th th th SEAMS - GMU 2011 Yogyakarta - Indonesia, 12 - 15 July 2011 th th Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications th Mathematics and Its Applications in the Development of Sciences and Technology. Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications th Department of Mathematics Faculty of Mathematics & Natural Sciences Universitas Gadjah Mada Sekip Utara Yogyakarta - INDONESIA 55281 Phone : +62 - 274 - 552243 ; 7104933 Fax. : +62 - 274 555131 MATHEMATICS AND ITS APPLICATIONS IN THE DEVELOPMENT OF SCIENCES AND TECHNOLOGY International Conference on Mathematics and Its Applications The ISBN 978-979-17979-3-1
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
ththth
SEAMS - GMU 2011
Yogyakarta - Indonesia, 12 - 15 July 2011th th
Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications
th
Mathem
atics and Its Applications in the
Developm
ent of Sciences and Technology.
Proceedings of the 6 S
EA
MS
-GM
U International
Conference on M
athematics and Its A
pplications
th
Department of Mathematics
Faculty of Mathematics & Natural Sciences
Universitas Gadjah Mada
Sekip Utara Yogyakarta - INDONESIA 55281
Phone : +62 - 274 - 552243 ; 7104933
Fax. : +62 - 274 555131
MATHEMATICS AND ITS APPLICATIONS IN THE DEVELOPMENT OF SCIENCES AND TECHNOLOGY
International Conference on Mathematics and Its Applications
TheISBN 978-979-17979-3-1
PROCEEDINGS OF THE 6TH
SOUTHEAST ASIAN MATHEMATICAL SOCIETY
GADJAH MADA UNIVERSITY
INTERNATIONAL CONFERENCE ON MATHEMATICS
AND ITS APPLICATIONS 2011
Yogyakarta, Indonesia, 12th – 15th July 2011
DEPARTMENT OF MATHEMATICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
UNIVERSITAS GADJAH MADA
YOGYAKARTA, INDONESIA
2012
Published by
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Universitas Gadjah Mada
Sekip Utara, Yogyakarta, Indonesia
Telp. +62 (274) 7104933, 552243
Fax. +62 (274) 555131
PROCEEDINGS OF
THE 6TH SOUTHEAST ASIAN MATHEMATICAL SOCIETY-GADJAH MADA UNIVERSITY
INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS APPLICATIONS 2011
Copyright @ 2012 by Department of Mathematics, Faculty of Mathematics and
Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia
ISBN 978-979-17979-3-1
PROCEEDINGS OF THE 6TH
SOUTHEAST ASIAN MATHEMATICAL SOCIETY-GADJAH MADA
UNIVERSITY
INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS
APPLICATIONS 2011
Chief Editor:
Sri Wahyuni
Managing Editor :
Indah Emilia Wijayanti Dedi Rosadi
Managing Team :
Ch. Rini Indrati Irwan Endrayanto A. Herni Utami Dewi Kartika Sari
Nur Khusnussa’adah Indarsih Noorma Yulia Megawati Rianti Siswi Utami
Soeharyadi (Indonesia), Subanar (Indonesia) Supama (Indonesia), Asep K. Supriatna
(Indonesia) and Indah Emilia Wijayanti (Indonesia). Most of the contributed papers
were delivered by mathematicians from Asia.
We would like to sincerely thank all plenary and invited speakers who
warmly accepted our invitation to come to the Conference and the paper
contributors for their overwhelming response to our call for short presentations.
Moreover, we are very grateful for the financial assistance and support that we
received from Universitas Gadjah Mada, the Faculty of Mathematics and Natural
Sciences, the Department of Mathematics, the Southeast Asian Mathematical
Society, and UNESCO.
We would like also to extend our appreciation and deepest gratitude to all
invited speakers, all participants, and referees for the wonderful cooperation, the
great coordination, and the fascinating efforts. Appreciation and special thanks are
addressed to our colleagues and staffs who help in editing process. Finally, we
acknowledge and express our thanks to all friends, colleagues, and staffs of the
Department of Mathematics UGM for their help and support in the preparation
during the conference.
The Editors
October, 2012
CONTENTS
Title i Publisher and Copyright ii Managerial Boards iii Editorial Boards iv List of Reviewers v Preface vii Paper of Invited Speakers On Things You Can’t Find : Retrievability Measures and What to do with Them ……............... 1 Andreas Rauber and Shariq Bashir
A Quasi-Stochastic Diffusion-Reaction Dynamic Model for Tumour Growth ..……................... 9 Ang Keng Cheng
*-Rings in Radical Theory ...………………………………………………..................................................... 19 H. France-Jackson
Clean Rings and Clean Modules ...………………………………………................................................... 29 Indah Emilia Wijayanti
Research on Nakayama Algebras ……...…………………………………................................................. 41 Intan Muchtadi-Alamsyah
Mathematics in Medical Image Analysis: A Focus on Mammography ...…............................... 51 Murk J. Bottema, Mariusz Bajger, Kenny MA, Simon Williams
The Order of Phase-Type Distributions ..………………………………….............................................. 65 Reza Pulungan
The Linear Quadratic Optimal Regulator Problem of Dynamic Game for Descriptor System… 79 Salmah
Chaotic Dynamics and Bifurcations in Impact Systems ………………........................................... 89 Sergey Kryzhevich
Contribution of Fuzzy Systems for Time Series Analysis ……………….......................................... 121 Subanar and Agus Maman Abadi
Contributed Papers Algebra
Degenerations for Finite Dimensional Representations of Quivers ……................................... 137 Darmajid and Intan Muchtadi-Alamsyah
On Sets Related to Clones of Quasilinear Operations …...……………………………………………………. 145 Denecke, K. and Susanti, Y.
Normalized H Coprime Factorization for Infinite-Dimensional Systems …………………………… 159 Fatmawati, Roberd Saragih, Yudi Soeharyadi
Construction of a Complete Heyting Algebra for Any Lattice ………………………………………………. 169 Harina O.L. Monim, Indah Emilia Wijayanti, Sri Wahyuni
The Fuzzy Regularity of Bilinear Form Semigroups …………………………………………………..…………. 175 Karyati, Sri Wahyuni, Budi Surodjo, Setiadji
The Cuntz-Krieger Uniqueness Theorem of Leavitt Path Algebras ………………………………………. 183 Khurul Wardati, Indah Emilia Wijayanti, Sri Wahyuni
Application of Fuzzy Number Max-Plus Algebra to Closed Serial Queuing Network with
Fuzzy Activitiy Time ………………………………………………………………………………………………..…………… 193 M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo
Enumerating of Star-Magic Coverings and Critical Sets on Complete Bipartite Graphs………… 205 M. Roswitha, E. T. Baskoro, H. Assiyatun, T. S. Martini, N. A. Sudibyo
Construction of Rate s/2s Convolutional Codes with Large Free Distance via Linear System
Approach ………………………………………………………………………………………………........…………………….. 213 Ricky Aditya and Ari Suparwanto
Characteristics of IBN, Rank Condition, and Stably Finite Rings ………....................................... 223 Samsul Arifin and Indah Emilia Wijayanti
The Eccentric Digraph of n mP P Graph ………………………………………………………………….….……… 233 Sri Kuntarti and Tri Atmojo Kusmayadi
On M -Linearly Independent Modules ……………………………………………………………………….......... 241 Suprapto, Sri Wahyuni, Indah Emilia Wijayanti, Irawati
The Existence of Moore Penrose Inverse in Rings with Involution …........................................ 249 Titi Udjiani SRRM, Sri Wahyuni, Budi Surodjo
Analysis
An Application of Zero Index to Sequences of Baire-1 Functions ………….................................. 259 Atok Zulijanto
Regulated Functions in the n-Dimensional Space ……………………....................….....................… 267 Ch. Rini Indrati
Compactness Space Which is Induced by Symmetric Gauge ……..........................………………... 275 Dewi Kartika Sari and Ch. Rini Indrati
A Continuous Linear Representation of a Topological Quotient Group …................................ 281 Diah Junia Eksi Palupi, Soeparna Darmawijaya, Setiadji, Ch. Rini Indrati
On Necessary and Sufficient Conditions for L into 1 Superposition Operator ……...………... 289 Elvina Herawaty, Supama, Indah Emilia Wijayanti
A DRBEM for Steady Infiltration from Periodic Flat Channels with Root Water Uptake ……….. 297 Imam Solekhudin and Keng-Cheng Ang
Boundedness of the Bimaximal Operator and Bifractional Integral Operators in Generalized
Morrey Spaces …………………………………...................................................................................... 309 Wono Setya Budhi and Janny Lindiarni
Applied Mathematics
A Lepskij-Type Stopping-Rule for Simplified Iteratively Regularized Gauss-Newton Method.. 317 Agah D. Garnadi
Asymptotically Autonomous Subsystems Applied to the Analysis of a Two-Predator One-
Prey Population Model ............................................................................................................. 323 Alexis Erich S. Almocera, Lorna S. Almocera, Polly W.Sy
Sequence Analysis of DNA H1N1 Virus Using Super Pair Wise Alignment .............................. 331 Alfi Yusrotis Zakiyyah, M. Isa Irawan, Maya Shovitri
Optimization Problem in Inverted Pendulum System with Oblique Track ……………………………. 339 Bambang Edisusanto, Toni Bakhtiar, Ali Kusnanto
Existence of Traveling Wave Solutions for Time-Delayed Lattice Reaction-Diffusion Systems 347 Cheng-Hsiung Hsu, Jian-Jhong Lin, Ting-Hui Yang
Effect of Rainfall and Global Radiation on Oil Palm Yield in Two Contrasted Regions of Sumatera, Riau and Lampung, Using Transfer Function ..........................................................
Expected Value Approach for Solving Multi-Objective Linear Programming with Fuzzy
Random Parameters …..................................................................................................... ......... 427 Indarsih, Widodo, Ch. Rini Indrati
Chaotic S-Box with Piecewise Linear Chaotic Map (PLCM) ...................................................... 435 Jenny Irna Eva Sari and Bety Hayat Susanti
Model of Predator-Prey with Infected Prey in Toxic Environment .......................................... 449 Lina Aryati and Zenith Purisha
On the Mechanical Systems with Nonholonomic Constraints: The Motion of a Snakeboard
on a Spherical Arena ………………………………………..……………………................................................ 459 Muharani Asnal and Muhammad Farchani Rosyid
Safety Analysis of Timed Automata Hybrid Systems with SOS for Complex Eigenvalues …….. 471 Noorma Yulia Megawati, Salmah, Indah Emilia Wijayanti
Global Asymptotic Stability of Virus Dynamics Models and the Effects of CTL and Antibody Responses ………………………………………………………………………………….……………………………………….. 481
Nughtoth Arfawi Kurdhi and Lina Aryati
A Simple Diffusion Model of Plasma Leakage in Dengue Infection …………………………..………… 499 Nuning Nuraini, Dinnar Rachmi Pasya, Edy Soewono
The Sequences Comparison of DNA H5N1 Virus on Human and Avian Host Using Tree
Diagram Method …………………………………..……………………………………………………………..…………….. 505 Siti Fauziyah, M. Isa Irawan, Maya Shovitri
Fuzzy Controller Design on Model of Motion System of the Satellite Based on Linear Matrix
Unified Structural Models and Reduced-Form Models in Credit Risk by the Yield Spreads …. 697 Di Asih I Maruddani, Dedi Rosadi, Gunardi, Abdurakhman
The Effect of Changing Measure in Interest Rate Models …….…....................…....................... 705 Dina Indarti, Bevina D. Handari, Ias Sri Wahyuni
New Weighted High Order Fuzzy Time Seriesfor Inflation Prediction ……................................ 715 Dwi Ayu Lusia and Suhartono
Detecting Outlier in Hyperspectral Imaging UsingMultivariate Statistical Modeling and
Numerical Optimization ………………………………………...........................................……………......... 729 Edisanter Lo
Prediction the Cause of Network Congestion Using Bayesian Probabilities ............................. 737 Erwin Harapap, M. Yusuf Fajar, Hiroaki Nishi
Solving Black-Scholes Equation by Using Interpolation Method with Estimated Volatility……… 751 F. Dastmalchisaei, M. Jahangir Hossein Pour, S. Yaghoubi
Artificial Ensemble Forecasts: A New Perspective of Weather Forecast in Indonesia ............... 763 Heri Kuswanto
Second Order Least Square for ARCH Model …………………………....................………………............. 773 Herni Utami, Subanar, Dedi Rosadi, Liqun Wang
Two Dimensional Weibull Failure Modeling ……………………..…..................……….......................... 781 Indira P. Kinasih and Udjianna S. Pasaribu
Simulation Study of MLE on Multivariate Probit Models …......................................................... 791 Jaka Nugraha
Clustering of Dichotomous Variables and Its Application for Simplifying Dimension of
Quality Variables of Building Reconstruction Process ............................................................. 801 Kariyam
Valuing Employee Stock Options Using Monte Carlo Method ……………................................. 813 Kuntjoro Adji Sidarto and Dila Puspita
Classification of Epileptic Data Using Fuzzy Clustering .......................................................... 821 Nazihah Ahmad, Sharmila Karim, Hawa Ibrahim, Azizan Saaban, Kamarun Hizam
Mansor
Recommendation Analysis Based on Soft Set for Purchasing Products ................................. 831 R.B. Fajriya Hakim, Subanar, Edi Winarko
Heteroscedastic Time Series Model by Wavelet Transform ................................................. 849 Rukun Santoso, Subanar, Dedi Rosadi, Suhartono
Parallel Nonparametric Regression Curves ............................................................................ 859 Sri Haryatmi Kartiko
Ordering Dually in Triangles (Ordit) and Hotspot Detection in Generalized Linear Model for
Poverty and Infant Health in East Java ……................................…………………………………………. 865 Yekti Widyaningsih, Asep Saefuddin, Khairil Anwar Notodiputro, Aji Hamim Wigena
Empirical Properties and Mixture of Distributions: Evidence from Bursa Malaysia Stock
Market Indices …………………....................................................................................................... 879 Zetty Ain Kamaruzzaman, Zaidi Isa, Mohd Tahir Ismail An Improved Model of Tumour-Immune System Interactions …………………………………………….. 895 Trisilowati, Scott W. Mccue, Dann Mallet
79
Proceedings of ”The 6th
SEAMS-UGM Conference 2011”
pp. 79 - 88
QUADRATIC OPTIMAL REGULATOR PROBLEM OF
DYNAMIC GAME FOR DESCRIPTOR SYSTEM
SALMAH
Abstract. In this paper the noncooperative linear quadratic game problem will be
considered. We present necessary and sufficient conditions for existence of optimal strategy for linear quadratic continuous non-zero-sum two player dynamic games for index
one descriptor system. The connection of the game solution with solution of couple Riccati
equation will be studied. In noncooperative game with open loop structure, we study Nash solution of the game. If the second player is allowed to select his strategy first, he is called
the leader of the game and the first player who select his strategy at the second time is
called the follower.A stackelberg strategy is the optimal strategy for the leader under the assumption that the follower reacts by playing optimally. Keywords and Phrases : Dynamic, game, noncooperative, descriptor, system
1. INTRODUCTION
Dynamic game theory brings three keys to many situations in economy, ecology, and
elsewhere: optimizing behavior, multiple agents presence, and decisions consequences.
Therefore this theory has been used to study various policy problems especially in macro-
economic. In applications one often encounters systems described by differential equations
system subject to algebraic constraints. The descriptor systems, gives a realistic model for this
systems.
In policy coordination problems, questions arise, are policies coordinated and which
information do the parties have. One scenario is noncooperative open-loop game. According
this, the parties can not react to each other’s policies, and the only information that the
players know is the model structure and initial state.
In this paper we will consider a linear open-loop dynamic game in which the player
satisfy a linear descriptor system and minimize quadratic objective function. For finite
horizon problem, solution of generalized Riccati differential equation is studied. If the
planning horizon is extended to infinity the differential Riccati equation will become an
algebraic Riccati equation.
80 SALMAH
2. PRELIMINARIES
The players are assumed to minimize the performance criteria:
,)()()()(2
1)()(
2
1),...,,(
0 1
21 dttuRtutxQtxTExKETxuuuJ
T n
j
jij
T
ji
T
iT
TT
ni
(2.1)
with all matrices symmetric. Furthermore and semi positive definite and positive
definite, where the players give control vector to the system
, (2.2)
( ) .
with
, x(t) descriptor
vector n dimension. While ui(t), i=1,…,n are control vector dimension which is done by i-
th player, i=1,…,n. Matrix E generally singular with rank
Below is definition for Nash equilibrium strategy.
Definition2.1.The pair (
) is called Nash equilibrium strategy if
(
) ( )
(
) ( )
for all admissible strategies , .
If the second player is allowed to select his strategy first, he is called the leader of the
game and the first player who select his strategy at the second time is called the follower. A
stackelberg strategy is the optimal strategy for the leader under the assumption that the
follower reacts by playing optimally.
Assumption which is needed will be given.
Assumption 2.1:Descriptor system (1) regular, impulse controllable and finite dynamic
stabilizable which satisfy
(i). | | , except for a finite number of ,
(ii). ( ) ( ) ,
(iii). ( | ) [ ]
3. NASH EQUILIBRIUM OF DESCRIPTOR GAME
The To derive necessary condition of opnimal Nash solution we need the Hamiltonian
functions as follow.
( )
(
)
( ),
( )
(
)
( ),
With Lagrange multiplier method as in [11] we get necessary conditions for objective
function to be optimal in the Nash sense are
,
,
i=1,2. (3.1)
Substitute these equations to (1) yields
Quadratic Optimal Regulator Problem of Dynamic Game for Descriptor System 81
( ) ( ) ( ) (3.2)
and
with i=1,2. (3.3)
The boundary conditions are
( ) and ( ) ( ). (3.4)
From necessary condition of optimal Nash solution we get optimal strategies for 2
players dynamic game are ( ) with satisfy (3.2) and boundary
conditions(3.4), i=1,2.
We can write in matrix form and get
(
)(
( ) ( )
( )) (
)(
( ) ( )
( )) , (3.5)
with boundary conditions (3.4) . System (3.5) can be written in descriptor form as
. If ( ) regular, system (3.5) will have solution (see [11]). We need the
following assumptions for equation (3.5).
Assumption 3.1: Descriptor system (3.5) is regular and impulse free i.e
( ) .
If Assumption 3.1 is satisfied then system (3.5) will be regular and impulse controllable.
For 2 players linear quadratic dynamic game define 2 generalized differential Riccati
equation as follow
with
with boundary condition
( ) (3.6)
The following theorem concern with relationship between the existence of dynamic
game solution and generalized Riccati differential equation (3.6).
Theorem 3.1: The two player linear quadratic discrete dynamic game (2.1), (2.2) has, for
every consistent initial state, a Nash equilibrium if the set of differenttial Riccati equation
(3.6) has a set of solutions on [0,T].
Moreover the optimal feedback Nash equilibrium is given by
( )
( ) ( ), ( )
( ) ( ), where x(t) is a solution of the closed loop system
( ) (
( )
( )) ( ) ( )
PROOF: Let the player choose strategy
( )
( ) ( ), ( )
( ) ( ), to control system (3.1), (3.2) with ( ) ( ) solusion of(3.6).
Define ( ) ( ) ( ), and ( ) ( ) ( ), we get
)()()()()( 111 txtKEtxtKEtE TTT ,
)()()()()( 222 txtKEtxtKEtE TTT .
From (2.2) we get
82 SALMAH
)()()()()( 22
1
22211
1
111 txtKBRBtxtKBRBtAxxE TT ,
Therefore we get
xtKExKBRBLxKBRBLxQAxLxKAtE TTTTT )()( 122
1
222111
1
11111111
xELxKBRBLxKBRBLxQAxLxKA TTT 122
1
222111
1
1111111
xKBRBLxKBRBLxQAxLxKA TTT
22
1
222111
1
1111111
xKBRBLxKBRBLAxL TT
22
1
222111
1
11111
)()(11
txQtxKAT ,
and with same reason we get
)()()(222
txQtxKAtE TT .
This two equation has solution.
For 2 players infinite time linear quadratic dynamic game the players satisfy system (1).
Objective function to be minimized are in the form
,)()()()(2
1),(
0
2
1
21 dttuRtutxQtxuuJj
jij
T
ji
T
i
i=1,2 (3.7)
with all matrices symmetric. Furthermore and semi positive definite and positive
definite.
Generalized algebraic Riccati equation for 2 players infinite time problem that related
with Nash equilibrium are
(3.8)
with .
It can be proved that Theorem 1 will also be satisfied for optimal control for infinite
time problem, therefore we get the optimal Nash have form ( )
( ) ( ), with are constant matrices, solution of (3.8).
4. STACKELBERG EQUILIBRIUM OF DESCRIPTOR GAME
To derive necessary condition of opnimal Nash solution we need the Hamiltonian functions
for the follower is.
( )
(
)
( ),
With Lagrange multiplier method as in [11] we get necessary conditions for objective
function for the follower to be optimal is
,
,
i=1,2. (4.1)
From the first equation of (4.1) we get
( ) ( ) ( ). (4.2)
From the second equation of (4.2) we get the optimal control for the follower is
. (4.3)
The boundary conditions are
Quadratic Optimal Regulator Problem of Dynamic Game for Descriptor System 83
( ) and ( ) ( ). (4.4)
For the second player as the leader define Hamiltonian
( )
(
)
( ). (4.5)
Let get deridative of (4.4) to ( ) we get
. (4.6)
Let get deridative of (4.4) to ( ) we get
. (4.7)
With the second player as the leader let the Hamiltonian is
( )
(
)
( )
(
),
or
( )
(
)
( )
( ) (
). (4.8)
Let get derivative of (4.8) to x we get
( ) ( ) ( ) ( ). (4.9)
Let get derivative of (4.8) to we get
(
). (4.10)
Let get derivative of (4.8) to we get
. (4.11)
Let get derivative of (4.8) to we get
. (4.12)
Substitute (4.9) to (4.12) we get
. (4.13)
Substitute (4.3) to (4.13) we get
. (4.14)
From necessary condition of optimal Stackelberg solution we get optimal strategies for 2
players dynamic game are ( ) with satisfy (4.2), (4.8) and (4.13) and
boundary conditions(4.4), i=1,2.
We can write in matrix form and get
2
1
12
1
121
21
2
1
0
00
0
0
000
000
000
000
x
AQQ
AQ
SSA
SSAx
E
E
E
E
T
T
T
T
T
(4.15)
with boundary conditions (4.4) . System (4.15) can be written in descriptor form as .
If ( ) regular, system (4.15) will have solution. We need the following assumptions for
equation (4.15).
Assumption 4.1: Descriptor system (4.15) is regular and impulse free i.e
( ) .
If Assumption 4.1 is satisfied then system (4.15) will be regular and impulse
84 SALMAH
controllable.
For 2 players Stackelberg linear quadratic dynamic game define generalized differential
Riccati equation as follow
with
.
with boundary condition
( ) , ( ) . (4.16)
The following theorem concern with relationship between the existence of dynamic
game solution and generalized Riccati differential equation (4.16).
Theorem 4.1: The two player linear quadratic discrete dynamic game (2.1), (2.2) has, for
every consistent initial state, a Stackelberg equilibrium if the set of differenttial Riccati
equation (4.16) has a set of solutions on [0,T].
Moreover the optimal feedback Nash equilibrium is given by
( )
( ) ( ), ( )
( ) ( ) Where x(t) is a solution of the closed loop system
( ) (
( )
( )) ( ) ( ) .
PROOF: Let , and . We get . Substitute this to
(4.2) and because we get the first equation of (4.16). Because
substitute to (4.9) and because we get the second equation of (4.16). Because
, substitute to (4.14) we get the third equation of (4.16).
Generalized algebraic Riccati equation for 2 players infinite time problem that related
with Nash equilibrium are
With . (4.17)
5. NUMERICAL EXAMPLE
We will give numerical example to find optimal Nash solution of game by try to find ARE
solution. Consider system
,0
1
1
0
01
10
00
0121
2
1
2
1uu
x
x
x
x
(4.18)
101 )0( xx , 202 )0( xx
.
For the cost function, given
.1,2,1,11
10,
10
01212121
RRRQQ
We will find solution of the generalized algebraic Riccati equation (4.18) to get optimal
Nash of the game.
Because , we have
Quadratic Optimal Regulator Problem of Dynamic Game for Descriptor System 85
)1,1()1,1( 11 LK ,
0)1,2(1 L,
0)2,1(1 K.
Because we have
)1,1()1,1( 22 LK ,
0)1,2(2 L,
0)2,1(2 K.
Substitute the result to first algebraic Riccati equation (3.8) will give the following equations.
,0)1,1()1,1(2
1)1,2()2,1(1)2,1()1,2( 211111 KLKLLK
(4.19)
,0)2,2()2,1()1,1()2,2( 1111 KLLK (4.20)
,0)1,2()2,2()2,2()1,1( 1111 KLLK (4.21)
0)2,2()2,2(1 11 KL. (4.22)
Take ( ) , we get ( )
and based on (4.21), we get
1)1,1()1,2( 11 aKK. (4.23)
Based on (4.22) we get
)1,1(1
1)2,1( 11 Ka
L . (4.24)
Substitute (4.23), (4.24) to (4.19) give
02)1,1(2
3)1,1(
2
1 2
11 KK. (4.25)
Because of the second algebraic Riccati equation we get
,0)1,1()1,1(2
1)1,2()2,1()2,1()1,2( 221222 KLKLLK
(4.26)
,01)2,2()2,1()1,1()2,2( 1222 KLLK (4.27)
,01)1,2()2,2()2,2()1,1( 1222 KLLK (4.28)
.0)2,2()2,2(1 12 KL (4.29)
Based on (4.29) and because of ( ) , give ( )
. Based on (4.27) and
(4.28) we get ( ) ( ) ( ) and ( ) ( ) . Let L2(1,2)=b we
get ( ) ( ) Because ( ) ( ) ( ) , ( ) ( ) and based on (4.27)
we get
)1,1(2
1)1,1()1,2( 2
212 KabKK . (4.30)
Based on (4.29), (4.26) and (4.27) we get
2
112 )1,1(12
1)1,1()1,2( KabKK
. (4.31)
Therefore we have
86 SALMAH
,1)1,1(
0)1,1(
1
1
1
aaK
KK N
(4.32)
Let K2(2,1)=c we can write
,)1,1(
0)1,1(1
1
1
2
Kabc
KK N
(4.33)
where ( ) and ( ) can be found from (4.24) and (4.30). Solution for ( ),
( )are ( )
. Take ( )
we get ( )
.
Optimal Nash control gain for the players is given by
aa
K N
13
4
03
4
1
3
4
18
1
3
4
03
41
2
abab
K N
. (4.34)
Now we will find the Stackelberg equilibrium of the game. Because we get
. From the first Riccati equation (4.17) we get (4.19)-(4.22). From the second
Quadratic Optimal Regulator Problem of Dynamic Game for Descriptor System 87
From (4.42) we get
( )( ) ( ) .
From (4.35) we get
( ) ( )
( ( ))
( ). (4.43)
From (4.40) we get
( ) ( ) .
From (4.39) and (4.41) we get
( )
.
From (4.39) and (4.43) we get
( ) ( ( )
( ( ))
( ))
( ) ( ) ( ) ( ). (4.44)
Therefore we can find ( ) from (4.44). Let ( ) . From (4.39) we can get
( ). Let ( ) . Then we get optimal Stackelberg for the game is given by
(
( ) ), (
( )
).
6. CONCLUDING REMARK
This paper consider 2 player non-zero-sum linear quadratic dynamic game with descriptor
systems for finite horizon and infinite horizon case. Necessary condition for the existence of a
Nash equilibrium and Stackelberg equilibrium have been derived with Hamiltonian method.
The paper also consider 2 couple Riccati-type differential equation for finite horizon case and
algebraic Riccati equation for infinite horizon case related with Nash equilibrium and
Stackelberg equilibrium.
References
[1] BASAR, T., AND OLSDER, G.J., Dynamic Noncooperative Game Theory, second Edition,
Academic Press, London, San Diego, 1995.
[2] DAI, L., Singular Control Systems, Springer Verlag, Berlin, 1989. [3] ENGWERDA, J., On the Open-loop Nash Equilibrium in LQ-games, Journal of Economic
Dynamics and Control, Vol. 22, 729-762, 1998.
[4] ENGWERDA, J.C., LQ Dynamic Optimization and Differential Games, Chichester: John
Wiley & Sons, 229-260, 2005.
[5] KATAYAMA, T., AND MINAMINO, K., Linear Quadratic Regular and Spectral
Factorization for Continuous Time Descriptor Systems, Proceedings of the 31st
Conference on decision and Control, Tucson, Arizona, 967-972, 1992.
88 SALMAH
[6] LEWIS, F.L., A survey of Linear Singular Systems, Circuits System Signal Process,
vol.5, no.1, 3-36, 1986.
[7] MINAMINO, K., The Linear Quadratic Optimal Regulator and Spectral Factorization for
Descriptor Systems, Master Thesis, Department of Applied Mathematics and Physics,
Faculty of Engineering, Kyoto University, 1992.
[8] MUKAIDANI, H. AND XU, H., Nash Strategies for Large Scale Interconnected Systems,
43rd IEEE Conference on Decision and Control Bahamas, 2004, pp 4862-4867.
[9] SALMAH, BAMBANG, S., NABABAN, S.M., AND WAHYUNI, S., Non-Zero-Sum Linear
Quadratic Dynamic Game with Descriptor Systems, Proceeding Asian Control
Conference, Singapore, pp 1602-1607, 2002.
[10] SALMAH, N-Player Linear Quadratic Dynamic Game for Descriptor System,
Proceeding of International Conference Mathematics and its Applications SEAMS-
GMU, Gadjah Mada University, Yogyakarta, Indonesia, 2007.
[11] XU, H., AND MIZUKAMI, K., Linear Quadratic Zero-sum Differential Games for
Generalized State Space Systems, IEEE Transactions on Automatic Control, Vol. 39