Hyperbolic equations with Dirichlet boundary feedback via

Appl. Math. Optim. 8, 1-37 (1981) Applied Mathematics and Optimization

Hyperbolic Equations with Dirichlet Boundary Feedback Via Position Vector: Regularity and Almost Periodic Stabilization--Part I*

I. Lasieckd and R. Triggiani 2.*

I Department of Systems Science, University of California, Los Angeles, CA 90024, and Mathematics Department, University of Florida, Gainesville, FL 32611

2Mathematics Department, Iowa State University, Ames, IA 50011, and Mathematics Department, University of Florida, Gainesville, FL 32611

Communicated by A. V. Balakrishnan

Abstract. A hyperbolic equation defined on a bounded domain is considered, with input acting in the Dirichlet boundary condi t ion and expressed as a specified feedback of the position vector only. Two main results are established. First, we prove a well-posedness and regularity result of the feedback solutions. Second, we specialize our equation to the case when the original differential operator with zero boundary condit ions is self-adjoint and unstable. Here, under certain natural algebraic condit ions based on the finitely many unstable eigenvalues, we establish the existence of boundary vectors, for which the corresponding feedback solutions have the same desirable structural property of a stable free system: They can be expressed as an infinite linear combinat ion of sines and cosines (special case of almost periodicity). A cosine operator approach is employed.

Glossary of Symbols

A = generator of a cosine operator A F = feedback generator of the feedback cosine operator

*This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-78-3350 (I.L) and Grant AFOSR-77-3338 (R.T.) through ISU.

**This research was performed while the author was visiting the Department of Systems Science, University of California, Los Angeles.

0095-4616/81/0008-0001 $07.40 01981 Springer-Verlag New York Inc.

2 I. L a s i e c k a a n d R. T r igg ian i

±iar: #r: ~(S):

C(t): CF(t )"

Ci:

D :

~(s): @(A):

bit) d r , d ' rJ

E(-,-,.): eF, n

~(t )} hr

I L C =

M~:

.(t) l p = q= N = 0(.): ~km:

S ~

s ( t ) =

simple zeros of denominator 1 - ~ ( s ) of b(s) (cf. (4.22)) cf. (4.14)

~_" 09 entire function with zeros only at { - t/3k}k= L of multiplicity one; cf. (4.25) (strongly continuous) cosine operator generated by A (strongly continuous) feedback cosine operator generation by AF; cf. Thm. 1.1' positive constant such that - c ~ i= 1 .... dim X u are the simple eigenvalues of A,; cf. Lemma 3.2. Dirichlet map cf. (2.1) cf. (4.27) domain of operator A

function and coefficients defined by (4.9) and (4.43)

el. (4.26) normalized eigenvectors of feedback generator AF: cf. Cor. 1.5

function and coefficients defined by (4.17)-(4.18)

almost periodic of the special class ILC: Definition l. 1. Banach space of summable sequences geometric multiplicity of eigenvalue --?~k of A.

function and coefficients defined by (4.13)- (4.15)

A uPDg: cf. (4.2) (--As)I/4-°QDg: cf. (4.1) ( - ~ , ~ ) < const I" I normalized eigenvectors of A : k = 1,2 . . . . ; m = 1 .... M k corresponding to eigenvalue - ~ k normalized eigenvectors of A u on X~, corresponding to simple eigenvalue - c 2, forming a basis on Xu. complex variable integral of cosine operator; cf. (2.4)

subindex s stands for "stable": it refers to the subspace X s subindex u stands for "unstable": it refers to the subspace X u

(,)-- (,)= II = [ ' ]k in z

(.)i =

eigenvalue of A dimension of space domain inner product on L2(f~ ) inner product on L2(F ) norm in L2(~ ) (or else absolute value of a number). ~-, qbkm ) =coordinate with respect to ~km; cf. Remark 3.5, p. 29 ( ' , +i) =coordinate with respect to q'i, cf. Remark 3.5, p. 29 cf. (4.23) Laplace transform

Hyperbolic Equations with Dirichlet Boundary Feedback: I 3

1. Introduction and Statement of Main Results

Let ~ be a bounded open domain in R ~ with boundary F, assumed to be an ( v - 1 ) dimensional variety with fl locally on one side of F. Here, F may have finitely many conical points [14]. Let A(~, ~) be a uniformly strongly elliptic operator of order two in ~ of the form

A ( ~ , 0 ) = ~] a,~(~)O '~ (1.0) lal<2

with smooth real coefficients a~, where the symbol 0, rather than the traditional D, denotes differentiation. In the present paper, the symbol D will denote the Dirichlet map, as defined below. We consider a vibrating system based on a with Dirichlet input applied on F, i.e.,

x,( t , ~) = - a ( £ , 8)x(t, ~) in (0, T] × a

x(O, ~) = Xo(~); x,(O, ~) = x,(~) , ~ ~ a

x (t, ~" ) = f ( t , f ) in (o, T] × F. (Dirichlet B.C.)

(1.1)

(1.2)

(1.3)

Here, f(t, ~) is the input function or control function, or forcing term, defined on (0, T] × F which influences the solution x(t, ~). It is assumed throughout that the operator A consisting of -A(~, 0) with Dirichlet zero boundary conditions generates a strongly continuous cosine operator o n L 2 ( ~ ) (cf. (2.3) and (2.4) for the precise definition). This assumption is necessary and sufficient for the zero input problem ( f = 0 ) to be uniformly well posed in Lz(f~ ) [7]. For background on cosine operator theory, see [26] and [7].

The Feedback System

The distinctive new feature of the present paper is that we demand the input function f ( t , f ) to be expressed in a feedback form, as a linear operator (of finite-dimensional range) acting only on the position vector x(t, (). More precisely, we choose the feedback operator F to be a continuous operator from L 2 ( ~ ) to a J-dimensional subspace of Lz(F ) of the form:

J def . f ( t , f ) = Y~ ( x ( t , ' ) , w j ( . ) ) g j ( f ) = Fx( t , . ) on (0, T] × F. (1.4)

j = l

Here, N(.) and gj(.) are fixed vectors in L2(~ ) and L2(F ), respectively, and the symbol ( , ) denotes the inner product in Lz(f~ ). For J = 1, we write w and g instead of w I and gl- The vectors gj are assumed linearly independent.

4 I. Lasiecka and R. Triggiani

boundary input [ hyperbolic solution x(t,.) t f(t,~) equation t

I <x(t ,) ,w>g I

Fig. 1. The feedback system.

In the present paper, we study two problems: (i) First we establish (in section 2), that the system (1.1)-(1.3), in the

feedback form demanded by (1.4) ("closed loop system") is well posed in appropriate function-spaces. The precise statement concerning existence, uniqueness, and regularity of solution is contained in Theorems 1.1 and 1.1' below.

(ii) Second, we investigate (in section 3) what in system theory terminology can be called a boundary feedback stabilizability (or stabilization) problem. A qualitative statement thereof may be given as follows: Assume preliminarily, that the (free) open loop system with f ( t , ~)----0 is not stable, in the sense that some (free) solutions x(t, x o, x l ) blow up in time (in fact, exponentially) because of the unstable eigenvalues (3.4) below (Example: - A ( ~ , O)=A+c 2 for suitably large c2). Then, find, if possible, (a) appropriate vectors gj EL2(F)--as well as their minimum number--and (b) the least conditions on the vectors wjEL2(f~ ), j = 1,..., J, as to guarantee that all corresponding feedback solutions x(t, x o, x~), due to initial data in naturally "large" subspaces, are abstractly valued almost periodic functions, and hence uniformly bounded in t, in the strongest possible norm. []

Should such boundary vectors exist, we can then say that the original unstable free open loop system, once operating as a boundary feedback closed loop system as dictated by (1.4), with such gj EL2(F ), has (qualitatively) the same oscillatory behavior as that of a free stable open loop system. In other words, the boundary feedback is used to restore the natural oscillatory character of hyperbolic equations.

Actually, even a stronger, more refined result will be obtained in Theorem 1.2 below and its generalizations, under the mild assumption (1.7) on the subspace of Z2(~), from which the vectors wj may be drawn. To state it, we need the following:

Definition 1.1. A (scalar or vector-valued) function z( t ) of the form

o o

z ( t ) = ~ aksinakt+bkCOSagt k = l

t E R

with a k reals and with the numerical sequences (lagl)~=l and (Ib~l)g~__l in e 1 (I I


being either the absolute value or the space norm):

oo

lakl + Ibk[ < oo k = l

will be called almost periodic of the special class ILC (infinite linear combination of sines and cosines). []

The following results concerning well-posedness, regularity, and almost periodic stabilizability of the system under study constitute the main contribution of the present paper.

Theorem 1.1. (Well-posedness and regularity.) Let the vectors w i satisfy

w j E ® ( ( c I - A ) ¼ + P ) C H ' + 2 P ( a ) j = l . . . . . J . (1.5)

Here, p is any positive number and c is a constant strictly greater than 12tl 1, so that the fractional power is well defined. (The inclusion c in (1.5) is always true).

(a) Under the same assumptions as in subsequent Theorem 2.1 for F, for each

x o E L 2 ( a ) and x, E H - l ( a )

the Dirichlet feedback system (1.1), (1.2), (1.3), with input f ( t ,~) given by (1.4), admits a unique solution x( t, x o, xt)EL2(O, T; L2(f~)).

(b) Such solution belongs, in particular, to

C([O,T];H½-2p(f~)) = C( [O ,T ] ; @( ( c I - A )¼ -P) ] (see(2.6))

provided that x o and x I are, respectively,

i P i ,

x o E H ~ - ( ~ ) = @ ( ( c I - A ) ~-°) and x ,E[@(cI - -A)~+°] ' (1.6)

([ ] '= dual of[ ] (with graph norm) with respect to the L2(f~)--inner product). [] An altogether different proof in spirit and technicalities will provide more

information for F allowed to have corner points [14, 15].

Theorem 1.1'. (Feedback cosine operator). With the vectors wj satisfying assumption (1.5), the feedback closed loop solutions x( t, x 0, x 1 ) of (1.1)-(1.3), (1.4) can be expressed simply as the following miM form

x( t , Xo,X,) = CF(t)X o + SF(t)x , , Xo,X, E L2(~ ) t E R


where C F defines a (feedback) strongly continuous cosine operator on L 2 ( ~ ) , and S F is the corresponding "sine" operator (cf. (2.4)). Actually, the feedback cosine operator extends~restricts as such to each ( f ixed) interpolation space, defined by Equation (2.37) below, between [ @ ( ( c I - A)3/4+°)] ' and @((cI - - A ) l / 4 - ° ) . []

The proof of the above regularity results are given in section 2. As to the almost periodic stabilizability problem, we special ize--as is moti-

vated in Remark 3 . 1 - - b y restricting ourselves to the case in which the operator A is self-adjoint, with (point) spectrum-satisfying condition (3.4), which asserts, in particular, the existence of ( K - 1) unstable eigenvalues. These eigenvalues are the preliminary condition for the stabilizability problem posed in this paper to be significant.

Remark 1.1. It is obvious a priori that a general solution of the stabilization problem can only be hoped for, when dimfl=p>~2; i.e., when a full function space is available on F for the sought-after vectors g/, that are required to "counteract" the possibly large number of unstable eigenvalues. The one- dimensional case p = 1, when only two scalar components gl and g2 are available at the degenerate boundary, is in general hopeless if the dimension of the unstable eigenspace (X, below) is larger than three. All this will be confirmed by our quantitative analysis in Appendix 3B.

The simplest solution of the almost periodic stabilization problem is for- mulated as Theorem 3.1 in section 3, and takes place when maximum freedom is assumed on selecting the vectors wj ~L2(f~ ). The most convenient choice is that they lie in a natural finite-dimensional "unstable" eigenspace (X, below): this corresponds to the situation in which the system's projections (3.12) and (3.13) are decoupled. Here, instead, we state our most general result, which corresponds to minimal assumptions on the vectors wj (in particular, it is postulated that wj ~X, ) , so that the projections (3.12) and (3.13) are coupled. In our statement, we first single out the technically simpler situation, when all the eigenvalues have geometric multiplicity equal to one: M k ~ 1.

Theorem 1.2 I. (Weak almost periodic stabilizabifity of class ILC when M k --1). Let dimf~ = u~>2 and let f~ either have 2 C~_boundary F, or else be a parallelepiped. Let the operator A be self-adjoint and let its (necessarily point) spectrum

IRecently this result and subsequent corollaries have been suitably extended by the authors to cover the full trace feedback case; i.e.

J a~,. ~ (x(t.)[r,w/)F~:j(~.) j = I

(instead of (1.3)-(1.4) of the present paper), where (,)r is the L2(I')-inner product, whereby now wjC L2(F), while xlr is the Dirichlet trace. This case is more challenging than the feedback term (1.3)-(1.4) discussed in this paper, and requires refining some estimates of section 4. For a parabolic counterpart of the structural assignment problem with full trace feedback, see "Structural assignment for boundary feedback parabolic equations: the case of Dirichlet trace in the feedback loop" by I. Lasiecka and R. Triggiani.

2This assumption on F is needed only to invoke Cor.2.2 in [24] to guarantee (3B.11) in Appendix 3B.


satisfy (3.4). Let the given vectors wj satisfy the full rank condition (3.10) and condition (3.11) at the ( K - 1) unstable eigenvalues { - ~ k }, 1 <<- k <~ K - 1; moreover, let

0 :~ (wj, (I)k)~ const k = 1,2 . . . . j = 1 . . . . . J (1.7) 3

v 4 + 0 )

where ~k is the (normalized) eigenvector corresponding to ~k so that the vectors ~s - - ( - As)¼ +Pwjs defined in the subsequent Equation (4.7) satisfy

c o n s t 0 : ~ ( ~ s , d ~ k ) ~ < K ~ k = 1,2 . . . . i = 1 . . . . , J (1.7')

(as one sees via (4.23)). Let the initial points x o and x 1 satisfy (1.6). Then there exist boundary vectors gj E L2(F), whose minimal number is dis-

cussed in Remark 3.4, such that the corresponding feedback solutions x( t, Xo, x 1) of (1.1)-(1.4), have the following property: the scalar function

I

( ( c I - - A)~-Px( t , x0, x , ) , y )

is almost periodic of the special class ILC for any y E L2(~); equivalently stated, the feedback solution x( t, x o, x l ) is weakly almost periodic of the special class IL C in the graph topology of ~(( cI - A)¼-P ) equivalent to H~ 20(~'~) (see (2.6)). []

Actually, the proof will show the following more precise result, which displays a more subtle structure of the feedback solutions.

Corollary 1.3. Under the same assumptions of Theorem 1.2, the suitable feedback solutions, which are claimed in the conclusion there, can be written as:

K--1 ( (CI-- A) '/4 °x( t , Xo, x , ) , y ) : ~,, bicoscit + b;sincit

i=1

+ ~ XrCOSart + x~sina~t r = l

(1.8)

Here the { - c2}K=I 1 are negative constants, which can be preassigned in any chosen interval (in particular, e.g., as in (3.18)) that replace the unstable eigenvalues

K--1 . {--2~k}k= l, the {a~} are a suitable sequence (see 4.39)(ii)) of positive constants having the same asymptotic behavior as the {~r): a~ - 2~ r ~ 0 as r ~ oo. Moreover, the coefficients (Xr} and {X'~) are all in l 1 and, along with the coefficients {bi}, {b~}, are exhibited in the proof as dependents on y, the initial data Xo, x 1 as in (1.6) and on the system parameters, including the sought-after vectors gj E L2(I" ) (see Equa- tions (4.64) and (4.68) which depend on the sequences {dr} and (d ; ) ; these

3Therefore wjE @((cl -- A)~ +o) consistently with (1.5).


sequences are then related to the sequences (ni}, (n'i} by (4.47), (4.49); these, in turn, are related to the initial points and the system's parameters via (4.15)-(4.16). 1

[]

Corollary 1.4. Under the very same assumptions of Theorem 1.2, the feedback solutions mentioned there are almost periodic in the (strong) graph norm topology of

6~( (c1_ A) ¼-p-~) equivalent to H'2-2°-~(f~)

for any e > 0 (see (2.6)). The lengthy and technical proof of Theorem 1.2 will take up all of section 4,

plus relative appendices.

Remark 1.2. In the approach given in section 4--which treats explicitly the case where (according to Remark 3.4) one can take J = 1 and thus only one suitable boundary vector g is needed-- i t is shown that: any vector gC L2(F), (i) which satisfies the finite moment problem (3B.7) given in the subsequent Appendix 3B, and (ii) such that the corresponding vector q = ( - A s ) ' - ° Q D g [see (4.3)] has sufficiently small lo~-norm and has all coordinates ( q , ~ r ) , r = K , K + l . . . . different from zero will work: see in particular, Appendix 4C.

Remark 1.3. The expansion at the right-hand side of (1.7) holds also for ( x ( t , x o, Xl), y ) , i.e., for the feedback solutions in the L2(~2)-weak topology, if it is assumed only x0E Lz(~), x 1E Lz(~ ).

A reformulation of this expansion in terms of spectral properties of the feedback generator is given next. To appreciate it, one should note that the feedback generator A F, corresponding to those special vectors ~ and gj as in Theorem 1.2, that produce oscillatory feedback solutions, cannot be a self-adjoint operator ~, so that an orthonormal basis in L2(f~ ) of eigenvectors of A F is out of question (with nonzero feedback). On the other hand, for any vectors wj and g j, the generalized eigen vectors of the corresponding feedback generator A F always span all of L2(f~ ) (cf. Remark 2.4). With the special vectors wj and gj of Theorem 1.2, the situation achieved for the corresponding A F falls in between.

Corollary 1.5. The following spectral properties hold for the feedback generator A F

corresponding to the vectors w and g of Theorem 1 2 (or subsequent corollaries): J J

2 ~: 1 2 ~ are the eigenvalues of such (i) the distinct constants ( - c i }i= 1 , and ( - a r }r= 1 feedback generator AF;

(ii) the corresponding (normalized) eigenvectors (eF, i}ir_-i l, ( F,r}~=l form a (Schauder) basis in L2(f] ) (nonorthogonal, when the gj 's or the wj's are not all

IMoreover, if x 0 =0 (resp. x I --0), the final expansion (1.7) contains only "sines" terms (resp. "cosines" terms). In fact, say, x o =0 implies n' r ~-0 by (4.16); hence ft(-- +- i%)=real, by (4.49); hence d'r =0 by (4.46) and (4.47); hence the expansions of Yu(t) and Xs(t ) contain only "sines", by (4.64) and (4.65).

*By Green's second theorem for all x, yC6~(AF) (and J = l ) : (AFX, y ) Ox

( X, A F y ) = ( x, w ) ( g , ~n r ) r -- ( Y , w ) ( g , ~n r ) r and theidenticalvanishingoftherighthandside implies either g =0 or else w =0.

Hyperbolic Equations with Dirichlet Boundary Feedback: 1 9

zero) so that the following expansions apply:

K--1 ov x : ~ 71i(X)er, i + ~ ~r(X)eF, r, X ~ L2(~'~ ) (1.9)

i - - I r - - 1

K - - 1

A r x = Y~ -- c~n~(x)eF,~ + -- a~g~(X)~e ,~ , x ~ ® ( A F ) (I.10) i = 1 r - - I

where the bounded finear functionals {~/n} and the eigenvectors { e F, ~} are biorthogo- nal sequences:

1 n = m n"(er'm)= 0 n # m

Thus

C F ( t ) x = ~ ~]i(X)COSCiteF, i-[- ~r(X)COSOtrteF, r i = 1 r : l

The general case, in which M k ~ 1, can be treated similarly. However, its detailed treatment would have considerably overloaded the presentation, particularly at the notational level. It is therefore analyzed only in the first part of the proof (section 3), while the more technical part of the proof (section 4) is restricted to the case M k = 1. As to the literature, we are not aware of any work that has treated our same problem of restoring the natural oscillatory character of a hyperbolic equation, originally unstable with zero B.C. (e.g., - A(~, O)= A + c 2 ), by means of a boundary feedback (i) applied in the Dirichlet B.C. and (ii) acting only on the position vector. There are, instead, works notably [23] and [25], that concern, however, the wave equation ( - A ( ~ , ~ ) : A), which is already oscillatory with zero B.C.: then a boundary feedback is considered, which (i) is applied in the Neumann B.C. (or a variation thereof) and (ii) acts on the trace of the velocity vector x t (dissipative B.C.). This way, damping is introduced in the system, and hence asymptotic decay of the energy to zero is indeed possible.

As to the general approach that we follow, we may say that the guiding ideas employed here in formulating and attacking the stabilizability problem come primarily from the authors' work: more precisely from [16, 29] on modeling boundary input hyperbolic equations, and from [30] on a similar problem for boundary feedback parabolic equations, where, however, only decay in norm, not exact structure, of suitable feedback solutions was sought. The present hyperbolic case, where the precise structural property of class ILC is achieved for suitable feedback solutions presents a far greater challenge, at both the conceptual and the technical level, than that parabolic counterpart, which accounts for the length of the present paper.

2. Well-Posedness and Regularity of Feedback Solution

The approach we take here in investigating the system's well-posedness and stabilization is based on a recent functional analytic formulation developed by the


authors [16, 29]. It consists of replacing the closed loop system (1.1)-(1.3), (1.4) with the corresponding abstract version as developed in [16, 29] and recalled below. To this end, the following background material is needed.

The "Dirichlet map" D is the linear map: L2(F)--,H½(~2 ), which solves the homogeneous elliptic problem corresponding to (1.1)-(1.3). It is defined by v = Dg, where

- A ( ~ , a ) v = 0in a; VJ r=g . (2.1)

D is continuous even for boundaries with corners: see [14] for domains f~ with finitely many conical points. It is believed in [21, Problem 1.1, p. 252] to be true also for all a belonging to the very general class 9]L. Let A be the operator Lz(~)D @(A)~ L2(~ ) defined by

( A h ) ( f ) = (2.2)

for h E @ ( A ) , where ®(A) consists of the closure in n2(~) of functions h in C2(~) that satisfy the boundary condition

hit = 0. (2.3)

Notice also that the definition of D in (2.1) always implies D g ~ @ ( A ) unless g=0. However, see (2.6) below.

As to the well-posedness problem (i) of the Introduction, the following regularity result for the ("open loop") system (1.1)-(1.3) should be kept in mind for comparison purposes. Moreover, it is also invoked in the proof of the present section.

Theorem 2.1. [16, §6] Let F be C 1. For XoE L2(~2 ) and x I E H l(f~), the solution x( t , Xo, x I ) of the system (1.1), (1.2), (1.3) due to Dirichlet boundary input f ( t , . ) L2(0, T; L2(F)) satisfies: x( t, x o, x l) E L2(0, T; L2(~)). The same conclusion holds, when f~ is a parallelepiped and the generator A is a normal operator. []

As mentioned in the introduction, we make throughout the standing assumption that A generates a strongly continuous cosine operator C( t ) on L2(f~ ), - o o < t < oo. See [7] [26] and references cited therein for general background on cosine operators. We then define the operator S(t) by

s(t)x = f0'c(,)x d,, x E L 2 ( a ) , - o o < t < oo. (2.4)

Moreover, at the price of considering a suitable A - c/, c >0, rather than A (which changes C(t) into c o s h - ~ t C ( t ) , and hence does not affect the local regularity of solutions, the object of the present section) we can always assume in this section without loss of generality that 0 E O ( - A ) and that the fractional powers of ( - A) are well defined.

Then, with x ( t )= x ( t , . ) ~ L2(f~ ), the abstract version on L2(f~ ) of problem (1.1)-(1.3), subject to the feedback condition (1.4), is the following abstract

Hyperbolic Equations with Dirichlet Boundary Feedback: I I 1

integral equation:

x(t)= c(t)x0 + s(t)x

J

+ (-- A)¼+°fo' ( - A)½S(t -- r ) ( - A)J-°D ~ gj(x(r) , wj> dr j = l

(2.5)

We refer the reader to the authors' work [16, 29] for further explanation and justification. Here, we only remark explicitly that the following relations hold:

r angeo fD C H~(~2)C H' : -~( f~)= o~((--A)~-P) , for anye = 20 > 0 ( 2 . 6 )

[the left-hand side inclusion is a result of the elliptic theory [20] [14], while for the right-hand side equality (set theoretically and topologically) we refer to [9; 19 Theorem 5.1; 20, p. 107; 15 Appendix B]]; and moreover

S(t)L2(~)C@((--, A)') } (-- A)~S(t)L2(~) continuous in t

(2.7a)

(2.7b)

[see 7, II, Theorem 2.3, p. 58], whose proof when A is normal is immediate by using the basis of eigenvectors. We shall also need below that

ac(t) at x = A S ( t > = e (2.8)

Therefore, well-posedness and regularity of (1.1)-(1.3), (1.4) is analyzed in terms of well-posedness and regularity of (2.5). Cosine operator theory provides for the resolvent operator [7]:

f0 o ~.R()~e,A)x = e-XtC(t)xdt,Re~ > Oao,X @ Lz(f~ )

fo R(X 2,A)x = e X~S(t)xdt,ReX > %, x ~ L2(a) .

(2.9)

(2.10)

We are now in a position to prove Theorem 1.1.

Proof of Theorem 1.1. We start with a remark

Remark 2.1. A fixed-point technique applied to (2.5) seems to produce only partial results, i.e., the operator defined by the right-hand side of (2.5) as acting on x ( r ) , 0 ~ < r < t , can be shown as in [16, §6.1] to be a contraction on L2(0, T; L2(f~)), provided that either the time interval T or else the system's parameters sup l&l, sup I 1, J--1 . . . . . J are suitably small. A further attempt to compute the n iterate of such operator in the general case, in the hope that it may well be a contraction for some n (as is the case for boundary feedback parabolic equations [31]) appears to run into serious computational difficulties. We therefore resort to Laplace transform techniques as shown below. []


(a) From (2.5) we obtain (for J = 1) for simplicity of notation)

I I I

(- A) -~-~x(t) = c ( t ) ( - A)-~-"Xo + S( t ) ( - A) ~-~x,

+ f o t ( - A ) ' S ( t - T ) ( - A ) ¼ °Dg(x ( r ) ,w)dT (2.11)

Under the given assumption (1.5) on w, we can write

( x ( r ) , w ) = ( ( - A ) - ¼ - ° x ( r ) , ~ ) "rE R (2.12)

for ~ = ( - A * ) ¼ + ° w , since ® ( A ) = @(A*) in our case [20] and, by interpolation, ® ( ( - A)°) = ® ( ( - A*) °) 0 ~ < 0 ~< 1. Our next step is to show that

(x ( t ) ,w) ~ L2(0, r ) . (2.13)

To this end, it is sufficient to establish that

e-Ut(x(t), w) E L2(0, oo)

for suitable positive constant u; or--equivalently, by (2.9), (2.12), and [5, p. 212] - - t o establish the existence of an L2(f~)-valued function 2(2t), Re ~ large enough, satisfying

, ~ / ,'-">-~ ( - A ) -~ °2(X)= C ( t ) ( - A ) ~-°x o + S ( t ) ( - A ) " °x I

I I I

+ (- A)~R(X 2, A)(- A)~-o Dg((- A) -~-~(x), ~)

i . e . ,

( ( _ A ) - ' - o 2 ( X ) , ~ ) [ 1 ' 2 ' - ( ( - A)~R(~ ,A)(- A)~-ODg,.>]

= + s ( t ) ( - A)- oXl, ~> (2.14)

as well as

[ ((-- A ) - ~ - ° 2 ( u + iv),~)12 dv < ~ , oO

X = u + iv, u large enough;

(2.15)

we have

const O~ < 0 ~< 1; (2.16) II(-A)°R(N2, A)II <~ i;~ll 2°(Re}t -Wo) Re~>~Oo

(~o o can be taken zero in the present case with ( - A) ° well defined). In fact, for 0 =0, (2.16) follows from (2.9) using the known fact that II C(t)x II ~< const II x II e '°°t,


and for 0 = 1 (2 .16 ) follows from the identity

AR(Iz, A) = I~R(tx, A) - I. (2.17)

Then, one interpolates. In particular for 0 = ½, we obtain

, , const If(- A)~R( x2, A)(- A)~-ODg,~)I ~< Re----£"

For all X with Re X large enough, say Re X >i r o > 0, we then obtain

I I

0<~<II-((-A)~R(X2, A)(-A) ~ °Dg,~) I

(r o depends on the preassigned constant 8). Hence, for all such ~ we have from (2.14) , ~

I {(-A)-~-P2(A),~)I ~< Cro I(C(t)(-A)-~-°Xo + S ( t ) ( - A ) " °x,,~)]

and the desired conclusion (2.15) is achieved via (2.9)-(2.10). Therefore, (2.13) is proved. It then follows, by virtue of property (2.7a) and the standard convolution result [2, p. 5], applied to the right-hand side of (2.11), tha t - -ac tual ly- - the left-hand side of (2.11) satisfies the stronger property

I

((- A)-~-.4t),~)= (x(t),w)~ c[0,r]. (2.18)

In any case (2.13) is enough, via Theorem 2.1 applied to (2.5), to conclude that x(t, Xo, xl)E L2(0, T; L2(f~)), as desired.

(b) We shall now prove that the same regularity result holds for 2(t): i.e., that (from (2.12))

I

<(-A)-~-o~(t),~)= (~( t ) ,w)~ C[O, TI. (2.19)

In theory (in particular (2.6) and (2.8)) yields

(- A)-'-o~(t) = - ( - A)~S(t)(- A)" % + C( t ) ( - A)-~-~x, I I

+ (-- A)~S(t ) ( - A) ~-°Dg(xo, w)

+ fot(- A)½S(t - ¢ ) ( - A)"-"Dg(2(r), w) d'r

fact, differentiating (2.11) by means of standard facts on cosine operator

(2.20)

whose right-hand side is well defined. We can then apply to (2.20) the same procedure applied before to (2.11); i.e., after taking the inner product with ~ and using the identity in (2.19) one obtains an integral equation in (2(t) , w) with the solution (2( t ) ,w)E L2(0, T), and hence, again by convolution argument, (2.19). Combining (2.18) and (2.19) we can write

( ( - A ) -¼ 0 x ( t ) , ~ ) = (x ( t ) ,w)C C'[0, T]. (2.21)


Remark 2.2. Without assuming compatibility relations, one cannot further dif- ferentiate (2.20), because of the third term at its right-hand side.

Having obtained (2.21), one then returns to (2.5) and integrates by parts, using (2.8):

j0'[ " ] x ( t ) = C ( t ) x o + S ( t ) x 1 + ~ C ( t - - ~ r ) D g ~x (~ ) ,w )d ' c

= C ( t ) x o + S ( t ) x 1 - D g { x ( t ) , w ) + C ( t ) D g { x o , W )

+ f ' C ( t -- ~) Dg{2( 'r) , w) d'r ao

Hence, using (2.7) and the assumed smoothness on x 0 and x~, we finally get

I I I ¢

(- A) i- x(t) = C( t ) ( - Al -°Xo + ( - A)'S(t)(-- A)-' Ox, I I

- - ( - A t ~ - ° D g ( x ( t l , w ) + C ( t l ( - A) ~ " D g ( x o,w}

+ fotC(t -- r ) ( - - A) ¼ °Dg{Jc('r), w) d'r

whose terms on the right-hand side plainly belong all to C([0, T]; Z2(~)). We conclude that

x(t) ~ C([0, TI; ®((- A)~-~))

as desired. The proof of Theorem 1.1 is thus complete. [] The proof of Theorem 1.1' is quite different in spirit and technicalities. The

first step of it consists in obtaining a differential version of the integral model in Equation (2.5). Note that in Equation (2.5), no further fractional power of ( - A) can be moved over to the right-hand side of the sine operator S(-). Therefore, any attempt to provide a differential version of (2.5) in the form: 2 =(A + H1)x in the space L2(~2 ) is bound to fail. (For a "factor" form 2 -- A ( I + H2)x in L2(~ ) see, instead, subsequent Remark 2.4.) What is needed is an extension of Equation (2.5) to a space larger than L2(a ), which we now proceed to identify, following [20; 151.

Henceforth throughout the end of this section only, we shall indicate the generator by - A rather than by A as done thus far: This will simplify the notation involving fractional powers, a heavy use of which will now be needed. With this convention, ( - A) generates afortiori an analytic semigroup [7].

Thus, since ( - A) and ( - A*) are maximal dissipative operators and 0 E p(A) (see statement above (2.5)), it follows that:

(i) A (resp., A*) defines an isomorphism from @(A) (resp., ®(A*)) onto L2(~ ) (see [27; p. 101]);

(ii) The operators A it and A *it are bounded over some interval It ]~<const. [11; Thm 5, p. 247].


From (i), by taking the dual operator to A* (resp. A), we also have that: t

A** (resp., A*) defines an isomorphism from L 2 (f~) ]

onto [®(A*)]' (resp., [@(A)]').

Since A** coincides with A of ®(A), A** can then be interpreted as an extension of A o n L2(~ ). In this sense, we extend (i) to:

A (resp., A*) defines an isomorphism from L2(~2)

onto [@(A*)]' (resp., [®(A)] ' ) . ~ (2.22)

In addition, by (ii) above, Theorem 1.15.3 in [27; p. 103] applies, leading to the important conclusion that the domains ®(A ") and ®(A*~), c~E[0, 1], can in fact be expressed as interpolation spaces:

@(A ~) = [@(A), L2(~)]l_ a} a ~ [0, 1] (2.23) @(A *~) = [@(A*), L2(~) ] I - - a

where [.,.]~ denotes as usual the appropriate interpolation space between ®(A) and L2(~ ). Moreover, for the operator A, as defined at the beginning of this section, the following identity holds [15], [18]: ®(A)= ®(A*). Hence, from (2.23), we obtain (See also [27; Thm. 5.4.2, p. 381; p. 384].)

®(A ~) = @(A*~), a ~ [0, 1]. (2.24)

We now apply the interpolation theory (see [19, vol. I. Thm. 5.1, p. 27]) to the right-hand side of (2.23) and use property (i), Equations (2.22), (2.24), and the duality theory (see [19; vol. 1. Thm. 5.2, p. 29]) to obtain that, for 0<~a~<l, the map

A : ® ( A ~) ~[L2(~) , [@(A*)] ' ] ,_ ~ =[@(A*),L2(~)] '~ = [@(A *0 ~))]'

~-[@(A' ~)]' (2.25)

is also an isomorphism. Such an extension of A, acting from @(A~)--,[®(A I ~)]' and viewed as an

unbounded operator on the basic space [®(A l-~)]', will be denoted by A,. Thus,

A~: @(A~) = @(A ~) --,[@(A 1 ~)]' andA~x = A x forx ~ ®(A~). (2.25')

Furthermore, for all 0 ~< a ~< 1, the maps

A *~,A~: @(A ~) = @(A *~) ~ L2(~ ), (2.26)

t y, denotes the dual space of Y.


are also isomorphisms (see [27; p. 101]). Consequently, taking the duals to A *~ and A ~, we have that, for 0 ~ < a ~< 1, the following maps are isomorphisms:

A*~,A~: L2(~ ) -*[@(A*~)] ' ~[6~(A~)] ' . (2.27)

Defining A ~ ~(A~) -1, as a consequence of (2.27), we then have that, for 0 ~< a <~ 1, the following maps are isomorphisms:

A* ~,A-~:[6~(A*~)]'=[@(A~)] '-0 L2(~ ). (2.28)

Similarly, from (2.26), taking the duals to A-~ and A *-~, we have that, for 0 ~ < a ~< 1, the following maps are isomorphisms:

A*-", A - " : L2(~ ) -o @(A *~) = @(A"). (2.29)

Finally, for all 0 ~ < a ~< 1, by virtue of (2.26), (2.28), we obtain the important topological equivalences:

I x I~Ao) = I A~x ILl,a), Ix [t~(Ao)r = [A-~x IL2(a). (2.30)

It follows from the above extensions, particularly from (2.27) and (2.25'), that as a differential version of Equation (2.5) with J = 1 (for simplicity) we can take the equation

2 = - J , x + J~+°A '. °Og<x,w) (2.31)

with a = k - p , xE@(A¼-°), 2~[@(A¼+°)] ', and A"-°DgEL2(~2) (cf. (2.6)). The following specializations of relations (2.30) will be used repeatedly

I x I~A¼ ~) = A¼-°x IL2(a); Ix ]t~(/]+o)l, = A ¼ °x IL~(a) (2.30')

We next extend the cosine operator C(t) and its time integral (2.4), the "sine" operator S(t). As we shall see below, we may likewise, by continuity, extend C(t) and S(t), originally defined on L2(~), to, respectively, Ca(t ) and S~(t) as bounded operators from [6~(A1-~)]' into itself.

Proposition 2.2. The extension operator - A~: 6~(A~)~[@(AI-")]', 0 ~ < a ~ < 1, of - -A (see (2.25')) is the generator of a strongly continuous cosine operator on [6~(A1-~)]' which is precisely the extension C~( t ) of the original cosine operator C( t ) defined on L2(~).

Proof That the extension C,(t) satisfies the D'Alambert identity ~, which char- acterizes a cosine operator function, is a consequence of L2(~ ) being dense in [@(At-~)] ' (since @(A ~) is dense in L2(~2)) and of a standard continuity

*The D'Alambert identity or "cosine operator functional equation" is: C(t + s)+ C(t- s)= 2C(t)C(s).


argument. Similarly, C,(t) and A " - 1 commute on [@(A 1-~)]' (see (2.28)):

Aa-'Ca(t)x =Ca(t)Aa- 'x = C(t)Aa-lx , x E [ @ ( A ' - ' ~ ) ] '

This, along with (2.30) and strong continuity of C(t) on L2(~), leads to strong continuity of C~(t) on [~(AI-~)] '. That the generator of C~(t) is - A ~ follows, say, from the known identity d2C,(t)x,/dt 2= -A~C~(t)x,, x ,E @(A), where x. -, x e ®(A~) = ®(Ao). []

Motivated by (2.31), we now single out the relevant case a = ¼ - 0 , and introduce henceforth the simpler notation: for a = ¼ - 0: /[~ ~ A; C~(t) = C(t); S~(t) -= S(t) so that (2.31) is rewritten as

5t = - -Ax + AJ+°A~-ODg(x,w 5 (2.31')

on the basic space [@(A~+P)] '. It is then natural to introduce the perturbation operator/5:

/sx = L (a)

v =-- -d~+OA'-PDgE[®(A~'+°)] ' (2.32)

and the (feedback) perturbed operator ~z[ v

(set theoretical equality) (2.33)

It is important to note that (i) P can be extended to a bounded operator on [ 6~(A ~ +°)]' when w ~ @(A ~ + °)

[as we know (cf. (2.24)) set theoretically equal to 6~ (A, ~ + p)]:

I <x,w>[=l (A-¼-°x,A*¼+Pw>l~l A*~+Pw IL2(a)I x I~(A~+°)j,

if the L2(~)-inner product is extended by continuity to the duality pairing on [®(A~+0)]'x ®(A~+.).

(ii) Otherwise when w~®(A~,+"), fi is unbounded as an operator on [@(A~ +0)]', and moreover--being of finite dimensional range--/5 is even unclosable there [12, p. 166, Prob 5.18].

Our overall strategy can now be qualitatively expressed as follows: under the standing assumption that wE@(A ~+°) (for which /5 is generally unclosable) prove that the operator .4~ generates a strongly continuous cosine operator (i) first on the space [@(A,~+°)] ', next (ii) on the space 6~(A¼-"), finally (iii) on the interpolation spaces ~ ®(A ~-0-°) 0 ~ < 0 ~< 1, from which L2(f~ ) is obtained when O= ¼+p.

IThe notational convention 6~(A-')=[®(A~)] ', s >0 is used.


(i) Generation of a cosine operator on [@(A~+P)] '. We shall make use of a suitable extension of Fattorini's perturbation result for cosine generators [8, Thm. 2.1]--which is the analog of a well-known perturbation result for semigroup generators [6, VIII. 1.19, Theorem 19, p. 631], [10, Cor. 1, p. 400]. We take the space E in [8] as E --=[@(A3,+P)] '. We have to verify that the following assumptions, as dictated by [8], hold:

(I) @(/5) _D @(4) and

S ( t ) x E @(/5), x E E, t sufficiently small

(II) lim PS( t )x = O, x E E t ~ 0

This will indeed be the case for w~®(AI+p) , as assumed. To see this, we note that (2.7) (a) implies a gain of ½ in the fractional power of A, i.e.,

} ~['2~(t )E continuous in t E R

(i)

(2.34)

(ii)

Thus, for all x ~ E , the expression

/sS( t )x = v ( ( S ( t ) x , w ) )

is well defined, where (( , )) is the duality pairing on [®(A', +o)],× ®(A'4 +0) that continuously extends the L2(a)-inner product; moreover, we have lim S ( t ) x ~0 ,

t ~ 0 as needed, by a well-known property of the sine operators. There is, however, another point that needs to be examined: Fattorini's

quoted result [8, Thm. 2.1] assumes further that the perturbation operator be closed, and this fails to hold in our case when w E @(A ¼+°) but w 62 @(A¼+°), as pointed out above. (Closedness of the perturbation is likewise assumed in the semigroup treatment of [6, Thin. 19, p. 631] but dispensed with in the more general result of [10, Cor. 1, p. 400] or in [12] for holomorphic semigroups.) Nevertheless, we can still invoke the conclusion of Fattorini's perturbation theorem; in fact, close examination of the proof of [8] reveals that closedness of the perturbation operator is used only to make the following assertions, as they apply to our present case:

(a) that the operator/5S(t) can be extended to a bounded operator on E, (b) that in applying the operator/5 on both sides of the known relation

R ( ~ 2 , - . d ) x = f0~e XtS( t )xdt , x E E

it is legitimate to move it inside the integral sign. These two facts can be easily verified directly to hold indeed in our case,


under the standing assumption that w ~ @(At+°). In fact, with x ~ E

[/~S(t)x Ir~ =l v]E I ( A ' S ( t ) A ~-Px, A*"+°w>[

~1 v IE ] A ' ~ ( t ) I ~ ( ~ ) I x I~ I A*~+~w I~(~)

by (2.34) and (a) is verified. Verification of (b) is immediate. We thus conclude that: A r defined by (2.33) generates a strongly continuous cosine operator Cv( t ) on the space E=[®(AI+P) ] ', as desired.

(ii) Generation of a cosine operator on ®(A¼-°). Consider an invertible translation of the generator

~,o-- ~ - 7,o~: ~(~ ,0) = ~(A'-O) --[~(A~+0)]' (set theoretically)

for some positive X 0, henceforth kept fixed, which moves the spectrum of AF strictly to the left of the complex plane. It then follows, in the vein of Proposition 2.2, with a = 1, that AF, 0 generates a strongly continuous cosine operator on the space 6~(Ar.0) equipped with its natural norm

(2.35)

But the set 6~(~Z~F. 0) coincides (set theoretically) with O~(~ZlF) , and thus with the set ®(A" o) (cf. (2.33)). The original norm of the latter is given by the equality relation in (2.30'). Plainly the two norms-- the left relation of (2.30') and (2.35)--are equivalent: in fact AF, O A - 1 is a bounded operator on [@(A]+P)] ' by the closed graph theorem, and thus

IAF, oX[L~(A¼+o)]' ~ IIAF,oA ~l[IAxli~¢~+o)r -_ I!AF, oA-'IIIAI-OxIL:<~)

I A' -Px I L2(.) = [AxII~<A~ +')], ~< 11AAF,~ IIIAF, oXlI~(AI +P)], (2.36)

where II II denotes the operator norm from [@(A~+P)] ' into itself. Thus, Z~F, O-and hence AF--generates a strongly continuous cosine operator on [@(AJ-P)] (with norm as in the equality of relation (2.6)) or in the left relation in (2.30').

(iii) Generation of a cosine operator on t ®( A ' -P - ° ) , 0~0~<1. We apply the interpolation theorem [20, Thm. 5.1, p. 27] with

dk [XR(X2, ] x : ~ : [+(A~+,)] ', Y : ~ : + ( A . ' - , ) . ~ : V ~

tThe notational convention: ®(A -s)=[®(AS)]', s>0 is used.


in the notation of [20] to, say, the known characterization ([7, II], [3], [26]) of the Hille-Yosida's type in [°~(A-]+P)]' and ~ ( A ¼+°) to obtain that for all X with Re ) ,>some w~>0, the operator ~r x is a continuous operator from the interpolation space (see e.g. [17])

[ [ ® ( A ~ + ° ) ] ' , @ ( A ¼ - ° ) ] l _ O = ® ( A l - ° - ° ) , 0<~0<~1 (2.37)

into itself and satisfies the analogous characterization

dX K [ ] ~(A¼ o °o, ( R e X - w ) k + ' M ° k ! d k . XR(X2, A) ~< , R e X > w ~ > 0

k = l , 2 . . . .

for some constant M 0 depending on 0, proportional to max(q , c2} (see final estimate in the proof of [20]), where c 1 and c 2 are the analogous constants in X and Y. Thus/~F,0 - - and hence AF--generates a strongly continuous cosine operator on @(A¼ -P 0), 0 ~ < 0 <~ 1, in particular on L2(f]), which is obtained when 0 = ~ - p. Theorem 1.1' is thus fully proved. []

Remark 2.3. In boundary feedback problems of the type studied here for hyperbolic equations and e.g. in [17] for parabolic equations, the assumption that the perturbation operator be closed is too restrictive or false. Perturbation theory of just* strongly continuous semigroups without closedness of the perturbation operator is indeed available (see [ 10], § 13.3-13.4, particularly Cot. 1, p. 400]) even though more accessible treatments like [6, VIII.I.19, Thm. 19, p. 631] do assume closedness. As to strongly continuous cosine operators, Fattorini's proof in [8] also assumes closedness, as is patterned after the semigroup treatment of [6]. However, closedness of the perturbation can be dispensed also in the cosine operator case, by combining Fattorini's proof with results of [10, §13.3]. More precisely, the following claim holds: let E be a Hilbert space, or an Lp space, 1 < p < oo, - A the infinitesimal generator of a strongly continuous cosine operator on E, and P a possibly unclosable operator:

E D ® ( P ) - - { x E E " lim P?~R(X,A)xex i s t s ) D @(A)-~ E. (*'

Then - A + P, with domain @(-- A + P ) = @ ( - A), generates a strongly continuous cosine operator on E, provided that

(i) PS( t ) is bounded on E (for t sufficiently small, and hence as in [8, p. 202], for all t ~ R) and IPS(t)]<~ Ke & for some constants K and ft.

(ii) P is A-bounded (or PR(X, A) is bounded on E) in addition to the assumption of [8] that

(iii) lim PS( t )x = 0, for all x E E. t$0

*As distinct from perturbation theory of holomorphic semigroups e.g. [10, §13.7], [11]. (*)If the linear operator P obeys only (ii) and Q(P)D @(A), this condition is always satisfied by

its unique extension [10, Thm. 13.3.1, p. 392]. The conclusion of our claim then applies to such an extension (still possibly unclosable).


Note, that (i) and (ii) are implied by t (i') P is A½-bounded (or PR(X, A~) is bounded on E). In fact (see (2.7) (a)):

PS(t) = pA-½/I½S(t) where I A½S(t)[<<- MveV'

(for this last inequality, see Equation (6.12), p. 98 in [7, II]). To substantiate our claim, it suffices in light of our comments in (a) and (b) above, to show that in applying P to both sides of

f0 ° R()t 2, - A)x = e-XtS(t)xdt, x E E, ?t sufficiently large

it is legitimate to move P inside the integral sign (this would be standard if P were closed). To show this we shall invoke [10, Lemma 13.3.2, p. 393]. Assuming that the reader has [10] at hand, we compute accordingly to an assumption of [10, Lemma 13.3.2]:

] ?tPR(?t ,-A)e-XtS(t)x le <-I PS(t)II ? t R ( ) t , - A ) l e - X ' l x IE

<- Ke B'. const e-X' l x le

and thus we verify that for X > fl the right-hand side of the above inequality is integrable on (0, oo), as desired. Note that ]XR(X,- A)l~<const, as - A , being a generator of a cosine operator, is afortiori a generator of an analytic semigroup. Our claim is proved. []

Remark 2.4. The viewpoint adopted in Theorem 1.1', being explicitly focused on the feedback generator, has further advantages. We illustrate one such advantage in connection with the analysis of the spectral properties of the feedback generator, which will--in turn--shed new light on the boundary feedback almost periodic stabilization solution of Theorem 1.2. Le t / l v denote the generator of the cosine operator Cr(t) on L 2 ( ~ ) , claimed by Theorem 1.1', which is the restriction of CF(t) on L 2 ( ~ ). Thus, (for J = 1):

A F = 1~ F [®(AF) where

@(AF) = (xEL2(~) : AFX~L2(~ ) a n d x l r = ( x , w ) g } (2.38)

On the other hand, on the basis of the differential version, Equation (2.31'), of the boundary feedback equation, one readily deduces that A v can be characterized

tNote that for our case of interest here: E = [ 6~(A ] +°)]', p =/~ and S = S, the standing assumption wEa~(A .]+p) achieves precisely (i'): ]IS.4-'2y]E<~]V]E](A-]-Py, A*¼+PW)]<~]VlE [ A*¼+"wll.vl~


more explicitly as:

A v : - A ( I - Dg(',w))

@( AF) = ( x E L2(f ) : x -- Dg(x, w) @(A)}

(i)

(2.34')

(ii)

One important consequence of the explicit factorization of A F in (2.34') (i) is the following

Claim. For all vectors g E L2(F ) and all vectors wE L2(f]), the operator A F in (2.34') has the following spectral property when A is self-adjoint:

m sp { generalized eigenvectors of A F } = L2 ( ~ )

(the closure of the span is in the Le(~2))-topology. The above claim simply follows from the general Theorem in [6, Vol. III, p.

2374], which is applicable to the operator A F in (2.34') since: (i) A is self-adjoint in L2(~); (stabilization problem) 001 A-m is of Hilbert-Schmidt type for sufficiently large positive integer m,

as it follows from the known asymptotic estimates of the eigenvalues {?~,) of A (cf. (4.23)): o

1 IIA roll2 S =

n : l ~2nm - - - E < ~ ( f o r 4 m > p )

n 1 n4m/r

(iii) the perturbation Dg( . , w) is compact in L2(f~ ). The content of the above claim should be compared and contrasted with

Corollary 1.5, which applies, instead, only to the special vectors ~ and gj as in Theorem 1.2.

3. Boundary Feedback Stabilizability when A is Self-Adjoint

3.1. Qualitative Statement

Since ~2 is a bounded domain, the resolvent operator R(X, A) is compact [6. p. 1740]. Hence, the spectrum o(A) is only point spectrum and consists of a sequence of isolated eigenvalues {-Xk} with no finite accumulation point: [?~k [--' ~ and corresponding normalized linearly independent eigenvectors (~km), k = 1 . . . . m----1,...,Mk, M k being the geometric multiplicity of -?~k. As is well- known from the general theory of cosine operators, the ( - ?~k} are contained in a

IWithout loss of generality, we assume A :~ well defined as a bounded operator on L2(~ ).


parabolic sector [7, Remark 5.6]

-(Im h) 2 + ¢o2}. {h2"Reh~<%} = )~: Re )~ ~< 4oa2

Therefore, at the right of any vertical line in the complex plane, there are only at most finitely many of them.

Remark 3.1. To describe our problem we preliminarily consider the case in which the generator A is a normal operator (this normality assumption covers all classical boundary value problems. Moreover, the test for any normal operator to generate a strongly continuous cosine operator is simply that the spectrum of A be in a sector of the type described above). With A normal, the eigenvectors (e~km) form a complete orthonormal system for L2(~ ) that we use as a basis on L2(~ ). The cosine operator C(t) and its integral S(t) [cf. (2.4)] are then given by

C( t )x = c o s ~ , E < x , ' I ' ~ m > % , . , - ~ < t < ~ , k 1 m--1

S(t)x= ~ sin ~Tt M~ =, m=,2

x

(3.1)

x L2(a) .

(3.2)

We then see from (3.1)-(3.2) that in order to guarantee boundedness in time for the solutions

x( t ) = c(t)Xo + s(t)x (3.3)

of the free system ( f ( t , ~ ) = 0 in (1.3)), we must require that all eigenvalues { - ?~k} be real and nonpositive, in which case (3.3) is an L2(f~)-almost periodic function and

i

IC(t)Xol lxol; I(-A) S(t)x,I Ix, I, t E R .

In other words, to have this stability of the free motion, we must restrict to the class of self-adjoint generators A with spectrum on the closed half line ( - oo, 0].

With the above remark in mind, in studying the stabilizability problem of the present paper, as qualitatively stated below, we confine ourselves to unstable self-adjoint generators, i.e., to self-adjoint operators with spectrum upperbounded above (so that they generate cosine operators) and with finitely many positive eigenvalues (unstable eigenvalues), say

" ' " < --~k K < 0 < - - ] k K _ 1 < " ' " < --~k 2 < --~kl. (3.4)


A canonical example is: - A(~, ~)= A + c 2, for c 2 sufficiently large. Notice that the assumption of A self-adjoint is automatically guaranteed, when the double- index coefficients a~ in (1.0) are real and symmetric. Equation (3.4) means that with zero input f ( t , f)=--0, some free solutions (3.3) of (1.1)-(1.3) will blow up in time, in fact exponentially. We then pose, in qualitative terms, the following:

Almost periodic boundary feedback stabilizability problem. Find, if possible, appropriate vectors g jE L2(F)--as well as their minimum number--and the least conditions on the vectors wj E L2(~), j = 1 . . . . J as to guarantee that all feedback solutions x ( t ) = x(t , x o, x~) of the closed loop system due to any pair of points x 0 and x 1 in a natural "large" class of initial conditions are abstract-valued almost periodic functions (and hence uniformly bounded in t ~ R ) - - i n the strongest possible topology.

A precise solution to this problem, under a very general condition on the subspace from which the vectors wj can be drawn, is given in Theorem 1.2 and its corollaries and generalizations. The proof of Theorem 1.2 is lengthy and technical and will take up all of section 4 plus relative appendices.

3.2. Simplest Solution of the Stabilizability Problem

In the present section, we choose therefore--as a preparatory stage for the more demanding situation of section 4 - - to state and prove a result of interest in itself, which concerns the simplest possible solution to the problem under study. This solution presumes unlimited freedom in choosing the vectors wj and is obtained precisely when all such wj can be constrained initially to lie in a natural finite-dimensional subspace (X, below). In this case, the proof is much more amenable than in the general case, for projections of the feedback solution on appropriate invariant subspaces (X u and X s below) of the underlying space Lz(f] ) result decoupled (cf. (3.12) and i f ) .

In preparation for the proof in this (and in the coming) section, and following a procedure introduced in [28] and carried out also, e.g., in [30], we let X = L2(~ ) be decomposed into two orthogonal subspaces X u and X s, corresponding to, respectively, the subsets ( - )~l . . . . . - ~/~-1} and ( - ~k, k >/K} of the spectrum o(A) of A (the subscripts u and s stand for "stable" and "unstable," respectively). Here, we appeal to the standard decomposition theorem as in [12, p. 178]. With P denoting the orthogonal projection of L2(a ) onto X, and Q = I - P , then Q ® ( A ) C ® ( A ) , X, and X s are invariant under A and hence under the motion [i.e., under the cosine operator C(t) (see Appendix 3A)]. As for the spectrum, we have o ( A , ) = { - ~ k 1 . . . . ' -- XK--I}' o (A , )= { -- Xk, k ~ K}, where A. is the restriction of A on X. and A, is bounded, in fact, finite dimensional. Finally, P and Q commute with A and hence with the cosine operator C(t) (see Appendix 3A). We shall henceforth use the notation x , = Px and x s = Qx.

The representations (3.1) and (3.2) then specialize to

Mk QC(t)x:Cs(t)x~-- ~ cos ~kt ~ (X,f~km)f~km (i)

~ : K , . : l (3 .5)


M, Q S ( t ) x = S s ( t ) x : ~ sin ~-k t X

k:K ~k m=l (x,f~km)f~krn~6-~((--As) ~) (ii)

K--1 Mk

ec(t)x=Cu(t)x= Y~ cos ~(~-k-kt Y~ (X,¢km)¢km (i) k : l m=l (3.6)

K-I s i n ~ t Mk P S ( t ) x = S u ( t ) x = ~ ~ <X,~km>f~km (ii)

k : l ~ m : l

Remark 3.2. For handy reference in the sequel, we collect here the following results. Let Af@- k ~ c i, then:

f j s i n ~ ( t - r) sin ci(r - o ) d r = ak C2

f o t S i n ~ ( t - ' r ) c o s c i z d ' r - ~ ( c o s c i t - c o s ~ t ) X~ - c~

cisin ~k t sin / t sin Af~-k (t - - r )s in ci'r d'r =

cit J0

s i n c i ( t - o ) - cisin Aft- k ( t - o) (i)

(ii) (3.7)

(iii)

i.e., convolving two different frequencies preserves the oscillatory character. By contrast we have

fotSinc(t - ~') sin c~'dr sinct cosct = 2 - -7 - - t - -5- -

fo sin ct t s ine( t - T)coseTdT -- Cosct ~_ t - - 2e 2

i.e., convolving the same frequency destroys the oscillatory character (resonance phenomenon). These remarks will play a crucial role in the development given hereafter in proving the desired stabilizability results.

In order to formulate our stabilizability results, we need to introduce the following important number

Definition 3.1. Let the integer ¢r(1 ~< er ~<dim Xu) denote the number of linearly independent Neumann traces

[~ (~km] , k . . . . . K - 1 ; m .. . . , M k 1 1 - - f fC l r

of the normalized eigenfunctions associated with the unstable eigenvalues (3.4).

26 I. Lasiecka and R. Triggiarfi

We next introduce the J × M k matrix W k defined by

=

(w, . ,Ok,) (wL.,Ok2) "'" (w~.,Ok~4~)

(w2.,Ok,) (w2.,Ok2) "'" (w2~,Ok.~) k = 1 , . . . , K - 1

associated with each unstable eigenvalue of A and the J × (dim X,) matrix

w = [ w,, w2,.:.,

(3.8)

(3.9)

Remark 3.3. It appears that no systematic study for general f~ has been undertaken regarding the linear dependence or independence of Neumann traces of eigenfunctions associated with Dirichlet homogeneous elliptic problems. When f~ is a parallelepiped, one can verify directly that such traces are indeed linearly independent, so that ~T=dimXu in this case, and condition (3.11) below is satisfied with just J = 1. When f~ is a sphere, instead, one has Cr < d i m X, and a correspondingly larger J is needed. We report a simple example at the end of Appendix 3B. We believe the linear independence of such Neumann traces to be the typical case.

Theorem 3.1. (Simplest solution to the stabilizability problem). Let f~ be as in Theorem 1.2. Let the generator A be self-adjoint, have compact resolvent, and satisfy (3.4). Assume that the vectors wj can be taken to lie in X,, and that, moreover, they satisfy the following full rank conditions

rankW k = M k k = 1 , . . . , K - 1 (3.10)

at the unstable eigenvalues, and, in addition, the condition

d i m X u ~< ~r + ~ w - 1 (3.11)

where f w is defined by

rankW = ~w (max(Mk, k = 1 . . . . . K - 1 ) < ~ e w < ~ m i n ( J , d i m X . )

for the matrix W in (3.9). Then there exist vectors g j ~ Lz(F), whose minimal number is discussed in

Remark 3.4 below, such that any solution x ( t ) = x ( t , Xo, Xl) of the Dirichlet feedback system (2.5) corresponding to any initial conditions xoELz(~),Xl~ H - 1(~2), is an L2(~)-almost periodic function. Moreover, provided that the initial conditions satisfy (1.6), the following functions

I I

(A.)'- ,x.(t) and ( - A s ) ' - % ( t )


are almost periodic on X~ and X s, respectively: this means, by (2.6), that x( t ) is an H" -~( f2 )-almost periodic function, e = 2 ~.

Remark 3.4. Condition (3.10) and the definition of fw in particular imply

J > ~ m a x ( M k , k = l ..... K - l ) and d~> ~W"

Moreover, the proof given in Appendix 3B will show that the largest multiplicity of the unstable eigenvalues is indeed the minimal number of boundary vectors gj required for the conclusion of Theorem 3.1, provided that the traces

{ Of~km ~k ~-- 1 K - 1 = 1 ]

m Mk, J

of the eigenfunctions are linearly independent, J i.e., er = dim X,. Otherwise, more vectors gj E Lz(F ) are needed. For instance, if M k ~ 1, 1 ~< k ~< K - 1 and 2 fT < dim Xu = K - 1, then J suitable vectors g j E L2(F ) that satisfy the moment problem (3B.7) in Appendix 3B, where J~>dim X u - ~v + 1 will suffice 3. A full analysis of the situation, amounting to a certain output stabilizability problem in X, is contained in the proof of Lemma 3.2 in Appendix 3B.

The projections of the solution x(t) of (2.5) onto X~ and X s are, Proof respectively (see (3.5)-(3.6)),

x . ( t ) = cu(tlxo~ + s~(t)x,~ J

- - fo iS t ( t - r ) 2 AuPDgy[(x~('r),wy.)+{xs(r),Wys)]d'r (3.12) J = l

-' f j ( '2 xs(t) = Cs(t)Xos + ss(t)xls + ( - As) "+° - As) s s ( t - , )

J

× E ( - A s ) " °QDgj[(Xs('rl,wjs)+(xu(r),wj,)] dr (3.131 j = 1

and are written here in full as they will be referred to in the general situation of section 4. In the present case we have, by assumption, wj = wj~ and wjs =0 , and so x:(t) and Xs(t ) become decoupled. In fact, xu(t) in (3.13) is the solution of

J

: = Auz + ~ AuPDgj(z,wju ) , z E Xu,z(O)= Xo~, z(O)= x,~ (3.141 j = l

i This is the case when ~2 is a parallelepiped. 2This is the case when f~ is a sphere. 3In the case of one-dimensional ~ where J and fr are at most two, the unstable eigenspace X,

cannot be of dimension greater than three for the problem to be solvable.


which can be more conveniently rewritten as

2; = ./T.z (3.14a)

where A, is a square matrix of size equal to dim X,, depending on the vectors AuPDg / and ~ , (besides A,). This can be seen by using in X u the basis of orthonormal eigenvectors ~km, k = 1 . . . . . K - 1 which make the matrix corresponding to the operator A u diagonal. As we are seeking to obtain stabilizing boundary vectors g j ~ L2(F), we find it convenient to consider the projections (3.12)-(3.13) after setting

af df , p j = A . P D g / ; q /= ( - A s ) a - ° Q D g j , j = 1 . . . J (3.15)

and thinking of the vectors p /and q /as - - fo r the time being--simply vectors in X u and Xs, respectively, without any connection with the vectors g /which generate them. The question of synthesizing p/ and q/ through an appropriate g/wil l be taken up later on. With (3.15) in mind, suitable almost periodic solution to (3.15a) for a suitable choice of the vectors p / i s obtained through the following lemma.

Lemma 3.2. Assume the same conditions (3.10)-(3.11) as in Theorem 3.1. Then, (i) there exist vectors pj, j---1 . . . . . J in X,, such that the corresponding matrix A , in (3.14a) has a set of eigenvalues arbitrarily close to any preassigned set of (dim X~)- complex numbers. Moreover, each vector pj can indeed be synthesized, as required by the left equality of (3.15), by any of the infinitely many vectors g j ~ L2(F), satisfying the moment problem (3B.7) in Appendix 3B. The minimal number J required is discussed in Remark 3.4.

(ii) In particular, these eigenvalues of A~ may be preassigned to be all distinct and equal to negative constants - c 2i, i -- 1, . . . , dim X~, in which case, the solution to (3.14), or, equivalently, to (3.14a), is an almost periodic function on Xu

z ( t ) = C . ( t ) X o . + S u ( t ) X l u (3.16)

where

d i m X u

Cu(t)Xo. = ~ (Xo~,~i)coscit~i (3.16a) i = l

di~X~ ff~(t)xl~ = (X,~, ~i) sincit+i" (3.16b)

i = l Ci

Here ~i is the normalized eigenvector of Au corresponding to the simple eigenvalue - c 2 and the (ffi), i = 1 . . . . . dim Xuform a basis on X,.

Proof of Lemma 3.2. See Appendix 3B. []


Remark 3.5. A vector in X~ will always be referred to the basis (q~i)dii-ml x". Instead, a vector in X~ will always be referred to the basis {~km}*~r. Also, we shall adopt henceforth the following short notation for the projection v on X~ or X~ of a vector in X.

if v E Xu, we set ( v ) i = (v, ~i), i = 1 .. . . ,dim X u

i fv E Xs, w e s e t [ v ] k m = ( v , C b k m ) , k = K , K + 1 . . . .

Continuing with the proof of Theorem 3.1, from now on let the vectors pj be the ones provided by Lemma 3.2. Then

Xu(t)= Cu(t)Xou+ffu(t)Xlu (3.17)

with C,(t) and ff~(t) satisfying (3.16a, b). Now let the preassigned constants c i provided by Lemma 3.2 be chosen all different from ~f~-,, say, for later conveni- ence:

~K ~ - -C2 ~ " ' " ~ - -C2 ~ 0 . dim X. (3.18)

By (3.17), the convolutions (3.7), and the convention of Remark 3.5, we can write

fo'Sin X~k-k ( t - -r ) (Xu(T) , w j , ) dT

dim)(. ~k (COSCiI__COS~kI) = i=l ~ (XOu)i(Wju)i~kk--C2

+ ( x , . ) , ( w j . ) i f~k s inci t - cisin~X-~ t

ci X k - c~

1 ) (3.19)

We rewrite (3.13) as

~ Mk J Xs(t) = c,(t)Xos + Ss ( t )x . + "2 "2

k=Km=l j = l

× (3.20)

By virtue of (3.5) and (3.18)-(3.19), we conclude that xs ( t ) is an almost periodic function on X s. Actually, if

i ! l i

X o s ~ @ ( ( - A , ) ~-°) and Xl, E @(( - -A , ) ' - ° - ~ ) = [ @ ( - A , ) ~ + ° ] '


as postulated in (1.6), then (see (3.5)(ii))

i i _L ! J 2 - - 4--p--~ ( - A s l a - O x s ( t ) = C s ( t ) ( - A s ) ~ - O X o s + S s ( t l ( - A s ) ( A s ) x,s

Mk J

+ 2 2 k = K m = l j : l

× fo 'S in~ ( t - - r ) ( x~ ( r ) ,N . . )d ' r (q j ,OPkm)~k . , (3.21)

the infinite sum in k being convergent by the estimate (3.19). Therefore, one similarly concludes that ( - As)~ PXs(t ) is an Xs-almost periodic function, i.e., by (2.7), that xs(t ) is an ® ( - As)I-P = QH '~ ~(~2)-almost periodic function.

We now tackle the synthesis problem for pj and qj. The proof of Lemma 3.2 provides suitable vectors g j E L 2 ( F ), J = l . . . . . J (among the infinitely many satisfying the moment problem (3B.7)), generating the appropriate vectors A2 ~pj. Once the gj are selected, the qj are then defined by the equality at the right-hand side of (3.15). The proof of Theorem 3.1 is thus complete. []

Appendix 3A

Claim 1. Since P and Q commute with A, then they commute also with C(t). In fact, it follows from the assumptions that P and Q commute with R(-, A)

[12, Theorem 6.5, p. 173]. Then the general formula ((2.9)).

fo XR(X2, A )x = e X'C(t)xdt ReX > some constant ~%,x ~ X

gives

0 ~ XPR(X 2, A )x -- XR(X 2, A)Px

f f [ ] = e -x ' P C ( t ) x - C ( t ) P x dt ReX > ~o 0

and the uniqueness of Laplace transform yields the desired conclusion for P

PC( t )x = C( t )Px 0 < t <- oc

hence also for - o o < t <0 since C(t) is even. Similarly, for Q.

Claim 2. In particular, Claim 1 implies that X u and X s are invariant under the motion, i.e., under C( t ).


A p p e n d i x 3 B

Proof of Lemma 3.2 +

Part (i). Write pj = A~PDgj. Using the basis {t~)km } of eigenvectors in X,, 1 ~< k ~< K - 1 ; 1 ~< m ~< M k, one can show by direct computa t ion that the square matrix A-,, of size equal to dim X, = M l + • • • + M K_ i, can be written as

A. = A + f f W ,

where

A =

- - /~ . l I !

-- ~212 0

-- XX_IIK_l

W = [W1,W 2 . . . . . W K _ , ] ' J X (dim X, )

= [ p , . . . . . p+]:

and I k is the identity on the e igenspace- -o f dimension equal to M k (geometric mul t ip l ic i ty) - -corresponding to ( - )~k) - One further computes the dim X u × J- matrix

A"WT =[V,W, ,V2W2 . . . . . XK-,WK , ] T ( - - 1 ) " n n = 0 , 1 , 2 . . . .

where here and only here throughout the paper T stands for transpose. It is then easily seen that the assumption (3.10) is the necessary and sufficient condit ion for the dim X u × J (d im Xu)-matrix

[ W T, A W r, A2W r . . . . . AdimX.-- I w T] (3B.1)

to be of full rank. In this case, rewriting

A~ = A + W r P T, (3B.2)

a known result in system theory, see, e.g., [33], guarantees the existence of a matrix if(real, if A and W are real) such that the matrix A u has an arbitrary set of eigenvalues.

For the problem under study, however, it is crucial to synthesize the feedback matrix f i in terms of the boundary vectors g j E L 2 ( F ) . To this end, further analysis is needed and, for simiplicity of notation, we carry it out when the unstable eigenvalues K-- {?~k}k=l have geometric multiplicity equal to one: M k = 1.

IThis is, essentially, the proof given in the Appendix of [30], where, however, hypothesis (3.11) was inadvertently omitted from the list of assumptions..


In this case, we write ~k instead of (I~kl. Also, throughout this appendix, the symbol ( . , . ) denotes the inner product on L2(F ). If gj E L2(I'), then

K--1 K--1

eDgj = E (Dgj,OPk)~k = E (gj, D*O~k)~k k=l k= l

where in the last step we have made use of formula (2), Chapter II in the thesis ~ [32]. Hence, the coordinates of the vectors pj = A~PDgj with respect to the basis (e~k} in X~ are:

0n FJ ..... F)] (3B.3)

Two cases have to be considered now. f 0(I) k I ]K--1

(a) First, if the traces t. j ~ F~jk~' of the eigenvectors happen to be linearly

independent, 2 then vectors gj in Lz(F ) can be found which synthesize, through the moment problem specified in (3B.3), the required coordinates of the stabilizing matr ix /~ Actually, for each given pj, there are infinitely many synthesizing vectors gj, as only the coordinates with respect to the first ( K - 1) traces are committed. In the present case (a), the minimum number J of boundary vectors gj required is plainly one, equal to the multiplicity of the unstable eigenvalues.

(b) Suppose now that the subspace

d f span v ~k is of dimension ~T < K - 1 0~7 r k = l

in which case 3 we assume that they are the first ~r traces independent: k = 1 . . . . . gr and hence that

that are linearly

0(I) k ~T 0 (I)j [ E "Ykj'-~-'~ k ~-- ~T -[- 1 . . . . . K - 1. (3B.4) a~

F j 1 F

Then, the sought-after stabilizing vectors Pl ..... pj are restricted to lie on a subspace @ of X, which is (i) isometrically isomorphic to ~ and (ii) of dimension e r •

1 O~k., [ 1i.e., of D * ~ k , , - X k 0~ Ir

2This is the case when ~2 is a parallelepiped. 3This is the case when ~ is a sphere.


Let C be a transformation from X, into itself such that C X , =range of C = ~. By imposing the conditions

C~ k = • k k = 1, . . . , l r

IT

C¢~k = E ~kjOj k = f r + 1 , . . . , K - 1 j = l

the transformation C is specified by the matrix (denoted with the same symbol) C given by

C =

"YIT+ 1,1 "'" ~/¢T + I,~T

YK 1,1 . . . YK-I ,e~

t

0 ~T

0

(3B.5)

Further, we demand that

pj = C97j j = 1 . . . . . ] (3B.6)

for some vectors

that, in fact, by virtue of (3B.3)-(3B.6) must satisfy

pj = --

g j , aO1

0~T

gJ' -an-n

/~ OOk r

k=l

~T aO, r YK-I,k ~'r/

k = l

~T

E "YgT + 1, k'B'j(k) k = l

~T

E ~K--l,kq"l'j(k) k = l


This, equivalently, requires that gj be any solution (among the infinitely many) of the moment problem

_ ( g j , OCk = j = 1 . . . . . ¢r" (3B.7)

Hence, from (3B.2) eigenvalues)

o A + W r ~7.]

-'-J I

can be rewritten as

and (3B.6) our requirement that the spectrum ( i .e . , the

E C - = {X:ReX<0}

o A + w r i c r e c - (3B.8)

which can be interpreted as the output s tabil izabil i ty problem of the system (with state space Xu)

p = A y + W r u (state equation)

o~ = C r y (output equation) (3B.9)

rank W r = ~ w (defined below (3.11)), rank C r = er, with output feedback

U "~- (3B.10)

A complete solution of the finite-dimensional output stabilizability problem is not yet available. Present theory ([1], [4], [13]) requires that

(i) the pair [A, W T] be a controllable pair (ii) the pair [A, C T] be an observable pair

(iii) dim X, ~<gr +gw - 1 to guarantee the existence of a J × ( K - 1) feedback matrix as in (3B.10) such that all the eigenvalues of

A + W r i c r


be negative and distinct. From the stabilizing ~ , j -- 1 . . . . , J the gj are obtained via (3B.7). Therefore it remains to verify (i), (ii), (iii). Assumption (i) was proved to hold in (3B. 1). Assumption (iii) holds by hypothesis (3.11) and requires taking J such that J~>dim X~ - ~r + 1) As for assumption (ii), if we partition the matrix C in (3B.5) in row ( K - 1)-dimensional vectors c I . . . . . cK-l, the characterization that the observability matrix

c

[ C , A C , . . . , Ad imX, -1C ] c 2

CK- 1

-X~cl ... ( - -Xl )~ - l c l

--X2c2 ... ( - -X2) r - l c2

--X~_~c~_~ ... ( ~-~ --X~_~) cK-1

be of full rank is equivalent to

r ankck= 1 k - - 1 . . . . . K - 1 .

This, by virtue of (3B.5) and (3B.4) is, in turn, equivalent to the requirement that

0 ~ r 07 4=0 k = e r + l . . . . . K - - 1 (3B.11)

which is guaranteed by recent results I [24; Cor. 2.2] based on deep uniqueness continuation arguments. The proof of part (i) of Lemma 3.2 is thus complete. Part (ii). As to the second part of Lemma 3.2, we can obtain the eigenvalues of A, to be the distinct negative numbers ( --t;i-2)i=~diml Xu, e i- > 0, by preassigning distinct negative numbers in the first place [E. J. Davison, personal communication]. Let then {q~} be the corresponding eigenvectors spanning therefore all of X,:

dim X u

Xou= E (XOu'~i)~i X o u ~ g u (3B.12) i=1

Next compute C,(t) by means of

i ~ 7n 2n e (t) = cos(-A ) t - - Aot

n=0 (2n)! " (3B.13)

Applying/T~ to x0~ in (3B. 12) and using (3B. 13) then yields the desired conclusion (3.16a), from which (3.16b) follows by integration. []

We illustrate next the application of Lemma 3.2 to a simple example, where the Neumann traces of the unstable eigenfunctions are linearly dependent.

t i n the case of one-dimensional ~, where J and er are at most equal to two, the unstable eigenspace X u cannot be of dimension greater than three for the problem to be solvable.

l for domains with C~-boundary; for parallelepipeds, the truth of (3B.I 1) is verified directly.


Example 3.1. Let ~2 be the two-dimensional unit disk. The first nonzero roots akin of the Bessel functions Jk('): Jk(akm) =-0 are

I I I I I I 0 Or01 all OL21 Or02 a31

see [22, pp. 458-60]. We can, therefore, choose a constant C2 SO that the only unstable eigenvalues of A + c 2 with zero Dirichlet B.C. are

I I I I I I

~5 ~k4 ~k3 ~2 ~kl

where

X4= __0122..~C2; X5 = __0/21.~_C 2.

Each such eigenvalue ?~k has multiplicity M k = 2, and the N e u m a n n traces of the corresponding eigenfunctions are (proportional to): sin k0,cos kO. Therefore, for fixed k, they are linearly independent. However, the eigenvalues ?~ and ?~4 originate from the same Bessel function JI, and therefore the corresponding eigenfunctions give rise to linearly dependent traces. We have, in this case,

d i m X , = 10 ~r = 8 2 ~< ~w ~< m i n ( J , 10).

Conditions (3.10) can be satisfied with suitable wj, while to satisfy (3.11) we need to restrict the wj's further so that C w ~> 3. We conclude that Lemma 3.2 applies in this case for three suitable vectors Wl,W2,W 3, chosen so that: r a n k W ~ 2 , k = 1,. . . 5, and rank W = 3. The minimum number J that works is therefore three.

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Accepted for publication October 13, 1980

Hyperbolic equations with Dirichlet boundary feedback via

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