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

  • Werner Krabs
  • Stefan Wolfgang Pickl
  • M. Beckmann
  • H. P. Künzi
  • G. Fandel
  • W. Trockel
  • C. D. Aliprantis
  • A. Basile
  • A. Drexl
  • G. Feichtinger
  • W. Güth
  • K. Inderfurth
  • P. Korhonen
  • W. Kürsten
  • U. Schittko
  • R. Selten
  • R. Steuer
  • F. Vega-Redondo

Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 529)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Werner Krabs, Stefan Wolfgang Pickl
    Pages 1-46
  3. Werner Krabs, Stefan Wolfgang Pickl
    Pages 47-92
  4. Werner Krabs, Stefan Wolfgang Pickl
    Pages 93-165
  5. Back Matter
    Pages 167-189

About this book

Introduction

J. P. La Salle has developed in [20] a stability theory for systems of difference equations (see also [8]) which we introduce in the first chapter within the framework of metric spaces. The stability theory for such systems can also be found in [13] in a slightly modified form. We start with autonomous systems in the first section of chapter 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 are 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 illustrate these results. We start with the classical predator-prey-model as being developed and investigated by Volterra which is based on a 2 x 2-system of first order differential equations for the densities of the prey and predator population, respectively. This model has also been investigated in [13] with respect to stability of its equilibrium via a Lyapunov function. Here we consider the discrete version of the model.

Keywords

Analysis Controllability Cooperative Games Non-cooperative Games Optimisation Stability Time Discrete Dynamical Systems and Games optimization

Authors and affiliations

  • Werner Krabs
    • 1
  • Stefan Wolfgang Pickl
    • 2
  1. 1.Department of MathematicsTechnical University DarmstadtDarmstadtGermany
  2. 2.Department of Mathematics Center of Applied Computer ScienceZAIK University of CologneCologneGermany

Editors and affiliations

  • M. Beckmann
  • H. P. Künzi
  • G. Fandel
    • 1
  • W. Trockel
    • 2
  • C. D. Aliprantis
  • A. Basile
  • A. Drexl
  • G. Feichtinger
  • W. Güth
  • K. Inderfurth
  • P. Korhonen
  • W. Kürsten
  • U. Schittko
  • R. Selten
  • R. Steuer
  • F. Vega-Redondo
  1. 1.Fachbereich WirtschaftswissenschaftenFernuniversität HagenHagenGermany
  2. 2.Institut für Mathematische Wirtschaftsforschung (IMW)Universität BielefeldBielefeldGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-18973-9
  • Copyright Information Springer-Verlag Berlin Heidelberg 2003
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-40327-2
  • Online ISBN 978-3-642-18973-9
  • Series Print ISSN 0075-8442
  • Buy this book on publisher's site