Analysis, Controllability and Optimization of Time-Discrete Systems and Dynamical Games

197 Pages·2003·16.51 MB·English

by  Claire J. Creutzfeldt, Prof. Dr. W. Trockel, A. Basile, A. Drexl, K. Inderfurth, C. D. Aliprantis, G. Feichtinger, W. Güth, P. Korhonen, U. Schittko, R. Selten

Lecture Notes in Economics
and Mathematical Systems 529
Founding Editors:
M. Beckmann
H. P. Kiinzi
Managing Editors:
Pro f. Dr. G. Fandel
Fachbereich Wirtschaftswissenschaften
Fernuniversitat Hagen
Feithstr. 140/AVZ 11,58084 Hagen, Germany
Prof. Dr. W. Trockel
Institut fur Mathematische Wirtschaftsforschung (IMW)
Universitat Bielefeld
Universitatsstr, 25, 33615 Bielefeld, Germany
Co-Editors:
C. D. Aliprantis
Editorial Board:
A. Basile, A. Drexl, G. Feichtinger, W Guth, K. Inderfurth, P. Korhonen,
W. Kursten, U. Schittko, R. Selten, R. Steuer, F.Vega-Redondo
Springer-Verlag Berlin Heidelberg GmbH
Wemer Krabs
Stefan Wolfgang Pick1
Analysis, Controllability
and Optimization
of Time-Discrete Systems
and Dynamical Games
Springer
Authors
Prof. Dr. Wemer Krabs Dr. Stefan Wolfgang Pickl
Department of Mathematics Department of Mathematics
Technical University Darmstadt Center of Applied Computer Science
Schlossgartenstrasse 7 ZAIK
64289 Darmstadt University of Cologne
Germany Weyertal80
50931 Cologne
Germany
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ISSN 0075-8450
ISBN 978-3-540-40327-2 ISBN 978-3-642-18973-9 (eBook)
DOI 10.1007/978-3-642-18973-9
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Preface
J.P. La Salle has developed in [20] a stability theory for systems of difference
equat ions (see also [8]) which we introduce in the first chapter within the
framework of metric spaces. The st ability theory for such systems can also be
found in [13] in a slightly modified form . We st art with autonomous systems
in the first section of chapte r 1. After theoretical preparations we examine
the localization of limit sets with the aid of Lyapunov Functions. Applying
these Lyapunov Functions we can develop a stability theory for autonomous
systems.
If we linearize a non-linear system at a fixed point we ar e able to develop
a stability theory for fixed points which makes use of the Frechet derivative
at the fixed point.
The next subsection deals with general linear systems for which we intro-
duce a new concept of stability and asymptotic stability that we adopt from
[18]. Applications to various fields illustrat e these results. We st art with the
classical predator-prey-model as being developed and investigated by Volterra
which is based on a 2 x 2-system of first ord er differential equations for the
densities of the prey and predator population, resp ectively. This model has
also been investigated in [13] with resp ect to stability of its equilibrium via
a Lyapunov function. Here we consider the discrete version of the model. If
we discretize the original model of interacting growth of populations in terms
of two first order differential equations we obtain a general discrete model for
interacting logistic growth of two populations. As a last example we present
an emission reduction model for the reduction of carbon dioxide emissions.
The next section of chapter 1 deals with non-autonomous systems. Def-
initions and elementary properties of such are presented. Again, we present
the stability theory based on Lyapunov's method for such systems. We regard
general systems and as a special case the linear case. As an application we
describe the temporal development of the concentration of some poison like
urea in the body of a person suffering from a renal disease and having to be
attached to an artificial kidney.
VIII Preface
The second cha pter deals with time-discrete cont rolled systems . Here we
begin with the autonomous case and introduce the problem of fixed point
controllability. If we regar d linear syste ms, we obtain the problem of null-
controllability. For t his we present an algorit hmic method for its solution.
Furthermore, we describ e t he problem of stabilizat ion of cont rolled sys-
tems. Then several applicat ions are presented . We pick up the emission re-
du ction model and concent rate ourself on t he cont rolled syste m which we
lineariz e at a fixed point. As a second example we treat the cont rolled prey-
predation model. Also a plan ar pendulum with moving suspension point can
be described by that modeling. We consider a non-linear pendulum of length
l( > 0) whose moment is cont rolled by movin g its suspension point with ac-
celerat ion u = u(t) along a hori zontal st raight line.
In t he next sect ion of chapte r 2 we regard the non-autonomous case and the
specific problem of fixed point controllability. Furthermore, t he general prob-
lem of controllability, the st abili zation of cont rolled systems and the problem
of reachability is t reate d.
The third chapte r deals with t he controllability of dyn amical games . These
are formul ated as cont rolled auto nomous dynamical syst ems which as uncon-
trolled systems admit fixed points. The problem of cont rollab ility consist s of
findin g cont rol functions such that a fixed point of the uncontrolled syste m
is reached in finit ely many time ste ps. For this problem a game theoretical
solution is given in terms of Par eto optima in the cooperat ive case and Nash
equilibria in the non- coop erative case.
For the emission redu ction model the non- coop erative treatment of t he cont rol
problem leads to the applicat ion of linear pro gramming for the calculat ion of
Nash equilibria and the cooperat ive treatment gives rise to the applicati on of
cooperat ive game theory.
In particular we have to investigate t he question und er which condit ions the
core of such a game is nonempty.
In this connect ion we also consider n-person goal-cost-games and present a
dynamical method for findin g a Nas h equilibrium in such a game.
After the treatment of evolut ion matrix games we come back to n-person
goal-cost -games which we transfer into cooperat ive n-person games . Here we
investigate the questi on under which condit ions the grand coalit ion is stable.
T he appendix supplies the reader with addit iona l information.
Sect ions A.I and A.2 are concerned with t he core of a general coop erative
n-person game and a linear production game, respectively. In Secti on A.3
necessary and sufficient condit ions for weak Pareto optima of non-coop erative
n-person games are given and Section A.4 deals with du ality in such games.
The book ends with bibliographical remarks.
Preface IX
The aut hors want to thank Silja Meyer-Nieberg for carefully reading the
manuscript and Gor an Mihelcic for excellent typesetting . He solved every tex-
problem which occured in minim al time.
Cologne, Wern er Krabs
May 2003 St efan Pickl
Contents
1 Uncontrolled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Autonomous Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Definitions and Elementar y Properties . . . . . . . . . . . . . . . 1
1.1.2 Localization of Limit Sets with the Aid of
Lyapunov Fun ctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Stability Based on Lyapunov's Method . . . . . . . . . . . . . . . 8
1.1.4 Stabili ty of Fixed Points via Lineari sation . . . . . . . . . . .. 13
1.1.5 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1.6 Applications .... ... .. ... . ... ... . .... .. ... .... ..... 21
1.2 The Non-Autonomous Case 32
1.2.1 Definitions and Element ary Properties . . . . . . . . . . . . . .. 32
1.2.2 Stability Based on Lyapunov 's Method . . . . . . . . . . . . . . . 35
1.2.3 Linear Syst ems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.2.4 Application to a Model for the Process of Hemo-Dialysis 43
2 Controlled Systems 47
2.1 The Autonomous Case 47
2.1.1 The Problem of Fixed Point Controllability 47
2.1.2 Null-Cont rollability of Linear Systems 57
2.1.3 A Method for Solving t he Problem of Null-Controllability 65
2.1.4 St abiliz ation of Cont rolled Systems 70
2.1.5 Applications 73
2.2 The Non-Autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.1 The Problem of Fixed Point Cont rollability . . . . . . . . . .. 80
2.2.2 The General Problem of Controllability .. . . . . . . . . . . . . 83
2.2.3 Stabili zation of Controlled Systems. . . . . . . . . . . . . . . . . . 86
2.2.4 T he Problem of Reachability . . . . . . . . . . . . . . . . . . . . . . . . 89
3 Controllability and Optimization 93
3.1 The Control Problem 93
3.2 A Game Theoreti cal Solution . . . . . . . . . . . . . . . . . . . . . . . . . 95
XII Contents
3.2.1 The Cooperative Case 95
3.2.2 The Non-Cooperative Case 99
3.2.3 The Linear Case 103
3.3 Local Cont rollability 106
3.4 An Emission Redu ction Model 107
3.4.1 A Non-Cooperat ive Treatment 107
3.4.2 A Cooperative Treatment 116
3.4.3 Condit ions for the Core to be Non-Empty 118
3.4.4 Further Conditions for the Core to be Non-Empty 122
3.4.5 A Second Cooperative Treatment 128
3.5 A Dyn amic al Method for Finding a Nash Equilibrium 136
3.5.1 The Goal-Cost-Game 136
3.5.2 Necessa ry Conditions for a Nash Equilibrium 137
3.5.3 The Method 139
3.6 Evolution Matrix Games 141
3.6.1 Definition of the Game and Evolutionary St ability 141
3.6.2 A Dynami cal Method for Finding an Evolutionary
St able State 147
3.7 A General Cooperative n-Person Goal-Cost-Game 151
3.7.1 The Game 151
3.7.2 A Cooperative Treatment 152
3.7.3 Necessary and Sufficient Conditions for a St able
Grand Coalition 153
3.8 A Cooperative Treatment of an n-Person Cost-Game 155
3.8.1 The Game and a First Cooperat ive Treatment 155
3.8.2 Transformation of the Game into a
Cooperat ive Game 157
3.8.3 Sufficient Condi t ions for a St able Gr and Coalition 158
3.8.4 Further Cooperative Treatments 160
3.8.5 Pareto Opt ima as Cooperat ive Solutions of the Game . . 162
A Appendix 167
A.l The Core of a Cooperat ive n-Person Game 167
A.2 The Core of a Linear P roduction Game 173
A.3 Weak Par eto Optima: Necessary and Sufficient Condit ions 177
A.4 Duality 179
B Bibliographical R emarks 181
References 183
Index 185
About the Authors 187