MATHEMATICS AND ITS APPLICATIONS IN THE …

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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

Hadrian Andradi

Supporting Team :

Parjilan Warjinah Siti Aisyah Emiliana Sunaryani Yuniastuti

Susiana Karyati Tutik Kristiastuti Sudarmanto

Tri Wiyanto Wira Kurniawan Sukir Widodo Sumardi

EDITORIAL BOARDS

Algebra, Graph and Combinatorics

Budi Surodjo

Ari Suparwanto

Analysis

Supama

Atok Zulijanto

Applied Mathematics

Fajar Adi Kusumo

Salmah

Computer Science

Edi Winarko

MHD. Reza M.I. Pulungan

Statistics and Finance

Subanar

Abdurakhman

LIST OF REVIEWERS

Abdurakhman

Universitas Gadjah Mada, Indonesia

Insap Santosa


Achmad Muchlis

Institut Teknologi Bandung, Indonesia

Intan Muchtadi-Alamsyah


Adhitya Ronnie Effendi


Irawati


Agus Buwono

Institut Pertanian Bogor, Indonesia

Irwan Endrayanto A.


Agus Maman Abadi

Universitas Negeri Yogyakarta, Indonesia

Jailani


Agus Yodi Gunawan


Janson Naiborhu


Ari Suparwanto


Joko Lianto Buliali

Institut Teknologi Sepuluh Nopember,

Indonesia

Asep. K. Supriatna

Universitas Padjadjaran, Indonesia

Khreshna Imaduddin Ahmad S.


Atok Zulijanto


Kiki Ariyanti Sugeng

Universitas Indonesia

Azhari SN


Lina Aryati


Budi Nurani


M. Farchani Rosyid


Budi Santosa

Institut Teknologi Sepuluh Nopember, Indonesia

Mardiyana

Universitas Negeri Surakarta, Indonesia

Budi Surodjo


MHD. Reza M. I. Pulungan


Cecilia Esti Nugraheni

Universitas Katolik Parahyangan, Indonesia

Miswanto

Universitas Airlangga, Indonesia

Ch. Rini Indrati


Netty Hernawati

Universitas Lampung, Indonesia

Chan Basarrudin


Noor Akhmad Setiawan


Danardono


Nuning Nuraini


Dedi Rosadi


Rieske Hadianti


Deni Saepudin

Institut Teknologi Telkom, Indonesia

Roberd Saragih


Diah Chaerani


Salmah


Edy Soewono


Siti Fatimah

Universitas Pendidikan Indonesia

Edy Tri Baskoro


Soeparna Darmawijaya


Edi Winarko


Sri Haryatmi


Endar H Nugrahani


Sri Wahyuni


Endra Joelianto


Subanar


Eridani

Universitas Airlangga, Indonesia

Supama


Fajar Adi Kusumo


Suryanto


Frans Susilo

Universitas Sanata Dharma, Indonesia

Suyono

Universitas Negeri Jakarta, Indonesia

Gunardi


Tony Bahtiar


Hani Garminia


Wayan Somayasa

Universitas Haluoleo, Indonesia

Hartono


Widodo Priyodiprojo


Hengki Tasman


Wono Setyo Budhi


I Wayan Mangku


Yudi Soeharyadi


Indah Emilia Wijayanti


PREFACE

It is an honor and great pleasure for the Department of Mathematics –

Universitas Gadjah Mada, Yogyakarta – INDONESIA, to be entrusted by the

Southeast Asian Mathematical Society (SEAMS) to organize an international

conference every four years. Appreciation goes to those who have developed and

established this tradition of the successful series of conferences. The SEAMS -

Gadjah Mada University (SEAMS-GMU) 2011 International Conference on

Mathematics and Its Applications took place in the Faculty of Mathematics and

Natural Sciences of Universitas Gadjah Mada on July 12th – 15th, 2011. The

conference was the follow up of the successful series of events which have been

held in 1989, 1995, 1999, 2003 and 2007.

The conference has achieved its main purposes of promoting the exchange

of ideas and presentation of recent development, particularly in the areas of pure,

applied, and computational mathematics which are represented in Southeast Asian

Countries. The conference has also provided a forum of researchers, developers,

and practitioners to exchange ideas and to discuss future direction of research.

Moreover, it has enhanced collaboration between researchers from countries in the

region and those from outside.

More than 250 participants from over the world attended the conference.

They come from USA, Austria, The Netherlands, Australia, Russia, South Africa,

Taiwan, Iran, Singapore, The Philippines, Thailand, Malaysia, India, Pakistan,

Mongolia, Saudi Arabia, Nigeria, Mexico and Indonesia. During the four days

conference, there were 16 plenary lectures and 217 contributed short

communication papers. The plenary lectures were delivered by Halina France-

Jackson (South Africa), Jawad Y. Abuihlail (Saudi Arabia), Andreas Rauber (Austria),

Svetlana Borovkova (The Netherlands), Murk J. Bottema (Australia), Ang Keng Cheng

(Singapore), Peter Filzmoser (Austria), Sergey Kryzhevich (Russia), Intan Muchtadi-

Alamsyah (Indonesia), Reza Pulungan (Indonesia), Salmah (Indonesia), Yudi

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 ..........................................................

365

Divo D. Silalahi, J.P. Caliman, Yong Yit Yuan

Continuously Translated Framelet............................................................................................ 379 Dylmoon Hidayat

Multilane Kinetic Model of Vehicular Traffic System ............................................................. 386 Endar H. Nugrahani

Analysis of a Higher Dimensional Singularly Perturbed Conservative System: the Basic

Properties ….............................................................................................................................. 395 Fajar Adi Kusumo

A Mathematical Model of Periodic Maintence Policy based on the Number of Failures for

Two-Dimensional Warranted Product ..…................................................................................. 403 Hennie Husniah, Udjianna S. Pasaribu, A.H. Halim

The Existence of Periodic Solution on STN Neuron Model in Basal Ganglia …………………………. 413 I Made Eka Dwipayana

Optimum Locations of Multi-Providers Joint Base Station by Using Set-Covering Integer

Programming: Modeling & Simulation ….................................................................................. 419 I Wayan Suletra, Widodo, Subanar

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

Inequality …………………………………………………………………….…………….…………..…….……………………..

515 Solikhatun and Salmah

Unsteady Heat and Mass Transfer from a Stretching Suface Embedded in a Porous Medium

with Suction/injection and Thermal Radiation Effects…………………………………………………………. 529 Stanford Shateyi and Sandile S Motsa

Level-Set-Like Method for Computing Multi-Valued Solutions to Nonlinear Two Channels

Dissipation Model ……………………………………………………………………………….....………………………….. 547 Sumardi, Soeparna Darmawijaya, Lina Aryati, F.P.H. Van Beckum

Nonhomogeneous Abstract Degenerate Cauchy Problem: The Bounded Operator on the

Nonhomogen Term ………………………………………………………………………..…………………..……………… 559 Susilo Hariyanto, Lina Aryati,Widodo

Stability Analysis and Optimal Harvesting of Predator-Prey Population Model with Time

Delay and Constant Effort of Harvesting ………………………………………………………..……………………. 567 Syamsuddin Toaha

Dynamic Analysis of Ethanol, Glucose, and Saccharomyces for Batch Fermentation ………….. 579 Widowati, Nurhayati, Sutimin, Laylatusysyarifah

Computer Science, Graph and Combinatorics

Survey of Methods for Monitoring Association Rule Behav ior ............................. 589 Ani Dijah Rahajoe and Edi Winarko

A Comparison Framework for F ingerprint Recognition Methods . . . . . . . . . . . . . . . . . . . . 601 Ary Noviyanto and Reza Pulungan

The Global Behav ior of Certa in Turing System …………………. ..................................... 615 Janpou Nee

Logic Approach Towards Formal Verif ication of Cryptographic Protocol . . . . . . . . . 621 D.L. Crispina Pardede, Maukar, Sulistyo Puspitodjati

A Framework for an LTS Semantics for Promela ….. . . . . . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . 631 Suprapto and Reza Pulungan

Mathematics Education

Modelling On Lecturers’ Performance with Hotteling-Harmonic-Fuzzy …................................ 647 H. A. Parhusip and A. Setiawan

Differences in Creativity Qualities Between Reflective and Impulsive Students in Solving

Mathematics …………………......................................……...................…………………........................ 659 Warli

Statistics and Finance

Two-Dimensional Warranty Policies Using Copula ...…………...................………………………………. 671 Adhitya Ronnie Effendie

Consistency of the Bootstrap Estimator for Mean Under Kolmogorov Metric and Its

Implementation on Delta Method ……................................………………….................................. 679 Bambang Suprihatin, Suryo Guritno, Sri Haryatmi

Multivariate Time Series Analysis Using RcmdrPlugin.Econometrics and Its Application for

Finance ..................................................................................................................................... 689 Dedi Rosadi

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


( ) 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)


. (4.11)


. (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


)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

Riccati (4.17) we get

,0)1,1()1,1(2

1)1,2()2,1()1,1()2,1()1,2( 221222 KLKLPLK

(4.35)

,01)2,2()2,1()1,1()2,2( 1222 KLLK (4.36)

,01)1,2()2,2()2,2()1,1( 1222 KLLK (4.37)

.0)2,2()2,2(1)2,2( 12 KLP (4.38)

From the third Riccati equation (4.17) we get

,0)1,2()1,1()1,2()1,1()1,1()1,1(2

122111 KKKKKP

(4.39)

,0)2,2()2,2()2,2()1,1( 21 KKPP (4.40)

,0)1,2()1,1()1,2()1,1()1,2()2,2()1,1()2,2( 22111 KKKKKPPP(4.41)

.0)2,2()2)2,2()(2,2( 21 KPK (4.42)

Take ( ) , we get from (4.22) ( )

. From (4.21) we get

( ) ( ) .

From (4.20) we get

( )

( ).

Take ( ) From (4.38) and because ( ) we get

( )

.

From (4.42) we get

( ) ( ). Take ( ) , from (4.36) we get

( ) ( ) .


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

No.1, January, 1994, 143-147,1994.

SALMAH

Department of Mathematics Gadjah Mada University

e-mail: [email protected]

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