Construction of Global Lyapunov Functions Using Radial Basis Functions
The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. A new method to construct such a Lyapunov function using radial basis functions is presented in this volume intended for resea
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Peter Giesl
Construction of Global Lyapunov Functions Using Radial Basis Functions
1904
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1904
Peter Giesl
Construction of Global Lyapunov Functions Using Radial Basis Functions
ABC
Author Peter Giesl Centre for Mathematical Sciences University of Technology München Boltzmannstr. 3 85747 Garching bei München Germany e-mail: [email protected]
Library of Congress Control Number: 2007922353 Mathematics Subject Classification (2000): 37B25, 41A05, 41A30, 34D05 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-69907-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-69907-1 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-69909-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 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 by the author using a Springer LATEX macro package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
SPIN: 11979265
VA41/3100/SPi
543210
Preface
This book combines two mathematical branches: dynamical systems and radial basis functions. It is mainly written for mathematicians with experience in at least one of these two areas. For dynamical systems we provide a method to construct a Lyapunov function and to determine the basin of attraction of an equilibrium. For radial basis functions we give an important application for the approximation of solutions of linear partial differential equations. The book includes a summary of the basic facts of dynamical systems and radial basis functions which are needed in this book. It is, however, no introduction textbook of either area; the reader is encouraged to follow the references for a deeper study of the area. The study of differential equations is motivated from numerous applications in physics, chemistry, economics, biology, etc. We focus on autonomous differential equations x˙ = f (x), x ∈ Rn which define a dynamical system. The simplest solutions x(t) of such an equation are equilibria, i.e. solutions x(t) = x0 which remain constant. An important and non-trivial task is the determination of their basin of attraction. The de
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