsional vibration can be written in the form of an evolution equation

in all appropriate Hilbert space t/2 = L2(0, L) x R as

~(t) + 2&A2~)(t) + A~v(t) = h[1.f~2g(t), (20)

where 9(t) = ~;m(t). The inner product of H~ is defined by

# L

and the first component of an element of I/2 corresponds to the torsional distribution

T(t, .) and the second component of Hz corresponds to ~(t, L). The operator As has the

same property as At, and has the eigenvMues { /& } dud the corresponding eigenfunctions

{,-,,(,,,)}. By cementing several strain gage foils at the root of tile beam, it is possible to

measure the bending nloment and the torsional moment at z = 0, i.e., yn(t,O) and

$o'(t,0). Therefore, the output of each system can be defined by z1(t) = V'(t, O) ~nd

z2(t) = 9'(t, 0), respectively.

The solution of (16) can be expressed in terms of the eigenfunctions ~b,,(.) =

(w(-), v,,(Z), V:~(L)) ~ as

~,(,,.) = £ ~,,,,(,).¢,,.(.), (21) 71.=1.

where u,~(t) are the solution of

~,,(t) + 26~A,,/t,~(t) + A,~,,(t) = -Zlv',~(O)y(t). (22)

In the same way, the solution of (20) can be expressed as

r / , = l

where v,,(t) are the solution of

~, , ( t ) + 2,h~,,,,:,,,(t) + ,,,,m,(O = -c.,'¢.,(0)g(t). (24)

141

In (22) and (24), f ( t ) = 0(t) and g(t) = ~ ( t ) are regarded as the control inputs for each

distributed parameter system. By defining variables ~t=(t) by ~,( t ) = - l lw, , [h, , ( t ) +

~A,,u, ,( t)] , where oJ,, v,',~. ~ ~' = - 6tA,, , let us consider two finite-dimensional modal equa-

tions

~ ( t ) = F , ~ ( t ) + B d ( t ) ,

~(t) = ~ ( t ) + ~J(t) ,

(25)

(26)

where zl( t ) = (ul(t), ~21(t), . . . , u~(t), ~21(t)) m is a 2/-dimensional state vector correspond-

ing to the first l modes of (16), and ~'l(t) = (ui+l(t), ~ l + l ( t ) , ' " , uxv(t),i2N(t)) r is a

2(N - /)-dimensional state vector corresponding to the subsequent (N - t) modes of

(16). The matrices in (25) and (26) are defined by

F1 = block diag(Ml,--- , Ml), Ba = [b~,---,bT] r ,

F1 --- block d iag(Mi+l , . ' . ,MN) , /~i = [bT+i,"',b~v] r ,

ii T

t ~,= -61A,= J ' ~,~ J "

The output can be correspondingly approximated as

N - 1

z l ( t ) y"(t, O) -': ~ ~,<,,(Od(O) = Cl~.~(t) + G~.I(t), (27)

where Cl = M , ' " , c,], G = [c,+~,.. . , cN], c,, = (y:~(O)l~':, 0). i f y~:(0) # 0, ,< =

1 , - . . , l , then b, ¢ 0 and c, # 0, n = 1 , . . . , l , and the linear system (F1,B1,C1) is

controllable and observable [2].

By constructing two kinds of observers [2]

~bl(t) = ( F1 - V , Ct)w,(t) + Gi[z,(t) - 0151(t)] + Bar(t), (28)

~l(t) = F~5~(t) +/}~/(t) ,

the state vector let(t) can be estimated with an arbitrarily high convergence rate. In

(28), the term Cl~ l ( t ) is introduced to avoid observation spillover.

The following results was proved in [2]: Given an arbitrary damping constant o"

such that /71At < ¢ < (1/2~1), if I is dlosen corresponding to ¢r and N is large and

b,~ # 0,~ = 1 , - . - , l , then a finite-dimensional controner f ( t ) = Kiwi( t) , where wi(t)

is the solution of (28) with an arbitrary initial condition, can be constructed in such a

way that the solution of the evolution equation (16) satisfies

ll,,(t)ll _< const.~-<"lt,,(0)ll.

142

This means that the original infinite-dimensional system is exponentially stabilizable

with the exponent ~ by using the finite-dimensional observer-based compensator.

Let rx(t) be a given reference input function for the rotation angle 0(t), and let

~(t) = o ( t ) - ~lCt), ~ lCt) = e(~) - +~(t) . (29)

Combining (25) with (29) y{elds

,~(t) = Al,~(t) + t%/(O + ~ l~ , ( t ) , (30)

where ~(~) = (~,(t) T, ~(~),,~l(t)) T, ann

[i '°°] I!'} [°l A l = 0 1 , / 3 1 = , " D I = 0 .

0 0 - 1

We consider the case where ~'t(t) is a step function. Since the pair (Ft, Bt) is control-

lable, it is easily seen that the pair (A~, ~ ) is also eont~oUable. Since our problem is

to regulate (30), a feedback control law

should be sought by nfinimizing the performance index

£ J(f) = [q~y2(t, L)+qz~12(t, L) + qaO2(t) (32)

+ q4~b) + "fb)le~'dt ,

where q~ > 0 (i = 1,---,4), r > 0, and o- > D. Because z~(t) cannot be measured, we

use the observer output wl(t) in place of Zl(t ) in (28).

If the rotation motor M 1 has an amplifier of speed-feedback type, the torque de-

veloped by the motor MI is givea by

T1(t) = ki(V,¢f1(t) - k~.w1(t)), (33)

where wl(t) = 0(t), V,~/x(t) is the input-speed reference voltage to the amplifier, and

k~ and k~ are gain constants. Substituting T~(0 in (8) into (33) gives

71 [ t + + ~l f(t )_ El v . . ~ ( t ) = (k~ + V ) i j o : ( t ) d t ..,(0)]

where wl(O) and z1(t) are known, and ](t) is given by (31).

143

The feedback control law for the torsional vibration can be obtained in the entirely

same malmer as discussed above. Namely,

g(t) = K,~l(t) + K~#.~(t) + Ks&2(t), (35)

where ~1 is the finite-dimensionM state vector corresponding to the first several modes

of (20), #.~(t) = W,(t) - r2(t), &2(t) = #. . ( t ) - ~:2(t), and r2(t) is a given reference

input function for the twist angle.

Acknowledgment. The authors wish to thank Mr. T. Murachi for his cooperation.

REFERENCES

[1] Y. Sakawa and Z. H. Luo, Modeling and control of coupled bending and torsional

vibrations of flexible beams, IEEE Trans. Automat. Contr., vol. 34, no. 9, pp.

970-977, 1989.

[2] Y. Sakawa, Feedback control of second order evolution equations with damping,

SIAM J. Contr. Optimiz., vol. 22, no. 3, pp. 343-361, 1984.

[3] Y. Sa-lmwa, F. Matsuno, and S. Fukushima, Modeling and feedback control of a

flexible arm, J. Robotic Systems, vol. 2, no. 2, pp. 453-472, 1985.

[4] L. Meirovitch, Analytical Methods in Vibrations. New York: McGraw-Iiill, 1967.

STRONG SOLUTIONS AND OPTIMAL CONTROL FOR

STOCIMSTIC DIF}~RENTIAL EQUATIONS IN

DUALS OF NUCLEAR SPACES *

Situ Rong Department of Mathematics, Zhongshan

University, Guangzhou, China

Introduction

Stochastic differential equations in a dual of a nuclear space is an appropriate

stochastic model for investigating a diffusion process, which is a limit of a

sequence of stochastic processes arising in very diverse fields such as ne~rophy-

siology £7], interacting particle diffusions [17] , and chemical kinetics C8].

Moreover, it is also an effective model for exploring some physical problems, such

as the random mo%ian of strings, etc. ([6,J, [2] ,[5])- In %his paper we obtain re-

suits on Girsanov theorem, existence of weak and strong solut,ions, martingale re-

presentation theorem, maximum principle to optimal s~ochastie cont=ol, and path-

wise uniqueness, stability of solutions for stochastic differential equations (S

DE) with discontinuous drift, which even can be greater than linear growth, and

with no~-Lipsehitzian diffusion coefficient in the duals of nuclear spaces.

Girsanov theorem and weak solutions

Let ~ be a real nuclear Frechet space with topology generated by a countable fami-

ly of increasing Hilbertian norms ~|.~n ). Denote the completion of ~ under norm

~'|n by H n, the strong dual of~ bye', the dual of H n by H -n. Then

= ~ H -n. = ~ n=l Hn' ~' ~=i

Now let us adopt the notation from ~6], [8~. A ~'-valued process is called a ~'

Brownian Motion process (BM), iff w t is a ~'-valued n2-martingale (i.e. wt[~I is

a real square integrable martingale for each ~ ~ ~) such %hat

l ° w 0 = O, p - a.s.;

2 ° <w>t[~,~] = t Q[~,~], for all ~, ~ e ~,

where Q @ ~' x ~', which is non-random, and <w~ t is a ~' x ~' - valued predictable

process, which exists, is increasing in t, and for all ~, ~ ~ ~

Yt[~,~] = wt[~]wt[~] - <w~t[~,%J

is a real local martingale satisfying yof~,q] = O, p - a.s. ~y [12J, [63 there

exists a 0 < q ( ~ such ~hat for each r ~ q w.(W) @ C(R+| H -r) (%he space of

strong continuous maps in H-r), and Q has a unique continuous extension to a nu-

clear form on H -r such that Q: H r x H r ~ R ~, and for all ~, ~ ~ H r

• This work is supported in part by the Foundation of Zhongshan University

Advanced Research Center.

145

for a unique non-negative trace-class operator % on H r , where (h~} is" a complete

orthonormal system (CONS) in H r . In the wholz paper we shall fix a special index

r (~- q), and call Q the covarian~e operator of w t, NoW let us introduce the defi-

nition of a stochastic integral with respect ~m (w.r.t.) a~ BM w t wi~.h Q. Set

2 = f(s,~v): R x CA -) L(~' ~'), which is &t-adapted, and E w = the totality of fs ÷ '

El0 ~ Q~q~, f;~] ds, ~, for all ~ ~, ~ ~ O, where f~ = f~(s,~): R+ x fi @ L(#)~) is the adjoin~ operator of fS such that

~, r1"~ ~:) = ( f s9 ) r,I'.1, for ~ l ~ ~- ~, q- ~- _~,. f ~t fsdws, t @ ~0,T], is a 2 the s~oehasti¢ ir~tegral I% = 0 Def~nitlon 1. For f ~ L w

~'-valued L2-mar~in~ale, which is strongly ~ontinuous w.r.t. ~, and satisfies 2

1° lit is linear in f 6 ~w)2 i.e. for all a, b - real numbers~ and f, g ~ T, w

Iaf+bg f big , t = aIt +

2° < ~ S t [ ~ , ~ -- I t ~ ; ~ , ~ ; ~ d~, for a n ~, ~ ~-~.. Remark I. By a category argument [6~ ~here exists a sufficiently large m -~ x"

(depending on f and T) suc,h ~hat I{ ~ C([O,T]; H-m), P - a.s.

5 ° if lh~ ")jr1 ~ ~ is any CINS i~ }~, then as 0 -~ t &- T

@e m 5%

2 where the right hand side is an ~-convergent series of the usual Ito stochastic

integrals.

We have %he following Girsanov type theorem.

Theorem I. Assume that

h: R+ x_~' ~#,, ~: R+ x~'_ ÷ L(~_", ~'),

for each T > O and sufficient large m -~ r there exis~-~ an index p ~ m, as t z T

1 ° b(ttu): R+ x H ~m .i, H -p, whi=h is jointly measurable, and

(u, b(~,u)) z_ ko(1 + ~2p ~=I gi {u})" for all ~ & H -m,

where for i = I, 2, ..., k

g t ( u ) = I + i n g i _ l ( U ) , gO(u) I + [u f2n° I a n - c o n s t a n t , = ~--p ~ -- O

2 ° Jff(t,u)V]_p " kolVl_p, for all u e H "m, v e H-P. I~ x t with ~' BM w t satisfies ~' - valued SDE

~t : Xo + ~o ~ ( ~ ' ~ o ) ~ w s " Xo ~ ~" ' o -~ t ~ ~., (~) t hen

~ t = wt - ~(~ Q;b(S'Xs)dS* 0 -~ t ~ T, is a ~' BM with the same covariance operator Q under the probability measure

d~ = Z~dp, where

Zt = exp(~t (b(s'xs)'dws)-p-½ IO [q~½b(s'xs))-2p ds)'

146

and p -~ m, m comes from remark I such that x. G C([O,T|; If'm), and p from the

*. II -p ~H -p is the dual of % such that above assumption; and Qp.

~[Qp~]-- (Q~)[~I, for all ~ ~ HP0 @ ~" H -p.

The proof of theorem 1 depends on the following

Lemma 1. If x t with ~' BM w t satisfies (1) and E Z T = I, then the conclusion

of theorem I holds.

proof. Denote b t = b(t,xt), Nt = it (bs,dWs)_p '

Then it is not difficult to derive ~hat

By Its formula for arbitrary ~ _~w,~ _ measurable, bounded function it yields %ha "" S

E(e~pK~(U t - Us)K~j~tl) : e~p(- A2/2 Q[V,$~(t - s)) E(Z ~). Hence if denote the expectation w.r.t, probability P by ~, then

$(exp(iA(U t -~s)[~S~'~) = exp(-A2/2 Q~,~J(t - s)).

From this w t is a ~'-valued L 2- martingale, and for all ~, ~£-~, 0 -~s _z t

<~)t[~,~] = t Q[%~]. Q.E.D.

The proof of theorem I can be accomplished by lemma I and the technique as [14].

We omit it here. In the following we always make the assumption

(A) there exists {hj} ~ J=l & ~' which is a common orthogonal system in Hm~ for all m A~ m -~ I. (Hence (hj~j= I f ~ is a CON~ in H m, if set h~ = hj/|hjlm).

Applying lemma 1 and a result from [6] we have immediately a theorem on %he exis-

tence of a weak solution for a ~'-valued SDE with discontinuous drift, which even

can be greater than linear growth as following:

Theorem 2. Assume ~hat conditions ~o and 2 ° in theorem I hold, and

5 ° ~: R. x~' -~ L(~', ~') is Jointly continuous,

4 ° ~'(s,u)v ~- H -m, if u, v ~ H -m,

~o Q(~*(s,u)~, ~-'(s,u)~) is continuous in u @ ~ for each ~ ~_~.

Then there exists a weak solution for ~-~lued SDE

x t : Xo + i t ~ . (S,Xs)Q.b(S,Xs)d s + ~ t ff(S,Xs)dWs ' x 0 : x ~" ~ ' , OZ-t-ZT.

In case, if

6 ° ~--I exists and I|~-I[I~(H-P,H-P) -z ko ,

then ~he following &'-valued SDE also has a weak solution:

x t : * I t * I t (s' s)dwo' Xo : x 0 t T .

Remark 2. Here we define

Q* ~ = Q~, a s ~- ~. H -~. (*)

Since the extension Qp Is unique, hence

Q~2 IH-pl = Q~I" as Pl 4 P2"

(*) is well defined.

Proof. ~y ~6) (I) has a weak solution. Applying theorem I one obtains the de-

sired result of theorem 2. (Note %hat O'Q*~'Ibp = ~b). Q.E.D.

147

Tanaka Formula and .Strong Solutions

Theorem 3. (Tanaka formula). Assume that

b i : ~+x_~, ÷ ~,, i=I, 21 ~- R÷x~' ~ ~C~',~'),

for each T > 0 and sufficient large m -~ r there exists an index p -~ m, as t ~- T

i ° IbiCt,x)l 2_ + tracC~,~Q*) -~ ~(Ixt_p), i = I, 2, as ~ c- II -m, where b(%,.)-~. • H -m @ H -p, (y(t,.) : H -m .~ L(H -p, H-P),

, * ~ .½h-P .½h-P trac(~Q) =~j=l (=Qp j "~Qp J )-p' and g: R I .~ R I is continuous;

2 ° for all x, y ~ H -m, as |xl p, ~yi_p _z N, N = I, 2, ...

ll~(t,~) - ~(t,y)Ik(~-p,H p ) ~ ~(t) pN(I~ - yI_p),

where 0 ~ pN(u) la strictly in~reasi~, ~onti~uo~s, ~N(0) = 0; kN(t) -~ 0, and

foranyO~T <4".

Ira'-valued processes x~, i = I, 2, satisfy

. x ) d s + 1 0 ~ ( s . x s ) d w s . t -~ 0. (2 ) then for each T > 0, there exist m _~ r, and p ~ m such ~,hat as 0 _It _z T

ix ~ - 2 I 2 t x I x2-i I x 2 b1(s,xl). xtl- R = l X o - X O l - p + [0 Zf.xs~Xs)l 7 2. I s - s l - p { ( X s - s'

-- S ' -- +

½ Z ~ t

2 2 I 2 -5 - z , ~. ~. ds. (~) =o)_piix s -:ol_p <xs, o,

Theorem ~ can be proved similarly by the technique as f13]. We omit if here.

Applying theorem 5 one can derive the pa%hwise uniqueness theorem for (2):

Theorem 4. Assume that all conditions 1 ° and 2 ° in theorem ) hold, and b I = b 2 = 2

b, x I = Xo, moreover, as x, y e I[ -m, Ixi_p, lYl_p z N, t ~ [O,T~

t 2 Z GN([ Yl_p) Ix Yl_p, $o ll¢(t,x) - ~( ,y)IIL(H-p,H-p ) - kN(t) x -

where GN(U ) > 0, as u > 0, it is increasing, concave and such %hat

1"O+ dUfGN(U ) = . , tT kN(t)dt ,~ . , ~(%) ~ O, t ~- O.

4 ° z( .7~ ) (~ - y . b ( t . ~ ) - b ( t . y ) ) _ p / l x - Y l_p - ~ k ~ ( t ) C , ( I x - Y l_n ) . where kN(t ) and GN(U ) have the same property as that in .50.

If two ~'-valued processes with the same ~' BM w t satisfy (27 on ~he same probabi-

lity space, then I 2

P(~: x t = xt, for all t -~ O) = 1.

Theorem 4 can be proved by ~he category argument [63 and the similar technique as

Remark ~. Each of the following GN(U ) satisfies condition in 50:

GN(U ) = u, GN(U ) = u In(11u), GN(U ) = u i n ( In ( f l u ) ) / . . . , etc.

Remark 4. b(t,x), which satisfies 4 °, can be discontinuous, e.g.

b ( t , X ) = - x / i x l _ p , as x 7 0; ~ ( t , x ) = 0, a~ x = O;

1 4 8

where we assume that (I) has a weak solution and m ~ r is such that x. ~ C'(~O,TJ;

H-m), and p ~ m. Then 4 ° holds for b(t,x) with such p and T.

Since the Yamada - Waganabe theorem s£ill holds for ~'-valued SDE (see ~61), so

applying it and by theorem 2 and 4 one obtains the following

Theorem 5. Assume that now I ° -- ~O in theorem 2 hold, and conditions 1 ° -- 3 °

in theorem 4 are fulfilled; moreover, 40 in theorem 4 holds for ~, where ~ = gQ*b.

Then SDE (2) with coefficients ~ and @ has a pathwise unique strong solutions.

In c~se 6 ° in theorem 2 holds, and 4 ° in theorem 4 is fulfilled for~, where~ =

Q*b, then SDE (2) has a pathwise unique stron~ solution with coefflolen~s ~and ~.

Example 1. Assume that O(t,x) satisfies 2 ° - 6 ° in theorem 2, then by theorem Z

(I) has a weak solution. For simplicity suppose that x t with w t satisfies (I).

Furthermore, assume that ~onditions I ° - 5 ° in theorem 4 for ~(h,x) are fulfilled.

Then by theorem 4 and the Yamada-Watanabe theorem (I) has a pa~hwise unique strong

solution, denote it by x t (with wt) again. Now fix a T • O, by the category argu-

ment C6] there exists a sufficiently large m w 0 such that the trajectories of so-

lutions x. ~ C([0,T|; H-m), P - a.s. T~ake an index p a m, and set

bCt,~) = - , x i [ ~ Q ;x - q;2x/iQ~:i_p = bl(x) + b2(x),

where N O ~ 2 is any natural number. Then b satisfies I ° in theorem I and 4 ° in

theorem 4 for this T and p. Indeed,

~½Xl2p * : - l x l ~ 7 I~ - l lpXl_ p ~ o , ( x , b ( t , x ) ) _ p - -

and as Ix l_p , lY l_p ~ N , x, y ~ H -m Ib1(x ) . h('y)l_p ~ I x - yl_p, ~ - constant;

(x - y, b2(x ) - b2(Y))_p = - lq~Xl_p - IQ~YI_p + ( ~ x , q~y)_p.

• ( I + I Q ? l _ p ) o.

Therefore theorem ~ is applied. (2) has a pa~hwise unique strong solution on t

JOLT] with such ~ and 5, where O is non-Lipsohitzian, non-monotonic, and b is

discontinuous, greater than linear growth in x very much; e.g. if e 0 ~ H -p,

eo = ~0 eo' |eo|-p = I, ~0 > O,

= n one has then set x n e0,

b(t,Xn) (n N0+I N0+I = - + 1) ~o Co' I~ ' ( t 'Xn) l -p ~ AOIXnl-p '

Stability Theorems

Applying theorem 3 one can derive stability theorems for solutions.

Theorem 6. Assume that for n = O, I, 2, ...

b n = bn(t,x,@: R+ x~' x• ~', Cr =(Y(t,x): R+ x ~' xf[ @ L(~',~')

for each T ) 0 and sufficiently large m • 0 there exists an index p -~ m such that

as x, y & H -m, 0 -~ t _z T, n = O, I, 2, ...

1o lbn(t,x,t,)12_p + l l l l ( t , x , t . ) [ i L ( H _ P , H _ p ) _Z ko(1 + i X l 2 p ) .

where bn(t...,.)~ ,-~ .> H-P, O ( t , ' , ~ ) . H "m ÷ L ( . -p, H-P);

149

2 ° 2 ° in theorem ) holds for all x, y ( E -m with k(t) = ~(t) andS(u) =~N(U),

which do not depend on N;

5 ° 5 ° in theorem 4 holds for all x, y 6 H -m with k(t) = kN(t ) and G(u) = GN(U),

which do not depend on N, and in addition, G is strictly increasing and conti-

nuous ;

4 ° 4 ° in theo=em 4 for bO(t,x,~) holds as x, y * H "m, where k(%) and G(u) have

%he same property as that in ~o;

5 O Ibn(t,x,.) - b0(t,x,.)l_p & Fn(t,.), limn.: E~ pn(t,•)dt = 0,

6 ° limn_~ E[x; 0 " x 0 l_p = 0,

n satisfy (2) with coefficients b n and 0", n = O, I, 2,.. If ~'-valued processes x t

and for each T ) 0 there exists a sufficiently large m ~ r such %hat n x. & C([O,TJ; H-m), for all n = O, 1, 2, ..., (*)

then for each T > 0 there exists an index p (~- m) such that 0

limn@ ~ E~x t - x0L p = O, as t -ZT.

Remark ~. If hn and ~ satisfy all conditions in theo=em 5, then (*) holds.

Remark 6. Beware of that b 0 satisfying 4 ° can be discontinuous. Moreover, if set

G(u) = fu l~.(I/u), as u < a,

[(ln(Va)-1)u+~, asu~a, where a is a ~onstant satisfying in(~/a) - ~ > O. Then G(u) satisfies the condi-

tions ~o and 4 ° . Besides, in this case f~u) = JG(U) ½ satisfies 2 ° .

Proof. Given T ) 0, then (*) holds. By I ° applying Ito formula in H -p and Gron-

wall inequality, we get that there exists a constant ~ -~ 0 such that

E Ix n 12 x ' t'-p ~, for all n = 0, I, 2, ..., as t ~ T.

Hence as t ~ T

c(~I~t- 0 T'herefore, by theorem 5 and Fatou lemma as t Z T

........ n 0 L t k(s) iimn.t G(E~xs n -xs01_p)dS limn@mE[xt - xt[-p kl ~0

- - ~ 0 = O, as t Z T. Q.E.D. I~ yields that limn~ E~x - x t [_p

We also have other stability theorems.

Theorem 7." Assume that

, 0": R+ x ~ ' for each T > 0 and sufficiently large m ~ r there exists an index p -~ m such %hat

as u, v £ H -m, t -~T

Z ko ~k+q(U)½ IV|_p; i ° ~(t,~)v ~ ~-m [~(t,~)v1_p

where ~k+1(u ) = gk+l(U) _ I, and gk+1(u) is defined in I ° of theorem I;

2 ° b(t,'): H -m -~ H "p, lh(t,u)~_p z k0(1 + ~U~_p ~k gi(U) ) ~k+l(U)~ ; - i=I

)O conditions ~o - 6 ° in theorem 2 hold.

Then SDE (2) with coefficients 0" and ~, where ~ = Q,b, has a pathwise unique

strong solution on t -~ 0 provided that ~ =~I = ~2, x0 = x; = x20 $ H -nO is non-

150

random. Moreover, if we denote this solution by x(t,x0) , then for each T'~ 0

there exists an index p (~ m O V r) such 1~hat for arbitrary ~ ~ 0

limlx0l_p,O P(suP0mt&T l~(t,x0)l_ p ~ ~) = o. (4)

Remark 7. Since by assumption ~(t,0) = 0) ~(t,O) = 0, 0 is evidently a pat hwlse

unique strong solution of (2) with initial value x 0 = 0. Therefore (4) means that

the solution of (2) is stable in probability on each finite inKer~al in terms or

[5].

The proof of theorem 7 needs some technique as ~hat of theorem I etc. We omit it.

Theorem 8. Assume that all conditions in theorem ~ hold, where, in addition, as-

sume that GN(U ) in )o and 4 ° of theorem 4 is increasing strictly) continuous; and

k N = ~(t) in )o and 4 ° of theorem 4 does not depend on t, which is a constant

depending on N only. Moreover, assume that

7 ° b( t .O) = O, G( t ,O) = O, for all t ~ O;

then the conclusion of theorem 7 holds.

Theorem 8 can be proved by theorem ~ and a little bit more discussion. We omit it

here.

Martingale Representation Theorems and Maximum~ Principle

Let us give now a real-valued martingale representation theorem with respect to

a ~' - diffusion process for our purpose.

Theorem 9. Assume that all conditions in theorem I and 5 hold, and 5 ° in theo-

rem 4 and 6 ° in theorem 2 also hold. If ~t is a real @; - adapted square inte-

~rabZe martingale, where x t satisfies ~' - valued SDE (~) with ~ and ~, where

= Q'b, then for each T > 0 there exists a sufficiently large m -~ r such that x. 6

C([0,T] ; H-m), and there exists an index p ~- m and there exists a ~ - adapted

H-P-valued process f(t, ~) satisfying

) t = + I t ( f ( s , . > , dws)_p, t C- CO,T , where

I dt E 1Q~f (t, ,-) -P

The proof of theorem 9 depends on the following W

Lemma 2. If N t is a real ~t - adapted square in~egrable martingale with N 0 = 0,

where w t is a~' BM with the covariance operator Q, then for any T > 0 and p -~ r W

there exists a ~t(&~), t _z T, which is ~t-adapted and H-P-valued satisfying

Nt : )t dws)-p' for an t o,Tj,

where A 2p

IT I Q ~ % l - ds < . .

The proof or lemma 2 and theorem 9, even it is simil~r to ~I0], needs more discus-

sion. For saving pages we omi~ it here.

Now we are in a position to discuss the stochastic optimal control problem. From

now on we shall always make the following assumpation:

(I) The conditions of theorem 5 (including 6 ° of theorem 2) for (r(~,x) itself

hold.

151

Applying theorem 5 we ob,tain that (I) has a pathwise unique st:rong solution xt, w

which is ~t - adapted. By the category argument for given T > 0 %llere exists a

sufficiently large m ~- r such %hat x(.) ~ C([O,TJ ; H'm). In ~he following we shall

fix this m. Consider now SDE (2) with drift =oefficient as ~ = Q.~b(~,x,u(t,x)),

where u: [O,T] x ~' ÷ V is Jointly measurable, and V is some Hilbert space. We

also make the following assumption:

(II) For given T > 0 and m & r there exists an index p =~ m such ~ha~ as t ~ T

b(%,.,.): H -mx V -~ H -p,

k Ib (~ ,x ,u ) l_p L ko O ÷ lx l_ p 71i=1 ~ i ( x ) ) , as x ~ t~ -~, u , V.

Then by theorem 2 (2) has a weak s o l u t i o ~ ( x t , w~) fo r ¢ o e f f i v i e n t s ~ and ~ on

probability space (/l,~,(~t),pU) as t ~ [0,T~, where pU is a probability measmme

defined by

u dP, dp u = Z T

u -ib(B,Xs,U(S,xs) Z T is defined in theorem I bu~ with b(t,x) substituted by ), and

w~ is a &' BM under pu such ~,hat (deno~,° ~ = ~-Ib) U

w t = w t - I t Q ? ( s , x ) ) d e . Denote now by~ the totality of u mentioned above, i.e.

w = {U = u ( t , ~ ) = u ( t , x t ( W ) ) : i t i s &t - adap ted , and u ( t , ~ ) (~" V,

for all t C- [O,Tq~

Introduce a meiotic d on ~ by setting

d(u,v) = (mx P ) { ( t , ~ ) (- [O,T] x ~ - u(t ,@) ~ v(t , ,~)3 , for u, v ( ' ~ , where m is the Lebesgue measure on [O,%~]. Then d is a complete ~is~ance on ~.

(See [I]). Consider the minimization of functional

F(u) = E u gT(x(.)) = ~t gT(x(')) dpu(~)'

where

gT (x ( . ) ) = ~T c ( t , x t ) d t + h(xT), among all u @~. Le~. us make the following assumption:

(III) c(t,x) and h(x) are real, Jointly measurable, and as t & ~O.T], x ~ H -m

l c l + lh l + l b ( t , x , u ) l _ p + It~(t ,x)l~L(H-p,H-p ) ~- k o -

We have the following Maximum Principle.

Theorem 10. Under assumption (I) - (Ill) %~hen

I) for any ~ ) 0 %here exists a control u e ~ such that

~(~) z i~u~F(~ ) + ~, and for all v ~-

;(v) -~ ~(u ~) - d(u~.v) ; w

2) %here exists a&t - adapted process )'t(e)), which is H-P-valued, and satisfies

and pU - a.s. for all t 6 EO,T~

152

such %hat for all v~-~( denote ~0t(b) = Zt, where Zt is defined in theozem I)

P0 ~t Q.~v) &p0 ~ ) ( i t , o*~u')_p-~ mx F - a . e . (5)~ ' p -p -p ,

f" - (5)~, then for all v (- )) if u £ with Ft verifies (5)I

;(v) -~ F (J ) - exp(2kIT) ~T, where k I = k02 Ilqpl]L(H-p H-p ), and k 0 comes from ( I I I ) and 6 ° of theorem 2;

4) if u ° is an optimal control, %hen 2) holds for ~ -- O;

~) i f u°6 ~ and ~Z(~) is a ~t " adapted process sue~h £hat o

t 0 0 O = U )~ o(g(xE.))l~t) = j ( o ) . [o(~s.dws)_p ' (~et w t w t u

o i.e. Ft is the integrand in the martingale representation for %he ~ o(g~(x(.))l~ w ) ~ _

u

and if for all v (- ~ m x p - a.s.

(~t(~), Q; ~(t,xt(~),vt(~)))_ p -~ (~t((o), Q~ ~'(t'xt(w),uZ(~)))_p,

%hen u ° is an optimal control.

To prove theorem 10 one has to apply the Ekeland lemma [I ] and theorem 9. But for

applying the Ekeland lemma one needs the following

lemma 5.. F(U): (~,d) ~ V is a continuous map under assumption (I) - ( I l l ) . Remark 8. The proef of lemma 3 of Elliot and Kohlman~ £I] is imcomple~e;. Since

the following incorrect fact was used: d - convergen=e implies convergence a.e.

with respect to m x P. A counter example is given by [16}. Hence i% is doubtful

that their lemma ~ is true under condition

0~-1(t,x)b(t,x,u) z ko(1 + Ix [ ) , as t C- [O,TJ, x (- R n, where l~ is a n - dimensional vector, and CF is a n x n mat=ix.

Lemma 3 can be proved as [15]. To show our method let us Just prove 5) of theorem

10 only.

Proof of 5)- Note %hat for any v (-

IT 1,t, dwV)_ T --v o ( --Io (Y~, dwt - * bt)-~' -- * @ --

where k I = ko 2 [IQp~IL(H-P,H-p). Therefore by (~)5 u &

F(U ~) _~(T E~t(~v)~t(~u £)-Idt ~ F(u £) - e2kIT(T , " Hence

~(~) = ~(~(x(.))) = ~v(E ~(~(~(.))I~)) -~ ;(u ~) - ~ ~E. ~E.~.

REFERENCES

[I] Elliot, R.J. and M. Kohlmann. (1980). The variational principle and stochas-

tic optimal control. S toehastics, 3, 229-241.

~2) Funaki, T. (1980). Random motion of strings and related s@ochastio evolu-

tion equations. Nagoya Math. J. , 89, 129-~93.

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~3] Hazminskii, R.Z. (1980). Stochastic Stabilit[ of Differential Equa~dons.

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[42 Ikeda, N. and S. Watanabe. (1981)o Stochastic Differential EQuations and

Diffusion Processes. North-Holland.

[5] Ito, K. (1984). Foundatlons of Stochastic Differential Equations in Yn[i-

nite Dimensional Spaces. SIAM, Philadelphia.

[6] Kallianpur, G., Mitoma, I. and R.J. Wolpert. (preprin~). Diffusion equa-

tions in duals of nuclear spaces.

[7| Kallianpur, G. and R. Wolpert. (1987). Weak convergence of s~ochastio neu-

ronal models. Stochasti@ Methods in Biology, Lecture Notes in Bioma~hematics

70, Eds. M. Kimura, G. Kallianpur and T. }[ida, 116-145.

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[?] Korezlioglu, H. and C. Martias. (I788). Stochastic integration for operator

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fusion model of interacting particles° Hiroshima Math. J. 11, 415-425.

SOME N E W RESULTS ON A P P R O X I M A T E CONTROLLABILITY

FOR S E M I L I N E A R SYSTEMS

H. W° SUN

GUNANGDONG INDUSTRY COLLEGE,GUANGZHOU,CHINA

Y. ZHAO ZHONGSHAN UNIVERSITY, G UNANZHOU, CHINA

ABSTRACT

In this paper,the existence of the local or global solutions for an abstract semitnear time

--variant system is discussed under weaker conditions. The approximate controllabity of the

system is obtaied in two case respectively and the results have improved that in some papers

published recently. Some examples are also given to illustrate the applications of the

results.

I . INTRODUCTION

In this paper,we consider an abstrat semiUear system in a reflexive Banach X as

follows

I' x( t )=U( t ,O)~ '0 - t - oU(t,s)(N(z(s),u(s)) + Bu(s))d.s t ~ O (1 .1)

where x0E X ,U( t , s ) is a mild evolution epcerator,control space U ks another reflexive 13a-

nach space, u ( ° ) E LP([O, c~) , U) (1 ~ p ~ - l - o o ) , B E L ( U ) X ) ,N(x) u) is a X--valued

function from X X U to X.

Denote linear system correslxxling to (1 .1 ) as N~--O by

f'n(t,8)Bu(s)gs x(t) = V(t,O)zo + (1 .2 )

In the case of time--invariant systems,U(t,s) is reduceed to be a Co semigroup S( t - - s )

and the controllability of the system was discussed by some papers (4) - - ( 1 0 ) ,but the condi-

tions needed are restricted somewhat. That is, in addition to uniformly Lip condition for N

( x , u ) , B U = X ( s e e ( 4 ) ) or that N(x ,u ) is uniformly bounded(see(5)~ (8 ) ) or that S(t)

is a compact(see(10)) is proposed. The results of these papers are mainly applied to discuss

the approximate controllability with out regard to terminal time. We attempt to obtain the ex-

istence of the solution under weaker conditions and the results of approximate controUabilty

under two cases: N is uniformly Lip or uniformly bounded for time--variant system(1.1)

155

(N1):

that

1I. THE EXISTENCE OF SOLUTIONS

First, we list some hypotheses which will De used in the sequel.

(H1) • U( t , s ) is mild evolution operator on X and for any given T < ~ + o o , (TS>0),exist

M(T)~>0 such that

max 11 u(t,s) tl M(T) (2. 1)

O ~ s ~ t - ~ T

(H2) : (H~) holds and(i) U ( t , s ) ( 0 ~ < s - ~ t ~ q - c ~ ) is a compact opreator on X;

(ii) there exists M]> 0 such that

supllU(T,s) ll<~M ( 2 . 2 )

0 ~ s < t < c o

N(x,u)----N(x) is unfiormty Lip with respcct to x, that is,there exists K ~ 0 such

lIA'(xj) - - N(a ) l l ~ KIIx, - - a'zll for ~., ,x2 E X (2 .3 ) (N2)= for any given uELa(E0,c,o) ; U ) , N ( x , u ( t ) ) / ~ F ( f , x ) satisfies

( i ) F ( . , x ) E L ~ ( [ 0 , o o ) , X ) for x E X

(ii)for any given c2>0 and T~>0,thcre exists L(c ,T)~>0 such that

l l m t , z , ) - p(t, )ll L( ,,T)IIx, - - z ll t E (2. 4)

for any xl ,x2E Bc(0)= {x :xEX, llxll~<clcx (N3): For any given u E L * ( E 0 , o o ) l U ) , N ( u ( t ) , x ) ~ - - - F ( t , x ) satisfies (i) in (N2) and

(i)it is continuous at any x E X unfiormly for t E [0 ,T]~

(ii) it is L p integrable o n [ 0 , T ] for any x E X ,

(ill)it maps bounded subset in[-0, oo] ~< X to bounded subset in X.

(N4): N ( x , u ) is uniformly bounded with respect to x E X and u E U jointly.

In what follows,we present a series of results on the existence of local or glol~ solu-

tions for (1 .1 ) under different hypotheses.

Theorem 2. 1 If (HI ) and (N1) hold,then for any given u ELr (E0 ,o o )~U ) (1 .1 )

has unique solution x E C ( ~ 0 , o o ) ~ X ) .

Theorem 2. 2 I f (H2) and (N2) hold,then for any given u E L " ( [ 0 , o o ) ; U ) . There

exists t C [ 0 , o o ) such that (1. 1) has unique solution x E C ( [ 0 ~ t ) ~ x ) . Moreover,if i

~=oo and thes solutions o f ( 1 . 1 ) can not be extended to the right side from,t , then

Corollary 2. 1 If (H1) , (N2) and (N4) hold,then for any given u E L*([0. co) ~U) (1.

1) has unique solution x E C ( E 0 , o o ) ~X).

Theorem 2. 3 I f (H2) , (N3) hlod. The conclusions of theorem2.2 hold.

Corollary2. 2 If (Hfi) , (N3) and (N4) hold ,The conclusions of corollary 2. 1 hold.

The thorem 2. 1 can be proved using generalized Banach contraction mapping principle

by means of the methods similar to that in (1) .

156

Proof of theorem 2. 2:For any t>~to>~-0, ( 1 . 1 ) can be rewritten as fotlows

z(t) = U(t,to)z(to) + f ( t ) - - U(t,to)f(to) + j 'U( t , s )F( ,~ , z ( s ) )d~

S? ~(t,t0,z(t0)) + ( t , s ) l ~ ( s , x ( s ) ) d s (2.5)

where f ( t ) = U (t , 0)x0.

? Since F ( t ) ~ ( 0liP(s, 0)II'ds)~ is continuous with respect to t , so for any given T ,~o o , l t

exists M~(T)>O) such that

,~xtl~(t)]l ~ M,.(T) (2. 6) t E E0,7'~

Also,it exists M~(T)~0 such that

I1.¢(') II.<Eo.~.3.x) ~< MAT) (Z. 7) beaeuse fE C(E0,T],X) Consequently ,we have

II~(t,to.x(to) )11 ~< Ms(T) (1 + M(to + 1) + M(to + 1)llx(to)ll) ~ N(llx(t)ll) (2. 8)

for t E [ t o , t o + l ] by (2. 1) and (2. 7) Let

,re(t0) = ZN(llx(to)lt) (z. 9) and take

6 = min(1,(N(llxoll)/M(to + 1)(L(K( t0) , to + 1)K(to) + M,( t + 1)) q)

(2. 10) where l / p + l / q = 1.

Define mapping Q: Y = c ( [to, to+5] ~ t3K(,,) (0))--,-C ([to, toq-5] ~ X) as foUows

Q(z) (t) = .q(t,to,z(to) ) -4- )ds (2. 11)

According to (2. 4 ) , (2. 6 ) , (2. 8 ) , ( 2 , 9 ) and (2. 10) ,we see thatltQ(x) (t)ll~<K(t0) ~or

any X E Y which implies that Q maps Y into itself.

On the other hand,for x j , xzEY,one can deduce that

ll(Q'(z,)(-) - - Q ' ( ~ O ( . ) II, ~ ( ( M ( t -I- 1 ) ) L ( K ( t ) , T + 1 ) ' / u l )

( II x , ( . ) - ~ ( . ) II , ) _~_m tl ~ , ( ' ) - z~ ( - ) 11 , As n is taken large enough we have m<~l ,So Q has fixed point in Y by Banach generized

contraction principle and (1 .1 ) has unique solution E Y.

"File reainder can be proved by the methods similar to that i n ( l ) .

The corollary 2. 1 follows theorem 2. 2 and (N4) immediately.

Proof of theorem 2. 3=We start to work with (2 .5 ) and assume that it exists x(t0)

For any given O~;>0,1et BO(to) = S ( o ; x ( t o ) ) C X and t E [ t o , t o + l ] . By (N3) it exists

N(to)~>0 such that

II p(t ,x) il <~ N(to) :or ( t ,x) E [o,to -Jr- 1"1 )< Bp(to)

157

Noting that g ( t , to ,x (to)) is continuous at t = 0 and g( to , t0 ,x( to))- -~x(to) one can see

that there exists 5' ~ 0 such that

ttg(t,to,x(t~)) - - x(to)tl~ ~ ~ (2. 12)

Let

~ - m i n ( 1 , 6 ' , p / 2 .MN(~) ) ( 2 . 13 )

where M is as in (H2) .

Let

Y = C([to,to + ,~],X),l% = (z.z E Y,z(to) = ~'(to),z(t) E/3p( to) for t E [t~,to + ,~])

It is easy to check that Y0 is a bounded convex subset in Y. Define mapping P:Y--~Y as

follows

S'oU(t,s)R(s,x(s) )da (Px)(t)) = g(t,to,z(to)) + (2. 14)

One can work out that (2. 14)

for any x E YoCY. One can see that P is a continunous mapping from Yo to Yo by

(N3) and P Y 0 = Y ( P x : x E Y 0 ) is compact in X and consists of equicontinous family in Y by

regular conputation and estimation.

Hence ~/is a compact subest in Y by Arzeia--Ascoli theorem. Furthermore, P has a fixexl

point in Yo by Scharder*s fixed point theorem. It implies that(2. 5) has unique solution x E

Yo.

The remainder can proved by methods similar to that in (1) .

The corollary 2 .2 follows theorem 2. 3 and (N4) immediately.

1~. THE APPROXIMATE COMTROLLABILITY O F ( 1 . 1 )

- - T H E CASE OF N ( x , u ) BEING UNIFORMLY BOUNDED

Introduce the "blocking 't system correopoding to (2. I ) as follows

I:,U (t,s) Bn(s )ds y(t) ---- y(t,t0,y0,u) = U(t,to)yo + t ~ to (3. 1)

and express the soiution o f ( 1 . 1 ) by x(t)-----X(t,Xo,U).

Let

S':U(t2,s)p(~)ds C(tl,t2)'~(.) = : / 1 ( [ 0 , c o ) , X ) - - , - X ( 3 . 2 )

Where tz:>tl ,ti , tzE ['0 , co)

K(to,T,yo) ~- (¢r:¢r = y(T,&,yo,u) ,u E Le([to,T) ~U) ) K,v(T,zo) = (¢r = x(T,x,u) ,u E /_?(0,T;U))

Definition 3. 1 (3. 1) is called approximately controllable on['t0,T] with initial value y ff

K(t,T,y0)------X; ( 1 . 1 ) is called approximately controllable on i t , T ] with initial value x0

if KN(T,x0)=X

158

As well known we have

Lemma 3.1 K ( t o , T , y o ) = X iff R ( G ( t o , T ) B ) = X

Now, introduce some hypotheses as follows

(B~) : R ( G ( h , t z ) B ) = X for t 2 > h ~ 0

(B2) : II G(h, t~)N(x ,u) II . = 0 ( 1 ) with respect to ( t 2 - - h ) ( t 2 > h ~ 0 ) uniformly for x

EX and uEU.

(F) ." (i. I) has unique solution x6C([O,oo);X) for nay given uELP([O,oo)IU)

where N(x( • ) ,u( - ) ) E L ' ( [ 0 , c ~ ) iX)-

Theorem 3. 1 Assume that (B1) , (B2) and (F) hold,then K ~ ( T , x 0 ) = X for any

gaven T > 0 and x06X.

Proof • It is only necesary to prove that for any given ~IEX and eS>0,there exists

u 6 L P ( , T ; U ) such that

11 x(T,x,,,u) - - 11 II < e (3. 3) Let t . = ( 1 - - 2 - " ) T so that t.---~T as n--.-cx~. We have that

(i)There exists v 0 ( ' ) E L P ( 0 , T ; U ) such that

y(T,lo,xo,,,o(.)) - - ~llx < -~

by(B1).

(ii) After taking x, = x ( h ,xo,vo( • ) ) ,there exists vl ( • ) E LP(% ,T;U)such that

I l y (T,~, ,~ , , , , ( - ) ) - - '~11 < -~

by (B1). Let

by

p,o(t) 0 < t < t~ tt(t) %

and we can see that ut( • ) E I2 (0 ,T~U)

(iii)After taking Xz=x(t2,xo,u( • )) , there exists Vz( • )EL~( t ,T~U) such that

I1 y(T,~,::,,,:(.)) - ,? 11 < ~-

(B1). Let

u2(t) = / uj(t) O ~ < t < t z u2(-) 6 L~(0,T~U)

(iv) Repeating above procedures we have that v.( • ) E LP(t,T~ U) such that

e u(T,t,,~:,~,,) - - ~? tl <

and

= [,~ ,(t) u . ( t ) b , . ( t )

o ~ < t < t , ~<t<T,

~.( . ) E L,(0,T~U)

( 3 . 4 )

( 3 . 5 )

1 5 9

(v)According to (B2) ,we have

l imC(&,T)N ( z ( , ) , u ( - ) ) = 0 I , r .*oo

so that there exsts N such that

IIG(&,T)N(x(') ,u,) II < (3. 6) as ~ > N.

(vi) Taking u.+,( • ) Q {u.( • ) } C L ? ( 0 , T ; U ) we have

II = ( T . . o , , . + , ( - ) ) - - , , 11 < 11 x (T ,xo ,un+, ( . ) - - y(T,t~+,,x~+,,v~+l II + 1t v ( T , t ~ , + , .~,,+,,,,.+,) - - ,~ i1

By(1. 1 ) , (3. 1 ) , a n d ( 3 , f ) w e can dcducc that (3. 3) hold as u ( * ) = UN+I( " )which

completes the proof.

Theorem 3.2 If ( H 1 ) , ( N 4 ) and (B1)hold then KN(T,x0)~-X for any given x 0 E X

and T > 0 .

Proof : Noting that (B2) is valid as (N4) hold,the conclusion follows from

corollary 2. 1 and theorem 3. 1.

Similary, one has

Theorem 3. 3 l f ( H 2 ) , ( H 3 ) , ( H 4 ) a n d (Bi) hold then I ~ ( T , x 0 ) = X for any given

x0E X and T > 0 .

Example 3. 1(8): Let X = L Z ( 0 , n ) , e , ( x ) = x / 2 x ~ sin n x , n = l , 2 , . . . , w h e r e { e , } i s

a family of orthogonal basis in X and tx~ c o ¢~o

v { , , , . = z ' . . o . , z , ~ < ~ / , 1 1 . 1 t . = (z',,.~) '/~ m~2 ~2 m ~ l

Consider a hcat~conduction equation as follows

a:/(t ,z) a~y(t,z) at - - & 2 -{- F ( t , y ( t , z ) ) -}- Bn( t ,x ) t E (0 ,T)

z E ( 0 , ~ ) , ( 3 . 7 )

y ( t , 0 ) = y( t , , r ) = o t E [0 ,T3

y(0 ,=) = ~o(=) x E [o,~r3 Imt A y ~ ~ (y")so that ( ~ A ) generates a compact semigroup of contraction S( t ) .

The mild solution of ( 3 .6 ) is expressed by

y( t ) = 8(t)yo -} - f :S (t - - ~) (F(y(,~) ) -}- BuO) )da (3. 8)

t,ct

Bu = 2rile1 "+ ~u.e. ( 3 . 9 ) w, u 2

It follows from above hypotheses that(H2) and (B1) hold. If F (y ) is continious and bound-

ed uniformly for x,then (N3) and (N4) hold so that ( 3 . 7 ) is approxcmately controllable

for any given T > 0 by theorem3. 3 without uniformly Lip condition even local Lip condition

for F. Say,

160 {:1 -~ ~ • uy,sin el (3 ,10 / F(y) /~ (Z.~/.e.) = ~ Y' ¢= 0

m - - ]

y~---- 0

This re.suit can not be obtained from ( 5 ) , (8) o r ( i 0 ) .

Example 3.2 If S( t ) is only a semigroup in example 3. 1 and o o o o

F (y , u ) = P ( ~y,e, , ;z'~e,) = cos~e, + cosulzez

instead of F ( y ) , it is easy to chech that (H1) , ( H 2 ) , (N4) and (B1) hold. So,the system is

approximalely controllable by Theorem 3. 2 without uniformly Lip condition for F ( x , u ) ,

This can not be obtained f rom(4) , (8) ,or (10) .

VI. THE APPROXIMATE CONTROLLABILITY OF(1 .1 )

- THE CASE OF N ( x , u ) SAT1SF1Y1NG UNIFORMLY Lip CONDITION

In this c a s e , N ( x , u ) = N ( x ) satisfies(Nl) and (1.1)becomes

x ( t ) = U(t,O)xo -Jr- ( t , s ) ( N ( x ) q- Bu(s ) )ds (4. 1)

Theorem 4. 1 If (N1) and (B1) hold,then (4. 1) is approimately controllable for any

given xoE X and T > 0 .

Proof:Similar to the proof in theorem 3. I from the begining to the end of ( iv). We

have(3. 4) and (3. 5). Consequently, we have

II x(t ) -11 II < II y(t ,t . ,x. ,v.)-~111 + II , .o(t ,s)Nx(s))ds II ~< II y ( t , t , , x~ ,v . ) - - , l I[

q--M(T)( It N(0) II + K II ,~ II ) ( t - - t . ) + M ( t ) , II x ( s ) - n II ds

for t E [ t . , T ] where K is Lip constant and

M ( T ) = - m a x 11 u(t,,~) II

Furthemore,by Growall's inequality one has

Ii :~.(t) - ,~ il <~ e""~'~'E II ( , A t . ~ . . ~ . . , , . ) - ,~ tl + M ( T ) ( II g ( O ) 11 + K il ,~ It ) ( t - - t.)3 ( 4 . 2 )

Taking t = T in ( 4 . 2 ) , i t follows that

"~ ~ "2 q- M ( T ) ( II N ( 0 ) I [ , [t ) (T - - &) )

So,there exists N > 0 and u ( t ) = us+l( t ) such that

lt~ (T) - - ,;ll --~-~ ~,(~=(_e.2 + 2 ) = e~'(~'~'~

that complets the proof.

161

In example 3. 1 if F (y ) satisties ( N I ) ,the system is approximately controllable without

any other conditions. This rasult can not be obtained f r o m ( 5 ) - - ( 1 0 ) .

REFERENCE

1. Pazy. A. , Semigroups of linear Operators to Partial Dfferential Equations, Lecture

Notes 10, Dept. of Math, University of Maryland, 1974.

2. Curtain. R. F. , and Prirchard . A. T . , lnfinte Dimensional Linear Syetems Theory,

Springer--verlag, New York, 1978.

3. Russell. D. b. Controllability and Stabilizability Theory for Linea Partical Differential E-

quations; Recent Progress and Open Question. SIAM Rievew 20(1978) PP639--739

4. Henry, J ; Etude ole La Controlabilite Eole Certains Equtains Pareboliques N o n - Lin-

eaires. These,detat ,Paris,June, 1978.

5. Zhou. H. X, A note on Approxmate Controllability for Semilinear one dimensional Heat

Equation ;Apph math. option 8 (1982) ,pp275--285.

6. ~ ,Applroximate Controllability for a class of Semilinear Abstract Equation,SIAM,J.

Control and Option, (21) (1983)pp551--565.

7. ,Controllability Properties of Linear and Semilinear Abstract Control Systems, I bid

22(1984) , pp405----422.

8 . - - , Approximate Controllability on Aastract Control systems with a Nonliear Ditur-

banee, IFAC 84.

9. H. Zhou,andY. Zhao. A Survey of Conrtollability Theory for Nonlinear systems,Con-

trol Theory and Applications,Vol. 5. No2. 1988.

]0. Naito. K. Cntrollability of Semilinear Control Systems dorninted by the linear part,

SIAM,J,Control and Optim, 25 (1987)pp715--722.

OPTIMAL CONTKOL FOR A CLASS OF SYSTEMS AND ITS APPLICATIONS IN THE P O W E R FACTOR O P T I M I Z A T I O N OF THE N U C L E A R REAGTORT

Wang Miansen Kuaag Zhifeng Department of Mathematics , Xi'rm Jiaotong University

Xi'an, Shaa~txi, 710049, O]fina Zhu Guangtlan

Institute of Systems Sciences, Acadimia Si~fie,~ Beijmg 100080, China

1. In t roduc t ion

On the background of tile power factor optimization of the nuclear renctor, tile optimal control for a class of s~mtems governed by the eigenequation

I

has been presented in [1]. Under the variance index: [1] shows the existence of optimM control and gives the corresponding optiraality eondi¢$ons. Bu~ these resuks can% be gpplied to those s)-stem~ i , which the neutron density iv relative with the direction of the speed. In this paper, we put these results i , to so general ca~e that they can be applied to all stable transport .~ys~ems.

We start from a example. Example 1. Gon~ider the following energy~ependant ,~teady-~tate transport equation in a

slab ~ t h generalized rethxive boundary conditions:

ON{x,v,i,) ~. I ?~'"' ? ' .

" ~ ' ~ - + ~'~(* ' " )~ (* ' " '~ ' } = ~./o ./_,/~(.~,,,.,,',t,,/}xCx.,,'./)a,,',O,' l u l I ,~ I< ,,, l l, I < i , ~ e (o ,~ . , ) , ,,,,, < +0o

N(-, , , , , , I , ) = o'(,,,1,)~'(-a, t,, -e ) , 0 _< I' -< t (L2} ~V(a,v,-t0 =/3(~, tdN(a,v , S0, 0 _< ~t < I (t.3]

where N(x, i~,#) denotes the neutron density with the veh, city v and direction # ,xt the point x; ~.{z.,~) is the macro-absorpfio,, cross section; k(x,v, V',l *,/} is the transfer kernel; o(x, t,) and /J{.r, v) are the reflexive coefficients.

If we suppose ~hat D(x,v) is composed try two parts: ~[x,v) ----- Do(x,v} + ~e(x,v), where E,,(x, v) is a fixed part, attd ~c(x, v) is a controllable part, realized by regulating the control bars, and denote the following:

~ t = (-a,a) x (0, v.,), f}z = (-1, 1),~ = i}t x 1~ O

L-= ~,t, a-~- ' - + v ~ , ( ~ , ~ ) .

D{L) = {~ eL~{D) I io is aboolatcly co.ti..uous in x,L~ e L" (f~), p satisfies bottitdar~j conditions (1.2} a.d (1.3}}

= [ ~ " [ ' K. k{x, t,, v', #, #') dr'did

dO J --1

. . = ~ o ( ~ . , , , ) . , . ( ~ c ) = . ( , . ) = L z(n)

#Tiffs work is supported by the National NaturM Science Fundation of Olfina

163

then above system becomes 1

Accordh~g to the reactor theory[ ~l, ia order tha t reactor operator safety and efficiently, the equation (1A) should be in a critical state (i.e. A(a) = r = 1) meanwhile the power factor

Jo{e 0 = max (IC~)(x, u, p) {1.5) n (h, ~o)

takes the minimum and total power mains constant (i.e. (h, ~) = p). This is the so-called power factor opimization problem. For ~mplicity, instead of the index (1.5}, we adopt the variance index

,£ where 'L m(~}) denotes the Lebesgue measure of fL

From tiffs example, we present the optimal control problem for a cla~s of ~Tstem.~. Let's state our problem more precisely as follows.

Let ~t C R n and f~2 C R m be bounded Lebesgue measnrable d,m~ains, t] = f~t x fl~, h2(i~} be a real Hilbel~. space with ivaer product

{f , ~} = f~ f(~'),j(x)dx

K and L be llacar operators in LT{fi). Consider the ~Tstems with the state {A{a),~c.(a}}

(k, ~) = v (t.7)

where A(a) is the critical eigeic~oJue of the operator T(a) = (L + ~)- i I t" and a is th,~ contr,,llable variatioa. The control set is defincd by

U = {or E L"'~(~'l,) I 0 < a(x) < a{x) <_ b{x)a.e.inni}

with the desiga,ated function a(~.) and b(=) in LC°(f~,) and b(.) # ma(.). (]i'.~.~n the c,~st funclJonal

'L J(a} = ~ I ~(,71 - ~A./'~(a) [2 dO

We consider the following problem (P): Given a positive number r, we want to find a control a~ ~ U such that

"~("°) = " J("°) = 22~ .v(,,)

ht the case that ~ = fi t , the problem has been .~olved in [1]. But fl 7~ f]t in ex.~mple 1. and their methods is of useless.

164

2. Exis tence of the O p t i m a l C o n t r o l

In lids section, we prove the existence of the optimal coutrol by use of compact imbedding theorem [~t. Firs~ we give following hypertheses:

(HI) f h , f l are the domains of the class J~ , , J~{ i - ,~ < ~ < 1,1 - ~ < o, < 1}[ 31 , respcctivcly;

(H2} p is a positive number, h i s a positive function in L2[1]}, L is a closed linear operator, and K bounded linear operator under the aorta II" Ih = fn l . ( 4 1 d~;

(tI3) for any a E U,(L + a) - l exists and is a positive operator, T(a) = (L + c r ] - i K is a positive compact operator in LZif2}; and the critical eigem~due A{a) of 7"(0) exists.

L e m m a 2.1 For any ~r E U, there exixts a unique positive fimction wlfidl satisfies (1.6) and II.7).

P r o o f This is a direct ¢olL~equence of (H3) and Theorem 2.1 in [4]. Denoting Vv = {a 6 U I A{a) = r} as the admissible control set, we have L e m m a 2.2 Uo is an infinite set, if A(a(~.)) > r > A(b(x)).

P r o o f For any al and az in U, since

as o l { x ) > oz{x) a.e. in ill

By Theorem 4,2 in [41, A(eyt) < A(az), If we define )h(t) = A(la(x)), (t >__ t), then At(t} is a noJdncreasing contimmus function 15" Lemma 3.2.

Therefore, there exists a sequence {6k}l°°,hk ---, 1 +, such that A1 (6k} = A(~a{x)} > r and for every 6k, we eon~ider the noninereasing continuous function:

A2{t) = A(6ka{z) + t(blx) - ~.alz))) , t ~ [0, 1]

I t ' s east to know that there exSs'ts a to E (0,1) or c%o E U such that A{ako) = r and ako ~ erk' o. if Crk ~ ak'. Thus the Lemma is right.

Define a subset in LZ(f~):

W = { ~ E L ~ ( f } ) l ' ~ > O , there exists a ~ U such that ~ io the

solaHou to L~+a<~ = 1K~-~ (h,~) : p } r

Then by Lemma 2.2, we have L e m m a 2,3 tf A(a(x}) > r > a(b(x)), then W is a. infinity set. Note W isn't a convex set here, but W is convex in [1]. It 's easy to know the problem (P) is equivalent to the tmninfizatlou problem:

{ i n f J(~) s . t . ~ ~ W

where J (,~) is defined as

/(~) = ~ I ~ ( 4 - M ~ I ~ , l .

L e m m a 2.4['1 J (~ ) is a continuous cow:ex functional in LZ(12). L e m m a 2.5 J (~ ) is a weakly lower semi-continuous hncxioaal. This is tlle conclusion of Lemma 2.4 and Cur. 1.8.6 in [5]. L e m m a 2.6 U is a bounded weakly dosed subset ill L2(fl l) .

P r o o f Obviously~ U is bounded. It 's enough to show that U is strongly dosed because of tile convexity.

165

Let {~,,} be an arbitrary strongly convergent sequence in U c L~(f~t) attd a . --* ao strongly in LZ(flt). Then, by Cauchy inequeliW~ for any ~ > 0

~ ( [ ~ u [ a . ( x ) - ao{x)I ~ dx)~ >- .,u [ - , l a . ( x j - ao(x) [ dx

-> [ I a . (x ) - ao(z) I dx > * , , , (n~( l o-,. - ,~o l> *)). ./9

L e t n -* oo, m([ ' / l (t a n - o'0 t > _ ~)) --~ 0. B y F . ILie~-~ T h e o r e m , t h e r e e x i s t s a c o n v e r g e n t

subsequence {an. } C {an }, such that

Since a(x) < am,(x ) < b(x), and hence a(x) < co{x) < b[x) or ao e U. L e m m a 2.7 W is a weakly dosed subset in LZ(f~). P r o o f Let {iP,*} be a weakly convergent sequence in W, and ion - ' ion wealdy in L~{f~). By

the definition of W, there exists a sequence {or.} in U, such that

(2.2)

t2.a)

It 's obvious tl~at (h, ~o) = p from (2.3). By Lemma 2.6, there exists a subsequeuce {a,,~ } c {#n} .such that en~

L~(~ , ) . I~ec ~ltse

compact , TM ,

- , or0 weakly in

tim embeddiv6 operators It :L2(f2) --~ Ll(i2), a ltd 12 : L2(f l t ) - - Ll(fh} are

'~,) - - ~o slronyly ila L I (I-I) {2.4)

a,,: --* ao strongly in L x ( ~ ! ) {2.5}

By the l,ro,:ess of Lemma 2.6, there exists a subsequence of {a,, r }, written s'flU {a,,~ }, such that (2.1) is right. Then [ (em.(x) - a o ( x ) ) ~ o ( x , y } [--, 0 a.e. (in l'~) and [ {a, ,~(x)- cro(xJj,pn(a',y } [_< 2b(x) I 'f0(x, Y) I, by Lebe~gue Theorem,

~.XFt~t

o~ H (0.~ - - 0 I v 0 I i , - 0, . s k -~ oo. h o,~ (2.2),

a~ k - ~ o o

1 - K ~ , ~ o - a o ~ o ~troagly i~t g l ( ~ ) r

Combining (~.a), (2.0) ~nd (~2), we h~,'e

~oo E D{L) L~o = l-K~o -oo~o 7

t2.6}

s(, ~o~j G W. That is W is weakly closed. L e m m a 2.8111 Each minimizing sequence of J in W is bounded. T h e o r e m 2.1 IfA(a(x)) > r > A(b(x}~, there exists foo ~ W such that l ( ~ ) = i a f ~ w J(~) . P r o o f Let {~.} be a minimizing sequence of J in W, that Ls ilm . . . . J ( ~ . ) = i n f l o w J{~).

166

A¢cordlng to Lemma 2.7 and Lemm 2.8, there exists a weakly convergent subsequence {~n,.} c {~.}, sudt that ~o,,~ --, too E W weakly in L~(fl}.

By Lemma 2,5, J(to0) = inf~aw J{?). Corol lary 2./. If A(a(x)) > r > A(b(x]), then the optimal solution ¢o {P} exists.

3. Necessary Co]tditlons for Op t ima l Cont ro l

In [t], the authors gives the necessary conditim,s for the optimal control by use of Duboviskii- Milyutin Theorem [6]. In their process, the main difiicidty is to prove, the conclusion that " A(#} a1~d ~(oj are Frechet differe~ti~b¢ mappings from U i~to R as~d L2(fl), re.spect, ively'. But the conclusion is not an immediate consequence of [7] as they say. In this sectio., first, we give a strict verification of the conclusion, then give the same n(:cessa<~" condition by use of generalized Kahu-Tucker Theorem[N.

In tlds section, we further assume that (Ft4) L is densely defined, so the adjoint operator L* exist.~ and is unique. (HS) for any aa E U, there exists a neighbourhood A(q,~) of cr.~ (it mas" not be contained in

l!) such eha.t (EI3) is satisfied in A(#o), and II (z + ~ ) - ' II- M for any ~ in A(o~) . L e m m a 3.1 For any oa~ E U, there exists a tteighbourhood A(ffa.) of ff~ axed a commom

Jordan turve P such that the eigen-projection operator Pa ¢.nnecting with A(~) cem be represented 1,y

. /rRa(z)dz for an9 a in A(r&,] P~=

and P#L2(f~) is the eigenspace of T{a) corresponding to the eriticM dgen-va]ue A{a). And Po is Fr&he~, differentiable operator, where ~a (z) is the resolvent of T(a).

P r o o f According to the Theorem IV.1.5 and it's notes in [8 l, we have

}1T(a) - T (a , j 11=tl (L + .)-'K - (L + .~0-' K 11

-<11 zc Ill1 (L + ~ ) - ' - (C + ,~o,)-' II_< II K II 11 (L + + J - ' II z 11 o - , , . 11

- o , a~ II ~ - " o II--" o

and

i[ Ro.(z} - Ra,.(z) H=H (z - T(#}) - l - (z - T(#,}} -~ II

< II (= - T ( ~ . ) ) - ~ ll~tl r ( . ) - T ( ~ . ) II - - 0, .~ - 1 - [I ( z - T ( a ~ ) ) - ' [[li T ( # ) - T ( # ~ , ) [[

iI ~ - ~ , , I t - 0

Therefore there exists a neighbourhood A(o~)) of Üe, and a bounded hmction ill(z) such that l) R~iz) ])_< M(z).

Similar to the definition in [9], T{a) is a strongly stable approximation to T(#~,). By tile sinfilar process with proposition 5.6 in [9], we can conclude that there exists a common

Jordan carve P such that

29# =2-~i R#{z)dz for every a in A{g,~}

and P#L z (f~) is tile eigenspace of T(#).

167

Let. $ denote l~(z)(L + a~,)- ' a (L + a,)}- ' K R ~ . ( z ) , since

- 1 r P"+" - P'" - ~ J~r $dz

-'/,. Z . . . . 2rri (Ra.+a(z) -- Ro ° {zl)dz - ~-1 Sdz

= 2x'-~ R , r , , + , , ( z } ( L + a ~ , ) - ' o l L + a , , + a ) - l K R ~ , ° l z l d z - ~ . Sdz

- I t + ~ i t r R , , ( z ) ( L + # ~ ) - l a [ i L + at, + o ) - ' - (L + ac ,} - ' ]KR, , , ( z )dz

thus

[I P¢.+~ -- P~,, - ~ ' Sdz [1 / [1 o" I[-" 0 as [1 ~ [1"* O.

So P# is Fr&het different(able oper.'~tor, for a,), is art arbi trary point. T h e o r e m 3.1 For any aa ~ U therc exists a neighbourhood A{a~) of #o such that the

solution to (1.6} and (1.7} can be represented by

P

and 1:)# i.~ Fre'chent defferentiable in A(a~). P r o o f By lcmma 3.1,

P (1 ,p ,~( , , , , ) ) ' v ' v ( ' ' )

is the positive solution to {1.6) and (I.7}. By Lemma 2.1.

I)

Since Pg is Fr~chet differentiable in A{a~), i t is ea.~T to show ~,[a) is, too. L e m m a 3.2 A,~ is a continuous functional fr~)m U into R.

P r o o f By [71, for ~ .y bounded l inear operator T, the spectrum set a{T) i.~ upper .~emi- .:ontinuous function, that is for any e > 0. there exists a fi > 1), ,~u.:h that

. , , c ~ . ( s ) , t ; ~ ' ( ~ , a l r ) ) < ~, ,,. II s - r I1<

Filet given n 0 < he) < 1, we ea.n take

,~' <_ , , , i , , (~ /1 t ( L + # ~ ) - ' It, (~ + ~)6/11 (L + #~)-~ II ~ II K II)

F~o,~ (z.i), II T(a)-T(a~,) If< a, a~ II # - ~ o II< 6'. Since a(~) e o(T{o)), distiA(a),o(T(o.))) < z .

Gonsidering the compactness of oCT(a,-,)), we know there exists A* 6 a(T(#a)) such that I ala} - A* 1< e. Thus (since a ( . ~ ) >l A* 1)

a(#) - A(~o) _< a ( ~ ) - I A* 1=1A{,,) I - 1 A" t<l a(~) - A* I<

B:, th~ ~a~.e proee~, a(#,.) - X{#) < ~-. a . d so I a ( ~ . ) - A(,,) I< #, a.~ II #,, - o II--" 0. or A(o,) is eol t t i l t l lOl l~/ ~l.t 8rt~.

168

L e m m a 3.3 For any o~ ~ U, the Gateaux derivative A~ of A[a) at a~ exists and is given by

where A. = k (a~), to~ = ~o(a~,) and q,a is an eigenfimetion of (L" + ~.a,)-~g * associated with A~, i . e .

P r o o f Denote T{tJ = T(a~ + t#), then it 's ea~T to show that T(t} is a holomorphie function at a ndghbourhood of 0.

By [7], A{t) = A{a~ + ta) is analytic at 0, that is A(~) is Gateaux ditferentiable a~ c,, in any direction ~ in L'~"(f~).

According to Lemma 3.2 ia [l], we obtain (a.2). T h e o r e m 3.~ For any aa ~ U there exists a neighbourhood A(a,,) such that A{a) is

eontinuvusly Fr&het ditterentialflc in A{o~,). P r o o f Combiniatg (3.2} with Lemmg 3.2 and Theorcm 3.1, and co~tsidering q~ sharing thc

same properties with laa, we can conclude the conclusion after a complexive estimation. Similar to Lemma 3.3 in {1], we obtain L e m m a 3.4 The Fr&het derivative ~b,, of ~(a) at a~ satisfies

&, ( I f ~ ,, , .¢,,,}

( ~ , , h) = o (3.41

where the notation is as Lemma &3. L e m m a 3.5 For a~ ~ U{a0}, J{~ro) is Fr&het ditferentiable at oo and the derivative is given

by J'(o-,.,,}(o-) = ( ~ , , - M~, , , , ,b, . , . ( . , , - ) )

where ~'~, ~ are defined as Lemma 3.4, U{oo) is a neighimurlmod of ou.

Proof ICs easy tu show J'(p~)(~) = (~ --~iVa.,~). Because J{a) = (J o ~)(a},

Theorem 3.3 If A(a(x)j > r > A{b{x}}, and ~0 is tile optimal solution to (P}, *hen there. exist rl _> 0, r2 E R, not all zero~ and adjoint state ¢~, E D(L*) such that

1

(h, ~.1 = p

r

f ~ { o - ao)~o(~ + r~q, ol& >_ 0 f o r e w r y a c U

P r o o f After the preparation above, the Kuhn-Tueker theorem can be applied to (P}, and if ¢o is the solution,then there exist rl >__ 0,1q E R, which are not equal to zero simultaneously, snell that

r, (~o - M~0, V~(~ - ao)) + #~ ((~ -- ~0)~o,¢,o) > 0 V a ~ V {3.5)

169

In order to simplify. {3.5), we introduce the adjoint state ¢, which is described ~ ,

5 % + ao¢ = I K * 9 - r~ i~0 - M ~ o ) + oh r

M ~ o , ~Po ), so where o' is a constant ~o be defined. Similar to [1], o = v (t°° -

r

{K~o, ¢0)

(¢,, K~0)

Let (¢,, Kv~,,)

r~ = tt, + (K~oo, % ) '

ta.~)

then {3.5) becomes r2((~ - ~ o ) ~ 0 . ¢ o ) + {(~ - ~o)~0. ¢') -> 0, that is

( ( ~ - ~ o ) ~ o , ¢ , + ~ q ) o ) > 0 , w e e

Similar to {1], we have T h e o r e m 3.4 We ca Jr take rt = i in Theorem 3.3, and the last formula can be replaced 1~"

f fn (~(x) - ao(x))~ot:~,y)9(~.,y)dxdv >_ o, V~ ~ U ° tXf~2

,,.h~,'e V ° = {. ¢ e I ( ~ o . ¢'o) = {~oV., Oo)}.

4. A p p l i c a t i o n s

Let 's re turn to the example 17 we give tile following hypertheses: 1} ,~(v,tz),fl(v,p~ are positive measurable functions and 0 < ¢~{v,p) < 1,0 < /3Iv, tO < 1,

2) ~ ~,c, (x, v }, k(x, d , v, t t', #) are nonnegative bomlded measurable timct.lons and

~o* = essinft ,2c,{x,~,) > 0

3; k (~ ,~ ' , , , , / , i , ) > o ~.e. i~ [-~,~1 x (o,v,,j x (o,~.,I x [ - , , l ] x [ - i , i ] a,,d there exist Xl,X2 (~ [ - - a , a ] , x I < x z a l ld Vl , t l 2 ~ (O, vm],Vl < v2, s l tch t h a t k(x,v*,v,ld,lO > ko > 0 n.c. in

[x, , .~l x [,,,,,,~] x [",,",t x t - t , q x l - t , q. With the above hyperthesess, basing oa {10], we can imitate the pro,:e~s in [111 to show that

all t, he ¢onditlons stated in ~eet~m 2 altd seetiolt 3 axe sa,t-lsfied. St) we obtaia T h e o r e m 4.1 If A{cl) > 1 > A(c2), the opti-tM eolgrol e:dst.s: where et = a(x,¢,Jv, aitd

,'~ = bix, o)~,.

L e t S = { ~ . 6 L ' ~ ( l ~ , ) t O < a ( x , v ) <__~c{x,,,) <b(x , , , ) a.e. in n , } , w e h a v e T h e o r e m 4.2 The optimM macro-absorption cross section ~t. ° E U i.* given by

k ( x, v*, v, I t'. p ) ~>,, Ix., v'. Id Jdvt d l d ,.~, °~°!!(i~.' ~ + ,,1~, (~ , , )+ ~/ ' (~ , , , ) )vo~, , , , ) = , ,

,po t -a , v, I'] = ~.(o, It)~oo t -a , v, - M

~,,{a, v,-I~) =/Jlv , l t )~ola, v ,#) (h, ~,,} = v

/ (,,~c(x, t , ) , ~ - v v',.~c°'~a:, v))(-Cb(x, v,i,) + r~¢o(x,v,it))dxdvdtt > 0 V~c(a:, t,) 6 S {4.11

170

where r~ ~ R, and ¢ is directed by (4.2):

- ~;1~ Ox

v,. I iclx, v, v , t , td~l,(x,v, t~,)dvtdl, r i (~ . , c ,_M~o)+ ~(~o t ,_Mvo ,~oo) h = . , I P

~ ( - a , v, - # l = o(t,, !,) O(-a , v, ~)

(~, v, t,) = i3(,, , t,),/,(a, v, -~,)

(4.2)

and 00 is a solution to the homogeneous equation of (4.2). T h e o r e m 4.3 In Theorem 4.2~ we can take rl = 1, and {4.1] can be replaced Iry

ft~ {vile(x, v) - t'~e ° (x, v))¢p,~(x, v,/tl'g~(x, v, p))dxgvdp > 0 VH,: E S U

where S o = {~° e S I (v~c~o,~t') = (vSc%~o,¢'ol}

Sintilar to example 1, the obtained resul~.s in section 2 and section 3 can be applied to the 3-dimensionM stable transport equation ~-ith isotropk ~eattering:

{ ,(l.#rad¢¢ii',v,~}+,y~(~',v)¢,ii',,,,~) = ~. dr' l:(e',v, Vl,¢,i~,v',5'ld~'

¢,l¢,v, fi)=0, ¢~rv, ~ . f i<0

where V is a. bmmded convex domain ill R 3.

References Ill De.'dng Feng and Guang~.ing Zhu: J. Math. Pures et. AppI. [984{2), page 169-186 [2] A. M. Wernbcrg, and E. Wigner: Tile physical Theory of Nela¢ron Chain Reaet,)r, Univ. of Ckicago Press 1958 [3] Maz.. la. V. G.: Sobo|ev Space, Berlin-Springer, 1984 (Trans. h'onl Rus,~ion) [4] Marek. I.: SIAM J. Appl. Ma.th. 1970(19) 607-628 [5] Balakri~tmann A. V.: Applied Functional Analysis~ Splinger-Vcrlag 1976 [6] Zeidler, E.: Nonlinear Functional Analysis and its Application,~ [7] Kato. T.: Pez~urbation Theory of Linear Operators~ Spriliger-Verlag 1984 [8] Taylor A. E.: httroduction to Functional Analysis, N. Y. 1980 [9] C, ha.telin F.: Spectral Approximation of Linear Operators, N. Y. 1983 [10] Song Dcgong~ Wang Miaasen and Zhn Guangtiazt: Systems Science and Mathematical Sciences~ (;lfina, 1990{2), 102-125 [I1] Ylng Minzhu and Zhn Guangtian: Acta Mathematica, Scientia, China, Vo|. I, No. 1, 1981, 1-12

SINGLE I N P U T C O N T R O L L A B I L I T Y FOB.

S P E C T R A L S Y S T E M S IN B A N A C I t SPACES

3INGBO Wu

Department of Computer & System Scicnces

Nankai UniversiW, Tianjin, China

Consider the discrete-time .system defined by

Xn+l = Axn q" BUn

where A is a bounded linear opera~or on a ]3anach space X a n d / 3 is a linear operator from the input space G m to X. This system is denoted by {A, B}. We say that the system is controllable if X = Vk°°=o ran AkB. If A is a se,~lar type spectral operator, then {A,B} is called a scalar type spectral ssrstem. ][at the case where A is a normal operator on a Hilbert space, the system {A,B} is called a normal system. We can define self- ~djoint system in an analogous way. We say that the normal system {A, 13} is bilatera~y controllable if X = Vk~=o ran A*k'Att?.

In [7], Wonham showed that if X is tiiLite dimensional and A is cyclic, then the controllability of {A, B} implies that {A, b} is controllable for some b Eran 23. There have been many papers concerned with the single input controllability for infinite dimensional state space. In [5], Fuhrmaam proved that this property holds if A is self-adjoint. Feintuch [4] pointed out that Fuhrmann's proof in the slef-adjoint case remains valid for reducCive nolanal operators. Lubin [6] gave an example to show Chat this result may fa~ in general for normal operators.

Recall that a controllable ,system {A, B} is dngle input controllable if {A, b} is controllable for some b ~ran Z?. The main purpose of this note is to discuss tile following problem: when is a controllable scalar Wpe spectral system on a Banach space single input controllable? The main results will be given in Theorem 5 and Corollary 6.

Throughout, by an operator we mean a bounded linear tran~fmxnation unless it is otherwise s~a~.ed. The Banach algebra of operators on X is

172

denoted by L(X). The a-algebra of Borel subsets of the complex plane is denoted by E;. Let A be a scalar type spectral operator with resolution of the identiW E(.) . For fixed xo E X, denote by ~ (xo) the closed subspace spanned by E(a)xo for a l t a E ~3. We say tha t Xo is a cyclic vector of E(.) if X = ~(z0) and that x0 is a cyclic vector of A if X = V~°=oAkzo. For normal operator A, we say tha t xo is a s~,ar-cyclic vector of A if X --- W o o ~ * k ~ l • to,t=0 ~ ~ x0. Clearly, xo is star-cyclic for A f fand oldy if it is cyclic for E(.) . Given Xo E X, there exists x~ E X* such tha t x~E(a)xo > 0 for all a E :E and such tha t x ;E(a )x0 = 0 implies E(a)x0 = 0 (cf. Theorem 3.1 of [1]). Wri te / to( ' ) = z~E(-)x0. For any bounded Borel function f , the integrM

/ s(f) = / ( a ) E ( d a )

e.xists in tile uniform operator ~opology. When f is unbounded, define an = {AII/(A)t < n}, n = 1 , 2 , - . . . TILe unbounded operator S(f) has domain

/2 D(S(f)) = {x[ nlim¢~ ' f(A)E(dA)x exists}

and is given by the formula

S(f)x = lira I f(A)E(dA)x, x ~ D(S(f)). ? ~---r ¢20 j ~ tr

The oper,~tor S(f) (no¢ necessarily bounded) corresponding to a Borel function f is a densely defined closed operator. For x0 E X,

~t(~o) = {s(/)~ol~o E D ( s ( / ) ) }

(cf. [21, §4). Note tha t A has the tingle-valued extej, sion proper~5, (cf. Tl, eo=om ~3~ of t31) Do~ote by p~(~) tl,e loc~ ro~ol~ont of A ~ • = d by aA (X) t, he local spectrum of A at x.

Lemma 1. Let X be a/3anach .space altd let A b c a ~calar type ~pecCraI operator with resolution of tile identiW E(-). Assume tha t x0, ill X, is

then S(f)xo is cyclic for E( . ) if and only if f(A) ¢ 0 almost evelTwhere with respect to go.

Proof. If f(A) = 0 on a Borel subset a0 of tile complex plane with /to(no) > 0, then E(eo)xo # 0. I t Mlows ~aa~ ~(sro)zO ~ ~i (0 (:)XO) ann

173

so s ( / ) ~ o is not cynic for E( . ) . Oo,,~ersely, ~s~ume tl,~t f(A) # 0 ~ .e .

with respect to Po. Let er be a closed subset of the complex plane such that If(A)] is bounded above and possesses a positive lower bound on a. Define g by letting g(A) = f ( A ) - ' for A ~ a and 9(A) = 0 elsewhere. We I I & V ~

E(a)Xo -- S(9f)xo = S(g)S( f )xo .

Using the countable addifiviW of E(.) in the strong opera?,or ¢opol%%, together with the cyclicity of Xo, we have X c .M(S(f)xo) aztd so S( f )xo is cyclic for E(-).

Lemma 2. Let the assumptions of Lemma 1 be satisfied. If f and ~j are Bore1 functions such tha t xo E D(S ( f ) xo )AD(S (g )xo ) and such tha t x = ~ ( s ( / ) ~ o ) v ~ ( s ( g ) ~ o ) , th¢,~ ~(x) = er~ (s ( / )~ , ) u er~(s(g)~o) and there exists a complex number cr such tha t S ( f + a g)xo is cyclic for E(.) .

Proof. Since X = J~t(S(f)xo)V3t(S(g)xo), we have

x = [E(erA (S(f)Xo))X]V[E(erA (S(g)Xo))X] (cf. Theorem 5.33 of [3t) and so er(A) : aA(S( f )xo) UaA(S(g)xo). For fixed ~, we write

a, : {A ~ a(A)lf(A ) ÷ eg(A) = 0}.

For o, # fie i~ is clear ~ha~ f(A) = g(A) = 0 on er a. N tr z. Suppose tha t .o( .~ner,) > o. We have E(ero, ner,)~o # o. It foao.s-o,~t E(er~ner,)~o ¢ ~ ( s ( y ) ~ o ) V ~ ( S ( ~ ) ~ o ) . B~t thi~ is ~ contradiction ~nd so ~ o ( ~ nerO) = O. We have first t~o(a~) _> n - l # o ( a ( A ) ) for a tiaite set of subscripts and tl~en/to (a~) > 0 for a countable set of subscripts only. This implies tha t tl,~.~ exists ~, ~, such that ~0(er~) = 0 and l,e~,ce f(A) + o,j(~) # 0 ~.e. with respect to P.0. Using Lcmma 1, S ( f + ag)xo is cyclic for E(-).

Theorem 3. Let the assumptions of Lemma 1 be satisfied. If X = VzeY 3t (x) where Y is a finite dimensional subspace of X, then X -- ~t (y0) for some yo E Y.

Proof. There exist unit vectors x l , . . . , x , , in Y such tha.t X = f / l

such tha t Xk -=. S(fk)xo, k = 1,- .- ,.m. Using Lemma 2 m - 1 times, we have er(X) = ' " • ,.,kmler A (Xk). Further~ there exi~ al~. • , a m - i such tha t

• } ~ / X ~' "¢r .M(S(f, ÷ oqA + ..-4- a~.,_,f.~)Xo) = Y ~ 1 ( k ) " - - .A.,

174

This completes the ln'oof by letting yo = xl + ~k= l ozkxk+l.

The proof of the following theorem is straightfor~vard and will be omit- ted.

Theorem 4. Let {A, B} be a bilaterally contrdlable normal system. If A has a star-cyclic vector, then there exists b Eran B such tha~ {A, b} is bilaterally controllable.

Theorem 5. Let {A, B} be a controllable scalar type spectral system. Suppose that A has a cyclic vector. If the resolution of the identity E(.) for A leaves invariant every closed subspace of X invariant under A, then there exists b Eran B such that {A, b} is controllable.

Proof. Note that a. cyclic vector of A is necessarily cylcic for E(.). By Theorem 3, there exists b Eran B such that X = ~4 (b). Write Y = V,~oA'~b. It is clear that Y is invariant under A and so by the assumptions it is invariaz,t under E(.). l~ence Y = X and so {A,b} is controllable.

0orollary 6. Let {A, B} be a controllable scalar t)Te spectrM system. Suppose that A has a cyclic vector. If ~(A) is nowhere dens° and p(A) is connected, then there exists b Eraai B such that {A; b} is controllable.

]Proof. Let Y, a closed subspace of X, be invaria,nt under A. Then Y is invariant under p(A) where p is a polynonfial. Define

c~(f) = f~(n)f(),)E(d)t)

.h~re / ¢ a ( . (A) ) . Note that ~(f) ~ ~ bi¢onti,,~o~s ~l,obr~ isomorphism from O(~(A)) into L(X) (a. P r o p o ~ o n ~.9 of [~]). By Lavre,ltie~'~ theorem (of. Theorem 5.37 of [3]), Y is invariant under a(f) fox' all f E U(a(A)). Using Proposition 12.13 of [3], Y is invax:iant under E(.). This completes the proof by using Theorem 5.

References 1. W.G. Bade, On Boolean algebras of projections and algebras of

operators, Trans. Amer. Math. Soc. 80 (1955), 345-360. 2. W.G. Bade, A multiplicity theory for Boolean algebras of projections

in Banach spaces, Tl'ans. Amer. Ma~h. $oc. 92 (1959), 508-530.

175

3. H.R. Dowson, Spec.tral ¢heo13r of li,lear operators, Academic Press, London, 1978.

4. A. Feintueh~ On single input controllability for infinite dimen,ional linear sTstems , J. Math. Anal. Appl. 62 (1978), 538- 546.

5. P.A. Fuhrmann, Some results on cont.rollal>ility, ILicerche di Auto- maCica 5 (1974), 1-5.

6. A.R. Lubin, A note on single inpu¢ controllabilit3, for norton] ~TsZems~ Math. Systems Theory, 15 (1982), 371-373.

7. W.M. Wonha.m, Linear multivaa4~tble control, Lecture Notes in Eco- nomies and Ma¢hematieal Sys¢ems, 101, Springer-Verlag, Berlin, 1974.

Distributed Parameter Systems with Measure Controls *

Jiongmin Yong

Department of Mathematics, Fudan University, Shanghai 200433, China

§1. I n t r o d u c t i o n .

Let us first give a motivation of our optimal control problem which is to be studied in this paper.

Let X be a Banach space and A : 7)(A) C X --* X be the infinitesimal generator of some Co-semigroup e At on X. We consider a controlled evolution system

f &(t) = Am(t) + g(t, x(t), u(t)), t q [0, T], (1.1)

t ~(o) = z o .

The state of the system is usually understood as the mild solution of (1.1). Now, suppose we have another control action--an impulse control, i.e., at times t = ri, i _> 1, we make an impulse ~i to the state z(ri - 0). We refer {(ri, ¢i) [ i > 1} as an "impulse control". Thus, one has

~(n) = z ( n - 0) + ¢ , i>_. 1.

Then, the state z(-) formally satisfies the following evolution equation:

{ x ( t ) = A x ( t ) + g ( t , z ( t ) , u ( t ) ) + ~ f . , ( i S ( t - r l ) , t E [0, T],

z ( 0 ) = z0,

where 6(-) is the $-function. Similarly, we understand the state of the system z(.) as the mild solution of the system (1.2). It is more reasonable that, in general, the impulse ~i at time t = ri should also depend on the state z(rl - 0), which is the state of the system before making impulse ~i. Thus, it is more natural to consider the following state equation (compare with (1.2))

[ ~ ( t ) = (A~(t) + g(t, ~(t), =(t)))~t (1.3) ! + go(t, ~ ( t - 0), ~ ( t ) ) ~ ( t ) , t ~ [0, TI,

I, ~(o) = ~o,

where ~(t) = ~ ~xv,,oo>(t), t e [0, oo).

i_>1

Hence, in general we have the following type evolution equation:

(1.4) I dx(t) = Ax(t)dt + F(t , x(t - 0), u(t))d#(t),

z(o) = ~o, t E{O,T],

* This work was partially supported by the Chinese NSF under Grants 0188416.

177

with some £(Z, X)-valued function F and some Z-valued vector measure/~(.), which together with u(.) will be considered as control actions. Associated with (1.4), we are given a cost functional

j , T

(1.5) J(z(.), u(.), ~(.)) = ]o (f(t , x(t - 0), u(t)), ~(dt)),

with some X*-valued function f . In the case there is no impulse, or the measure p(.) is fixed and is absolutely continuous with respect to the Lebesgue measure, the problem is reduced to a classical semilinear distributed parameter systems with Lagrange form cost functional.

The control problem we will study in this paper is to minimize functional (1.5) over some class of admissible controls, subject to the state equation (1.4) and an end constraint for the state of the following type:

(1.6) (x(0), z(T)) e &2 C X x X.

In [14,15], a similar problem in finite dimensional spaces was studied. It is immediate that the major difference between this paper and [14,15] is that whether the coefficient of #(.) depends on the state z(.). Secondly, we are in infinite dimensional space and we have a general end constraint (1.6), which is different from the separated end constraint case (see [12] for comments). On the other hand, we should point out that the operator-valued function F(t, x, u) has to be assumed Frechet differentiab[e in z, which is a little more restrictive than [15] for the finite dimensional case (in [15], only the Lipschitz continuity in z was assumed). The main results of this paper consist of the study of the infinite dimensional Volterra-Stieljes integral equations and the Pontryagin type maximum principle for the related optimal control problems.

§2. Evolution Equations. Let us first introduce the so-called Young integral of operator valued functions with

respect to vector measures. To this end, let X and Z be Banach spaces and T > 0 be given. For any metric space V, we denote

BVo([O,T]; V) = (v(.) : [0, T] ~ V [ v(.) is of bounded variation, v(0) = 0}.

For any #(.) E BVo([O,T]; Z), there exists a unique vector measure associated with it. We denote it by/2(.). Next, by noticing the fact that any B V function #(.) has at most countably many discontinuity points, we may define

I ub(t) = ~ [~(T + 0) - u(T - 0)l + u(t) - ~(t - 0), vt e [0, Tl, (2.1) o<T<t

~, no(t) u(t) - re(t), w ~ [0, TI.

Now, we let/2,(.) and pc(.) be the vector measures induced by Mb(-) and #c('), respc- tively.

178

Delhd t ion 2.1. An operator-valued function F( - ) : [0,T] -* £(Z, X) is said to be uniformly Borel measurable if there exists a sequence of Borel measurable simple functions F,,(.) : [0,T] ~ £(X, Z), such that

(2.2) lira I I F . ( 0 - F(~)II~(=,~) = o, w E [O,T].

Then, we may define Bochner integral for operator-valued functions with respect to vector measures. Next~ we would like to introduce the extended Young integral.

Definl t ion 2.2. Let t~(') e BVo([0, T]; K) and F(.) : [0, T] ~ £(Z, X) be uniformly Borel measurable. We say that F(-) is/~-Young integrable if F(.) is pc-Bochner integrable and

(2.3) ~ IW(~-)ll~cz,x)l~,(r + o) - ~,(~- - o)1 < oo o<~<T

In this case, we define the Young integral of F(-) with respect to/~(.) by the following {' f. r(T),t ,CT) = F(T) (T I ( r ( l !

(2.4) + F(s)[t,(s + 0) - #(s)] + F(t)[l~(t) - #(t - 0)], Y0 _< s < t < T,

J ' F(r)dl~(r) = O, Yt e [0,T].

be the set of all /~-Young integrable functions F( , ) : We let Yt,([0, T]; £(Z,X)) 10, T] -+ Z:(Z, X). Next, we let

(2.5) p , (~ , t) = Is - t[ + I I~I(~) - [~l(t)l, vs , t e [o, TI.

Then, for any metric space V, we define

Cu([O, TI,V ) = {v(.): [0, T] ~ V I v(.) is uniformly (2.6)

continuous from [0,T] with metric p~(.) to V}.

We should note that, in general, [0,T] is not necessarily compact under metric p~,(.) (see [8] for relevant remarks for the scalar valued case).

Now, let/~(.) a BVo([O,T];K) and F( .) E Yt,([O,T];~.(Z,X)). Then, by (2.4), for any t e [0, T], one has

(2.7) F(r)d~(r) = ,q F(r)#~(dr) + ~_, F ( r ) [ . ( r + O) - . ( r - 0)]

+ F ( 0 ) t u ( 0 + 0) - u(0)] + F ( ~ ) [ u ( 0 - u(~ - 0)1,

We refer the above the indefinite Young integral. Let

2~ - {(t,r) E [O,T] x [O,T] I t >_ T},

179

and let G(., .) : £ ~ £(Z , X ) and consider the following integral

L' (2.8) g(t) = G(t, r )dp(r) , t • [0,T].

We have the following P r o p o s i t i o n Z.3. Let ~(.) • BVo([0, T]; K) and G(-, .) : ~ --* £(Z , X) . We assume the following:

(i) G(.,-) is uniformly Borel measurable. There exists a [pl-integrable function a(.) : [0, T] ~ R , such that

(2.9) llG(t, r)ll~<z,x) _< a(r), v(t, r) c &.

(ii) There exist functions wo : R + ~ R + and 8 : [O,T] --* /1% with the properties that wa is nondecreasin$, wa(O) = 0 and O(.) is [f~l-integrable, such that

(2.10) llG(t, r) - G(s, ,')ll~(z,x) _< wG(Pt,( t, s))O(r), V(t, r), (s, r ) 6 ~..

Then, the function g(.) defined by (2.8) is #.continuous and

(2.11) g ( t + o ) = a(t+O,r)~.(r)+G(t+O,t)[.(t+O)-~,(Ol, te[O,~,

Z' g(t - o) = a ( t - o, r ) e . ( , - ) - a ( t - o, t ) l . ( t ) - . ( t - o)1, t e [o, rl, (2.12)

where

(2.13)

(2.14)

G(t - 0, t) - v(~ , t), t e [0, T],

G(T + 0,T) = C(T, T).

In particular, i f there exists a constant C, such that

(2.15) w o ( r ) = Cr, Vr e R,

then, g(.) • BV0([O, T]; X). Now let us consider the following integral equation

I' (2.16) x(t) : ~(t) 4- G(t, r, x(r -- 0))d#(r),

We state the following assumptions. (I1) The function ~(.) E C~([0,T]; X). (I2) The function G : ~. x X --, £ (Z , X ) satisfies (i) There exists a constant L, such that

(2.17)

t • [0, r].

fiG(t, r, ~)ll~(z,x) _< L ( I 4- Ixl), V(t, r) • £, ~ • X,

180

(2.18) IIG(t, r , x ) - G(t,r,~)Uc(z,x ) <_LIx-~I, V(t,r) E 7,,=,~E X.

(ii) For any r E [0, T] and z E X, the map G(., r, z) is continuous on [r, T]. (I3) There exist functions w~ : [0 ,T]xl t + ~ I t + and {9 : [0, T] ~ R + with the

properties that wa(O, r) = 0,Vr E R +, o~c(t,. ) and (a~(., r) are nondecreasing and 0(.) is I#l-integrable, such that

UG(t, ~, x ) - a ( s , r, ~)llc(z,x) ___ wa (p.(t, s), Ixl)a(r), (z19) Vr 6 [0, T], t, s e [r, T], z e X.

Following result gives the existence, uniqueness and some properties of the solution to (zl~) . T h e o r e m 2.4. Let/~(.) E BVo([O,T];K) and let (11)-(12) hold. Then, (2.16) has u,ique soiutio, =(.) e C,([0,T}; X). I~ ~(.), ¢(.) e C~,([o, TI; X) and z(.) a n d ~(.) are the solutions of (2.16) correspondin$ to ~o(.) and ~(.), respectively. Then, for any fi > 1,

(z~o) I~(t) -~ ( t ) l < ~ [ s~p l~(~) - ~(r)ll~ ~"l( ' ) , t e [o, T], - - # - i o_<r<~

/ ' 4 (z.21) I~(t)l _ ~ ~--~_ 1 [ o ~ , I~(~)1 + Ll,l(~)]e "~l"l('), t e [0, T].

Moreover, ff (13) holds, then, for the unique solution x(.) o[ (2.16), one has

Ix(t) - z ( s ) l < I~o(t) - ~o(s)l + L(1 + sup Ix(~) l ) ( l~ [ ( t ) - I~t(~)) 0<r_<~

(2.22) + w~(p,(t, s), sup Ix(r)l) O(~')dl#l(r), o<~_<f

YO<s<t<T.

zn particular, Jr ~(-) e BV([0,T]; X) a . a

(2.23) ~ ( t , r) = t~(r), V(t, ~) ~ [0, T] × R,

for some nondecreasing function ~ :R + ~ + with ~(0) = O, then, x(. ) E BV([0, T]; X).

We should point out that in the case we will study, the function in(-) is not necessarily bounded variational and (2.23) does not hold (in general) either. Thus, we are not expected to obtain bounded variational solutions (even though they are/~-continuous). However, in finite dimensional case, we do have x(.) E BV([0, T]; X), provided (2.19) and (2.23) hold, which is not too restrictive. Also, we should note that assuming (I1) is just for simplicity. It can be replaced by

~o(.) E D([0, 7]; X) = {~: [0, T] --+ X I ~(t + 0) and ~(t - 0) exist for all t E [0, T]},

and we look for ths solutions of (2.16) in D([0, T]; X) instead. The results will be the same.

181

The rest of this section is devoted to the study of following linear equation:

[ ~(t) =eA'~o + eA('-r)B(r, ~(r - o),eu(T)) (2.~4)

I' + eA('-r)O(T)e~'(T), ~ e [0,TI.

We make the following assumptions: (A1) The operator A : ~ (A) C X -~ X generates a C°-semigroup e A' on X and

for some constants M _> 1 and w Ei~,

(2.25) IleA'll,-(x) _< Me ~'', t > O.

(A2) Function B : [0,TI --* I~(X x g ; X ) ~ (B : X × Z ~ X [ B is bilinear } is uniformly Borel measurable and [[B(T)[[~- is bounded by Lo.

(A3) v(.) E BVo([O,T];Y) and (~(-) E Y.([0,T]; L:(Y, X)), with Y being some Ba- nach space. T h e o r e m 2.5. Let (aI)-(a2). Then, exists a unique solution x(.) e Cu([O,T];X) of (2.16). Moreover, there exist well-defined evolution operators ~, 6) : A ~ £ (X) given as follows:

• (t,.)~o = eA('-')~o + [ C~('-~)B(T, O) S)~O) d/~(r)),

(2.26) 0 < . s < t < T , x o E X ,

• ( s - 0 , s) = I, 0 < s < T,

(z~z)

I t oft , ~)~o = f ~A('-')B(T, OfT - 0, S)~o, d0(T)),

I

O < s < t < T , x o E X , ® ( s - O , s ) = - I , 0 < s < T.

such ~hat and x(.) can be represented by

(z2s) ~(t) = )(t,O)=o + ( ) ( t , s) + 6)(t, s))~(s)a~,(s) , t E [o, T].

From (2.27), we see that

f o(s, s) = o, (2.29) e ( s + 0, s) = B(s , - / , #(s + 0) - u(~)).

Thus, if/~(-) is right-continuous at some point n E [0,T], then,

(2.30) e f t , s) = o, vo < , < t < T.

182

§3. Optimal Control Problem and M a x i m u m Principle.

In this section, we state our main results concerning the control problem of this paper. We let X and Z be two Banach spaces, K be a convex and closed cone in Z and U be a metric space. We define

/ / - - {u(.) : [0, T] ~ U I u(.) is Borel measurable },

and 2,4 = {#o(')} + ~4o([0, T]; K),

where

.Mo([0 , T]; K ) = {#(.) e BVo([O.T]; K) ] f~(E) = j~ O(r)lf~](dr ),

VE e B([0, T]), for some 0(-), with 10(r)lz = 1, I#l - a.e.},

and #o(') E BVo([O,T]; Z). We can prove that .M is convex. Next, let us make the following hypothesis:

(H0) The dual X* of X is strictly convex. (H1) The same as (A1). (H2) Maps F : [0, T] x X x U .-~ £ (Z ,X ) and ] : [0,T] x X x U ~ Z" satisfy the

follwoing: (i) For any (x,u) e X x U, the maps F ( . ,~;,u) : [0,T] ~ £ (Z ,X ) a n d / ( . ,x ,u) :

[0, T] -+ Z* are uniformly Borel me~surable; For any (t, =) e [0,T] x U, the maps F( t , . , u) : X -+ £ ( Z , X ) and l ( t , . ,u) : X --+ Z* are continuously Frechet differentiable; For any (t,x) E [0,T] x X, the maps F( t , x , . ) : U -+ £ ( Z , X ) and ./(t, z, . ) : U + Z* are continuous.

(ii) There exists a constant L > 0, such that

(3.1) HF(t,x,u)llqz, x) ~ L ( l +lxl), ( t ,x,u) e [ O , T ] x X xU,

(3.2) I IF(t ,x ,u)-F(t ,x ,u)U~(z ,x) <-LIx - x l , (t,u) E [0, T] × U, x , ~ e X.

(H3) Set g~ is convex and closed in X x X.

Now, for any Xo E X and a pair (u(-), #(.)) E / / x At, the response of the controlled system is defined to be the unique solution of the following integral equation:

(3.3) = eA' o + fo - 0), t e [0, T].

From §2, we see that the above equation makes sense. Moreover, by Theorem 2.4, we know that under (H1)-(H2), there exists a unique solution x(.) E C~([O,T];X) of (3.3) corresponding to the triplet (xo, u(-), #(.)). System (3.3) is a mild form of (1.4). Thus, sometimes, we refer the soultion ~:(-) of (3.3) the mild solution (1.4). We call x(t) the state of our system at time t, x(.) the trajectory of the system and x0, u(.) and /~(.) the intial state, continuous control and measure control, respectively. We see that the measurc control is an extended notion of the so-called impulse control ([2,16,17]). Assumption (H0) is technical for proving the maximum principle. For the case X is

183

reflexive or separable, we can always change the norm of X to another equivalent one so that (H0) holds (see [12] for comments).

Remark o*.1. It is very important to notice that by using the Young type integral in (3.3), the trajectory of the system inherits the discontinuity of the measure control /~(.). In fact, if we let a:(.) be the solution of (3.3) corresponding to (x0, u(.),/~(.)), then, we have

{ =(~) = x ( t - o ) + F ( t , x ( t - 0) , u ( t ) ) [ ~ ( t ) - ~ ( t - 0)1, ~ E (0, TI, ~:(t -I- 0) = x(t) -I- F( t , x(t - 0), u(t))[/~(( + 0) -/~(t)], ( e (0, T].

While, the Bochner type integral does not have such a property! As in [32], the Bochner (or Lebesgue) type integral was used and as a result, the trajectories had to be restricted to right-continuous functions. In our case, the trajectories are just of elements in the space D([0, T]; X) -- {v(-) : [0, T] --* X [ v(t -!- 0), v(t - 0) exist Vt e [0, T]}.

The payoff of our control problem is given by (1.5). The meaning of the right hand side of (1.5) is now clear from §2 if we regard the map .f as a Z:(Z,lR.)-valued function.

Now, we are ready to state our optimal control problem.

P r o b l e m C. Minimizing functional (I.5) over all (u(.), lz(.) ) E bl x ¢t4, subject to system

Next, let (x#(.), u#(.), ##( .)) be an optimal solution to Problem C. We define

(3.4) u ~ = 1,( . ) e u I ,(T) = ,,~(T), if ~,*(.) jumps at ~ }.

Note that F : [0, T] x X x U - , E(Z, X) , thus, the Frechet derivative F~ of F is a map from [0, T] x X x U to /} (Z x X; X). We let

B(r, ~, z) = F.(T, x # ( r -- 0), u#(r) ; ~)z

(3.5) = lim F(r, x # ( r - O) + e~, u#(r ) ) z - F(r , x # ( r - 0), u# ( r ) ) z

V(r,{, z) e [0,T] x X x Z.

Then, we may define B(., .)*: [0, T] x Z --+ Z:(X*) as follows:

(3.6) (B(r,z)'=',ll)=(B(r, Lz),='), V(r, Lz,= ' )e[O,T]xXxZxX"

As in §2, we let ~(., .), 0(-, .) and ~(.,-) be the evolution operators associated with the operator A and the bilinear form valued function B(r , ~, z). Then, we define (3.z)

/o = { q~(T, r)[F(r, =#(r - 0), u(r)) - F ( r , x # ( r - 0), u# ( r ) ) ]du#( r )

+ ~(T,r)F(r,~#O--O),u#(~))d(U(T)- U#O-)) ] (,,(.), U(.)) eU# × M},

and

(3.s) Q = {yl - ¢(T, 0)~o [(yo, y,) e a} .

184

We introduce the Z*-valued Hamiltonian: V(r, x, u, ¢0, ¢) E [0, T] x X x lq.x X °

(3.9) H(r, x, u, ¢0, ¢) = ¢ o / ( r , x, u) + F(r, x, u)*¢.

Our main result for the Problem C is the following T h e o r e m 3.2. (Maximum Principle) Let (HO)-(H3) hold Let (x#(.), u#(.) , ##( . ) ) be an optimal solution to Problem C. Let the set

~ - Q = { r - q i r e ~ , qeQ} be of finite codimensional in X ([7,11,12]). Then, there exists a pa/r (¢(.), ¢o) E D([O,T];X*)xR, such that

¢(t) =eA'(T--O¢(T) + / 7 ' ea ' (~- ' )B( r, dt~#(r))*¢(r)

(3.1o) + ¢0 cA' ( r -0 fx(r, x#(r - 0), u#(r))*d##(r), Vt e [0, T],

(3.11) ~h ° < 0, (~b(-), ¢0) ~ 0.

(3.12) o2" ~ H(T, ~#(T - 0), ~# (~), ¢°, ¢(~)), d ,# (T))

2"

= ~(')~max fo (H(" ~#(T - 0), ~,*(~), ¢°, ¢(~)), ~.(~-) I,

(~.13) o r (H(r, x#(r - 0), ug( r ) , ¢o, ¢(r)) , d/~#(r) )

= max (H(r, x e ( r - O ) , u ( r ) , ¢ ° , ¢ ( r ) ) , d # # ( r ) ) , u(-)¢u*

(3.14) (¢(0), yo - x#(O) ) - (~b(T), yl - x#(T) )

- ( ¢ ( T ) , O(T, 0)(yo - x # ( 0 ) ) ) < 0, (yo, y~) E s2.

The proof is based on the Bkeland's variational principle ([6]) and the analysis of the variation of trajectories under spike purturbation of controls ([11,12]).

Remark 3.3. Condition (3.14) is the transversality condition for the optimal solution of our problem. The appearance of term

(~b(T), ®(T, 0)(yo - x#(0)) )

is unexpected. This is caused by a possible jump of the measure control #(.) at t = 0. It is interesting to notice that in the case

(3.15) ~,#(0 + 0) = ~*(0),

185

i.e., there is no jump of ~#(.) appeared at t = 0, then, from (2.29), we see that condintion (3.14) is reduced to the familiar form

(3.16) (¢(0) ,yo -- ~:#(0))-- (¢(T) , yl - z# (T ) ) <_ O, (Yo, Y,) e I1.

lZeferences

[1] G. Barles, Deterministic impulse control problems, SLiM J. Control ~ Optim., 23 (1985), 419-432.

[2] A. Bensoussan and 3. L. Lions, Impulse Control and Quasi-VariationM Inequalities, Bordes, Paris, 1984.

[3] L. D. Berkovitz, Optimal Control Theory, Springer-Verlag, New York, 1974.

[4] P. C. Das and R. R. Sharma, On optimal controls for measure delay.differential equations, SIAM d. Control & Optim., 9 (1971), 43-61.

[5] 3. Diestel and 3. J. Uhl, Jr., Vector Measures, AMS Providence, 1977.

[6] I. Ekeland,, Nonconvea: minimization problems, Butt. Amer. Math. Soc. (New Se- ries), I(1979), 443-474.

[7] H. O. Fattorini, A unified theory o] necessary conditions for nonlinear noneonvex control systems, Appl. Math & Optim., 15 (1987), 141-185.

[8] J. Groh, A nonlinear Volterra-Stieltjes integral equation and a Gronwall inequality in one dimension, Illinois J. Math., 34 (1980), 244-263.

[9] T. H. Hildebrandt, On systems of ffnear diffcrentio-Stieltjes-integral equations, Illi- nois d. Math., 3 (1959), 352-373.

[10] C. S. Hhnig, Volterra Stieltjes Integral Equations, North-Holland, Amsterdam, 1975.

[11] X. Li and Y. Yao, Maximum principle of distributed parameter systems with time lags, Distributed Parameter Systems, Lecture Notes in Control and Information Sciences, Springer-Verl~.g, New York Vol.75, 410-427, 1985.

[12] X. Li and J. Yong, Necessary conditions of optimal control for distriduted parameter ~ystems, submitted.

[13] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, Math- ematicM Theory of Optimal Processes, Wiley, New York, 1962.

[14] R.. W. Rishel, An e~;tended Poztryagin principle ]or control systems whose control law contain measures, SIAM J. Control & Optim., 3 (1965), 191-205.

[15] R. B. Vinter and F. M. F. L. Pereira, A mazimum principle .for optimal processes with discontinuous trajectories, SIAM J. Control & Optim., 26 (1988), 205-229.

[16] J. Yong, Optimal switching and impulse controls for distributed parameter sy~tem~, System Sei. & Math. Sci., 2 (1989), 137-160.

[17] J. Yong, Systems governed by ordinary differential equations with continuous, switching and impulse controls, AppL Math. Optim., 20 (1989), 223-236.

[18] W. H. Young, On integration with respect to a function of bounded variation, Proc. London Math. Soc.(2), 13 (1914), 109-150.

The Existence and the Uniqueness of

Optimal Control of Population Evolution Systems

Yu Jingyuan

(Beijing Insititute of Information and control,Beijing)

Gao Ling (Insititute of Population Research , People's

University of China,Beijing)

Zhu Guangtian (Insititute of System Science, Academia Sinica,Beijing)

Abstract In this paper, we give a conclusion of the existence and

the uniqueness of optimal control of the population evolution equation with the specific fertility rate of female ~(t) as parameter.

Recently the population control is an important topic for population workers . How to realize the population control is practically how to control population to approach the population distributed state expected by humanity. Controllability of the population system was made out in [2]. In [3] J.Yu etal sketched the necessary conditions for the optimal control with a eontral domain consisting of continuous functions . in this paper we shall come to the conclusion that the optimal control is exist uniquely for the control domain consisting of bounded measurable functions .

We shall discuss the equation of population evolution process

Op(r,t) ap(r,t)

at Or + - u(r,t)p(r,t) 0<r<r , t>0

p ( r , O ) = p o ( r ) O_~r_<rm ( 1 ) .

p(0,t)=~(t)/ Zk(s)h(s)p(s,t)ds 0St ST

Where t denotes time , r denotes age , r is the highest age

ever attained by individuals of the population , p(r,t) is

called age density function , U(r) is tile relative mortality function , p°(r) is an initial age density of the population,

~(t) is the specific fertility rate of female at time t, k(r)

is the female sex ratio at age r , h(r) is the fertility

pattern and [rl,r =] is the fecundity period of females .

187

Suppose that k(r),h(r) and H(r) are measurable functions on [ O , r ] , p ( t ) i s a

conditions (a) 0 S k(r) S 1 ,

measurable function o n [0,Tl,and

r ~ [ 0 , r m ]

0 < k S k ( r ) < 1 , r ~ [ r , r = l

(b) 0 < h(r) S h ° , r ~ (r ,~) r

I Xh(r)dr = I ; h(r) = 0 r ~ (r r ) £e 2

(e) /~(p)dp < ~ , r < rm " /o ~'~(p)dp = ~

(d) po(r) is measurable on [0,r m] and

po(r) > 0 , for r ~ [0, r z]

Where k, h are two positive constants .

For two given positive constants ~ <~ we define the £ X

control domain

U = { ~(t) I ~ S ~(t) S ~x a.e., ~(t) is measurable "7 2

T h e c o s t f u n c t i o n a l o f o u r p o p u l a t i o n c o n t r o l s y s t e m i s

~iven by r

j(~) = IT i m [plr,t;~l_pOlr,tl]Zdrdt for all ~ ~ U O O

w h e r e p ° ( r , t ) i s g i v e n a n o n - n e g a t i v e L 2 g o a l f u n c t i o n , o n [0,r ] x [0,T], plr,t;0) is the solution to the equaEion 111

e o n e s p o n d i n g t o ~ w h i c h h a s f o l l o w i n g r e p r e s e n t a t i o n

P

-I H(P)dP , for r Z t Po (r-t)e ~-t

F p(r,t;~) = I

L ~(t--r)l z k(s)h(s)p(s,t-r;/~)dse o

P~

for r < t

It is obvious that p(r,t;(~) > 0 and

p(r,t;~) ~- L2( [0,r m]x [0,T] )

p(r,t;~) may be expended into [4]

~ ( p ) d p

(2)

-Irr tp (p) dp p(r,t;p)=po(r-t)e -

oo -J'rb~(p>d p u

k = O ( 3 ) .

188

here m

J ~o(tl=~(t)l[ = k(slh(s) =-t~(oldppo(s-tlds , i

m

~k(t)=Slt)irZkls)hlsle o #k_~t_slds • i

If for s < 0 we define po(s)=0 , then ~k(t) ,

well defined on R and supp t~k} : t k q . ( k + X ~ l

Recall that the optimal control of the population

evolution system (l) is to find a ~ ~ U = U such that 2

J(~ ) = min J(~) (4).

I n t h i s p a p e r w e s h a l l p r o v e t h a t t h e r e e x i s t s s u c h a ~ ~

k=l,2 ....

k=l,2,...,are

U unigucly .

Lemma i. If [~ ~ U , then

p(r,t;~) > 0 ,

lot t ~ [O,T]

Proof . By (3) we have

" (pldp

p( r, t;~ ) =Po (r-t) er-t ~-I

and r ~- [O,r+t] n [O,r I .

>O for r -- ~ t

If r < t , then there exists a integer k with t-r

[kr,(k+l)r ]. Since the support set of ~. is [kri,(k+l)r z] ,

we have ~k(t-r) > 0. So p(r,t;~(' )) > 0.

Lemma 2 . If ~i" ~z ~ U, and in L z ([O,T]), ~i$ ~z " then in

L ~ ([O,r m] x [O,T])

p(r,t;~i) % p(r,t;~z) .

Proof . If the conclusion is false , i.e. there exist ~

and ~z in U " /91 ~ /~z ' such that

p(r,t;/~ I) = p(r,t;~ z)

it follows for all t ~ r

189

p(r,t;/~ i) - p(r,t;0z)=

= ~l (t-r)$rZk(s)h(s)p(s"t-r;/9~)dse £

(o)do -Io~ r

- ~z (t-r)/ Zk(s)h(s)p(s,t-r;~z)dse r r £

r (o)dP = [~l (t-r)-/~z ( t -r) ]IrZk(s)h(s)p(s, t-r;~z )dsel°H

l

By Lemma 1 and k(s) ~ k ° > 0 , s e [r ,r 21 ,

P

12k(s)h(s)p(s ' t - r ;~z ) d s ~ $ ; u ( p ) d o r > 0 i

hence

= 0

/~i(t) = /~z(t) , a.e. for all t > 0 .

It contradicts our supposition . Lemma 2 is proved .

Lemma 3 . Let ~n ~ U, n = 1,2,.... If ~n converge to ~ in

L z([O,T]) , then ~ ~ U and

LZ--]im p(r,t;~ n) = p(r,t;~*) Tb~ ~60

Proof. By the formula (2) , p(r,t;/~n) and p(r,t;/~ ) can

he extened into

-Ir_t H ( p ) dp m ~ n~ -I ; H~p~dp P(r ' t ;~n)=Po(r-t)e + ~ ~k ( t - r )e

k = o

p(r,t;(? )=Po(r-t)e r-tN(P + ~ k(t-r)e k=o

(m)

respectively . So it is only need to prove that for any k

(t) L z- converges to ~k (t) . Since

(o)dp <"~ (t)=~n(tllrZ kls)hls);l~-t" pols_g)d s •

£

1 9 0

. *' ( p ) dp % (t)=/~ (t)$ =k(s)h(s)e x=-L~ Po (s-t)ds

L z

j ,T [ . (m) 4~ O ~O (t) --~O (t) ]2dt=

P

: ¢T(~n(t)-"o (t))Zx [/Zk(s)h(s)er ~L

--< C ITo [~r, (t)-/~ (tl]2dt "

L x

If for some k > 0 . ¢~n~>k Ck*

[qSk+ ( t ) - -¢ ( t ) ]Zdt < o +

>0 .

-I:_LM~P~dP po(s-t)ds]Zdt

• then

. r --I: jucp~dp b ~ r~) ( --<2/:[13n(t)-/~ (t) ]2X[/ 2k(s)h(s)e t-s)dsl2dt

k i

, r. - . £ : p ~ p ) d p , ~2/:[/~ (t)Iax [I~Zk(s)h(s)e (¢<n~(t-s)ds-~kk (t-s))ds]:,dt

I

-J '~ ,J ,p , dp . ,,,, . f f ~,.,'~ t p ( t ) - ~ ~2max[/ Zk(s)h(sJe ~k (t-sJdsJ=dt *(hJ]=dt r £

2 h:, II / q * l l 2 tl II 2 o k -~k L~

<- 2 h ° ( r z-r~ ) II #(n)k (t)llZL z II ~n (t)-~ (t) II x52

+ 2 h 2 II /~*II:, II ¢~n~ ~* II:, o k k L:,

Since Con'(t) > 0 Note

c,~) h z z ~,n~ II ,.~k (t) II z S II II a o 2 k-i L:,

_< < h2k+1 z k + t - ~ II Poll :, = C < ~ .

o 2 L~

So

II~ ~n~ (t)-¢k+i(t)ll 2 _< k+i T2

n--> co

_<2oh o '~-r.'""o -"*"" +2h" ''"*"~''*:~'~:":, - - " o -~ k . _:, ..... >0. L

191

Hence , LZ--lim ~k+i } = ~k÷i (t) .

The lemma is proved

Denote A = ~p(r,t) ~ Lz([O,r ] x [O,T] # there exists a

~ U such that p(r,t)=p(r.t;~)~.

Lemma 4 . A is a bound closed convex set in LZ([O,r ] x m

[O,T 1).

Proof . We firstly prove convexity . Let

Px(r't) ~ A , then there are ~, " ~z ~ U such that

p (r,t) = p(r,t;~ ) ,

pt(r,t) = p(r,t;~ 2) .

For 0 < ~ t {

c s . t - r , a s J + i

r - d o l l (p) dp + (l-~)~z(t-r)lw~k(s)h(s)pz(s,t-r)dse

£

(r,t) ,

r < t

-$~_t~ (p) dp Po (r-t)e r Z t

" (p)4p ~(t-r)/~Zk(s)h(s) [ap (s,t-rl+(l-~lpt(s,t-r) ldse -$°H

i r < t

here

~(t)=

r ]"

~i (t)$ Zk(s)h(s)pL (s,t)ds+(l-a)/~ (t)$ Zk(s)h(s)pz (s,t)ds

~z k I (slhls)[a~ls,t)+ll-~)P2ls,tllds £

By Lenuna i , pi(r,t) , pz(r,t) > 0 , for all (r,t) ~ [0,i~]

x [O.T] . Therefore

192

r 2

I k(s)h(s)[~p (s,t-r)+(l-~)pz(s,t-r)|ds > 0 I"

&

for all t ~ [0,T] hence /9(t) is well defined . Obviously

S ~(t) < ~ , for all t ~ [0,T] Z

So ~ ~ U . It shows a~+(l-a)pz ~ A ,hence A is a convex set.

Now we prove that A is closed .Let p (r,t) ~- A , p (r,t)

L z- converge p (r,t). According to the definition of the set A . there exist ~ ~ U .n =1,2 ..... with p, (r,t) = p(r,t;fi ).

Therefore there exists a subsequence ~n [ of ~r, ~ and ~ in L

U such that /~n weakly converge to k

By

pn{r,t) = t

-2 r (p) dp

po(r-t)e ~-t~ r ~ t

i, (p)dp /~( t-r)IrZk(s)h (s)p(s,t-r;~)dse I°~

L

r < t ,

The term on the left side weakly converges to p(r,t) as i --> ~ and one on the right side converges to

llence

r

-Ir_tU (p) dp 6 Po (r-t)e r Z t

" (O)dp . r

/9 (t-r)l 2k(s)h(slp(s,t-r;flldse I°~ P

r < t ,

p(r,t) = p(r,t;/~ ) ,

It shows that A is closed .

It is obvious that A be bounded .llence Lemma 4 is proved.

t93

Now we define a new cost functional

~(~) = I T I '~ [p(r,t;~l-p°(r,t)]~drdt 0 0

and consider the new problem of the optimal control

p~-A ( 5 ) tP ~- A

By Lemma 2 , the existence and the uniqueness of

solutions to (4) are equivalent to those to (5) .

Lemma 5 . ~ is a strictly convex functional on A °

Proof . Let p, " Pz E A , 0 < ~ < 1 , we have

j.

3 (c~p+ (i-~)pz) = re'So oml~P,+(1-~)P~ -p° ]=drd t

r = ITI u~[~( o o o P~-P )+(l-~)(Pz -p°Ilzdrdt

: fT$o om[aZ(P= -P°)z+(1-a)z(p= -p° )= ]drd t +

÷ ITIo O m [ 2~ ( i--= ) ( Pl -p° ) ( PZ -p° ) ] drdt

: i;io m[~ ( Pl-P° )z + ( l-a ) (p~-p°)z ] d rd t -

- 2T2o o m [ a ( 1 - ~ ) (Pi-Pm)zjdrdt

--< ITIo om[~lPl -P°lx+(l-~)lp2-p°)zldrd t

= ~ ( p ) + l l - ~ ) 3 ( p 2) (6)

The above equality holds if and only if p~ = ~ almost

everywhere .

From this the strictly convexity of ~ is at once obtained .

Lemma 5 is proved .

194

Lena 6 . Let p~ ~ A . , = 1,2 ..... J p~ }

to p , then

~(p ) ~ limn inf ~(~ )

Proof . Since 7(pn) = II pn-p°llz , Pn

p . S o

7 (p) = Ilp*-p ° 112 S liming II pr- p L ~

= limingn 2 1~,) .

L e m m a 6 i s p r o v e d .

weakly converge

weakly converge to

ii 2

Theorem . There exists a unique optimal control for the

population evolution system (I) .

Proof In order to establish the existence of the

optimal control , it is sufficient to show that there exists

a p (r,t) 6 A such that

U(p ) = inf U (p) p ~A

Let ~, ~ A , n=l,2, .... be a minimizing sequence of J , i.e.

lim ~(P ) = inf ~ (p)

Since A is a bounded closed convex set in L~([0,rm] x [0.T]),

so A is also bounded weakly closed , hence weakly compack

there is a subseqnence ~pn ~ of ~pn} which converges to p %

A . Dy Lemma 6 ,

(P) S liming ~ (pn) = inf ~ (p)

This shows that the existence holds

The uniqueness is implied in the result of Lemma 5 .The

t h e o r e m i s p r o v e d .

195

Reference

[I] . Song Jing , Tuan Chihsien and Yu Jingyuan , Population

Control in China , New York , 1985 . [2] . Song Jian , Yu Jingyuan ,Liu Changkai ,Zhang Lianping,

Zhu Guangtian , The specturc properties of the population evolution operator and the controllablity of the population system (in Chinese), Sciences Sinica A , 2,1986 .

[3] . Yu Jingyua, , Guo Baozhu Zhu Guangtian Optimal control of population system, Control Theory and Applications (in Chinese), 1,1989.

[4] . Yu Jingyuan , Zhu Guangtian , Guo Baozhu , The asymptotical properties and the controllability of the population evolution process in the L p space ,System Sciences and Mathematics , No.2, 1987.

R E A C H A B I L I T Y F O R A CLASS OF

N O N L I N E A R D I S T R I B U T E D SYSTEMS G O V E R N E D

BY PARABOLIC V A R I A T I O N A L I N E Q U A L I T I E S

Y. Zhao and Y. Huang ~' Zhongslaan University tGuangzhou China

W.L. Chan ¢' The Chinese University of Hong Kong thong Kong

Abstract

In this paper the reachabilily for a class of distributed control problems governed by parabolic variational inequalities is considered. The negative reasult for the complete teachability of the system is obtained. Sufficient conditions for approximate reachability of the system are discussed undcr two kinds of hypotheses.

I. INTRODUCTION The reachability of linear or scmilincar distributed parameter control systems has been

surveyed in['8-]. But the reacllability of the followingsystem governed by the parabolic vari- alional incquality tto our knowledge t has not been treated~ ( I , 1) [<y' (t) -I-Ay (t) ,y ( t ) --z~>-t-qg(y ( t ) ) --c0(z) ~<~Bu (t) ty ( t ) - - z ~

ty (0 ) =Y0 a.e. t E E0 tT-] ,zE V where V and H are real separable Hilbert spaces,V C H C Wand denote the norms of H,

V and V'by I • It II • II and II • II ,respectively and < • t " ~> the pairing between V and W.

Assume that A is a linear continuous symmetric operator from V to V r and satisfies the coercive condition :

(1 ,2 ) < AU, U > ~J ,L'il r/II ~ V u E Y for some w ~-0, tp. V--~ R is a lower--semicontiuous convex ftinctional. B is a linear bounded operator from U to H where U is a Hilbert space, y0E V.

Let :3c?: V --~ W be the subdifferential of ~ s o that ( 1 ,1 ) is equivalent to

,.~. Research ~'upported partly bY the Foundation of Zhongshan Univdrsity Advance Re,search Centre.

,g. R~ea.arch Supported by UPGC Direct Grant,CLIHK,

197

(1,3) I ¢ ( t ) + A ( v ) + ap(u(t))-)-Bu(O y ( 0 ) = y0

We have following result Theorem 1.1 Under the above hypotheses if YDE V such that W ( y 0 ) < + o o , t h e n

(1 ,3 ) has a unique solution

Y E WJ'~(0,T~H) N L2(0,T~ V) for every given u E~jI .--LZ(0,T;U). Moreover, (y0,u)---~ y is a Lipschitz mapping from H >(L2(0,T~U) to C(0 ,T~H)ALZ(0 ,T~H) and (1 ,4 ) y' ( t) = (Bu(t ) - - Ay( t ) ~ 3¢p(y(t))°a. e. t E (O ,T ) Where M ° denote the minimum element of tile set M with respect to the norm in H.

In this sequel we will prove the negative result for complete reachability and the approximate reachability under two kinds of hyprtheses for (1 ,1 ) or equivalently (1 ,3 ) motivated by the worksE4],E5]and[lO]. 2. THE REACHABILITY OF (1 .1 ) OR (1 .3 )

Assume that hypotheses of theorem 1.1 hold throughout this section. Let

K ( yo ) ( t ) = { y = y ( t , yo , u ( " ) ) : y is the solntion o f (1.3) correapomliu9 to u ,u rnus orer the space L2(O,T~U) }

be the reachable set of (1 ,3) with initial value Y0 at moment t E [ 0 , T ] . Definition 2. 1 (1 .3 ) is called approximately reachable if there exists t E ~0,T 3

such that K ( y o ) ( t ) = H . Generally,as well know K ( y 0 ) ( t ) ( t E E0,T3) is not a closed subset in H even for lin-

ear parabolic distributed parameter systems. Similar result is also true for (1 ~ 3). Theorem 2. 1 If the level set

(2 ,1) (Y E/-!. I:tl ~ + < A y , U > + q,(u) ~ } w

is compact in H,then y,--,-y in C(0,T~H) as u.--.-u in L2(0,T;U)whcre y, are solutions

of (1 .3) corresponding to u,. Moreovcr,y is the solution o f (1 .3 ) corresponding to u and

K ( y o ) = U u ( t , y o , u ) O<~t~T u E L z ( O , T , U )

is containcd in the sum aggregate of countable compact subsets in H. Consequently,K(y0) has dense complcmcntary set in H when dim H = o o .

w

Proof. Let u,---~u and y, bc corresponding solutions of (1 .3 ) , i . e. (2 ,2 ) ] y ' , ( t ) + Ay.( t) + ~ ( y . ( t ) ) - ) - B n . ( t ) , a . e. t E [O,T']

y,(O) = y~ Acting on both sides of (2 ,1 ) by y ' , and y, rcpectively and integrate over [-0,TJ we obtain that

2-'J'ol:,,',12d.~ + 2 - ' < A u , , u , > + ~o(,j.) (2,3)

~ < z - ' <

;: T (2 .4 ) lu.(o-:~0t~+ llu.(,*) II~d*~<c olB,~,(OI2'/.~ II follows from ( 2 , 1 ) , ( 2 , 3 ) and (2 ,4 ) that {y.} is a precompact subset in C ( 0 , T : H )

198

by the Arzela--Ascoli theorem. So,there exists a subsequence of {y,}, stiU denoted by {y,) ,such that (2. 55 y, (t)----y(t) in H uniformly for t E [ 0 , T ] .

Furthermore,by a well known method in I'2"] we can see that y ( t ) is a solution of(1. 3).

Let

K.(y0) = [3 y( t , yo ,u ) O <<. t <~ T

II. II <~, one can see that K.(y0)is precompact in H by the above result so that K(y0) is contained in

the sum aggregate of countable compact subsets in H because oo

K(y0) C Ug,(:/0). =--1

Moreover,if dim H = o o then K(y0)has dense complementary set in H by Baire's Category lemma.

In order to discuss the appro ximate teachability we assume that ( e , ) B ~ = L2(O,T~H) where BZ/:= {y : y ( t ) = B u ( t ) , u Q L 2 ( 0 , T ; U ) } (F2) D(A)CD(3q0 and 3ep satisfies the growth condition as follows (2 .7 ) Itzll . ~<c(l~o(~)l + Izl + 1) a . e . z Q D(cp),z Q 13~p(x).

THeorem 2. 2 I f (F1) and (F2) hold then (2 ,8 ) K(yo)( t ) -~ H for every 9~ven t E [0 ,T~.

To prove the theorem it is sufficient to prove that K(y0)(T)----H and we have to prove three lemmas first.

Let us introduce a linear system as follows.

(2 ,9 ) [ y' (~) + Ay(t ) = Bu(t) ( y ( O ) = ~.~o

Lemma 2.1 For any given zr Q H and n > 0,there exist u~ 6 W J'z(0 ,T: U) and y~ (t) sa t i s fy ing(2 ,9)andly l (T)- -zr l<Ccj l 1 (c j s a c o n s t a n t ) , y ~ Q C ( 0 , T : V ) f ' ] W j'2

( 0 , T : H ) , y l ( t ) 6 D ( A ) , t 6 [ 0 , T J . Proof: According to the results of theorem 1.1 and (F,) there exist u26LZ(O,T:U) and y2(t) satisfying(2,9) such that (2. 10) ty2(V) -- z-rl < ,i Since WI 'z (0 ,T;U) is dense in L2(0 ,T :U) , so there exists u IEWI '2 (0 ,T ; U ) such that

l'ol ,(t) - < ,f (2, 11 )

Consequently, we have

(2,125 ly , (D - - yz(t) l ~ JLI o lm( t ) - - u2(t) IZdt ~ Lrl z

by the results of therom i . I so that Iy,(T) -- z¢l ~-~ lyJ(T) -- y2(T) l + Iy2(T) -- zrl < (1 + LU2)rl

where L~-0 is the Uipschitz constant.

The other conclusions of the lemma can be deduced by the orem 4.3 inE23 Lemma 2. 2 There exists z 6 L~(0 ,T:H) such that z ( t ) 63¢p (y l ( t ) ) where yl

given by lemma 2. 1. is

199

Proof: According to the results of lcmma 2. 1 and (F2) one has y j ( t ) E D ( A )

CD(3q0 , t E [0 ,T~. Define a set valued mapping P( - ) :R---~2 v' as follows [ ~ ( v l ( 0 ) ) fo~ t ~ 0

(2 ,13) P(*) = 13cp(yt(*)) :for o < g ~ T t~o(yj ( T ) ) for $ ~ T

One knows that for every t E R , P ( t ) is a closed convex subset in W by the defintion of &v and it follows from the completeness of V ~ that P( t ) is complete.

On the other hand we can also prove that P is quasi--upper--sernicontinuous,L e . , (2 ,14) ~>ocl {P(D :t E N.(to) } = •P(N,(lo)) C_ P(to)

, > 0

where N,(to) is a e--neighbourhood of toER. In fact ,for w 6 N P ( N , (to))and e~ • 0 ,we *>0

have w E P(N,.(to)). So,there exists t . E N, ( to ) such that w E P( t . ) and tn-,-to,If to ~ E 0 , T ] then (2 ,14 ) holds obviously by (2 .13 ) . When t o E E 0 , T ] , W e have (2 ,15) yl(t,) "* Vj(to) in V because y j E C ( 0 , T : V ) . Hence w E&p(yt ( t0)=P( t0) [2"] . So ,P is measurable [7 ] and there exists a measurable sclection zl( • ) :R---*V J such that z l ( t ) E P ( t ) by the selection theorem [ 7 ] since V ~ is separable and R is locally compact. Let z ( t ) be the restriction of zl (t) on ['0 ,T ] and one can see that q)(Yl ( • ) ) is absolutely continuous with Yi E W~'2(0, T: H ) taken into account as given in lemma 2. 1. Consequently, z E L2(0 , T. H ) by (V2).

blow introduce a system as follows [y' (0 + Ay(t) + ~ ( y ( t ) + y,(t))-~-Bu(t) ,a. e. t E [O,T]

(2. 16) /y (0) = yo

where yl is given in lcmma 2. 1 Lemma 2. 3 There exist u~LZ(0,T:H) and y ( t ) satisfying (2. 16) and [y (T) [

cz*| (cz is a constant). Proof. For every given u E L z ( 0 , T ; U ) the existence of the corresponding solution y

( t ) of (2 ,16) is given by thorem 1.1 and lemma 2. 1 and y satisfies the results of theorem 1.1.

Since B"--~-----L2(0,T.U) by (Fl) and V C H C W ,B is dense in L 2 ( 0 , T ; W ) a n d there exists u E L2(0,T;U)such that

T

(2. 17) tl Ba - z( t ) 11 ~.dt ~ 0 0

by lemma 2. 2 where z( t ) is given in lemma 2.2. After subtracting z ( t ) on both sides of (2. 16) and taking the scalar product on both sides by y ( t ) which is the solution of 42. 16) corresponding to u ,we can deduce that

t,~(t) 1 ~ + 01t,)(Otl2at ~< - ~(t)11. I1.~(0t1~ by (1 .2 ) and monotonicity of &p. Consequently,we have

t,9 (T) I < c,~. Now we come to the proof of theorem 2 . 2 . According to lemma 2. l ~ l e m m a 2.3 . for

any given zTEH and q > 0 there exist u j E W t ' 2 ( 0 , T I U ) and u E L 2 ( 0 , T ~ U ) such that (2. 18) ly , (T) - - zrl <cz,I, [Y(T) I <c2,1 where y t ( t ) and y ( t ) are the solutions of ( 2 . 9 ) and (2. 16) corresponding to ul and

respectively. Taking u( t ) = u j ( t ) + u ( t ) we can see that the solution of ( 1 . 3 ) correspionding to u is

given by y ( t ) = y l ( t ) + y ( t ) and

200

I v (T) - - z,,I < [y,(7 ' ) - - z, l + I~(T) I < (c, + cDu so the proof is complete.

The approximate reachabil.ity of the systems with boundary of distribution ~ boundary control can be obtained by the similar method above via selection theorem.

(3. 1)

Where An (3.2)

Leg

3. DEGRADATION OF (F1) In practical problems there exist many systems which are approximately reachable do

not satisfy (Fz) in section 2E9]. So,in this section we attempt to discuss the approximate reachabiltity of ( 1 . 3 ) by introducing comdition (Fr~) which is weaker than(F1) ,but (F2) has to be supplied.

Throughtout this section we assume that q: is a lower--semicontinous convex functional from H to R.

Introduce smoothing system corresponding to( l . 3) as follows y' (t) + A,,y(t) + Vqa,(y( t ) ) = Bu(t) a. e t E (O,T) u ( O ) = vo

is the restriction of A in H,this is, Any = Ay V Y E D(A, ) = {y E V:Ay E H}

~,(~) = i , , f { I z - y l 2 / 2 ~ + ~ ( y ) , g E Hi ~ > 0 Lcmma 3. 1 E0 Assume that there exists constant c ~ 0 independent of ~ such that

( A y , V g X y ) ) ~ - - e ( I + l V ~ 0 , ( y ) l ) ( 1 - k [y[) V y E D(An) and y0E V. Then,for any given u E Lz(0 ,T; U) and e ~ 0 (3. 1) has unique solution y , E

W~'2(E0,T] ~H) NC( [ ' 0 ,T1 ~V) , y ' , G L2(0,T~D(Au))- Moreover,we have y,--~ y bt C(O,T;H) ~ Lz(O,T;V)

~, -~y ' in L2(0,T~ H) u ,

Auy~---~Ay &L2(0,Ti H)

as e---,-0 where y E W " 2 ( [ 0 , T ~ ; H ) [ ~ C ( [ - 0 , T - ] ; V ) is the solution o f ( 1 . 3 ) with the same control u.

One can see that Augenerates a semigroup S ( t ) on H since An is linear symmetric maximal monotone operator in H. So,the mild solution of(3. 1) can be expresed by

S" (3. 4) ~/,(t) = s(t)yo + os(t -- s)(Vrp,(y,(s)) + Bu(~))~

Let ¢'t2

(3. 5) fJ(tt ,tz)lff " ) = J,,,s(tz - s).p(s)da; bz(O,T ; t f ) --~ H. V t2 ]> t~,tj,tz E (O,T)

In order to discuss the approximate reachability of(3. 1) and ( 1 . 3 ) the following hypotheses are proposed (lZ' l) R(G(tl , t2)B) -~ n for auy t2 > tl ~ O (F r 2) (&p)O is ultif~mfy bonMed,tlmt is,

I (a,,)o:~l ~ M Y Y 6 H where (3q')°:H--.-H is the minimum element of 3q~.

~(&P)°(x) E &p(x) x E n (3. 6) / I (o~)~( , ) I = ~,ff{ lul ,,~ E 0 v ( , ) }

201

Theorem 3. 1 Under the hypotheses of lemma 3. 1 and (F' 1), (3. 1) is approximately reachable for every y , > 0 .

Proof. Noting thatVq), satisfies uniform Lipschitz condition['3,the conclusion follows from (F ' , ) by means of the results inF10-l. Theorem 3. 2 Under the hypotheses of lemma 3. 1, (F ' t) and ( F ' z ) , ( 1 . 3 ) is approximately reachable.

To prove theorem 3.2 , the following lemma is derived first. Lemma 3. 2 Under the hypotheses of theorem 3. 2 there exists a constant c independent of u such that

(3.71 [t/,(t) - - y(t)1 2 H- [~ tl y,(t) - - y(t) [[ Zdt < e8

Where y, and y is the solution of (3. I ) and (1 . 3 ) with the same control u respectively.

Proof: For any given e , k > 0 , i t follows from(3. 1)that ( 3 .8 ) ¢ , ( t ) - - y'~(t) + A,ye(t) -- A,y~(t) + V(p,(y,( t ) ) - - V(px(ya(t)) = 0

:t,(01 - - y~(0) = 0 ~. e. t • [ 0 , T ] where 31, and y) is the solution of(3. 1) corresponding to e.and 7~ respectively.

Multiplying (scalarly in H) (3. 8) b y ( y , ( t ) - - y x ( t ) ) o n e has ld

(3 .9 ) ~ l y , ( t ) - - U~(t)12 + ~, I1 ,Z,(t) - - ,Z,(t) II =

+ < V~o,(y,(O) - - V~o,(:/ ,(t)) ,y,(t) -- y~(t) ><.~ 0 as (1 .2 ) is taken into account.

NotinB that [I]

V%(y,) = e-'(y, - - (t + eo~)-':p E a~o((t + ea~)-'y) integrating (3. 9) on E0,T3 we have

(3. lo) ! I Y~(t) - ~J~(t) I~ + wJ"o I1:1"('~) - '~(~) 112'~ z

) )ds

by the monotonicity of 3q~. On the other hand we have [w3

(3. 11) I v % ( z ) 1 ~ I ( a p ) ' ( z / I v x E n It follows from (3. 101,(3 . 11) and (F'z) that

- ,,~(t) 12 + .[: II y.(o - , , ( t ) II ,dr ~< c(, -,t- x) ¥ ~,z (3. 12) I , ,(t) > 0

where c is a constant independent of u. Lcting X--,-0+in (3. 12) one has (3. 7) by lemma 3. I . Proof of theorem 3. 2: For any given z r E H and i t>0 it follows from (3. 7) that

(3.13) iv,(7') - - v ( 7 ' ) l ~ - ~ ¥ uE L2(0,T)U)

as 0 < ~ 0 = r~/4c. On the other hand,for such eo there exists ~ • L Z ( 0 , T ; U ) such that

(3. 14) I,),0(7') - ~,1 ~< )~/2 by theorem 3. 1 where y,0is solution of (3. 1) corresponding to u. Consequently, it follows from (3. 13) and (3. 14) that

202

ly(T) - - zrl < iy,,(T) - - ~ (T) I -t- lY,0(T) - - Zrl ~< ~/2 -1- '1/2 --- which completes the prooL

4. AN OPEN QUESTION If (1 .1 ) holds for z E K where K is a closed convex subset of H instead of z E V , w h a t

are the conditions for approximate reachable?In fact,in this case we are faced with a obstacle problem and

99(z) ~ , K ( z ) ~ { O z E K z E K

One of the difficulties is that we do not have D(A) C D(IK).

REFERENCES [1] . Barbu, V. Nonlinear Semigroups and Differential Equations in Banaeh Space,Noord- hoff Leiden, Netherland, 1976. ['2]. Barbu. V. Optimal Control of Variational inequalities,Pitman,Boston, 1984. [3 ] . Friedman, A. , Partial differential Equations of Parabolic Type, Prentice - - Hall, New York,1964. [4] . Henry. J. Etule de La Controlabilite de Ccrtains Equations Paraboliques Nonlineares. These. Paris, VI ,June, 1978. I-5]. Hou. S .H. Controllability and feed back systems, Nonlinear Analysis,Theory, Methods

Application ,Vol. 9 ,No. 12 ,pp1487-- 1493,1985. [ 6 ] . Lions. J , Optimal Control of Systems Governed by Partial Differential Equations, Springer-- Verlag, New York, 1971. [7] . Teo. K. L,and N. V. Abmed, Optimal Control of Distributed Parameter Systems,Nor~ --Holland, Amsterdam, 1981. [8] . Zhou. H. X,and Y. Zhao. A Survey of Controllability Theory of Nonlinear Systems, Control Theory and Applications, Guangzhou, China, No. 2, ppl - - 14,1988. ['9]. Hong Xing Zhou, Approximate Controllability For a Class of Semilinear Abstract E- quations,SIAM. J. Control and Optim. Vol. 21 ,No, 4 ,July1983. [10] . H. W. Sun and Y. Zhao, Some New Results of Controllability for Semilinear Systems, To Appear.

A N A L Y S I S OF T H E B O U N D A R Y S I N G U L A R I T Y OF A S I N G U L A R O P T I M A L C O N T R O L P R O B L E M "

Wei-Tao Zhang De-Xing Feng

Institute of Systems Science, Academia Sinica, Beijing, China.

A b s t r a c t : In tills paper, we consider a singular optimal control problem with cost function containing a small parameter e. Using the boundary layer theory developped by Lions in [1], we give some estimates of the singular optimal control u~ in Sobolev space. On the basis of the interior estimate obtained in [4], we analyze the boundary singularity of u,. According to the generalized Pohozaev

[s], we obtain the estimation of 11 ~{{L2(r). identity

K e y words : Singular optimal control, boundary singular, boundary Layer theory.

§1. P r o b l e m S t a t e m e n t

Let r /be an open bounded set in ]~"(n > 2) with the boundary r being differentiable n - 1 dimension manifold. Consider the control system described by the following elliptic equation { -AyC v) = f + v infl, (1.1)

y(v) = 0 o a r ,

with f E HI(~) , v E U = L2(ft). Take the cost function as follows

J'(') = fo Ivu(,) - z~l 2 d~ + , fo ~' a~, (1.2)

where Zd = ( Z l d , ' " , Z , , d ) , Zid E H~(f~), i = 1 , . . . , n , 0 < e << 1. The problem is to find u~ E U satisfying

J~(u,) = !n~ J,(v). C1.3)

*This research is supported by the Nationa| Natural Science Foundation of China and Partially by tile Institute of M~thematic8 (Open), Ac~demi.~ $inic~.

204

It is well known that there exists a unique optimal control u, which satisfies the variational inequality

J:Cu~)(v - u~) > 0 V~ e ~1, (1.4)

where J~(u~) is the Freehet devivative of J, at u~. By using the boundary conditions y(v)]r = 0, y(u,)]r -= 0 and Green's formala, it

follows from (1.4) that

-[o(ay(u,l-divZ~l(~(,,l-YC~,°)),~+~/o,,,(,,-u, le~>o v,,eU. (1.s)

Now define the adjoint state p= = p(u,) by

- /Xp(u , ) = - ( / X y ( u ¢ ) - divZd) in n, (1.6)

p(u,) = 0 o n t .

Then substituting (1.6) into (1.5) yields

p(u,) "t- eu, = 0. (1.7)

Taking v = u~ in (1.1) and substituting (1.1) into (1.6), then using (1.7), we obtain

-~Ap(u , ) + p(~,) = ~F in n ,

p (~ ) = 0 o n r , (1.s)

with F = f + divZd. In the remaining parts of this paper, we shall give some estimates involving u, in §2,

and in §3 we shall analyse the boundary singularity of u,. Finally in §4, we shall obtain the estimate of o~

§2. S o m e E s t i m a t e s o f u~

Denote a(u, v) = fn V u . V v dx, b(u, v) = f• uv dx, ( f , v) = f• f v dx, p(u,) = p,. Then (1.8) is equivalent to

~ ( p , , ~) + b(p,, ~) = ~(F, ~) W e H~(~) , (~.1)

which can be also written

e.a(p,,v - p,) + b(p,,v - P c ) = e(F,v - pe) Vv e H~)(a). (2.2)

b(~F,, - ~F) = ~(F , , - ~f) V, e L~(~). (2.3)

If F E H01 (fl), taking v ---- cF in (2.2) and v -- p~ in (2.3), then combining these two equalities, we have

HP, - er]{~tn) _< e{{f{lH,(n), (2.4)

205

lip, - ,Wll,=l,,) < - ' + ' / ~ I IF I I - . c ° ) , where and hereafter e always represents a constant independent of e.

We also have

b(p~ - eF, p~ - eF) = b(p,p,) - b(p,, eF) - b(eF, p, - eF).

Taking v = p, in (2.1), it follows that

b(p,,p,) <_ b(p,, ~F).

Substituting (2.7) into (2.6) yields

lip, - ,FII~(.) _< 'llFlln'(o).

T h e o r e m 2.1 I f F E Hl(f~) and F ¢ Hd( f l ) , then we have

l iP , - e'FliLa(n) < C"+I/411FIIH'(O}"

(2.s)

(2.6)

(2.7)

(2.8)

(2.9)

Proof. Set G, (F) = p, - EF, then G,(F) is a linear mapping. From (2.5), we know (7, e L(H~(12); L~(f~)). Moreover, from (2.8), it can be seen that G, e L(L2(t2); L2(fl)).

According to [1], for each F e Hl(f l ) , F can be represented in the form F = Fo+Fx with F0 E L2(i2) and F I E Ho~(~) which satisfy

IIFolIL'Co) <-- ¢.V'IIFII,~.(°),

ItF, I I . ' ( . ) -< ~ - " I I F I I . ' ( . ) , (2.10)

where )~ and a are constants with 0 < A < 1, a > 0. By using (2.10), it follows that

p, -- e f = G,(Fo) + G , ( F ) , (2.11)

Then using (2.5) (2.8) and (2.10), from (2.11), we obtain

liP, - EFHL,(n} --< ceA"ilFItul(,) + cel+l/ 'A-"HFIlu'(n). (2.12)

Thus taking • = e, a = 1/4 in (2.12), (2.9) is derived, which finishes the proof. T h e o r e m 2.2 f f F E H l ( f l ) \ Hlo(ll), then we have

I1,,, + FII,..(.) < e,V'IIFII,,I¢.I, (2.1a) It v (,.,, + F)IIL,(r0 < cC3/'IlFIIH,(.j. (2.14)

Proof. (2.13) can be directly obtaincd from (1.7) and (2.9). In order to prove (2.14), taking v = p, in (2.1), we have

(2.1s)

2 0 6

IIp, IIL~(O) - ,IIFII~*¢,~. (2.16) Substituting (2.9) and (2.16) into (2.15) yields

II v p, IIL~(,) -< ce~/~llFIlu'cn) • (2.17)

Then from II V (p, - '~)IIL=(.) < [t W P, II~=(.) + ellFIl,,'(.) and (2.17). w e obtain

l{ V (P, - eF)ll/;(n) < ces/allFllhr,(fl), (2.18)

from which (2.14) is immediately obtained. The proof is then complete.

§3. Ana lys i s o f t he B o u n d a r y S i n g u l a r i t y o f u,

Denote by d(z, F) the distance from x 6 [2 to F. Using Weingarten map (see chap.2 in [29, we can prove

Ad(x, F) = - tl, (x), (3.1)

where ILls (X) is the mean curvature of the parallel hypersurface F6 which passes through x, in other word, F6 = {xlx E fl, d(x, F) = 6}. Since r is regular, in some interior neighborhood of F, the following estimation holds

lD:d(x,F)i _< e, (3.2) ac, l +-..+¢~.

where D e = ~ .

Let A 6 [A0, co) with A0 > 0 small enough (0 < A0 << 1), fl0 be a fixed positive constant, fl be a positive number to be defined with ~ < rio. Denote by vp the outward unit normal to F at p, then we can define a positive function tip ) such that pl =

~ - - 1 {t p - t(p)Vp e r and p~ ¢ p. Set es = mi%er t ( p ) . Let q = (e 0 - 1 ) - ~ , then ~o l o g ( 1 + ~ - , o ) is an incroa~ing function of ~ e ( o , ~ ) . Let h = {~1~ ~° Iog( l+~-~° ) < ~}, (O, E2) = /'1 Cl (O, el), Ea = min(1,E2), and denote F~(c) = c~log(t + c-P), p~(c) =

{xlx 6 f l ,d(x,F) > g(e)} for some positive function g(e). For a _1 log 1/3 + e p ' f la(') =

e 6 (0, Co), from (3.2) and [4], we may construct an infinitely differentiable function ¢ with

¢(z) = 1, z e fi~nc,)+p~C,¿,

1 ~ ¢1 -< c~- ' (¢ + c").

(3.3)

where c is a constant greater than I.

T h e o r e m 3.1 Denote I(A) = fd(~.r)_>~'log{l+,-~)] V u,] 2dx, a(e) = 1/2 log, b (where e is the constant appearin9 in (3.8}, hence lim,-~o a(e) = 0). Suppose fl > ~, then

207

I -- a(e), we have

I(~) < etlF)l~,(.).

iO U } - - ( , ) < ~ < }, ~ hav~

I(A) _< e,]-2~llFllff,,Co ).

iii) I] )~ >_ }, we have

(3.4)

(3.5)

_ a 2 z(A) ~ ~, ,IIFII.,(~). (3.6)

Proof. Setting w, = p, - eF, from (2.1), it follows that

,aCp,, ~) + bC~,, v) = o. (3.r)

Using (3.3) and taking v = wee in (3.7), we obtain

Define

- c ]n w, V w, • V ¢ dz, 1'4

I,

I~ = - d [ VY. V(w,¢) dz. a l l

Using (3.3), we have

In order to further eatimate the right hand of (3.9), set

z~, = .x-~÷a fn I v w, ll,~,l dx.

We have

zl, _< .1-~(f. I v ~,1'~ a~)¢(fo Iw, l~¢ dx) ½. Noticing that

~-~xy <_ 6x 2 + ~--~c-ZXy 2,

where 5 is an arbitrary positive constant, then

I~, < e& fn l v w, t~¢dx + --TK-Jn Iw, l:¢dz.

Moreover,

(3.9)

(3.1o)

(3.11)

(3.12)

(3.13)

2O8

By substituting (2.9) and (2.18) into (3.13), it follows from (3.13) that

xl~ < c:+~-~÷~llFIl~,,(,). (3.14)

Taking cd~ --- ¼ in (3.12) and from (3.14), we obtain

°/o & <-~ Ivw, l'+c:+[-~+PllF.IIS,tu)+:~'-'~f I~,J'¢dx (3.15) J f l

Next we estimate 12, first Iz can be written

I, =-:fo v w,. vFe - : f w, v F.v,e . (31 )

Denote

I,,=-e'f~b~Tw,.Fdx, I22=-e'fw, vF.vCdx, then /2 = 121 + I~2, using (3.11), we obtain

Z,I < : : ½ ( & l V w,l~¢d:~)V~:/'(Zo I V Yl'¢d:~)½

< : ( : , 6 : , , i v w,t~¢ dx + ,5 : , I v Ft~¢ dx) (3.17)

< 6~fo 1'7 ,,,,1~¢ d:~ + ~11 v FII~,(,-,).

&, -< c: -~ f . Iv Fll~,I¢ d~ + .'-~+" f . 1V FII~.I d.. (3.18)

For further estimating I22, set

&2, = ~ : -~ '£ I v Fllw, l~dx,

we have

I~n < ce~-~e=--~(ffl I,~,lZCd:~)'e',-~(fr, t V FI2¢dx)½ 1.1+~ r V F12¢ dz) < .~-~(~:c~+1) f. i~,1~¢ d~ + ~ ~.

< .~-~x f, tw.t'¢ d~ + -~11 v FI}~,(.) (3.19)

From (2.9), it follows that

&~ < .a-~'+' /"+,~t lFI l~, ( . ) (3.20)

x in (3.17) and using (3.19) (3.20), we deduce Taking 6 = i

I, < ~ fn I V w'l '¢dz + e'3IIFI]~'(n} + c"-')' fn lw'l'dpdx-l-ce'-'+'/'+#ll'F]l~t'(n} (3.21)

209

Substituting (3.15) (3.21) into (3.8) and using e "~' < e'~'(m2 :> m,) , we have

fo jvw.l=¢a~ + (1-e.'-'~) fo Jw.l"¢d= <_-~llFIl~,,., + oe+V~+, IFIl~m°,, (3.22)

7 l + f l > 3, w e I f A o < A < l - a ( e ) , t h e n w e h a v e l - c e ~ - : a > O . I f 2 + ~ - A + f l > 2 + i - ~ _ then have p > ~s . . . . Therefore with A0 < A < 1 - a(e), and fl > ~, from (3.22) it follows that

f, lv ~,1'¢ d~ < ~'IIFII~,(.) (a.23) Since ¢(x) = 1 in 12Fa(d, using (1.7) and taking account of [V u,] 2 -< 2([ ~7 (u, +

F)t 2 + I V Fi ' ) , from (3.23), we doauco (3.4). Using (2.9) (2.18), we obtain

=~,-2~ f. lw.l~¢ d~ < ed+'/~-~llFlt~.,(.), (3.24)

~ fo i v ~,l~¢d= <_ ce2+¼11Fll~qn)- (3.25) Finally, using (3.24) (3.25), we deduce (3.5) (3.6). The proof is then complete.

§4. E s t i m a t c o f I I~ l lL ' (~)

T h e o r e m 4.1 Let 13 be a strictly stars~tapped domain. i) I f F e H'(n)\H~(12), then we have

Ou~ 1107~ llL,(r ) = o (~ - ' )

il) I f F E II~((1), then we have

11~ll (') L2(r) = o e-~

Proof. From (1.7) (1.8), we have

{ -eAu~ + u~ = -F in fl,

~ ---- 0 o n r .

According to (2.9), we obtain

lira Ilu. + FIIL~(n) = o. ~"-*0

We now consider the following equation

-ZXu = g(z, u)

~----- e on r,

in D,

(4.1)

(4.2)

(4.3)

(4.4)

(4.~)

210

where c is a constant (c = 0 or c # O) and g : ~2 x IP. ~ ~ is a continuous function, differentiable with respect to z.

Using the method similar to [3], we can prove the following generalized Pohozaev identity for (4.5),

g f t "

dx

- ~ ds (4.6) JO i=l ~Xi

-2 £

where G(x, u) = f~ g(x, t)dr, G(x, c) = f~ g(x, t)dr, u is the outward unit normal to r . Applying (4.4) and (4.6) to (4.3), we obtain

Since f] is strictly starshapped, i. e. (x. u) > 0 on F, from (4.7) we obtain (4.1) (4.2). The proof is then complete.

R e f e r e n c e s

[1] Lions, J. L., Perturbations Singuli~res dams les Probl~rnes auz limites el en GontrMe Optimal, Springer, 1973.

[2] Hicks, J. N., Note on differentlable geometry, D. Van Nostrand Company, Inc. Toronto, 1965.

[3] Pohozaev, S. I., Eigenfunction of the equalion Au+~f(u) = 0, Soviet Math. Doklady 6, 1965, 1408-1411 (translated from the l~ussian Dokl. Akd. Nauk USSR 165, 1965, 33-36).

[4] Zhang Weitao, Analysis of boundary layer singularity, J. Sys. Sci. & Math. Scis., 4(2), 1984, 81-96.

ANALYSIS OF THE PARABOLIC CONTROL SYSTEM

WITH A PULSE-WIDTH MODULATED SAMPLER*)

Hong Xing ZHOU

Department of Mathematics, Shandong University

Jinan, Shandong, 250100, P.R.C.

I. INTRODUCTION

In design of distributed parameter control systems one of im-

portant problems is to choose controller and actuator. As the dimen-

sion of an industrial controller in actual applications is finite it

restricts us to consider the distributed parameter system with a fi-

nite-dimensional output. In industrial process control systems on-off

actuators have been in engineer's good graces because of the cheep

prize and the highreliability. For example, time-proportional switch

actuator is applied usually in the temprature control system of a

large-power electric furnace and it is a typical pulse-width modula-

ted sampler° In this paper we will be concerned with the parabolic

control s~stem coupled with an finite-dimensional dynamical control-

ler and a pulse-width modulated sampler.

From the point of view of Engineering, a pulse-width modulated

sampler can be approximately seen as an equivalent pulse-amplitude

modulated sampler and the lumped parameter pulse-width modulated con-

trol system could be analysed by the classical theory of sampled-data

control systems (e.g. Z-transformation method etc.[3B). Because it is

impossible to make the sampling period very small in an electric-mag-

netic actuator some essential and important properties will be negle-

cted in the analysis by classical methods. Therefore, from the point

of view of control theory some rigorous theory to analyse pulse-width

sampled-data control systems is advanced, e.g. the direct analysis

method ~5,6,7], discontinuous control system theory [4] etc..

Here we will be concerned with a class of control systems go-

verned by an abstract parabolic differential equation

~(t)=Ay(t)+Bu(t)+f(t) (1.1)

z(t)=cy(t)

*) This work was supported by The State Natural Science Foundatio~

of China.

212

where the state y(t) takes values in a reflexive Banach space X: y(t)

~X, t~0, A is the infinitesimal generator of an analytically compact

semigroup S(t), t~O, on the state space X; u(t) is an q-dimensional

control: u(t)eR q and BgX[Rq,X} --- the space of all bounded linear

operators from R q into X; f(t) is a step disturbance of the system:

f(t)=f-1(t) with f~X. In (1.1) z(t) is the p-dimensional output of

the system and C is a given bounded linear operator from X into Rq°

In System (1.1) we assume that tee control signal u(t) is obta-

ined from an q-dimensional pulse-width modulated sampler with an in-

put signal v(t) which is the output of some dynamical controller

v(t)=Jv(t)+Kz(t) (1.2)

where J and K are q~q and q~p matrices respectively. In general, the

matrix J is fixed by dynamical characteristics of the controller and

the matrix K called to be feed-back matrix will be chosen and tuned

by the designer. The output u(t)=(u1(t),u2(t),...,Uq(t))' and the

input v(t)=(v1(t),v2(t),...,Vq(t))' of the pulse-width sampler satis-

fy the following d~namic relation:

ui(t)={~ig n ~n~

fvi(nT)

~m:~sign vi(nT )

nT4t~(n+{~a~I)T

(n+[~nd)T~t<(n+1)T

Ivi(nT)l~ I

[vi(n~)l~1 (1.3)

i=1,2 .... ,q, n~I + ~{0,I,2,... } •

where T~O is the sampling period o£ the pulse-width sampler. We deno-

te the relation described by (1.3) as u(t)=F(v)(t). System (1.1)-

(1.3) is called to be a parabolic pulse-width sampling control system

and briefly to be written as Parabolic PM System.

The purpose of this paper is to generalize the theory for the

lumped parameter control system with a pulse-width modulated sampler

to the parabolic control system. Section 2 gives a strict definition

o£ the steady-state of the system and proves the existence of the

steady-state of the system. Section 3 is devoted to the analysis of

the steady-state stability for a parabolic pulse-width sampled-data

control system. An example to illustrate the theory is presented at

last.

2. STEADY-STATE ANALYSIS

DEFINITION 2.1. In parabolic PM System the q-dimensional vector

), defined by (1.3) is called the duration ratio ~n=(~nl, n2, ...~nq

213

of the pulse-width sampler in the n-th sampling period, nel +.

If we defined a closed cube Q in R q as

~={a=(al,a2,...,aq)'~Rq: [ail~1, i=1,2,...,q} (2.1)

then we have ~n£Q for all n~I +.

DEFINITION 2.2. In System (1.1)-(1.3) if there exists an q-dim-

ensional vector ~=(~I,~2,...,~n)'~ and a corresponding periodlcly

rectangular-wave control signal u(t)=u(t;~) defined by

~sign ~i nT~t<(n+l~d)T ui(t)=ui(t ;

~i)=~0; (n+l~l)T~t< (n+1)T (2.2)

i=1,2,...,q, n~I +

such that the closed system (1.1)-(1.3) has a corresponding periodic

trajectory y(-)=y(.;~): y(t+T;a)=y(t;~), t~O, then the control (2.2)

is called to be a steady-state control (with respecZ to the distur-

bance f'1(t)), the periodic projectory y(.) is called to be a steady-

state corresponding to the steady-state control u(.) and the constant

vector ~Q defining the steady-state control (2.2) is called to be a

steady-state duration ratio.

LEMMA 2.3. Let the control signal u(t) in the dynamical system

(1.1) be a rectangular-wave signal u(t;~) with a period T defined by

(2.2) for a given ~e~ . Then there exists a unique solution y(t;~)

with the period T of (1.1) corresponding to the given control u(t;~)

under the assumption

jQne@(A), ~n=2n~/T, n=0,±I,±2,... , (2.3)

where ((A) is the resolvant set o£ the operator A.

COLLORARY 2.4. Consider the following open-loop system with an

input u(t;~) defined by (2.2) for a given ~e~ =

y(t;~)=Ay(t;~)+Bu(t;~)+f'1(t)

z(t;~)=Cy(t;~) (2.4)

v(t; ~):Jv(t ; ~)+Kz(t ;~)

where the operator A satisfies Assumption (2.3) and the matrix J

satisfies

j~n~(J), n=0,i1,±2, .... (2.5)

Then System (2.4) has a unique output v(t;~) with the period T corre-

sponding to the given vector ~e~ or the given periodic control

u(t;~) defined by (2.2).

214

LEMMA 2.5. Under Assumptions (2.3) and (2.5) consider a nonli-

near mapping <~ from ~cR q into R q defined by

~(~)=(i_eJT)-1TeJ(T-t)Kay(t;~)dt, ~e~ , (2.6) O

where y(t;~) is the periodic solution of (1.1) corresponding to a

given open-loop control u(t;~) defined by (2.2) for a given ~Q .

Then there exists some constant Mo>O such that

~(~)- ~(~)I( ~< MoaK~-~II for any ~.~ . (2.7)

PROOF. Define two qxq matrices

P(t, =)=KCS(t)R [1 ,S(T)] S(T-~)B, P(t)=KCS (t)B. (2.8)

Let ~ and & be given in ~ arbitrarily. Then

KC (y(t ; ~)-y (t ; ~) )=KCS ( t ) (yo-Yo) t

+KC f S( t -=)B[u(w; ~)-u(w;~x)] dz o

t = ] P ( t -~ ) [u (~ ; ~) -u( : ;~)~ d~+ / P( t -~ ) [uC~;~) -u (~ ;;)J d~ .

O O

Since S(t) is a Co-semigrou p there exist constants Ma>0 and k such

that

~< Ma e-At, t)O. (2.9) l is( t) l l By the boundedness of the operators B and C there exists some cons-

tant MI>O such that

~P(t,~)JJ~ MIIIKH ~'A(t-~), UP(t)II % M1~KJJe-~t, t~.~0.

Therefore the k-th component of the vector-valued function KC(y(t;~)-

y(t;~)) has the followin 6 estimation:

I [KC(y ( t ; °~ ) -Y ( t ;~ ) )Jk l4

2M n o' t (2. o ~.-i

When ~>0, without loss of any generarity, let 0<~¢<~ t, we

have

~T eX- ~ elXl o ~IT

%~<0, for example, ~l<O<~ l and ~l ~ >~I ' we have

I~IT ~e xz lUl(Z ;~l)-Ul(~;~l)l d~2 I eX~ d~-< 2eI%ITT["I-~I l o 0

Thus, from (2.10) it is known that there exists some constant M2>O

dependent of T and ~ such that

When

215

UKC(y(t;~)-y(t;~))~ 4M2~Ke-~t~ l~-&~ (2.11) Suppose that

lJeJt ~ ~ Mje-Qt , t~O (2.12)

where Mj~O and ~ are constants. By substituting (2.11) and (2.12)

into (2.6) the conclusion (2.7) follows immediatly if we take

Mo=M2Mj~(I-eJT)-I~ Te ~T, p=max(Ixl,I~I) . (2.13)

From Lemma 2.5 it is easy to prove the following theorem.

THEOREM 2.6. Suppse that the operator A and the matrix J sati-

sfy Assumptions (2.3) and (2.5) respectively and the matrix K and the

constant M o defined by (2.13) satisfy

Mo~KH <I (2.14)

Then Parabolic PM System (1.1)-(1.5) has a steady-state for any given

feX.

5. S~EADY-STATE STABILITY AND EXAMPLE

DEFINITION 3.1. Parabolic PM System (1.1)-(1.3) is called to be

steady-state stable with respect to the disturbance f(t)=f-1(t) if

there exists some ~ such that

lim ~ = ~ (5.1) n

n ~

In the sequel we need some assumptions and notations. At first

we assume that the constants x and ~ in (2.9) and (2.12) are positive

i.e.

~S(t)~Mae-&t , ~eJtl4Mje -&t , &tO, Q,O. (3.2)

This assumption is not a severe restriction since in general i~dus-

trial processes like heating, diffusion and etc and controllers are

initially stable.

Denote two qxq matrices

teJ(t - t G(t,~)= ~ r)KCS(=-?)Bdc , G(t)=G(t,O)= ~eJ(t-r)KCS(=)Bd= o

ASSUMPTION (G). For the matrix G(t) and the step constant dis-

turbance f¢X we assme that (I) there exists an inverse matrix

If- Ta(t)dt] -I and o

[I-~O(t)dt]-lj-IKcA-if~Int a ; (3.5)

(2) there exists some constant g~O such that

~j-IKcA-lf+ ~G(t)dt ~-v II ~ ~, ~ v~Rq-~ (3.4)

216

where ~=ProjQ(v) is the projection of v on the closed convex set 0 .

REMARK 5.2. Obviously, for any given f~X there exists some con-

stunt M K such that both of (3.3) and (3.4) are satisfied as ~KI~M K .

LF~MA 5.5, Let G(t) be an q~q matrix function continuously on

[0,~) and satisfy aG(t)~WMe -×t (t~O) where M>O and A>0. Then for any

given £>0 there exists some T*>O such that

TG(t)dt- ~TG(1T-~T)~<g , O<T<T*,

holds uniformly for ~=(~i,@2 .... , %)', O~@j~1, j=1,2,...q.

THEOREM 3.4. Parabolic PM System (1.1)-(1.3) is steady-state

stable under Assumption (3.2) and Assumption (G).

PROOF. The output v(t) of the dynamical controller (1.2) is

v(t)=eJtvo+ teJ(t-C)KCy(c)dc= Ve(t)+Vc(t) (3.5) O

where

Ve(t )=eJtvo+ Jo eJ( t-W)KC [S(r) (Yo+A-If)-A -I f ] d c t

Vc(t)= ;eJ(t-c)KCd~ / S(c-?)Bu(~)d?= ~ G(t,?)u(?)d~ . o o

It is easy to see tIlat under Assumption (3.2) one has

lim Ve(t)=j-IKcA-If . (3.6)

Since n.T n - f ( t + OT n - !

Vc(nT)= I G(nT,?)u(q)dT=X J G(nT,7)u(?)dT: Z TG(nT,IT+OlT)" 1 ,

from (3.5) we have

O= 7----* [Ve (nT) +Vc (nT)-v (nT)] n=O (3.7)

= z~ [Ve(nT)+ ~ TG(kT-OnT)~n-V(nT)] n=O k= I

where ~n=(On1,~n2 .... ,~nq)' , O~nj.~I~nj[~<1 , j=1,2, .... q, neI +. By

I.emma 3.3 there exists some qxq matrix E(T,n) such that

DO Oo

Z. TG(kT-@nT)= ]oG(t)dt+E(T,n) , (5.8) k=1

lira [[E(T,n)[[=O uniformly for n~I +. (3.9) T÷O

For any given T>O from (3.7) we have

O= lira [veCnT)+ ~. TGCkT-OnT)~n-vCnT) 1 n*m k=1

217

=j-IKcA-If+ lira ([ ~G(t)dt+E(T,n~ ~n-v(nT)} (3.10) n ~

Let ~>0 be the positive number in Assumption (O). It is not

difficult to prove that t~ere exists some T*>O and positive integral

N*=N*(T*,S) such that

v(nT)=~ n , O<T<T* , n~N* ° (3.11)

Substituting (3.11) into (3.10) we have

lim [I- [G(t)dt-E(T,n)]~n= j-IKcA-If , O<T<T* . n ~ o

Obviously, by (3.9) one has

lira [~- J G(t)dt~= J-IKcA-If

and Oo

lim c~ =~=[l- # G(t)dt]-Ij-IKoA-1£ . (3.12)

REI,~K 3.5. The above theorem shows that Panabolic PM System

(1.1)-(1.3) has the steady-state stability by tuning the feed-bank

matrix K.

EXAMPLE 3.6. We now consider a heat conduction temprature con-

trol system with an one-dimensional control and an one-dimensional

output

~y(t,x)= ~2y(t,x) +b(x)u(t)+f(x~1(t), t~O, O,x~L ~t ~x z

y(t,O)=y(t,L)=O, t~O (3.13)

L

z(t)= # o(x)y(t,x)dx t~o O

where b(x) and c(x) are in the state space X=L2(O,L). The dynamical

controller is designed as a proportional controller with an inertia

~a~(t)=v(t)+kz(t) (3.14)

where k~O is the gain of the controller.

By the semi~ro~p t~eory [2] (3.13) can be written as the abs-

tracl form (1.1) in the state space X=L2(O,L) as we define

D(A)= (y~L2(O,L): y,y"aL2(0,L), y(0)=y(L)=O)

(Ay)(x)=y"(x), y~D(A) .

The semigroup S(t) generated by A has the form

[s(t)y] (x)= ~ e-~t(y. ~) ~n(~) (3.15) n=1

218

where

A~(n~ /L ) 2 , ~n (X )=~7~s in nxx/L

L (y,~.)= ly(~)¢nC~)d~

O

Obviously

~s(t)~ ~ e- From (3.15) one has

X,t

n=1,2,...

, t~O ( 3 . 1 6 )

(~, %! (~[1,s(~)]~)(~)= Z iL,e_~.~,,,, %(x)

n= 1

The functions P(t,~) and P(t) defined in Lemma 2.5 are

k ~ -Xn(t+T- ~)

P(t'r)=~-~ n~=1 1_e-X.T

k ~ -X t P(t)=~- ~" e n (b,~n)(C ' ~n ) .

n=1

The constant M 2 in (2.11) can be obtained as M2=4T](b,c~e xIT. Denote

p = rain (%1' I /~a) (5.17)

then the constant M o in (2.13) is e~zosen as Mo=4T 2 {(b,c) le ~T. The

sufficient condition that there exists a unique steady-state with 2 respect to any step disturbance f(x)1(t), f~L (O,L), is

4k~-~ ~2 I(h,c)[ ~ < I (3.1~)

For the condition of the steady-state stability of PM System

(3.13),(3.14),(1.3), at first, we give the explici~ expression of

G(t) for the concrete case:

k t -(t-~)/ra

Therefore

-Anb -t/r a g e e ( b , % ) ( o , % )

ra n=1 !-x r~ n

IQ(t)dt= k Z e n=1

(b, %)(0, %) .

219

Since

j-IKcA-If=k~ I (£,$n)(C,~n) n=1 -~n

the condition (3.3) becomes

I 11_k ~ I l k~ lTn (~' ~n) (c' *n)l ~ = n=~ --~n(b, ~) (c, ~n) I (3.19)

and the condition (3.4) becomes

(e, *n) n=1 An

where [vl>1. If we assume

k~n=1~1 [l(f'#~)(C'¢n)l+l(b'÷n)(C't)l}~1 (3.21)

then both of (3.20) and (3.19) hold and PW System (3.13),(3.14),(1.3)

is steady-state stable by Theorem 3.4.

REFERENCES

I. R.F.Curtain & A.J.Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, 1978.

2. A.Pazy, Semigroups of Linear Operators and Applications to PDEs, Springer-Verlag, 1983.

3. B.C.Kuo, Degital Control Systems, Holt, Rinehart & Winston Inc., New York, 1980.

4. V.I.Utkin, Discontinuous control systems: State of the art in theory and applications, Preprints of 10th World Congress on Automa- tic Control of IFAC, Vol. l,pP.75-94, 1987.

5. Hong Xing ZHOU, Steady-state analysis for control systems with a pulse-width sampler, Acta Automatica Sinica, 13(1987), No.l, pp.66 -69.

6. ---, Steady-state stability for pulse-width modulated sampled-data control systems, Control Theory & Appl., 4(1987), No.3, pp.57-65.

7~. ---, Analysis of pulse-width modulated sampled-data control systems, J. System Science and Math. Science, 8(1988), No.l, pp.11- 18.

Control Theory of Distributed Parameter Systems and Applications: Proceedings of the IFIP WG 7.2 Working Conference Shanghai, China, May 6â€“9, 1990

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