Analysis, Controllability and Optimization of Time-Discrete Systems and Dynamical Games
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
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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 Department of Mathematics Technical University Darmstadt Schlossgartenstrasse 7 64289 Darmstadt Germany
Dr. Stefan Wolfgang Pickl Department of Mathematics Center of Applied Computer Science ZAIK University of Cologne Weyertal80 50931 Cologne Germany
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. .. Bibliographic information published by Die Deutsche Bibliothek . Die Deutsche Bibliothek lists this publication in the Deutsche Nahonalblbhografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISSN 0075-8450 ISBN 978-3-540-40327-2
ISBN 978-3-642-18973-9 (eBook)
DOI 10.1007/978-3-642-18973-9 This work is subject to copyright. AII rights are reserved, whether the whole Of part of the material is concemed, specificalIy the rights of translation, reprinting, re-use of ilIustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permis sion for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by author Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper
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Dedicated to the people of Navrongo
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 introduc e 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-p
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