Lyapunov Functionals and Stability of Stochastic Difference Equations
Hereditary systems (or systems with either delay or after-effects) are widely used to model processes in physics, mechanics, control, economics and biology. An important element in their study is their stability.Stability conditions for difference e
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Leonid Shaikhet
Lyapunov Functionals and Stability of Stochastic Difference Equations
Leonid Shaikhet Department of Higher Mathematics Donetsk State University of Management Chelyuskintsev str. 163-A 83015 Donetsk Ukraine [email protected]
ISBN 978-0-85729-684-9 e-ISBN 978-0-85729-685-6 DOI 10.1007/978-0-85729-685-6 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2011930099 © Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Hereditary systems (or systems with delays, or systems with aftereffect) are systems, which future development depends not only on their present state but also on their previous history. Systems of such type are widely used to model processes in physics, mechanics, automatic regulation, economy, biology, ecology etc. (see, e.g., [6, 8, 9, 11, 25, 92, 97, 119, 120, 122, 123, 131, 152, 178, 189, 210, 263]). An important element in the study of such systems is their stability. As it was proposed by Krasovskii [149–151], stability condition for differential equation with delays can be obtained using appropriate Lyapunov functional. By that the construction of different Lyapunov functionals for one differential equation allows to get different stability conditions for the solution of this equation. However the construction of each Lyapunov functional required a unique work from its own author. In 1975, Shaikhet [226] introduced a parametric family of Lyapunov functionals, so that an infinite number of Lyapunov functionals were used simultaneously. This way allowed to get different stability conditions for considered equation using only one Lyapunov functional. At last in the 1990s the general method of Lyapunov functionals construction was proposed by Kolmanovskii and Shaikhet for stochastic functional-differential equations and developed late
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