pro ceedin g of ciam - gadjah mada university · proceeding of conf. on industrial and appl. math.,...

EditorLS FaInsJalPhFa

rs: .H. Wiry.R. Pudj

aculty of Mathstitut Teknololan Ganesha 1

hone : +62 22ax : +62 22

Pro ConfeIndus 6th – 8thInstitut

yanto japrase

hematics and ogi Bandung 10 Bandung, I2 2502545 2 250 6450

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h July 201t Teknolo

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Natural Scien

Indonesia.

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10 ogi Bandu

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SSN: 977-2

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208-70510-00-8

Proceeding of Conf. on Industrial and Appl. Math., Bandung‐Indonesia 2010

i

Electronic Proceeding

Conference on Industrial and Applied Mathematics

6 - 8 July 2010

The Committees of the conference Scientific Committee

Larry Forbes (University of Tasmania, Australia) Robert McKibbin (Messey University, New Zealand) Susumu Hara (Nagoya University, Japan) Edy Soewono (ITB, Indonesia) Chan basaruddin (University of Indonesia, Indonesia) Roberd Saragih (ITB, Indonesia)

Organizing Committee

L.H. Wiryanto (Chair) Sri Redjeki Pudjaprasetya Novriana Sumarti Andonowati Kuntjoro Adji Sidarto

Technical Committee

Jalina Wijaya Agus Yodi Gunawan Nuning Nuraini Janson Naiborhu Adil Aulia Lina Anugerah Ismi Ridha Ikha Magdalena

Maulana Wimar Banuardhi Indriani Rustomo Pritta Etriana Adrianus Yosia Rafki Hidayat Intan Hartri Putri Yunan Pramesi Haris Freddy Susanto


ii

Introduction This proceeding contains papers which were presented in Conference on Industrial and Applied Mathematics. The editors would like to express their deepest gratitude to all presenters, contributors/authors and participants of this conference for their overwhelming supports that turn this conference into a big success. While every single effort has been made to ensure consistency of format and layout of the proceedings, the editors assume no responsibility for spelling, grammatical and factual errors. Besides, all opinions expressed in these papers are those of the authors and not of the conference Organizing Committee nor the editors. The Conference on Industrial and Applied Mathematics is the first international conference held at Institut Teknologi Bandung-Indonesia, during July 6 – 8, 2010; hosted by Industrial and Financial Mathematics Research Division, Faculty of Mathematics and Natural Sciences ITB. The research division has continuing research interests in financial mathematics, optimization, applied probability, control theory and its application; biological, physical modeling and the application of mathematics in sciences, fluid dynamics, and numerical methods and scientific computing. The conference provided a venue to exchange ideas in those areas and any aspect of applied mathematics, in promoting both established and new relationships. Permission to make a digital or hardcopies of this proceeding for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. Editors:

L.H. Wiryanto S.R. Pudjaprasetya Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung Jalan Ganesha 10 Bandung, Indonesia. Phone : +62 22 2502545 Fax : +62 22 250 6450


iii

Table of Content Page

The committees of the conference i

Introduction ii

Table of Content iii

Research Articles:

1 An adaptive nonstationary control method and its application to positioning control problems, Susumu Hara

1-8

2 Some aspects of modelling pollution transport in groundwater aquifers, Robert McKibbin

9-16

3 Jets and Bubbles in Fluids – Fluid Flows with Unstable Interfaces, Larry K. Forbes

17-25

4 Boundary Control of Hyperbolic Processes with Applications in Water Flow, M. Herty and S. Veelken

26-28

5 Isogeometric methods for shape modeling and numerical simulation, Bernard Mourrain, Gang Xu

29-33

6 FOURTH‐ORDER QSMSOR ITERATIVE METHOD FOR THE SOLUTION OF ONE‐DIMENSIONAL PARABOLIC PDE’S, J. Sulaiman, M.K. Hasan, M. Othman, and S. A. Abdul Karim

34-39

7 A Parallel Accelerated Over‐Relaxation Quarter‐Sweep Point Iterative Algorithm for Solving the Poisson Equation, Mohamed Othman, Shukhrat I. Rakhimov, Mohamed Suleiman and Jumat Sulaiman

40-43

8 Value‐at‐Risk (VaR) using ARMA(1,1)‐GARCH(1,1), Sufianti and Ukur A. Sembiring

44-49

9 Decline Curve Analysis in a Multiwell Reservoir System using State‐Space Model, S. Wahyuningsih(1), S. Darwis(2), A.Y. Gunawan(3 ), A.K. Permadi

50-53

10 Study of Role of Interferon‐Alpha in Immunotherapy through Mathematical Modelling, Mustafa Mamat, Edwin Setiawan Nugraha, Agus Kartono , W M Amir W Ahmad

54-62

11 Improving the performance of the Helmbold universal portfolio with an unbounded learning parameter, Choon Peng Tan and Wei Xiang Lim

63-66

12 Optimal Design The Interval Type‐2 Fuzzy PI+PD Controller And 67-71


iv

Superconducting Energy Magnetic Storage (SMES) For Load Frequency Control Optimization On Two Area Power System, Muh Budi R Widodo, M Agus Pangestu H.W

13 Dependence of biodegradability of xenobiotic polymers on population of

microorganism, Masaji Watanabe and Fusako Kawai

72-78

14 PROTOTYPE OF VISITOR DISTRIBUTION DETECTOR FOR COMMERCIAL BUILDING, Sukarman, Suharyanto, Samiadji Herdjunanto

79-85

15 APPLICATION ANFIS FOR NOISE CANCELLATION, Sukarman

86-93

16 THE STABILITY OF THE MECD SCHEME FOR LARGE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS, Supriyono

94-99

17 Real Time Performance of Fuzzy PI+Fuzzy PD Self Tuning Regulator In Cascade Control, Mahardhika Pratama, Syamsul Rajab, Imam Arifin, Moch.Rameli

100-103

18 APPROXIMATION OF RUIN PROBABILITY FOR INVESTMENT WITH INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM NET RETURNS AND MULTIVARIATE NORMAL MEAN VARIANCE MIXTURE DISTRIBUTED FORCES OF INTEREST IN FIXED PERIOD, Ryan Kurniawan and Ukur Arianto Sembiring

104-108

19 Stochastic History Matching for Composite Reservoir, Sutawanir Darwis, Agus Yodi Gunawa, Sri Wahyuningsih, Nurtiti Sunusi, Aceng Komarudin Mutaqin, Nina Fitriyani

109‐115

20 PROBABILITY ANALYSIS OF RAINY EVENT WITH THE WEIBULL DISTRIBUTION AS A BASIC MANAGEMENT IN OIL PALM PLANTATION, Divo D. Silalahi

116‐120

21 A Multi‐Scale Approach to the Flow Optimization of Systems Governed by the Euler Equations, Jean Medard T. Ngnotchouye, Michael Herty, and Mapundi K. Banda

121‐126

22 Modelling and Simulating Multiphase Drift‐flux Flows in a Networked Domain, Mapundi K. Banda, Michael Herty , and Jean Medard T. Ngnotchouye

127‐133

23 Calculating Area of Earth’s Surface Based on Discrete GPS Data, Alexander A S Gunawan , Aripin Iskandar

134‐137

24 Study on Application of Machine Vision using Least‐Mean‐Square (LMS), Hendro Nurhadi and Irhamah

138‐144

25 Cooperative Linear Quadratic Game for Descriptor System, Salmah

145‐250


v

26 ARMA Model Identification using Genetic Algorithm (An Application to Arc Tube Low Power Demand Data), Irhamah, Dedy Dwi Prastyo and M. Nasrul Rohman

151‐155

27 A Particle Swarm Optimization for Employee Placement Problems in the Competency Based Human Resource Management System, Joko Siswanto and The Jin Ai

156‐161

28 Measuring Similarity between Wavelet Function and Transient in a Signal with Symmetric Distance Coefficient, Nemuel Daniel Pah

162‐166

29 An Implementation of Investment Analysis using Fuzzy Mathematics, Novriana Sumarti and Qino Danny

167‐169

30 Simulation of Susceptible Areas to the Impact of Storm Tide Flooding along Northern Coasts of Java, Nining Sari Ningsih, Safwan Hadi, Dwi F. Saputri ,

Farrah Hanifah, and Amanda P.Rudiawan

170‐178

31 Fuzzy Finite Difference on Calculation of an Individual’ Bank Deposits, Novriana Sumarti and Siti Mardiah

179‐183

32 An Implementation of Fuzzy Linear System in Economics, Novriana Sumarti and Cucu Sukaenah

184‐187

33 Compact Finite Difference Method for Solving Discrete Boltzmann Equation, PRANOWO, A. GATOT BINTORO

188-193

34 Natural convection heat transfer with an Al2O3 nanofluids at low Rayleigh number, Zailan Siri, Ishak Hashim and Rozaini Roslan

194-199

35 Optimization model for estimating productivity growth in Malaysian food manufacturing industry, Nordin Hj. Mohamad, and Fatimah Said

200-206

36 Numerical study of natural convection in a porous cavity with transverse magnetic field and non‐uniform internal heating, Habibis Saleh, Ishak Hashim and Rozaini Roslan

207‐211

37 THE DISTRIBUTION PATTERN AND ABUNDANCE OF ASTEROID AND ECHINOID AT RINGGUNG WATERS SOUTH LAMPUNG, Arwinsyah Arka, Agus Purwoko, Oktavia

212‐215

38 Low biomass of macrobenthic fauna at a tropical mudflat: an effect of latitude?, Agus Purwoko and Wim J. Wolff

216‐224

39 Density and biomass of the macrobenthic fauna of the intertidal area in Sembilang national park, South Sumatra, Indonesia, Agus Purwoko and Wim J. Wolff

225‐234


vi

40 Intelligent traffic light system for AMJ highway, Nur Ilyana Anwar Apandi, Puteri Nurul Fareha M. Ahmad Mokhtar, Nur Hazahsha Shamsudin and Anis Niza Ramani and Mohd Safirin Karis

235‐238

41 Goodness of Fit Test for Gumbel Distribution Based on Kullback‐Leibler Information using Several Different Estimators, S. A. Al‐Subh , K. Ibrahim , M. T. Alodat , A. A. Jemain

239‐245

42 Impact of shrimp pond development on biomass of intertidal macrobenthic fauna: a case study at Sembilang, South Sumatra, Indonesia, Agus Purwoko, Arwinsyah Arka and Wim J. Wolff

246‐256

Proceeding of Conf. on Industrial and Appl. Math., Bandung-Indonesia 2010

145

Cooperative Linear Quadratic Game for Descriptor System

Salmah Department of Mathematics, Gadjah Mada University, Yogyakarta, DI Yogyakarta, Indonesia

(Email: [email protected])

ABSTRACT In this paper the cooperative linear quadratic game problem will be considered. In the game, either there is one individual who has multiple objectives or there are a number of players all facing linear quadratic control problem, will cooperate together to obtain optimal result. For ordinary system case the set of Pareto efficient equilibria can be determined for these games. In this paper the result will be generalized for descriptor system. KEYWORDS Dynamic, game, cooperative, descriptor, system. I. 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 cooperative 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. II. PRELIMINARIES 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. Formally, the players are assumed to minimize the performance criteria

,)()()()(2

1),...,,(

0

21 dttuRtutxQtxuuuJT

iiTii

Tni

(1)

with all matrices are constant, Tii QQ , and

iR positive definite. The inclusion of player

j’s control effort into player I’s cost function is dropped, due to the open-loop information structure, this term drops out in the analysis. The players give control vector to the system

)()()()( 2211 tuBtuBtAxtxE ,

0)0( xx

(2)

with 1)()()( ,, xmrni

rnxrn BAE , x(t)

descriptor vector n+r dimension. While ui(t), i=1,2 are control vector im dimension which

is done by i-th player, i=1,2. Matrix E generally singular with rank .nE The initial

vector 0x is assumed to be a consistent initial

state.

We recall the following result from theory of descriptor (see [7]) system for the differential algebraic equation

0)0(),()( xxtftAxxE

(DAE)

And the associated pencil

AE (3) System (DAE) is called regular if the characteristic polynomial )det( AE is not

identically zero. If the system (DAE) is not regular then any consistent initial conditions do not uniquely determine solutions (see [13]). If the system (DAE) is regular, the roots of characteristic polynomial are the finite eigenvalues of the pencil. If E is singular the pencil (3) is said to have infinite eigenvalues which is the zero eigenvalues of the inverse


146

pencil AE . In the next discussion we recall the Weierstrass canonical form. Theorem 1 If (3) is regular, then there exist nonsingular matrices X and Y such that

00and

00nT T

r

JIY EX Y AX

IN

(4) where J is a matrix in Jordan form whose elements are the finite eigenvalues,

k kkI R is the identity matrix and N is a

nilpotent matrix also in Jordan form. J and N are unique up to permutation of Jordan blocks. If (3) is regular the solutions of (DAE) take the form 1 1 2 2( ) ( ) ( )x t X z t X z t

where with 1 2[ ]X X X 1 2[ ]T TY Y Y , ( )

1 1T n r nX Y R , ( )

2 2T n r rX Y R and

( ) 1

1 1 1 1 00

1

2 20

( ) (0) ( ) (0) [ 0]

( ) ( )

tJt J t sn

iki

ii

z t e z e Y f s ds z I X x

dz t NY f tdt

under the consistency condition:

1

10 2

0

[0 ] (0)ik

ir i

i

dI X x N Y f

dt

Here k is the degree of nilpotency of N . That is the integer k for which 0kN and 1 0kN The index of the pencil (3) and of the descriptor system (DAE) is the degree k of nilpotency of N . If E is nonsingular, we define the index to be zero. From the above formulae then the solution

( )x t will not contain derivatives of the

function f if and only if 1k . In that case the solution ( )x t is called impulse free. In general, the solution ( )x t involves

derivatives of order 1k of the forcing function f if 3 has index k .

Next, let [ ]VW be an orthogonal matrix

such that image V equals the null space (

( )TN E ) of TE and image W equals the

null space of E . Then

1 1[ 0][ ]T TE E VW EV , where 1E is full

column rank. The next lemma characterizes pencils which have an index of at most one. Lemma 1 The following statements are equivalent: (i) pencil (4) is regular and has at most index one.

(ii)rank T

En r

V A

( ( ))Trank E VV A .

(iii) rank ([ ])EAW n r

( ( ))Trank E AWW . Since we do not want to consider derivatives of the input function in this paper, we restrict the analysis to regular index one systems here. The above discussion motivates then the next assumptions. Assumption 1 Throughout this section the next assumptions are made w.r.t. system (1):

1. matrix E is singular; 2. det ( ) 0E A ; 3. rank ([ ]EAW n r .

In the cooperative game, each player have a set of possible outcomes then one outcome (which in general does not coincide with a player’s overall lowest cost) is cooperatively selected. It is reasonable that the cooperative desicion is strategy that have the property that if different strategy is chosen, then at least one of the players has higher costs. Or, stated diferently, is solutions that the players outcome can not be improved by all players simultaneously. This strategy is called Pareto efficiency.


147

Definition 1 Let U is the set of admissible

strategies. A set of strategies is called Pereto efficient if the set of inequalities

),ˆ()( ii JJ i=1,2,...,N, where at least one

of the inequalities is strict, does not allow for any solution U . The corresponding point

NNJJ ))ˆ(),...,ˆ(( 1

is called a Pareto solution. The set of all Pareto solutions is called Pareto frontier. Usually there is more then one Pareto solution. It is noted that here we are not make definiteness assumptions for matrix

iQ . Here we follow the line in [5] which is

derive Pareto solutions for linear quadratic game with ordinary systems without definiteness assumptions for matrix iQ .

While in [4] there is more simple discussion about Pareto solutions of linear quadratic game with definiteness assumptions for matrix iQ .

III. THE FINITE PLANNING HORIZON

Let we consider the game (1), (2) under the assumption that T is finite. The game (1), (2) has a set of open-loop Nash equilibrium actions 1 2( ( ) ( ))u u if and

only if 1 2( ( ) ( ))u u are open-loop Nash

equilibrium actions for the game

)()(

)(

0

0

)(

)(

00

011

2

1

2

1 tuBYtx

tx

I

J

tx

txI T

r

n

01

2

122 )0(

)0(),( xX

x

xtuBY T

(5) where player i has the quadratic cost functional:

T

iTT

ni tx

txtxQtxtxuuuJ

0 2

12121 )(

)()()()(),...,,(

dttuRtu )(~

)( (6) From (5) it follows that

))()((0)( 22112 tuBtuBYItx Tr

))()(( 22112 tuBtuBY (7) Substitution of (7) into the cost functions (6) shows that 1 2( ( ) ( ))u u are open-loop

Nash equilibrium actions for the game (1), (2) if and only if 1 2( ( ) ( ))u u are open-

loop Nash equilibrium actions for the game

)()()()( 22211111 tuBYtuBYtJxtx ,

01

1 0)0( xXIx n

(8) with cost functionals 1 2( )iJ u u for the

players given by

)(

)(

)(

0

00

0

0

0

)()()({

1

1

2212

22

21

0

211

tu

tu

tx

BYBY

IXQX

YB

YB

I

tututx

ni

T

TT

TTnT

TTT

dttuRtu iiTi )()(

dt

tu

tu

tx

Mtututx i

TTTT }

)(

)(

)(

)()()({

2

1

1

0

211

(9) where

1

2

i i i

Ti i i i

T Ti i i

Q V W

M V R N

W N R

(10) with


148

1 1 1 2 2 1

1 2 2 2 1 2 2 2 2 2

111 1 2 2 2 2 11

21 2 2 2 2 2 21

12 1 2 2 2 2 12

222 2 2 2 2 2 22

T Ti ii i

T T T Ti ii i

T T T

T T T

T T T

T T T

Q X X V X X Y BQ Q

W X X Y B N B Y X X Y BQ Q

R B Y X X Y BQ R

R B Y X X Y BQ

R B Y X X Y BQ

R B Y X X Y BQ R

IV. SOLUTION OF COOPERATIVE GAME In the following discussion, the set of parameters, A, plays a crucial role.

}10),...,({1

1

N

iiiN andA

. The next two lemmas give a caracterization of Pareto efficient solutions. The first lemma give identification for Pareto efficient whithout convexity assumptions, while in second lemma states how to find Pareto efficient with convexity assumptions. The proof of those lemmas can be found in [4].

Lemma 2 Let )1,0(i , with 11

N

ii .

Assume U such that

N

iii J

1

)}(min{argˆ .

Then is Pareto efficient. Lemma 3 Assume that the strategy space U is convex. Moreover, assume that the payoffs iJ

are convex, i=1,...,N. Then, if is Pareto

efficient, there exist A , such that for all U ,

)ˆ()(1 1

i

N

i

N

iiii JJ

.

Note that if iJ are convex, then

N

iii J

1

)(

is convex too for an arbitrary A . In [5], characterization for the space of admissible control U is convex is given.

Next we will consider conditions for the cost functions iJ in (1) are convex. From [5] we

recall some preliminary result. Lemma 4 Assume that U is convex. We will consider the linear quadratic cost function

dttctu

txtptp

tu

txMtutxxuTsJ

TT

T TT

)}()(

)()()(2

)(

)()()({),,,(

221

00

(11) Subject to the state dynamics

01 )(),()()()( xsxtctButAxtx ,

(12) Where ipu, and ic aresuch that (11) and

(12) have solution,

RV

VQM T . Let

nx 0 . Then ),,,0( 0xuTJ is convex as a

function of u if and only if 0),,,0( 0 xvTJ

for all v where

dttv

tzMtvtzxvTJ

T TT

)(

)()()(),,,0(

00

with 0)(),()()( sztBvtAztz . (13) Note that in the second equivalence does not depend on 0x . In particular it follows from

that if J is convex for one 0x then is convex

for all 0x .

Lemma 5 Assume that U is convex. Consider the linear quadratic cost function (11) and (12). Then if ),,,0( 0xuTJ is for some 0x ,

),,,0( 0xuTJ is convex foar all 0s and for

all 0x .

Corollary 1 Consider the linear quadratic cost function (11) and (12). Then

),,,( 0xuTsJ is convex for all 0s and 0x

if and only if ),,,0( 0xuTJ is convex for all

(an) 0x . Which holds if and only if


149

0)0,,,0( vTJ for all v, where J is given by (13). From [5] we get the following theorem. Theorem 1 Assume that either T is finite or (A,B) is stabilizable. Then ),,,0( 0xuTJ is

convex for all (an) 0x if and only if

)0,,,0(inf vTJ exists. Note 1 In [5] we can find theorem that consider the existence of the following cost function.

dttu

txMtutxJ

T TT

)(

)()()({

0

with 0)0(),()()( xxtButAxtx ,

(14) The following Riccati equations plays important role in the next discussion.

0(

,))(())((

)()()(1

TK

QVtKBRVBtK

AtKtKAtKTT

T

;

(DRE)

QVKBRVKBKAKA TTT )()(0 1

. (ARE)

Let 21 BBB . Combining the results of Lemmas 2 and 3 and Theorem 1 and Note 1, we can derive existence and computational algorithms for both the finite and infinite horizon game problems. We just state for infinite horizon problem. Corollary 2 Consider the coocperative game (1,2) with T , seLU ,,2 and (A,B)

stabilizable. Assume (ARE), with

ii WVV and

iTi

ii

RN

NRR

2

1 has a

stabilizing solution iK , for i=1,2 respectively.

Then all Pareto efficient solutions are obtained by determining for all A ,

2211minarg)( JJuUu

, subject to

(12). The following theorem give procedure to calculate the Pareto efficient solution of the game with descriptor system. Theorem 2 Consider the cooperative game (9), (2) with T , seLU ,,2 and (A,B)

stabilizable. For A let

RV

VQMMM

T ~~

~~)( 2211 ,

Where

2211

~QQQ ,

],[~

22112211 WWVVV , and

222

2122

211

1111

~RN

NR

RN

NRR TT .

Furthermore let TBRBS 1~~ . Assume that

(15) below has a stabilizing solution iX for

1i , i=1,2 respectively.

0~

)~

(~

)~

( 1 QVXBRVXBXAXA TTT

. (15) Then the set of all cooperative Pareto solutions is given by

})))(()),(({( 21 AuJuJ .

Here

)()~

(~

)( 1 txVXBRtu Ts

T

t

sstAT dsscXeBR

Tcl )(

~ )(1

Where sX is the stabilizing solution of (15)

and, with sT

cl XSVRBAA~~~ 1 ,

the closed-loop system is

0)0(),()()()( xxtctSmtxAtx cl .

In case 0(.) c with these actions the corresponding cost are

000

~),( xMxuxJ i

Ti , where iM

~ is the

unique solution of the Lyapunov equation.


150

)~

(~

~}

~(

~~

1

1

Ts

Ti

scliiTcl

VXBR

IM

RVBXIAMMA

.

V. CONCLUSIONS In this paper we consider the solution of cooperative linear quadratic game for descriptor system. Using the theory of convex analysis we can get the result for finite and infinite horizon game. Basically, the result were obtained by reformulating the game as an ordinary linear quadratic differential game. Following the lines and combining the results of [5] and [7], similar conclusions can be derived. The above result cn be generalized straightforwardly for N-player game. REFFERENCES

[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, 1998, pp 729-762.

[4] Engwerda, J.C., LQ Dynamic Optimization and Differential Games Chichester: John Wiley & Sons, 2005, pp 229-260.

[5] Engwerda, J.C., “A Note on Cooperative Linear Quadratic Control”, CentER Discussion Paper, Tilburg University, The Netherlands, No 2007-15, 2007.

[6] Engwerda, J.C., “Necessary and Sufficient Conditions for Solving Cooperative Differential Games”, CentER Discussion Paper, Tilburg University, The Netherlands, No 2007-42, 2007.

[7] Engwerda and Salmah, “The Open-Loop Linear Quadratic Differential Game for index one Descriptor Systems”, Automatica, Vol. 45, Number 2, 2009.

[8] 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, 1992, pp 967-972.

[9] Lewis, F.L., “A survey of Linear Singular Systems”, Circuits System Signal Process, vol.5, no.1, 1986, pp 3-36.

[10] 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.

[11] Salmah, Bambang, S., Nababan, S.M., and Wahyuni, S., “Non-Zero-Sum Linear Quadratic Dynamic Game with Descriptor Systems”, Proceeding Asian Control Conference, Singapore, 2002, pp 1602-1607,.

[12] 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.

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