of Fuzzy Differentia quations and nclusions © 2003 V Lakshmikantham and R N Mohapatra ATICAL ANALUSI APPLICATIONS Series in Mathematical Analysis and Applications (SIMAA) is edited by Ravi P Aganva Institute of Technology, USA and Donal O'Regan, National University of Ireland, Galway, The series is aimed at reporting on new developments in mathematical ana applications of a high standard and of current interest Each volume in the series is dev topic in analysis that has been applied, or is potentially applicable, to the solutions of engineering and social problems Volume Method of Variation of Parameters for Dynamic Systems V Lakshmikantham and S.G Deo Volume lntegral and Integrodifferential Equations: Theory, Methods and Applications edited by Ravi P Aganval and Donal O'Regan Volume Theorems of Leray-Schauder Type and Applications Donal O'Regan and Radu Precup Volume Set Valued Mappings with Applications in Nonlinear Analysis edited by Ravi P Aganval and Donal O'Regan Volume Oscillation Theory for Second Order Dynamic Equations Ravi P Aganval, Said R Grace and Donal O'Regan Volume Theory of Fuzzy Differential Equations and Inclusions V Lakshmikantham and R.N Mohapctra Volume Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations I/ Lakshmikantham and S Koksal This book is part of a series The publisher will accept continuation orders which may be at any time and which provide for automatic billing and shipping of each title in the se publication Please write for written details © 2003 V Lakshmikantham and R N Mohapatra y Differentia quations and Inclusions V Lakshmikant and Taylor &Francis Group LONDON AND NEW YORK © 2003 V Lakshmikantham and R N Mohapatra First published 2003 by Taylor & Francis 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Taylor & Francis Inc, 29 West 35" Street, New York, NY 10001 Taylor & Francis is an imprint of the Taylor & Francis Group O 2003 V Lakshmikantham and R N Mohapatra All rights reserved No part of this book may be reprinted or reproduced or utilise form or by any electronic, mechanical, or other means, now known or hereafter in including photocopying and recording, or in any information storage or retrieval s without permission in writing from the publishers Every effort has been made to ensure that the advice and information in this book and accurate at the time of going to press However, neither the publisher nor the can accept any legal responsibility or liability for any errors or omissions that ma made In the case of drug administration, any medical procedure or the use of tech equipment mentioned within this book, you are strongly advised to consult the manufacturer's guidelines British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested ISBN 0-415-30073-8 © 2003 V Lakshmikantham and R N Mohapatra Conten vii Preface Fuzzy Sets 1.1 Introduction 1.2 Fuzzy Sets 1.3 The Hausdorff Metric 1.4 Support Functions 1.5 The Space En 1.6 The Metric Space (En,d) 1.7 Notes and Comments Calculus of Fuzzy Functions 2.1 Introduction 2.2 Convergence of Fuzzy Sets 2.3 Measurability 2.4 Integrability 2.5 Differentiability 2.6 Notes and Comments Fundamental Theory 3.1 Introduction 3.2 Initial Value Problem 3.3 Existence 3.4 Comparison Theorems 3.5 Convergence of Successive Approximations 3.6 Continuous Dependence 3.7 Global Existence 3.8 Approximate Solutions 3.9 Stability Criteria © 2003 V Lakshmikantham and R N Mohapatra CON 3.10 Notes and Comments Lyapunov-like Functions 4.1 Introduction 4.2 Lyapunov-like Functions 4.3 Stability Criteria 4.4 Nonuniform Stability Criteria 4.5 Criteria for Boundedness 4.6 Fuzzy Differential Systems 4.7 The Method of Vector Lyapunov Functions 4.8 Linear Variation of Parameters Formula 4.9 Notes and Comments Miscellaneous Topics 5.1 Introduction 5.2 Fuzzy Difference Equations 5.3 Impulsive Fuzzy Differential Equations 5.4 Fuzzy Differential Equations with Delay 5.5 Hybrid h z z y Differential Equations 5.6 Fixed Points of Fuzzy Mappings 5.7 Boundary Value Problem 5.8 Fuzzy Equations of Volterra Type 5.9 A New Concept of Stability 5.10 Notes and Comments Fuzzy Differential Inclusions 6.1 Introduction 6.2 Formulation of Fuzzy Differential Inclusions 6.3 Differential Inclusions 6.4 Fuzzy Differential Inclusions 6.5 The Variation of Constants Formula 6.6 Fuzzy Volterra Integral Equations 6.7 Notes and Comments Bibliography Index © 2003 V Lakshmikantham and R N Mohapatra Preface In the mathematical modeling of real world phenomena, we encounter two inconveniences The first is caused by the excessive complexity of the model As the complexity of the system being modeled increases, our ability t o make precise and yet relevant statements about its behavior diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics As a result, we are either not able t o formulate the mathematical model or the model is too complicated to be useful in practice The second inconvenience relates t o the indeterminacy caused by our inability to differentiate events in a real situation exactly, and therefore to define instrumental notions in precise form This indeterminacy is not an obstacle, when we use natural language, because its main property is the vagueness of its semantics and therefore capable of working with vague notions Classical mathematics, on the other hand, cannot cope with such vague notions It is therefore necessary to have some mathematical apparatus t o describe vague and uncertain notions and thereby help t o overcome the foregoing obstacles in the mathematical modeling of imprecise real world systems The rise and development of new fields such as general system theory, robotics, artificial intelligence and language theory, force us t o be engaged in specifying imprecise notions In 1965, Zadeh initiated the development of the modified set theory known as fuzzy set theory, which is a tool that makes possible the description of vague notions and manipulations with them The basic idea of fuzzy set theory is simple and natural A fuzzy set is a function from a set into a lattice or as a special case, into the interval [O, 11 Using it, one can model the meaning of vague notions and also certain kinds of human reasoning Fuzzy set theory and its applications have been extensively developed since the 1970s and industrial interest in fuzzy control has dramatically increased since 1990 There are several books dealing with these aspects © 2003 V Lakshmikantham and R N Mohapatra viii PRE When a real world problem is transformed into a deterministic value problem of ordinary differential equations, namely or a system of differential equations, we cannot usually be sure th model is perfect For example, the initial value may not be known and the function f may contain uncertain parameters If they are est through certain measurements, they are necessarily subject to error analysis of the effect of these errors leads to the study of the qua behavior of the solutions of (*j If the nature of errors is random we can discuss, instead of (*), random differential equations with r initial data However, if the underlying structure is not probabilis cause of subjective choices, it would be natural to employ fuzzy diff equations For the initiation of this aspect of fuzzy theory, the ne calculus of fuzzy functions has also been investigated Consequen study of the theory of fuzzy differential equations has recently been g very rapidly and it is still in the initial stages Nonetheless, there exi ficient literature t o warrant assembling the existing results in a unif so as to understand and appreciate the intricacies involved in incorp fuzziness into the theory of differential equations as well as t o pave t for further advancement of this important branch of differential equa an independent discipline It is with this spirit that we see the imp of the present monograph Its aim is t o present a systematic acc recent developments, describe the current state of the useful theory the essential unity achieved in the theory fuzzy differential equation clusions, and initiate several new extensions to other types of fuzzy d systems In Chapter 1, we provide the preliminary material of fuzzy set providing necessary tools that are relevant for further development ter is dedicated to the description of the calculus of fuzzy functi Chapter 3, we devote our attention t o investigate the basic theory o differential equations The extension of the Lyapunov-like theory of s forms the content of Chapter Chapter investigates several new a investigation relative to fuzzy dynamic systems by providing some results so that further advancement is possible Finally, in Chapte introduce fuzzy differential inclusions and investigate properties of s sets, stability and periodicity in the new framework suggested by Hiill This new approach has the advantage of preserving the properties tions corresponding t o differential equations without fuzziness As w illustrate in Chapter 6, the original fornlulation based on the Hu © 2003 V Lakshmikantham and R N Mohapatra PREFACE derivative totally changes the qualitative behavior of solutions when the initial condition is given more uncertainty by fuzzification However, it can be preserved if the initial level sets are chosen suitably Some of the important features of the monograph are as follows: (1) it is the first book that attempts to describe the theory of fuzzy differential equations; (2) it incorporates the recent general theory (still in the pipeline) of fuzzy differential inclusions; (3) it exhibits several new areas of study by providing initial apparatus for future development; (4) it is a timely introduction t o a subject that is growing rapidly because of its applicability t o various new fields in engineering, computer science and social sciences Actually the first five chapters of the monograph were written three years ago and because of various circumstances such as serious health problems and other unavoidable situations, the book could not be typed until now This enormous delay turned out t o be a blessing in disguise, since the new approach suggested by Hiillermeir, namely, developing the theory of fuzzy differential inclusions, is a better framework compared t o the earlier one utilizing the Hukuhara derivative Me hope that these two different approaches of considering fuzzy dynamic systems will generate other possible settings that may lead to a better understanding of incorporating f~~zziness into various dynamic systems We are immensely grateful to Professors Hiillermier, Diamond, Sieto, and Seikkala for providing the material related to fuzzy differential inclusions and Mrs Donn Miller-Kermani for typing the manuscript efficiently in a short time © 2003 V Lakshmikantham and R N Mohapatra 6.6 Fuzzy Volterra Integral Equations N o t e that a s(s2 b) + - a bs aslb s b' + apply it t o (6.6.8) and take the inverse transform t o obtain T h a t is, the solution set S ( t ) consists of the fuzzy set with ,O-levels the i n tervals c o s ( d - p t ) ) ( - P ) ( - cos( J W t ) ) ['(I ( - P ) ( ,O) + Recall that a n n-dimensional tem x f ( t ) = Ax(t) + Bu(t), open loop linear control sys- x ( ) = xo' has a solution which can be written in t e r m s of the variation of constants formula as rt Here, @ ( t )i s the state transition matrix, o r m a t r i x exponential, satisfying the m a t r i x differential equation N o w , suppose that the matrices A, B have fuzzy n u m b e r entries and the initial condition xo E En i s fuzzy T h e n (6.6.9) can be considered as a family of integral inclusions, provided that some ,meaning can be ascribed t o @ ( t ) Following the earlier discussion, w e interpret (6.6.10) as the family of differential inclusions where Q D , Ap denote level sets T h a t is, zf V i s a m a t r i x i n t h e set of matrices A p , a ( t )= {Y ( t ) : Y'= VY,Y ( ) = I) © 2003 V Lakshmikantham and R N Mohapatra Chapter Fuzzy Differential Inclusions 164 The Y ( t ) are found in the usual way: find a basis of vector solutions v l ( t ) , v ( t ) , : v , ( t )from the eigenvalue-eigenvector problem for V and form the matrix Z ( t ) = [vl v2 v,] T h e n Y ( t ) = Z ( t ) Z ( o ) - l Since Ap i s a n interval matrix? that is, has compact real intervals as Were, 4,A denote ordinary entries, U belongs to the interval [Ap,Ap] matrices whose elements are, respectively, the lower and upper end points of the real intervals (see Neumaier [go] for notation and theory) This gives a method for evaluating @ p ( ) as a n interval matrix by a simple extension of the results i n Sections 5.2 and 5.6, the @ p ( t ) form the level sets of a ~ n x -valued n function I n the case where is a nonnegative matrix, that is, all elements of the matrix are nonnegative, or Ap is a nonpositive matrix, the computation i s especially simple If C - A i s a nonnegative matrix, write C A Let A be a nonnegative matrix and suppose that C X ( t )= A X , X(0) = I , > then Y ( t )2 X ( t ) , t I n particular, zf A = with A V A and < < X ( t )= &(t), A If [A, A] i s an interval matrix XI ( t ) = V X ( ~ ) ,X' ( t ) = A X ( t ) , X ( ) = X(0) = X(0) = I < then g(t) X ( t ) < ~ ( t )t 2, Proof This is a simple consequence of the result mentioned earlier for integral inequalities, since t,he matrix differential equations are equivalent to Y ( t )= I + t CY (s)ds 1+ 6' AY ( s ) d s , and the function X t- A X is monotonic nondecreasing in the partial order induced by the positive orthant Clearly, a similar result holds for nonpositive matrices In the case where the interval matrix is not of these types, the interval matrix function will in general, have end points corresponding to matrices internal t o the internal © 2003 V Lakshmikantham and R N Mohapatra 6.6 Fuzzy Volterra Integral Equations 165 matrix A and can be estimated numerically by solving the matrix DEs on a grid As a numerical illustration, consider the system where n = and A, B, t 0, xo are given by > with a scalar control law given by the fuzzy-valued function Expressing the p-levels of A as an interval matrix: this interval system matrix is a stable family, because every characteristic polynomial is quadratic with positive coefficients So, for example if o = [go , 60o] Applying Lemma 6.6.2 using the MATLAB /3 = 0.0, a functions EIG and INV for the eigenvalue-eigenvector and matrix inversion calculations where = -1.5768, g = -4.6232, and where rT = -1.1789, C = -2.6211 (INV was not really needed because the end point matrices of the interval are symmetric.) Now, turning to the formula (6.6.9), with both the control system equation and @ interpreted in the differential inclusion sense (6.6.11), and writing xp(t) = [gp( t ) ,zg( t ) ]the , following is obtained © 2003 V Lakshmikantham and R N Mohapatra Chapter Fuzzy Differential Inclusions where Since the integral is a coiwolution, when the Laplace transform is taken X p ( = $ o ( s ) [ ~ ~ o+p@ ] p(s)BU3(~) [& (6.6.14) up Here, qs ( s ) = ( s ) ,J ~ s ) ]U3 ( s ) = [Up( s ) ( s ) ] are respectively, the transforms of @ p ( t ) , u p ( t ) A straightforward, but tedious calculation from (6.6.14) when p = 0.0 gives As we have seen, Hiillermeir [40] suggested a different formulation of the fuzzy initial value problem based on a family of differential iiiclusions at each P-level, /3 1, namely, < x'(t) E ( ~ ( t , z ( t ) ]x (~0 ) = [xo]', where [G(.,.)lo : R x Rn + K z However, Hiillermeir does not prove that S ( z o T) and A(zo,t ) are fuzzy sets and moreover requires that [ G ( t ,x)lP be not only bounded but also continuous and Lipschitz in x with respect t o d H The results presented in Section 6.2 are taken from Diamond and Watson [25]where to formulate the problem of fuzzy differential inclusions the notion of quasi-concavity is employed t o obtain regularity of solution sets See also Diamond [18] Section 6.3 essentially lists the needed results for differential inclusions from Deimling [17] In Section 6.4, fuzzy differential inclusions are discussed and periodicity and stability results are presented, which are taken from Diamond [19] The variation of constants formula considered in Section 6.5 is from Diamond [21] See Rzezuchowski and \$'asowski [I031 for the results on continuous dependence and parameters and initial values of solutions of differential equations with fuzzy parameters via differential inclusions Finally for the results related t o fuzzy Volterra integral inclusions given in Section 6.6, see Diamond [20]which depends on the corresponding results on integral equations that are taken from Corduneanu [12].For allied results, see also Diamond [22] © 2003 V Lakshmikantham and R N Mohapatra [I] Aubin, J.P and Cellina, A., Differential Inclusions, Springer Verlag, New York 1984 [2] Auinann, R.J., Integrals of set-valued functions, J Math Anal Appl (1965), 1-12 [3] Banks, H.T and Jacobs, M.Q., A differential calculus of multifunct i o n ~ J, M A A 29 (1970), 246 272 [4j Bernfeld, S and Lakshmikantham, V., A n Introduction t o Nonlinear Boundary Value Problems, Academic Press, New York 1974 [5] Bobylev, V.N., Cauchy problem under fuzzy control, B USEFAL (1985), 117-126 (61 Bobylev, V.N., A possibilistic argument for irreversibility, Fuzzy Sets and Systems 34 (1990), 73-80 [7] Bradley, M and Datko, R., Some analytic and measure theoretic properties of set-valued mappings, SIAM J Control Optim 15 (1977)) 625-635 [8] Buckley, J J and Feuring, T., Fuzzy differential equations, Fuzzy Sets and Systemu 110 (2000), no 1, 43-54 [9] Castaing, C and Valadier, M., Convex Analysis and Measurable Mult