Theory of Fuzzy Differential Equations and Inclusions

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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 1

Method of Variation of Parameters for Dynamic Systems V. Lakshmikantham and S.G. Deo Volume 2

lntegral and Integrodifferential Equations: Theory, Methods and Applications edited by Ravi P. Aganval and Donal O'Regan Volume 3

Theorems of Leray-Schauder Type and Applications Donal O'Regan and Radu Precup Volume 4

Set Valued Mappings with Applications in Nonlinear Analysis edited by Ravi P. Aganval and Donal O'Regan Volume 5

Oscillation Theory for Second Order Dynamic Equations Ravi P. Aganval, Said R. Grace and Donal O'Regan Volume 6

Theory of Fuzzy Differential Equations and Inclusions V. Lakshmikantham and R.N. Mohapctra Volume 7

Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations I/. Lakshmikantham and S. Koksal

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y Differentia quations and Inclusions

V. Lakshmikant and

Taylor &Francis Group LONDON AND NEW YORK


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

1 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 . . .

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2 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 . . .

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3 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 . . . . . . . . . . . . . . . . . . . . . 4 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 . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

........ . . . . . . . . . . . . . . . .

5 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 . . . . . . . . . . . . . . . . . . . .

6 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 2 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 4. Chapter 5 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 do 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.


Chapter 1

Fuzzy Sets 1.1 Introduction An exact description of any real world phenomenon is virtually impossible and one needs to accept this fact and adjust to it. The inexactness of the description is not a liability but is a blessing because it makes for greater efficiency. To specify imprecise or vague notions, Zadeh introduced the concept of fuzzy set theory. A fuzzy set is a membership function which describes the gradual transition from membership to nonmembership and is a subjective one. Spaces of such fuzzy sets are function spaces with special properties. Since the monograph of Diamond and Kloeden [24]provides a good exposition of fuzzy set theory outlining the background and covering a broad aspect of topological properties of spaces, we have included minimal background material sufficient to deal with the theory of fuzzy differential equations and inclusions. In fact, the contents of Chapter l are adapted from their book. Section 1.2 considers fuzzy sets, Zadeh's extension principle and the necessary spaces. Section 1.3 is devoted to the Hausdorff distance between subsets of Rn and its properties. Support functions form the content of Section 1.4. Theory of metric spaces of normal, upper semicontinuous, fuzzy convex fuzzy sets with compact support sets on the base space En, is discussed in Section 1.5. This section includes the representation theorem which is also important in the development of fuzzy differential inclusions. Section 1.6 deals with the properties of metric space (En,d) proving its completeness and properties of the metric d. Finally, Section 1.7 provides notes and comments.


Chapter 1. Fuzzy Sets

2

The idea of a fuzzy set was first proposed by Lotfi Zadeh in the 1960s, as a means of handling uncertainty that is due to imprecision or vagueness rather than to randomness. Fuzzy sets are considered with respect to a nonempty base set X of elements of interest. The essential idea is that each element x E X is assigned a membership grade u(x) taking values in [O, 11, with u(x) = 0 corresponding to nonmembership, 0 < ~ ( x < ) 1 to partial membership, and u(x) = 1 t o full membership. According t o Zadeh a fuzzy subset of X is a nonempty subset {(x, u(x)) : x E X ) of X x [0,1] for some function u : X i [O, 11. The function u itself is often used for the fuzzy set. For instance, the function u : R1 -t [O, I] with

provides an example of a fuzzy set of real numbers. There are of course many other reasonable choices of membership grade function. The only membership possibilities for an ordinary or crisp subset A of X are nonmembership and full membership. Such a set can thus be identified with a fuzzy set on X given by its characteristic function X A : X -+[0, 11, that is, with

Metric spaces of fuzzy sets provide a convenient mathematical framework for diverse applications of fuzzy sets. They are essentially spaces of special kinds of functions from a base space X to [0,1],where X is a metric space. The a-level set [uIa of a fuzzy set u on X is defined as [u]~={xEX : u ( x ) > a ) foreach

a€(Ojl],

(1.2.3)

while its support [uI0 is the closure in the topology of X of the union of all of the level sets, that is

[.I0=

u

[.la.

(1.2.4)

a€(O,lI

The union, intersection and complement of fuzzy sets can be defined pointwise in terms of their membership grades. Consider a function u : X + [O, 11 as a fuzzy subset of a nonen~ptybase space X and denote the totality of all such functions or fuzzy sets by 3 ( X ) . © 2003 V. Lakshmikantham and R. N. Mohapatra

3

1.2 Fuzzy Sets The complement uC of u E F ( X ) , the union u u A v of u, v E F ( X ) are defined, respectively, by

V

v and the intersection

for each x E X . Clearly uC,u V v, u A v E F ( X ) . The Zadeh extension principle allows a crisp mapping f : X1 x X2 -+ Y ; where X I , X2, and Y are noneinpty sets, t o be extended to a mapping on fuzzy sets f : 3(X1) x F(X2) 3 ( Y ) -)

where

for y E Y. Here f - l ( y ) = {(x1,x2) E X I x X 2 : f ( x l , x Z ) = y ) may be empty or contain one or more points. The obvious generalization holds for mappings defined on an N-tuple XI x . . . x X N where N 2 1, with the wedge operator being superfluous when N = 1. The definitions of addition and scalar multiplication of fuzzy sets in F ( X ) involve the extension principle and require the base set X t o be a linear space. For the addition of two fuzzy sets u,v E F ( X ) , the Zadeh extension principle is applied to the function f : X x X -t X defined by f (xl, 2 2 ) = x1 x2 to give

+

(uk)(x)=

sup XI +x2

=x

u(x1)Av(x2),

(1.2.9)

for all x E X , while for scalar multiplication of u E F ( X ) by a nonzero scalar c, the function f : X -+X defined by f (z)= ex is extended t o

for all x E X. Obviously, both u ?- v and ck belong t o F ( X ) . The totality of fuzzy sets 3 ( X ) on a base space X is often too broad and general to allow strong or specific enough results to be established, and therefore various restrictions are often imposed on the fuzzy sets. In © 2003 V. Lakshmikantham and R. N. Mohapatra

4


particular, a fuzzy set u E F ( X ) is called a normal f~lzzyset if there exists at least one point xo E X for which u(xo) = 1, so the 1-level set [u]' and hence every other level set [u]" for O < a < 1 and the support [uI0 of u are all nonempty subsets of X. For technical reasons, the level sets are often assumed t o be compact and, when X is a linear space, also convex. In fact, the convexity of the level sets of a fuzzy set u is equivalent to its being a fuzzy convex fuzzy set, that is, satisfying u(Xxl

+ (1 - X)x2) 2 u(xl) /\ u(x2)

for all

XI,

x2 E X ;

X

E [0, 11.

(1.2.11) In the case of fuzzy numbers, that is fuzzy sets u : R -t [0, I], fuzzy convexity means that the level sets are intervals. We shall consider the following three spaces of nonempty subsets of R n . (i) Cn consisting of all nonempty closed subsets of Rn, (ii) Kn consisting of all nonempty compact subsets of R n ; (iii) KE consisting of all nonempty compact convex subsets of Rn. We then have the strict inclusions

Recall that a nonempty subset A of Rn is convex if for all a1 a2 E A and all X E [0, I], the point (1.2.12) a = Xul (1 - X)a2

+

belongs to A. For any nonempty subset A of R n , we denote by coA its convex hull, that is the totality of points a of the form (1.2.12) or, equivalently, the smallest convex subset containing A. Clearly

with A = coA if A is convex. Moreover coA is closed (compact) if A is closed (compact). Let A and B be two nonempty subsets of R n and let X E R.We define the following Minkowski addition and scalar multiplication by

and XA = {Xu: a E A). Then we have © 2003 V. Lakshmikantham and R. N. Mohapatra

1.3 T h e Hausdorff Metric

5

.I. T h e spaces Cn, K n and ICE are closed under the operations of addition and scalar multiplication. In fact, these two operations induce a linear structure on C n , Kn and ICZ with zero element (0). The structure is that of a cone rather than a vector space because, in general, A + (-1)A

# {O}.

le 1.2.1. Let A = [O, 11 so that (-l)A = [-I, O]! and therefore

Thus, adding -1 times a set does not constitute a natural operation of subtraction. Instead, we define the Hukuhara difference A - B of nonempty sets A and B , provided there exists a nonempty set C satisfying

From the preceding example, [-I, 11 - [-1,0] = [0,1] and

[-I, 11 - [0,1] = [-1,0].

Clearly, A- A = (0) for all nonempty sets A. From (1.2.16), a necessary condition for the Hukuhara dzflerence A - B t o exist i s that some translate of B i s a subset of A, B {c} C A

+

for some c E Rn. W h e n it exists, A - B i s unique. However, that the Hukuhara diflerence need not exist is seen from the following example. Example 1.2.3. (0) - [O, 11 does not exist, since no translate of [0, 11 can ever belong t o the singleton set (0).

Let x be a point in Rn and A a nonempty subset of Rn. The distance d(x, A) from x to A is defined by d(x;A) = inf{llx

-

all : a E A}.

Thus d(x,A) = d(x,A) 2 0 and d ( x , A ) = 0 if and only if x E closure of A in Rn. © 2003 V. Lakshmikantham and R. N. Mohapatra

(1.3.1)

A, the

Chapter 1. Fuzzy Sets We shall call the subset

< E)

S,(A) = {n: E R n : d(x, A)

(1.3.2)

an E-neighborhood of A. Its closure is the subset

In particular, we shall denote by

S;2 the

closed unit ball in R n ,that is

which is obviously a compact subset of Rn. Note that for any E > O and any nonempty subset A of R n . We shall for convenience sometimes write S(A, E ) and S(A,E) for S, (A) and S, (A). Now let A and B be noiiempty subsets of Rn. We define the Hausdorff separation of B from A by

or, equivalently,

We have d&(B,A) triangle inequality

> O with d k ( B , A) = 0 if and only if B C A.

Also, the

holds for all nonempty subsets A, B and C of Rn. In general, however

We define the Hausdorff distance between noneinpty subsets A and B of R n by (1.3.8) dH (A, B) = max{dk(A, B ) , d k ( B , A)}, which is symmetric in A and B . Consequently,

>

(a) dH(A,B) 0 with dH(A,B ) = 0 (b) d H ( 4 B ) = ~ H ( BA); , (c) dH(A. B) dH(A>C ) dH (C,B);

0 , then

Conversely, if {A" : 0 5 a 5 1 ) is a family of subsets of Rn satisfying (1.5.1)-(1.5.3) then there exists a u E En such that

and

Proof.

If u E En, we have from the definition

where [uI0= cl U,,(o,l~ [u]". Since u is normal, u maps Rn onto I = [O, 11. Also [u]"is a compact subset of Rn for all a E I and for any nondecreasing . also know u and [u]"are convex sequence a, + a in I ; [u]"= n,21 [ u ] " ~We by Lemma 1.5.1. This proves the first part of Theorem 1.5.1. We next prove the converse. For x E A', define I, = [a E I : x E A"] and let a0 = sup I,. We claim I, = [0,ao].If a0 = 0 , there is nothing to prove and hence suppose that a0 > 0 and let ,O E ( 0 ,a o ) . Then there exists PI E [P, cro) such that pl E I,. Thus x E AP1 which implies by (1.5.2), x E A@and P E I,. By definition, 0 E I, and heme [0,a o )5 I,. Now let a, converge to cro in I, monotonically. Then x E A". for each i = 1 , 2 , . . . and therefore by (1.5.3), x E AaO.Thus a0 E I, and [O, ao]2 I,. Finally, since P E I, implies that P 5 ao, we have I, C [0,a o ] . Hence I, = [0,ao]as was asserted. Let a E [ O , l ] . If x E [u]", then u ( x ) 2 a > 0 and so x E AO and u ( x ) = sup I, = a0 a. Hence x E A"0 and consequently b y (1.5.2) x E A", that is, [u]"C A". Conversely, if x E A" , then u (x) = sup I, = ao a and

>


>

1.5 The Space En

13

hence x E [u]". This implies A" C [u]". Combining the results, we obtain [u] = A" for all a E (0, 11. Defining [uI0 as above, we find that u maps Rn onto I and is upper semicontinuous (usc) since [u]" are closed. Furthermore [uI0 is compact and hence bounded. It is also convex. Thus u satisfies the requirements except convexity. To prove convexity, let x , y E [u]" with min[u(x);u(y)] = y 2 a . (1 - A)y E [u]? for any Then, x, y E [uj?, which is convex and so Ax X E [0,11. Hence

+

and therefore u is fuzzy convex. Hence u E En and the proof is complete. An advantage of only requiring u E En t o be upper seinicontinuous and not continuous is that the nonempty compact subsets of Rn can then also be included in En by means of their characteristic functions. By a straightforward application of Theorem 1.5.1, we obtain

osition 1.5.1. If A E E z , then XA E En. We shall also need the following result. See Castaing and Valadier [9] for a proof.

.

Let {Ak} be a sequence in Pk(Rn)converging to A. Further, let dH(Ak,A) -+ O as k -+ oo. Then

In the context of fuzzy sets we call a subset of Rn, or more precisely its characteristic function X A , a crisp subset of R n . The endograph end(u) of a fuzzy set u E En is defined as end(u) = {(x, a ) E R n x I : u(x) 5 a } .

(1.5.4)

It is a nonempty closed subset of Rn x I. Restricting t o those points that lie above the support set, we obtain the stzpported endograph, or sendograph for short, of u (1.5.5) send(u) = end(u) n ([u1° x I), which is a nonempty compact subset of Rn x I. In fact, send(u) = U{[ulY x {a} : a E I).



14 Let A € K z j

E E n . Then

SOXA

e n d ( x a ) = (Rnx { 0 ) )

[xA]= a f o r a l l a

E

I;

U ( Ax I )

We shall define addition and scalar multiplication of fuzzy sets in En levelsetwise, that is, for u , v E E n and c E R\{O)

+

+

(1.5.6)

= c[u]"

(1.5.7)

[u v]" = [u]" [v],

and [CU]"

for each a E I . osition 1.5.2. E n i s closed under addition (1.5.6) and scalar multiplication (1.5.7).

Proof. We apply Theorem 1.5.1 to the families of subsets { [ u + v]" : a E I ) and { [ c u ] " ) . Properties (1.5.1) and (1.5.2) follow from those for {[u]" : a E I ) and { [ v ] " : a E I ) , definitions (1.5.6) and (1.5.7), and the closedness of Kz under set addition and scalar multiplication. Now let { a i ) be a nondecreasing sequence in I with ai 1' a in I. Then, by (1.5.6) and (1.5.7), by property (1.5.3) for { [ u ] " : cu E I ) and {[v]" : cr E I ) , and b y Proposition 1.3.2,

so both families of subsets also satisfy property (1.5.3). Hence, by Theorem 1.5.1, u v E En and c u E E n . In Section 1.2, the Zadeh extension principle was used to define the addition and scalar multiplication of fuzzy sets. That is.

+

( u T v ( z ) = sup min{u (z),v ( y ) ) z=x+y


(1.5.8)

1.5 The Space En

15

and (c?L)(x)= u(x/c).

(1.5.9)

In En, these are equivalent to the level set definitions (1.5.6) and (1.5.7) respectively.

If u, v E En a n d c E R\{O), then

-

u+u=u+u Proof.

and

Z=cu

Let a E (O,l]. Then

and so definitions (1.5.7) and (1.5.9) coincide. Now suppose tliat ( Z L $ v) (x) 2 a . By the definition of the supremum, there exist z k E [ u ] ~ ( ' - ' / ~. )Yk E [G]"('-'/~)for k = 1 , 2 , .. . such that 21, yk = z and so

+

Since [uja('-'Ik) -+ [ u ] ~[,u ] ~ ( ' - ~ -/+~ )[u]" with respect to the Rausdorff metric d H , by the compactiiess of all of these sets there exists xk3 --+ 53 and yk, -+ ij. Hence, xkJ yk+3 + Z ij. But xkj ykj = z SO z = J: Y E [u]" [v]" and {Z : (~Tu)(z) CY} [uIa [via.

+

+

+ r

+

+

c

+

>

Conversely, if J: E [u]" and ij E [u]", so that ~ ( 2 ) a and v(Q) with z = Z 5, (uTv) (z) 2 inin{u(?), v(ij)) a ,

+

> a , then

>

+

and so [uIQ [vIQC { z ( u T v ) ( z ) 2 a ) . Thus we have shown that

so definitions (1.5.6) and (1.5.8) coincide. The concept of support function of a nonempty compact convex subset of Rn, introduced in Section 1.4, can be usefully generalized to the fuzzy sets in En. Let u E En and define s, : I x Sn-' -+ Rn by

for (a, p) E I x Sn-l, where s(., [u]") is the support function of [u]". We shall call s, the support function of the fuzzy set u. Note tliat the supremum in © 2003 V. Lakshmikantham and R. N. Mohapatra


16

(1.5.10) is actually attained since the level set [u]" is compact and so can be replaced by the maximum. Moreover, for u. v E En u =v

if and only if

s , = s,,

(1.5.11)

since the support function on ICz uniquely characterizes the elements of Kz; see Proposition 1.4.1. The following properties follow directly from those of the support functions of the level sets.

En.T h e n the support function s , i s uniformly bounded o n I x Sn-l; -4. Let u E

(i)

(ii) Lipschitz i n p E

sn-I unzformly o n I; and

(iii) for each a E I ,

Proof.

Using inequalities (1.4.2) and (1.4.3)

since [u]"C [uI0and ilpll = 1 , and

Thus (1.5.1) and (1.5.2) hold. Property (1.5.3) is a restatement of (1.4.9). In addition, we obtain the following dependence on the lneinbership grade. osition 1.5.5. Let u E En. T h e n s , (., p) is nonzncreasing and left continuous i n a E I for each p E Sn-l.

s u ( P ,P ) = s U ( P ?

[ ~ l 5~ S)( P , [ u I ~ ~ = ) ~ u ( aP),

b y (1.4.4): so s,(., p ) is increasing for each p E nondecreasing sequence ai a in I ,


Sn-l.In addition,

for a

17

1.6 The Metric Space (En, d)

Let u E E' have level sets [u]" = 1-1 +a71 -a] for a E I . Then, as So = {-l,+l),the support function s, is given b y s,(a,il)= 1 - a for all a: E I .

A fuzzy set u E En is called a Lipschitzian fuzzy set if it is a Lipschitz function of its membership grade in the sense that

for all a , P E I and some fixed? finite constant K. In view of Proposition 1.5.4,this is equivalent to the support function s,(.,p) being Lipschitz uniformly in p E Sn-l. The subset EE of convex-sendograph f~rzzysets consists of Rn x I. Hence u E EF if and only if u : R n + I is a concave function over its support [zL]', that is if and only if

for all x,y E [us0and X E [0,1].Note that a fuzzy convex fuzzy set is not necessarily a convex-sendograph fuzzy set. We shall denote by E& the subset consisting of those u E EE for which the uppermost level [u]'is a singleton set. The fuzzy sets u in E' are often called fuzzy numbers. The triangular fuzzy numbers are those fuzzy sets in El for which the sendograph is a triangle. A triangular fuzzy number IL E E' is characterized by an ordered E R~with 21 5 2, 5 x, such that [uI0 = [ 5 1 , ~and r] triple (x2;x,,x,) [u]'= {x,),for then

for any a E I. In addition

so all the triangular fuzzy numbers are Lipschitzian.

Since En is a space of certain functions u : R n for a metric on En is the function space metric


-

I an obvious candidate


18

which measures the largest difference in the membership grades of the two fuzzy sets 21, v E En over all points x in the base space Rn. Note that d(cu, cv) = d(u, v)

(1.6.2)

for all u, v E En and c E R\{O}. where cu, cv E En are interpreted as the scalar multiplication of fuzzy sets (1.5.9), rather than the usual multiplication of a function by a scalar. Let DH denote the Hausdorff metric on Rn+'. For any u E Eni the sendograph send(u), defined by (1.5.5). is a nonempty compact convex subset of Rn x I c Rn+'. The sendograph metric D , on En is defined in terms of the Hausdorff metric on the subspace send(En) = {send(u) : u E En) of ICn+l, that is D, (u, t i ) = DH (send(u),send(z1)) (1.6.3) for all u, v E E7'. It is certainly a metric since send(u) =send(v) if and only if u = v in En. From the properties of the Hausdorff metric DH on jCn+l, it follows that

for all u, v, w, XI' E En and

for all u, v E En and w E E;, that is with send(w) convex. The most commonly used metrics on En involve the Hausdorff metric distance between the level sets of the fuzzy sets. They are, in fact, function space metrics applied to functions 4 : I -+R + defined by

for cr E I, where 21,v E E n . In view of (1.5.12) and Proposition 1.5.5, these functions are left continuous and hence measurable on I . The supremum metric d on En is defined by

for all u, v E En and is obviously a metric on En. The supremum in (1.6.7) need not be attained, so cannot be replaced by the maximum.

-1.Let u, v

E

El be defined o n level sets by


1.6 The Metric Space ( E n 9d )

and [u]"= { 0 ) , [v]"= [ 0 , 2 ( 1- a ) ] for

1 < a 5 1, 2

-

T h e n s u p { $ ( a ) : a E I ) = 1, but this is not attained. From the properties of the Hausdorff metric listed in Propositions 1.3.2 and 1.3.3, we get ~ ( c uC ,U ) = Icld(ujV )

and

d(u

+ w, v + w')

5 ( u ,v ) + d(w, w ' )

for all c > 0 , and all u, v,U I , w' E En. In view of the identity (1.5.12) relating the Hausdorff metric distance between level sets and the distance between their support functions, an alternative expression for d is given by

for all u , v E E n . The spaces ( E n jD m ) and ( E n ,d ) are complete metric spaces. The proof for showing ( E n , D m ) is complete is too long and we refer to Diamond and Kloedeii [24].We shall therefore provide the proof for ( E n ,d ) only. .I. ( E n jd ) is a complete metric space

Proof. Let { u k ) be a Cauchy sequence in ( E n jd). Then { [ u ~ ] " )for , each a E I ; is a Cauchy sequence in (ICE; d H ) , which is complete, so that there exists a 6, E K z for each a E I such that

This convergence is. in fact, uniform in a E I . We shall show that the family { C , : a E I ) satisfies conditions (1.5.1) and (1.5.3) and so there exists a u E En such that [ u ] , = 6" for a E I . Since the 6, E KF for a E I , condition (1.5.1) is obviously satisfied. Consider O /3 < a 5 1. Then

C,% and consequently

for all x E C,

o is also a metric on En. In fact, by definition (2.2.2) we have for all a E ( O , 1 ]

Since u0 and v0 are conlpact, then Ail' and consequently d(u, v), is finite. Furthermore, since dH is a metric, d obviously satisfies the axioms of a metric. Let ( 2 1 , ) be a sequence in En. We say that {u,) converges levelwise t o u E E7'if for all a E (0. I]

Finally denote

G = {u

E

En : t~ is concave).

In other words, if u E G then

for all x ; y E uO,X E I. If u E G, then by Lemma 2.2.3, u0 is convex and from the concavity of u it follours that also send(u) is convex.

.

The following implications for convergences i n En hold

true: ( 1 ) Convergence in (En,d) implies levelwise convergence.

(2) Convergence i n (En,d ) implies convergence i n (En)N ) .

Proof- The implication (1) is trivial. For the proof of (2), let E > O be arbitrary and let u,v E En such that d(u, v) < E . Choose any (x, a ) E send(u). If a > 0 then x E u". Since dH(ua, v") < E there exists a y E v" such that /Ix- yll < E. This proves that (x,oc) E N ( s e n d ( v ) , ~for ) (y.~E ) send (v) . © 2003 V. Lakshmikantham and R. N. Mohapatra

2.2 Convergence of Fuzzy Sets

27

If, on the other hand, a = 0 then x E uO. We show that dH(uO;vO)< E. Hence we can find a y E v0 such that Ilx - yll < E and as before we conclude that (x, a ) E N(send(v), E ) . Let {a,) be a decreasing sequence of real numbers converging t o zero. Then {uan) is a nondecreasing sequence of subsets of a compact set uO. Hence it has a metrically convergent subsequence and thus by Lemma 2.2.2, it also converges. By (2.2.3) the limit can be expressed as

Since {a,) converges t o zero and {uan) is nondecreasing, we see that the limit equals uO. So we have lim dH (uan,uO)= 0

n-00

and similarly for v. Since dH is a metric we have

Passing to the limit yields

The preceding argument shows that send(u) c AT(send(v),E). Similarly, we can prove that send(v) C iV(send(u),E ) and consequently N(send(u), send (v)) < E. It follows that the identity mapping

is continuous and hence convergence in (En,d) implies convergence in En,H ) . we next show by examples that the implications (1) and (2) cannot be reversed.

u(x)=


1 zjf O < X < 0 elsewhere

l ,

Chapter 2. Calculus of Fuzzy Functions

28 and

I++

~n ( x ) =

if O L : x < 1 , 0 elsewhere,

H(send(u,),send(u)) = 0 . B u t where n = 1 , 2 , . . .. T h e n clearly, lirn,,, dH(u:, u l ) = 1 for all n = 1 , 2 , . . ., so that {u,) does not converge levelwise and hence neither i n ( E n ,d ) .

v(x)=

1 zf x = 0 , 0 elsewhere

and

where n = 1 . 2 , . . .. T h e n vg = [O: 1 - al/"] and

It follows that for all a E ( O , l ] , dH(va,v,Q)--, 0 as n s u p d H ( v a ,u:) = 1 for all a>O

--+

oo. However,

n = 1,2,. . . .

Hence (u,) does not converge i n ( E n ,d ) . W e also see that d H ( v Ov:,) = 1 for all n = 1 , 2 , . . . and h.ence by a theorem of Kloeden [52] {v,} does not converge i n ( E n ,H ) . However i f we confine to G t h e n t h e implication ( 1 ) o f Theorem 2.2.1 can b e reversed. Before proving t h a t , we demonstrate some auxiliary lemmas.

-2.4.Let {u,) be a sequence i n

converging levelwise to u 6 G .

Then

u:) = 0 . lirn d~ (uO.

n-cc

From (2.2.5) we deduce that lirn d H ( u a ,u O )= 0 .

a-o+

Let

E

> 0 b e arbitrary and b y (2.2.6) choose an cr > 0 such t h a t


2.2 Convergence of Fuzz.y Sets For this cr choose an no such that for all n

If now n

> no we have

> n o then

which implies that uOc N(u,", E )

c ~ ( u :E ),.

Conversely we prove that there exists an

nl

(2.2.7)

such that if n

> nl then

On the contrary, suppose that

for infinitely many indices n. Kow by the assumption, u: intersects l V ( u O E, ) for all sufficiently large indices n. Then, by taking a subsequence if necessary, we may assume that (2.2.9) holds true for all n = 1 , 2 ,. . . and. since u0 is compact, there exists a sequence (2,) such that 2, E u: and d(z,; u O )= E for all n = 1 , 2 , . . .. Now choose a point y E u l . Since { u i ) converges metrically to u1 then there is a sequence {y,) converging to y such that y, E u; c u: for all n = 1 , 2..... Since {y,) converges t o y and 11y - zn 11 E for n 1, then there exist a sequence {t,) in I and an integer n2 such that

>

whenever n 2 n 2 ; where z,, = X, Now for n n2 we have

>

where p =diam(uo)

+ t n ( y n - xn).

< m , and consequently,


>

30 But for n

Chapter 2. Calculus of Fuzzy Filnctions

> n2,

and according t o the concavity of u,,

But this contradicts the fact tliat { u i ) converges metrically to uo. Hence (2.2.8) holds true and the leinrna follows from (2.2.7) and (2.2.8).

Let u E G and a E I be fixed. Then the function g(P) dH(uP,ua) is continuous at a .

=

Proof. Recall that we have already proved the continuity of g at a = 0 (see (2.2.6)). Let a > 0 and choose a nonincreasiiig sequence {ak)converging t o a. Then {uffk)is a nondecreasing sequence with uak c u0 for all k = 1: 2, . . . . Since uo is compact, the sequence {ua" has a metrically coiivergent subsequence and hence by Lemma 2.2.2 it converges t o the limit

Since uak c ua for all k = 1, 2 , . . . and uff is closed, we have B c tia. Conversely suppose that x E u a \ B . If u(x) > a , then x E uff"or all k sufficiently large and consequently x E B , which is impossible. Thus u(y) = a for all y E u a \ B . Since x E u a \ B and B is closed, then &I = p(x, B ) > 0. Choose y E B such that ~ ( y > ) a , which is possible since u1 c B . By the convexity (1 - t)y E ua and of u" we can choose a t E ( O , 1 ) such that z = tx 1l.z - XI/ = (1 - t)llx = +&I. Hence z E u a \ B and by the preceding paragraph u(z) = a. Applying the concavity of u we obtain

+

which is impossible. So B = ua, which proves that g is right continuous a t a. On the other hand, let {ak)be a nondecreasing sequence converging to a . As before, we see tliat { z i a k ) converges metrically t o B. But now , ~ = uan and consequently uak c ua71 for all k 2 n and hence c I ( U ~ uffk) B = n,"==, uffn.Thus by the representation-theorem 1.5.1, B equals u". This proves the left continuity of g and the lemma is proved. © 2003 V. Lakshmikantham and R. N. Mohapatra

2.2 Convergence o f Fuzzy Sets As a corollary we have:

.I. The function g ( P ) = d H (wR,u") is continuous. The triangle inequality for d H yields

Proof.

The desired result follows by Lemma 2.2.5 if we let y approach

P.

. Let {a,}

be a sequence of real nvmbers converging t o a E I . T h e n under the assumptions of L e m m a 2.2.4 lim d H ( u a , u ~ n=) 0.

nice

Proof. Suppose on the contrary that there are an c {u::" } such that

d ~ ( u " , u E f ' )2 E

for all

i

=

> 0 and a subsequence

1 , 2, . . . .

Since by Lemma 2.2.4, { u i } converges metrically to u0 we have u:" N ( ~ O >1) for all i sufficiently large. Furthermore, since c l ( N ( u o ,1)) is com" ) a metrically convergent subsequence. Thus pact it follows that { u ~ ~has we may without loss of generality assume that { u : ~ ~converges ) metrically t o a compact set A with d H ( u a , A ) = M E . We divide the rest of the proof into two cases.

>

.

Let the sequence { a , ) be nondecreasing. By (2.2.6) and the levelwise convergence of {u,} to u we have

Then by Lemma 2.2.1 we can find an x E u" and a y E A such that llx - y 11 = d H ( u 0 .A ) = Dl. Since {u:) converges metrically to u" and { u Z n } t o A for a11 n 1 1 we can choose x , E u: and a y, E u,an such that lim x , = x

n-00


and

lim y, = y. 7%-00

32


+

+

z , = z = $ ( x y) and by ( 2 2 . 1 8 )

Denote z,, = 1 ( z , yn). Then lirn,,, and the choice of x and y we have

1 p ( z i 21") = llx - z I I = --M 2 and by the concavity of u,

Now choose a /? < a such that d H ( u p ,u") < ; M , which is possible Lemma 2.2.5 and let no he an integer such that for all n 2 n o , we have

/jz,

-

1 zI/ < -hf

8

and

a,

2 p.

Since u p c N ( u 0 ; $&I) and p ( z , el") = ;M then

combining these inequalities, we obtain

>

for all n no. However by (2.2.11), z , E u,B for n 2 n o . But this is irnpossible since d , (d, u f ) converges to zero as n -+ oo. This proves Case 1.

. Assume that the sequence {a,) is nonincreasing. As in Case 1, we can show that

A

c ua.

(2.2.12)

Since d H ( u a :A) = PI > 0 we get ua\A # 0. Now let y E ua\A be such that u ( y ) = /? > a. Then there is an index k such that


2.2 Convergence o f Fuzzy Sets

33

Since u$ c uEm for all m 2 k and the sequence { c l ( ~ , > , u & ~ ) ) is nonincreasing it follows that

This contradiction proves that u(y) = a for all y E uff\A. Since {uk) converges metrically to u1 we have

Identical reasoning as in the proof of Lemma 2.2.5 yields a contradiction. Thus according to Cases 1 and 2, for every E > 0 there exists an n(e, a ) such that d ~ ( u ~ , u< ; ~E )

>

for all n n ( ~a), , which proves the lemma. We are now ready to prove:

.

Let {u,) be a sequence in S converging levelwise t o u E G. Then {u,) converges to u i n (G,d ) . Proof. Let a E I and {a,) be a sequence in I converging t o a. Define a sequence of functions f, : I -+ [O, GO) by

Then by the triangle inequality, we have

and hence by Lemmas 2.2.5 and 2.2.6, liin,,, f,(a,) = 0. So { f,) converges contiiiuously to zero and hence, since I is compact, it converges uniformly. But this is equivalent to convergence in (G, d). 2 . 1 . The functions f, are even continuous. In fact

and Lemma 2.2.5 gives the desired conclusion. © 2003 V. Lakshmikantham and R. N. Mohapatra


34

Example 2.2.1 shows that even in G convergence in levelwise convergence.

(G, N )does not imply

Denote

GI

=

{u E G : u1 coinprises only one point).

Then we have: -3. Let {u,) be a sequence in GI. If {u,) converges t o u E in (GI, H) then it also converges levelwise to u.

Gi

Since {un) converges in (GI,H) by a theorem of Kloeden [52] Proof. we get that for each 7 > 0 there exists an integer n(7) such that for all 0 < a 1 and n 2 n(7) we have

0 such that llxlj M for all x E Fo(t) and t E T. But this implies that d(G(s),G(t)) 5 M ( s - t ) ,

0 the II-difference of u(t + h) - u(t), v(t + h) - v(t) exists, and we have for t E I, m(t

+ h)

-

m ( t )= d[u(t + h), v(t + h)] - d[u(t),v(t)].


3.4 Comparison Theorems Using the triangular inequality for d , we get

+

d[u(t h ) ,v(t

+ h)

+

Id[u(t+ h ) ,u ( t ) h f ( t ,u ( t ) ) ]

+

+ h)],

+d[u(t) h f ( t :u ( t ) ),u(t ,


Chapter 3. Fundamental Theory

56 are solutions of (3.4.1), we find that 1

~ + m ( t= ) lim sup -[m(t + h ) - m(t)] hi0

0 , we find that

If J = [to,oo);we see that limtioo d [ u ( t ) v, ( t ) ] = 0, showing the advantage of Theorem 3.4.3.

We shall prove an existence and uniqueness result under an assumptio~imore general than the Lipschitz-type condition considered in Section 3.2 by the method of successive approximations. © 2003 V. Lakshmikantham and R. N. Mohapatra

3.5 Convergence of Successive Approximations

.I. Assume that f E C [ R + x E n j E n ] and

r ( s ,to, w o ) ) d s


64

Since lim,,o- r ( t , to, wo) exists and is finite by hypothesis, taking the limit as t l , t2 t p- and using the Cauchy criterion for convergence, it follows from (3.7.3) that limt,p- u(t, to, uo) exists. We then define u(P, to,wo) - limt,pu(t, to, uo) and consider the initial value problem

By the assumed local existence, we see that u(t, to, uo) can be continued beyond P, contradicting our assumption that /3 cannot be continued. Hence wo exists globally every solution u(t, to,uo) of (3.7.1) such that d [uo, on [to,m) and the proof is complete.

01
0; is said to be an E-approximate solution of (3.7.1) if v E C[R+,En],v(to,to, vo, E ) = vo and

In case

E

= 0;

~ ( tis) a solution of (3.7.1).

3.8.1. Assume that f E C[R+ x E n , En]and f o r t

> to,u, v E E n ,

where g E C[R$, R+]. Suppose that r ( t ) = r ( t , to, wo, E) is the maximal solution of (3.8.2) W' = g(t, w) E, w(tO)= wg 2 0,

+

existing for t 2 to. Let u(t) = u ( t , t O , u o )be any solution of (3.7.1) and v(t) = v(t, t o rvg; E ) is an E-approximate solution of (3.7.1) existing f o r t 2 to. Then d[u(t),v(t)l 5 r(t,to,ulo,€1, t 2 to, (3.8.3) provided cl[uo,vo] 5 wo. © 2003 V. Lakshmikantham and R. N. Mohapatra

65

3.9 Stability Criteria

Proof. We proceed as in the proof of Theorem 3.4.2 with m ( t ) = d[u(t),v(t)], until we arrive at ~ ' m ( t ) 5 limsupd hi02

This implies using the definition of approximate solution and (3.8.2) the differential inequality

and m(to) 5 WO.The stated estimate follows from Theorem 1.4.1 i11 Lakshmikanthain and Leela [61]. The following corollary provides the well-known error estimate between the solution and an E-approximate solution of (3.7.1). The function g(t, w) = Lw, L > 0, is admissible in TheCorollary 3.8. orem 3.8.1 to yield d[u(t,to, uo), ~ [ tto,, uo, €11

5 d[uo,vo]eL(t-tO)+ Proof.

L

(3.8.4)

Since (3.8.2) in this case reduces to

it is easy to obtain the estimate (3.8.4) by solving the linear differential equation (3.8.5).

Before we proceed further to investigate stability results of fuzzy differential equations, let us note the following fact. In view of Corollary 2.5.1, the solutions of fuzzy differential equations have, in general, the property that diam[x(t)la is nondecreasing as time increases. Hence the formulation we have been working with is not suitable to reflect the rich behavior of solutions of ordinary differential equations. © 2003 V. Lakshmikantham and R. N. Mohapatra


66

Consider the following example. Let a E I' have level sets [ala = [a?,a:] for a E I = [O, 11 and suppose that a solution x : [0,T ] -+ I' of the fuzzy differential equation

dx dt

-- ax,

on

(*I

has level sets [x(t)la= [ x y ( t )x, g ( t ) ]for a E I and t E [O, TI. The Bukuhara derivative also has level sets

&I

g(t)

for a E I and t E [O, T ]and by the extension principle, the fuzzy set f ( x ( t ) )= a x ( t ) has level sets

for all a E I and t E [O, TI. Thus the fuzzy differential equation (*) is equivalent t o the coupled system of ordinary differential equations

for a E I. For a = x l l ) E £I, the fuzzy differential equation (*) becomes

and the system of ordinary differential equations (**) reduces t,o

dx" 1 dt

dxg dt

--

-xy

for a E I. The solution corresponding to an initial value xo E El with [xOlN= [ x & ,x&] for a E I is given by 1

xP ( t ) = ( x & - xg2)et © 2003 V. Lakshmikantham and R. N. Mohapatra

+ 1 (x&+ ~ $ ) e - ~ ,


for a E I and all t 2 0. Thus for xo = x{,,) the solution x(t) = ~(,,,-t) -3 ~ { o as } t -+ 00. On the other hand, when [xojff= [a - 1 , l - a] for a E I, the solution has level sets [x(t)lff= [(a- l ) e t , (1 - a ) e t ] = (1 - a)et[-1, 11

>

for all a E I and t 0. In particular, diam[x(t)lff= 2(1 - a ) e t , and hence the solution becomes fuzzier as time increases. This shows that the stability results considered in this section and in Chapter 4 are of limited applicability. If the stability definitions are not with respect to the zero element of En but relative t o any given solution d ( t ) E En, then the corresponding stability criteria are perfectly in order. Nonetheless. in order t o avoid complexities in formulating such definitions and the results. we have chosen to present the usual definitions and results for convenience, fully realizing their limited usefulness in this setup. However, in Section 5.9, we describe a new concept of stability for fuzzy differential systems which includes, as a special case, the stability results, in the sense of Lyapunov, relative to a given solution. Moreover. the results presented would also cover orbital stability as well as other new notions between Lyapunov and orbital stabilities. We shall discuss some simple stability results. We list a few definitions concerning the stability of the trivial solution of (3.7.1) which we assume t o exist. nition 3.9.1. The trivial solution u = 0 of (3.7.1) is said to be (Sl) equi-stable 2f, for each E > 0 and to E R+, there exists a positive function 6 = 6(to,E) that is continuous in to for each t such that

01

d [uO,

< 6 implies d [ u ( t )0,1 < t , t 2 to,

where u(t) = u(t, to, uo) is the solution of (3.7.1);

(S2) uniformly stable, zf the 6 in (Sl) is independent of t o ; (S3) quasi-equi-asymptotically sta.ble, if for each t > 0 and toeR+, there exist positive do = bo(to) and T = T(to,E ) such that d [uo,61

< do implzes d [ ~ ( t )i)], < t , t 2 t o + T ;



68

(S4) quasi-uniformly asymptotically stable, i f 60 and T i n (S3) are independent of to; (S5) equi-asymptotically stable, if (§I) and (S3) hold simultaneously; (S6) uniformly asymptotically stable, if (S2) and (S4) hold simultaneously; (S7) exponentially asymptotically stable i f there exists a n estimate

Corresponding to the definitions (S1)-(S7), we can define the stability notions of the trivial solution w = 0 of the scalar differential equation

where g E c[R$, R] with g(t, 0) = 0. For example, the trivial solution of (3.9.1) i s equi-stable, i f given E > 0 and to E R+, there exists a d = 6(to,E ) > 0 that i s continuous i n to for each E such that

I wo < 6

implies w (t, to, wo) < E ,

t

2 to,

where w ( t , to,wo) is any solution of (3.9.1) existing o n t

2 to

0

We are now in a position to prove some simple criteria for stability. Assume that

(i) f E C[R+ x s(p),E n ] , s ( ~ = ) [WE" : d [ u , ~ < ] f o r h > 0 , t E R + , u E s(p),

, f ( t , 6 ) = Q and

(ii) g E C [ R $ ,R],g(t,0) z 0. T h e n the stability properties of the trivial solution of (3.9.1) imply the corresponding stability properties of the trivial solution of the fuzzy diflerential equation (3.7.1) respectively. Proof. Let the trivial solution of (3.7.1) be equi-stable. Then, given E and to E R+, there exists a positive 6 = &(to,E ) with the property

O<wo0

(3.9.3)

3.10 Motes and Comments

69

where w(t,to,wo)is any solution of (3.9.1). We' claim that with these E , 6, the trivial solution ~ ( t=j ~ ( tr) 0 of (3.7.1) is equi-stable. If this is false, there would exist a solution u(t) = u ( t , t o ,uo) of (3.7.1) with d [uo, < 6 and tl > to such that

01

d [u(tl),t)] = t

and

d [u(t),0]

< t < p,

to 5 t

< tl.

For [to,tl], using condition (3.9.2), Corollary 3.4.1 yields the estimate

proving the claim. One can prove similarly the other concepts of stability and we omit the details. For example, if g(t, w) = -aw,cr > 0, one gets exponential asymptotic stability, since by Corollary 3.4.1, we get

The results of Section 3.2 are adapted from Kaleva [42, 431 although the proof uses the metric t o simplify matters. The example is also from Kaleva [42, 431. The existence theorem in Section 3.3 is due t o Kieto [83, 821. See also Kaleva [43, 441 and Kloeden [53] for the local existence result parallel to Peano's theorem. Unfortunately, their results are not valid as described. See Friedman et al. [33, 341 for discussion and counterexamples. The results on various comparison results given in Section 3.4 are taken from Lakshinikantham and Mohapatra [66]which depend on the well-known theory of differential inequalities. See Lakshmikantham and Leela [61]. The general result relative to the convergence of successive approximations described in Section 3.5 is due to Lakshmikantham and Mohapatra [66]. which is adapted from a similar result for differential equations in a Banach space. See Lakshmikantham and Leela [60]. The work presented in Sections 3.6 and 3.7 is taken from Lakshmikantham and Mohapatra [66], which is modeled on the corresponding results in ordinary differential equations. Refer to Lakshmikantham and Leela [61]. The results dealing with approximate solutions and the error estimate are adapted from similar results in Lakshmikantham and Leela [61]. Finally, the simple stability criteria presented in Section 3.9 are modeled on the corresponding results in Lakshmikantham and Leela [59]. See for the example, Diamond and Kloeden [24]. © 2003 V. Lakshmikantham and R. N. Mohapatra

70

Chapter 3. Fundamental

For allied results, see Bobylev [5, 61, Chen et al. [lo], Kande [47, 481, Kwun et al. [55, 561, Nayak [77], Nayak and Nanda [78], and Wu [85], D. Park et al. [87, 881, J.Y. Park, et al. [95]-[92], Pears Keumaier [go], Seikkala [105], Song et al. [107]-[110], Wu et al. [11 Zhang et al. [119, 1201, and Zhou and Yu [122]. See also, Ding [ Kandel [46] and Lakshinikantham and Mohapatra [67].


4.1

Introduction

In this chapter, we investigate essentially stability theory via Lyapunov-like functions. We shall also initiate development of fuzzy differential systems utilizing generalized metric spaces. In Section 4.2 we prove a comparison result in terms of Lyapunov-like functions which serves as a vehicle for the investigation of the stability theory of Lyapunov. Section 4.3 establishes results on stability parallel t o the original theorems of Lyapunov in the present framework. We provide in Section 4.4 nonuniform stability crikria employing the method of perturbing Lyapunov functions, under much weaker assumptions. Section 4.5 considers the various boundedness notions parallel to those of stability and offer sufficient conditions for the concepts of boundedness t o hold. In Section 4.6, we embark on initiating the study of fuzzy differential systems, the consideration of which leads to generalized metric spaces, in terms of which one proves comparison results utilizing the concept of vector Lyapunov functions. Section 4.7 discusses the method of vector Lyapunov functions and stability criteria. Since in this setup, one gets a comparison differential system, the study of which is sometimes difficult; we provide certain results to reduce the study of comparison systems t o a single comparison equation. In Section 4.8, we consider the linear fuzzy differential systeni and its perturbation and develop the variation of parameters formula. We also discuss a simple periodic boundary value problem for nonhoniogeneous linear fuzzy differential systems. © 2003 V. Lakshmikantham and R. N. Mohapatra

Chapter 4. Lyapunov-like Functions

72

Consider the fuzzy differential equation

u1 = f (t,u ) ,

to) = uo,

(4.2.1)

01

where f E C [ R + x S ( p ) ,E n ] and S ( p ) = [ti E E n : d [u, < p ] . We assume that f ( t ,6) = Q so that we have the trivial solution for (4.2.1). To investigate stability criteria, the following comparison result in terms of a Lyapunov function is very important and can be proved via the theory of differential inequalities. Here the Lyapunov function serves as a vehicle t o transform the fuzzy differential equation into a scalar comparison differential equation and therefore it is enough t o consider the stability properties of the simpler comparison equation. eosem 4.2.1. Assume that

(ii) D + V ( t ,u ) r limh,o+ sup where g E C[R:, R].

[V(t+h,u+hf ( t ,u ) ) - V ( t , u ) ] j g(t, V ( t ,u ) ) ,

Then, if u ( t ) is any solution of (4.2.1) existing on [to,oo) such that V ( t o ,u o ) 5 W O , we have

where r ( t ,to,wo) is the maximal solution of the scalar dzflerential equation w1 = g ( t , w ) ,

w ( t o )= wo

2 0,

(4.2.2)

existing on [to,oo). Proof. Let u ( t ) be any solution of (4.2.1) existing on [to,oo). Define m ( t ) = V ( t ,u ( t ) )so that m(to)= V ( t o u , o ) 5 wo. Now for small h > 0,

+V(t

+ h, u ( t )+ hf ( t ,u ( t ) )- V ( t ,u ( t ) ) ,

+V(t

+ h,


2 4 )

+ h f ( t ,~ ( t ) ) )V ( t ,~ ( t ) ) , -

4.2 Lyapunov-like Functions using the Lipschitz condition given in (i). Thus

D+m(t) =

0 which is assumed to exist. Hence employing the properties of d[u,v], we see t,hat

Consequently

and therefore

1 h

+

lim sup - [d[u(t h ) ,u ( t )

h+O+

+ h f ( t ,u ( t ) ) ] ]

since u ( t ) is the solution of (4.2.1). We therefore have the scalar differential inequality D+m(t) 5 g ( t , m ( t ) ) , m(t0) I wo, t to,

>

which by the theory of differential inequalities (see Lakshmikantham and Leela [61]) implies

m ( t ) lr ( t ,to, wo),

t 2 to.

This proves the claimed estimate of the theorem. The following corollaries are useful. © 2003 V. Lakshmikantham and R. N. Mohapatra

Chapter 4. Lyapunov-like Fu

74

Corollary 4.2.1. The function g ( t , w ) to yield the estimate

=0

is admissible in Theore

Corollary 4.2.2. If, i n Theorem 4.2.1, ,we strengthen the assump D + V ( t , u ) to u)), D + V ( t , 4 5 - C [ w ( t , 7-41 + d t ,

w,

where w E C[R+x S ( p ) ,R+],C E K = [a E C[[o,p ) , R+] : a(w)i s inc i n w and a ( 0 ) = 01, and g ( t , W ) is nondecreasing i n w for each t E R we get the estimate

whenever V(to,uo) F wo Proof.

Set L ( t , u ( t ) )= V ( t ,u ( t ) )+ J:~ C[w( s ,u ( i ) ) ] d sand note t

using the monotone character of g ( t , w ) . We then get immediately b orem 4.2.1 the estimate

where d t ) is any solution of (4.2.1). This implies the stated estima A simple example of V ( t ,21) is d [ u ,01 so that

4.3

Stability Criteria

Having necessary comparison results in terms of Lyapunov-like func is easy to establish stability results analogous t o original Lyapunov for fuzzy differential equations. Let us start with the following result on eyui-stability. © 2003 V. Lakshmikantham and R. N. Mohapatra


75

Assume that the following hold:

1 D + v ( t , u ) r limsup -[V(t h-O+ h

+ h , u + hf ( t , u ) ) - V ( t , u ) ] 5 0;

(4.3.1)

(ii) b(d[u, 01) 5 V(t, u) 5 a ( t , ~ [ z L01), for (t, u) E R+ x S(p) where b, a(t, .) E K: = [a E C[(O,p), R+]: a(Oj = 0 and a ( w ) is increasing in w] . Then the trivial solution of (4.2.1) is equi-stable. Proof* that

Let 0

< E < p and to E R+ be given. Choose a 6 = 6(to,E ) such a(to,6)

to such that d[u(tl),61 = E

and

d[u(t),01

2 E < p,

to I t I tl.

(4.3.3)

By Corollary 4.2.1, we then have

Consequently, using (ii), (4.3.2) and (4.3.3), we arrive at the following contradiction

Hence equi-stability holds, completing the proof. The next result provides sufficient conditions for equi-asymptotic stability. In fact, it gives exponential asymptotic stability.

. Let the assumptions of Theorem 4.3.1 hold except that the estimate (4.3.1) be strengthened to

Then the trivial solution of (4.2.1) is equi-asymptotically stable. © 2003 V. Lakshmikantham and R. N. Mohapatra


76

Proof* Clearly the trivial solution of (4.2.1) is equi-stable. Hence taking e = p and designating do = &(to,p), we have byTheorem 4.3.1, d[uo,Q]

< 60

implies

d[u(t), 61 < p,

t

> to.

Consequently, we get from assumption (4.3.4), the estimate

Given e see that

> 0, we choose T

= T ( t o ,e) =

8In

+ 1. Then it is easy t o

b(~)

The proof is complete. We shall next consider uniform stability criteria.

and b(d[u,01) 5 V(t, u) 5 a[d[u,o]), a , b E K. Then the trivial solution of (4.2.1) is uniformly stable.

(4.3.6)

Proof. Let 0 < e O such that a(b) < b(e). Then we claim that with this 6: uniform stability follows. If not, there would exist a solution u(t) of (4.2.1), and a t2 > tl > to satisfying d[u(tl),01 = 6,

d[u(tz),01 = e

and

6 5 d[u(t),Q] 5

E

< p.

(4.3.7)

Taking 17 = 6, we get from (4.3.5), the estimate

and therefore, (4.3.6) and (4.3.7) together with the definition of 6; yield

This contradiction proves uniform stability, completing the proof. © 2003 V. Lakshmikantham and R. N. Mohapatra

4.4 Nonuniform Stability Criteria

97

Finally, we shall prove uniform asymptotic stability.

Let th.e assumptions of Theorem 4.3.3 hold except that (4.3.5) is strengthened to

T h e n the trivial solution of (4.3.1) is uniformly asymptotically stable. By Theorem 4.3.3, uniform stability follows and so for Proof. = So ( p ) . This means that designate

d [ u o ,01 < do

implies

d [ u ( t ) 0,1

E =

p, we

< p: t 2 t o .

w7

In view of uniform stability, it is enough t o show that there exists a t* such that for to t* j to T , where T = 1

O such that

Since the trivial solution of (4.4.1) is equi-stable, given = &*(to, E ) > 0 such that

$

> 0 and

to E R+, we can find a d* O < wlo < 6 "

implies

So w l ( t , t o , w l o ) < -, 2

trto,

(4.4.5)

where w l ( t ,to,w l o ) is any solution of (4.4.1). Choose wlo = V ( t ou, o ) Since Vl(t,u ) a o ( t ,d [ u ,o]), we see that there exists a S1 = &(to,c) > 0 satisfying

to such that

d [ u ( t l ) , ~=]62,

d [ u ( t 2 ) 0,1 = E

and

62 I d [ u ( t ) 01,

5 E Ip

(4.4.8)

for t l 5 t 5 t 2 . We let 17 = Sz so that the existence of a V, satisfying hypothesis ( A 3 ) is assured. Hence, setting

we obtain the differential inequality

which yields

where w2o = V l ( t l u(tl))+Vv(tl, , u ( t l ) ) ,r 2 ( t ,t l ,w20) is the maximal solution of (4.4.2). We also have, because of ( A l ) and (A2),



80

u o ) ,where r l ( t ,to,w l o ) is the maximal solution of (4.4.1). with w10 = Vl(to, By (4.4.5) and (4.4.6), we get

Also, by (4.4.41, (4.4.8) and ( A 3 ) , we arrive at

Thus (4.4.10) and (4.4.11) and the definition of wzo shows that w2o < bo which, in view of (4.4.3), shows that ~ : ~ t(l ,tw~z o.) < b(e). It then follows , ) 0 and (A3), from (4.4.9), V l ( t u

>

This contradiction proves equi-stability of the trivial solution of (4.2.1) since (4.4.7) is then true. The proof is complete. The next result offers conditions for equi-asymptotic stability.

Let the assumptions of Theorem 4.4.1 hold except that condition ( A 2 ) is strengthened to ( A 2 * ) D f T G ( t : u ) 5 - c ( w ( t , u ) ) + g l ( t , V i ( t , ( u ) ) c, E K.w E C[R+x S ( p ) , R + ] , N d [ u l , u a ] , N > 0 and D f w ( t , u ) is bounded w ( t , u i )- w(t,uz)l above or below.

to.

We shall show that, for any solution u ( t ) of (4.2.1) with d [ u OO] , < do, it follows that limt,m w ( t , u ( t ) ) = 0 , which implies by the property of ~ ( ut ),, lirnt,, d [u(t),61 = 0 and we are done. Suppose that limtim s u p w ( t , u ( t ) ) # 0. Then there would exist two divergent sequences { t i ) , { t y ) and a a > 0 satisfying

(a) w ( t i , u ( t i ) ) = :, w ( t y , u ( t y ) ) = a and w ( t , u ( t ) )2 © 2003 V. Lakshmikantham and R. N. Mohapatra

5: t E (t;,ti),or

4.4 Nonuniform Stability Criteria

81

( b ) w(ti,u ( t i ) )= a , w(t;, u ( t y ) )=

$ and w ( t ,u ( t ) )2

5 , t E (ty,t i )

Suppose that D+w(t,u ( t ) )j 11.1.Then using (a) we obtain

which shows that t/i/- t: 4.2.2, we have

2&

Since wlo = V l ( t o u, o )

ao(to,d[uo,Q ] ) 5 ao(to,60) t to. We thus obtain

(4.4.5) w l ( t ,t o , w 1 0 )

< 6 * ( p ) ,we get from

For sufficiently larger n, we get a contradiction and therefore lim sup,,m w ( t ,u ( t ) ) = 0. Since w ( t ,u ) 2 bo(d[u,01) by assumption, it follows that limt,, d [ u ( t )01, = 0 and the proof is complete. The following remarks are in order.

.I. The functions g l ( t , w ) = g2(t,w ) = 0 are admissible in Theorem 4.4.1 so the same conclusion can be reached. If V l ( t ,u ) = 0 and g l ( t , w ) 0, then we get uniform stability from Theorem 4.4.1. If, on the , ) -. 0 , g 2 ( t , w ) 0 and V l ( t ,u ) 2 b(d[u.o ] ) , b E K ,then other hand, V v ( t u Theorem 4.4.1 yields equi-stability. We note that known results on equistability require the assumption t o hold everywhere in S ( p ) and Theorem 4.4.1 relaxes such a requirement considerably by the method of perturbing Lyapunov fu~~ctions.

--

--

-

4.4.2. The functions g l ( t ,w ) 5 g a ( t 7w) 0 are admissible in Theorem 4.4.2 to yield equi-asymptotic stability. Similarly, if V v ( t ,u ) r 0, gz(t,w ) E 0 with & ( t , u)2 b(d[u,Q]), b E K ,implies the same conclusion. If V l ( t .u ) = 0 and gl(t,w) = 0 in Theorem 4.4.1, t o get uniform asymptotic stability. one needs to strengthen the estimate on D+T/;l(t,u ) . This we state as a corollary.


82


-

A s s u m e that the assumptions of Theorem 4.4.1 hold with Vl( t ,u ) = 0, gl ( t ,w) 0. Suppose further that oi-v,(tl

4 i -C[W(4 ~1, + g2(t1V,(t741,( t ,4 E R+ x S ( p ) n S c ( v ) ,

(4.4.12) where w E C[R+x S ( p ) ,R+],w ( t , u ) 2 b ( d [ u ,01); c, b E K: and y 2 ( t , W ) i s nondecreasing in w . T h e n the trivial solution of (4.2.1) i s unzformly asymptotically stable.

Proofs The trivial solution of (4.2.1) is uniformly stable by Remark 4.4.1 in the present case. Hence taking E = p and designating do = d ( p ) , we have d[uo,O]< S o

d [ u ( t ) , 0 ]to.

To prove uniform attractivity, let 0 < e 0 be + 1. Then the number relative to E in uniform stability. Choose T = C(6) we shall show that there exists a t" E [to, to T ] such that w ( t * ,u ( t * ) )< b ( 6 ) for any solution u ( t ) of (4.2.1) with d [ u o ,01 < 60. If this is not true, w ( t , u ( t ) ) 2 b ( S ) , t E [to,to TI. Now using the assumption (4.4.12) and arguing as in Corollary 4.2.2, we get

+

+

0I q t o

+ T ,u(to + T I )I

+ T: t o , ~ 2 0 -)

~2(t0

lr+T

U I ( ~U, ( S ) , M S .

This yields, since r 2 ( t ,to,w 2 0 ) < b ( p ) and the choice of T ,

+

which is a contradiction. Hence there exists a t* E [to,to T ] satisfying w ( t * , u ( t * ) )< b ( 6 ) , which irnplies d [ u ( t * ) 01, < S. Consequently, it follows, by uniform stability that

d [ u o ,O]

< So

implies

d [ u ( t ) 01, < E ,

t 2 to + T ,

and the proof is complete.

We shall, in this section, investigate the boundedness of solutions of the fuzzy differential equation U' =

f ( t ,u ) , u ( t O = ) UO,


(4.5.1)

4.5 Criteria for B o u n d e d n e s s

83

where f E C[R+x En,En]. Corresponding to the definitions of various stability notions given in Section 3.9, we also have boundedness concepts, which we shall define. T h e solutions of (4.5.1) are said t o be ( B l ) equi-bounded, if for a n y a P ( t o ,a ) > 0 such that d [ u o ,61

0 and to

implies

E

d [ u ( t ) ,01

R+, there exists a ,B =

< P, t 2 to;

( B 2 ) uniform-bounded, if ,B in ( B l ) does n o t depend o n t G ; ( B 3 ) quasi-equi-ultimately bounded for a bound B , if there exists a B and a T = T ( t o a , ) > 0 such that d [ u o ,01

< a: implies d [ u ( t ) ;01 0

> to + T ;

( B 4 ) quasi-uniform ultimately bounded, if T in ( 3 ) i s independent of to; ( B 5 ) equi-ultimately bounded, if ( B l ) and ( B 3 ) hold simultaneously; ( B 6 ) unzform-ultimately bounded i f ( B 2 ) and ( B 4 ) hold simultaneously; ( B 7 ) equi-lagrange stable, if ( B l ) and ( S 3 ) hold. ( B 8 ) unzformly Lagrange stable, zf ( B 2 ) and ( S 4 ) hold. Using the comparison results of Section 4.2, we shall prove simple boundedness results.

4.5.1. i l s s u m e that

to such that d[u(tl),b]= p

and

d [ u ( t ) , ~I ] p,

to t o + T , d [ u ( t ) Q] , 2 R.

Then

This contradictioii proves ( B 5 ) and the proof is complete. Finally we shall offer a result providing nonuniform boundedness property utilizing the method of perturbing Lyapunov functions.

where gl E C[R$, R];

(iii) the scalar dzjjferential equation

are equi-bounded and u n i f o m l y bounded respectively. © 2003 V. Lakshmikantham and R. N. Mohapatra

86


Then the system (4.5.1) is equi-bounded. Proof. Let B1 > p and to E R+ be given. Let al = a l ( t o , B1) = m a x ( a O , a * ) ,where a 0 = max[&(to,uo) : uo E cl{S(Bl) f l Sc(p))] and a* Vl (t, U ) for (t, U) E R+ x dS(p). Since equation (4.5.5) is equi-bounded, given al > 0, and to E R + , there exists a ,Bo = Po(to,a i l ) , such that

>

wl (t, to, w1o) < Po

t

t to

(4.5.7)

provided wlo < al, where w l ( t , t a ,wlo) is any solution of (10). Let a(B1) ,O0, then uniform boundedness of equation (4.5.6) yields that

+

a2 =

provided w2o < a;l, where w2(t,to, w20) is any solution of (4.5.6). Choose B 2 satisfying b(B2) > l ( a i 2 ) . (4.5.9) We now claim that uo E S(B1) implies that u ( t jto, uo) E S ( B 2 ) for t > to, where ~ ( tto, , uo) is any solution of (4.5.1). If it is not true, there exists a solution u(t, to, uo) of (4.5.7) with uo E S(B1), such that for some t* > to, d[u(t*,to, uo),01 = B2. Since B1 > p, there are two possibilities to consider:

>

(2) there exists a 2 to such that u(2, to, uo) E dS(p) and u(t, to, uo) E Sc(p) for t E [E. t*].

If (1) holds, we can find tl > to, such that

+

Setting m ( t ) = Vl (t, u(t, to, uo)) V2(t, ~ ( tto,, uo)) for t E [tl, t*],then using Theorem 4.2.1, we can obtain the differential inequality


87

4.6 Fuzzy Differential Systems where Thus

?2(t,

t l ,v O )is the maximal solution of (4.5.6) with

~2

( t i ,t l , vo) = vo.

q ( t * ~, ( t t*o ,,U O ) ) + V2(t*,~ ( t t*o , ~ o ) )

(4.5.11)

Similarly, we also have

where yl (t.to,u o ) is the maximal solution of (4.5.5). Set wio = Vi ( t o , uo) < 0 1 . Then Vl(t1,u ( t 1 , t o , u o ) ) L ~ l ( t 1to. . W o : u o ) ) 5 Po since (4.5.7) holds. Furthermore, V2(tl, u ( t l , t o . u O ) )5 a ( B 1 ) and (4.5.10). Consequently, we have

Combining (4.5.8): (4.5.9), (4.5.10) and (4.5.13): we obtain

which is a contradiction. If case ( 2 ) holds: we also arrive at the inequality ( 4 . 5 . l l ) , where tl satisfies (4.5.10). We now have, in place of (4.5.12))the relation

Since u(t,t o ,uo) E a S ( p ) and K(5,u ( t ,to:x o ) ) 5 a* we get the cont,radiction (4.5.14). This proves that

>2

< 0 1 , arguing as before,

for any given B1 > p, to > 0, there exists a B2 such that uo E S ( B 1 ) implies u ( t jto,u o ) E S ( B 2 ) ,t 1 t o . For B1 < p, we set B2(t0,B 1 )= B2(t0,p) and hence the proof is complete.

Recall that we have so far been discussing the fuzzy differential equation


88


where f E C[R+ x En,En],which corresponds, without fuzziness, t o scalar differential equations. To consider the situation analogous to differential systems, we need to prepare appropriate not,ation. In this section, we shall therefore attempt t o consider the fuzzy differential system, given by U' = F(t,U),

U(to) = Uo,

(4.6.1)

where F E C[R+ x EnN,EnN], U E E ~E~~ ~ = ,(En x En x . . . x En,iV times), U = (ul,u2, ..., uN) such that for each i , 1 i AT,u, E En. Note also Uo E E ~ ~ . We have two possibilities to measure the new variables U , Uo. F, that is,

<
1, skmust be an A-matrix, which means I - skis positive definite, where I is the identity matrix. For details of generalized spaces and contraction mapping theorem in this setup see Bernfeld and Lakshmikantham [4]. page 226. Moreover, in order t o arrive at the corresponding estimate (3.4.3) of Theorem 3.4.1, for example, one is required t o utilize the corresponding theory of systems of differential inequalities, which demands that G(t, w) have the quasi-monotone property, which is defined as follows: wl

5 wl

and

wli = w2i


for some i , 1

i

1 ,

implies

4.6 Fuzzy Differential Systems

89

G , ( ~ , w I ) < G , ( ~ , z L 'w~1) ,,w 2 E R I V . If G ( t ,w ) = A w , where A is an N x N matrix, then the quasi-monotone property reduces t o requiring a,:, 0, i # j . The method of vector Lyapunov-like functions has been very effective in the investigation of the qualitative properties of large-scale differential systems. We shall extend this technique t o fuzzy differential systems (4.6.1), where, as we shall see, both metrics described above are very useful. For this purpose, let us prove the following comparison result in terms of vector Lyapunov-like functions relative t o the fuzzy differential system (4.6.1). We note that the inequalities between vectors in R~ are t o be understood as componentwise.

>

0,

>

j=1

i#j

cL:~

Choosing Q(w) = d,w, for some d , > 0, we see that (4.7.10) is satisfied by Go@,v) = -yv, for some 7 > 0 in view of (4.7.11). Consequently, the trivial solution of (4.7.7) is exponentially asymptotically stable which implies that the trivial solution of (4.6.6) does have the same property.

Let us consider the linear fuzzy differential system


4.8 Linear %riation of Parameters Formula

95

where A is an N x N matrix of reals and U = (ul, 212, ..., u N ) such that for each i, 1 5 i 5 N;ui E E n . Note also Uo E E ~ " . See Section 4.6 for notation. We shall also consider the following fuzzy differential system which is a perturbation of (4.8.6), namely,

We recall that where F E C[R+x EnN,EnN].

:. is the generalized metric and D[U, V] E R embedded in El by the correspondence a

-+

a(t) =

Also, the real numbers can be

1 ift=a, 0 elsewhere.

Then we can generalize multiplication by a real number and for any real number a , we get

We also know that [u+v]" = [u]"+ [uIa, u,v E En. Thus we can write (4.8.1) in the expanded form

1, I - Q k is positive definite where I is the identity matrix. Then the generalized contraction mapping theorem (see Bernfeld and Lakshmikantham [4], p. 226) assures the existence of the unique solution to the IVP (4.8.1). One can verify easily that U(t) = eA(t-tO)vo is the unique solution of (4.8.1). Consequently; the variation of parameters formula relative to the IVP (4.8.2) takes the usual form

We shall employ the variation of parameters formula (4.8.6) later in Chapter 5, in discussing the bcuiidary value problem. Let us now consider the fuzzy linear homogeneous system with periodic boundary condition

where a E C[[O,2 ~ 1E, ~ ~The ] unique . solution of the corresponding IVP

is given by using (4.8.6)

Hence if (4.8.7) is solvable, we must have

and this is possible if we assume that [I- e2A"]-1 exists so that we can solve for Uo, that is


4.9 Notes and Comments

4.9

Notes an

97

Comments

The comparison theorem and t,he useful corollaries in terms of Lyapunovlike functions described in Section 4.2 are taken from Lakshmikantham and Leela [59]. Section 4.3 contains stability theorems parallel t o the original theorems of Lyapunov and are new. The method of perturbing Lyapunov functions and the nonuniform stability results of Section 4.4 are due t o Lakshmikantham and Leela [62]. The notions of boundedness and the sufficient condition for the boundedness concepts to hold in terms of Lyapunov-like functions, given in Section 4.5, are taken from Mohapatra and Zhang [76]. The description of fuzzy differential system and the corresponding comparison theorem in terms of generalized metric spaces developed in Section 4.6 are new and are modeled on the corresponding results in differential equations without fuzziness. See Lakshmikantham and Leela [59] and Bernfeld and Lakshmikantham [4]. The method of vector Lyapunov functions discussed in Section 4.7 is also new. This method is very popular and effective in applications. See Siljak [106], and Lakshmikantham, Matrosov, and Sivasundaram [64]. Section 4.8 incorporates new results on linear fuzzy systems including the variation of parameters formula and a simple criterion for periodic boundary value problems. See also Zharlg et al. [I201 for the solution of first-order differential equations in a special case.


5.1

Introduction

We initiate several interesting topics in this chapter, dealing with fuzzy dynamic equations which are yet t o be investigated. In Section 5.2 we introduce fuzzy difference equations. Since the study of difference equations has attracted many researchers, it is hoped that the investigation of fuzzy difference equations would be popular as well. Section 5.3 initiates the development of impulsive fuzzy differential equations. Impulsive differential equations have become popular and useful recently, and therefore the typical results provided in this section would be equally attractive. Functional differential equations or differential equations with delay are considered in Section 5.4 which is a well-known branch of differential equations. Consequently, fuzzy differential equations with delay should be an equally interesting area of research. Some typical results are incorporated in this section. The results of Section 5.5 are new and deal with the extension of the theory of hybrid systems to fuzzy differential equations. The contents of Section 5.6 investigate the existence of fixed points of fuzzy mappings via the theory of fuzzy differential equations. Section 5.7 attempts the development of boundary value problems for fuzzy differential equations. Finally, in Section 5.8, fuzzy integral equations are discussed.


Chapter 5. Miscellaneous Topics

100

Let N denote the natural numbers and N+ the nonnegative natural numbers. We denote by iV$ the set

with k E iVf and no E N . Let us consider the fuzzy difference equation given by un+l = f ( n i ~ n ) :un0 = U O , (5.2.1) where f ( n ,u ) is continuous in u for each n.Here u,: f E E4 for each n 2 n o , where (Eq,d ) is the metric space. Since we shall be using n for difference equations, we shall employ the metric space (Eq,d ) for ( E n ,d ) used earlier. This will avoid confusion. The possibility of obtaining the values of solutions of (5.2.1) recursively is very important and does not have a counterpart in other kinds of equations. For this reason, we sometimes reduce continuous problems to approximate difference problems. For simple fuzzy difference equations, we can find solutions in closed form. However, reducing information on the qualitative and quantitative behavior of solutions of (5.2.1) by the comparison principle is very effective as usual. We need the following comparison principle for difference equations. See Lakshmikantham and Trigiante [69] for details.

>

.I. Let n E N&, r 0 and g(n,r ) be a nondecreasing function in r for each n. Suppose that for n 2 n o , the inequalities

Proof. Suppose that the claim y, 5 z, for all n 2 no is not true. Then because of the assumption y,, 5 z,,, there exists a k E N,f, such that yk zk and yk+l > zk+1. It then follows, using the monotone character of g ( n , r ) in r 0 and the inequalities (5.2.2) and (5.2.3); that

This is a contradiction, which proves the claim. The following corollaries would be useful. © 2003 V. Lakshmikantham and R. N. Mohapatra

5.2 Fuzzy Difference Equations

101

-1. L e t n E N&,kn 1 0 andyn+l

< knyn+pn.

Then

Proofv Since k, 2 0 and g(n, r ) = knr + p n , the assumptions of Theorein 5.2.1 are satisfied. Take zn as the solution of zn+l = knzn+pn with zno = y,, which can be computed easily. Then the result follows from Theorem 5.2.1.

(Discrete Gronwall Inequality). Let n E N&, kn

20

and

Then n-1

n-1

/n-1

Proof.

\

n-l

n-1

1

/ n-1

The comparison equation is

The solution of this equation is the expression on the right-hand side of exp(ks), we get the final expression in (5.2.5). (5.2.5). Observing 1 k, Let us now discuss estimating the solution of (5.2.1) in terms of solutions of the scalar difference equation

+
0, no E N ,there exists a 61 = Sl(no,E) such that

0 5 zno < 61 implies

zn+l < b ( ~ ) , n

2 no.

Choose b = b(no,t) satisfying

Then Theorem 5.2.3 gives

which shows that

Choose z,, = V(no,u,,) so that we have

We then get b(d[un+l, 61) < b(~)9 n 2 no, which implies the stability of the trivial solution of (5.2.1). For asymptotic stability, we observe that

01

Since zn+l i 0 as n i 00,we get d[untl, i 0 as n i m. The proof is complete. As an example, take g(n,r ) = a n r where a, E R. Then the solution of

is given by n-1

We have the following two cases: © 2003 V. Lakshmikantham and R. N. Mohapatra


104

) 0 and to E R+, there exists a dl = &(to,E ) > 0 such that

0

< wo < dl

implies w ( t ,t o , w o ) to,

5.3 Impulsive Fuzzy Differential Equations

109

where w (t, to,wo) is any solution of (5.3.2). Let wo = a(d[uo,61) and choose ) that a(6) < 61. With this 6, we claim that if a 6 = 6 ( ~ such d[u0,0]< 6 then

d[u(t),Q] < E ,

t

2 to.

If this is not true, there would exist a solution u(t) = u(t, to,uo) of (5.3.3) tk+l for some k*, with d[uo,Q]< 6 and a t* > to such that tk < t* satisfying

0, we let Co = C [ [ - r ,01, E n ] . For any element 9 E Co, define the metric H [ p ,$1 = max-,<s 0. For any t t o , t E Jo, we let ut denote a translation of the restriction of u t o the interval [t-7; t ] ;specifically, ut E Co defined by

+

>

In other words, the graph of ut is the graph of u on [t- r ,t ] shifted t o [ - r , 0 ] . With this notation, we consider the fuzzy differential equation with finite delay

where f;u E En and f E C [ J x Co,E n ] ,J = [to,to the following existence result.

+ a ] . We can then prove

5.4.1. Assume that

for t E J and 9, $ E Co. Then the IVP (5.4.1) possesses a unique solution u ( t ) on Jo.

Proof. Consider the space of functioiis u E CIJo,E n ] such that u ( t ) = po(t), to - r I t I to and u E C [ J ,En] with u ( t o )= po(0) and po(t) E En for -7 t 0. Define the metric on CIJo,E n ] by

<

which means

Choose t* > 0 such that e-Pt*

I: i. Then

0, where K E C [ J x J x En,En], f E C[J,En] and J = [to,to a], to a > 0. We shall be content in proving an existence and uniqueness theorem concerning (5.8.1) via the contzaction mapping principle. .1. Assume that f E C [ J , En]: K E C [ J x J x En,En]and for ( t ,S, u ) , ( t ,S, V) E J x J x En,

Then there exists a unique solution u(t) on J for (5.8.1).

Proof* Let C [ J , En]denote the set of all continuous functions from J t o E n . Define the weighted metric H[u, v] = sup d[u(t),~ ( t ) ] e - ' ~ , J

for u , v E C [ J ,En],X > 0 t o be chosen later. Since (En,d) is a complete metric space, the usual argument shows that (C[J,En],H) is also a complete metric space. Define the mapping T by the relation (Tu)(t) = f (t)

+


K ( t , s, u(s))ds, t E J.


124

Then by Corollary 2.4.2, T u E C[J,E n ] . Moreover, assumption (5.8.2) yields, using the properties of the metric d ,

This, in turn, implies that

~ e - ' ~~ [ uv ] ,

C X t d[(Tu)( t ) ,( T v )( t ) ]

T

Thus choosing X = 2 L , we get

where f E C [ R , x EnN,E n N ]and uo E Enhi.We employ the metric space (E'"", d o ) where do[u,u]= d[ui,ui], ui, ui E En for each i , 1 5 i ( N . We also utilize the generalized metric space ( E n N ~where )

zEl

We need the followiiig known results. See Lakshinikantham and Leela [ G I ] . Hereafter, the inequalities between vectors in Rd are to be understood componentwise.

.I. Letg E C [ R + X R $ X R $R , d ]g, ( t , w , [) be quasi-monotone nondecreasing in zc; for each ( t ,[) and monotone nondecreasing in [ for each ( t ,w ) . Suppose further that r j t ) = r ( t ,to,wo) is the maximal solution of

existing on [to,m). Then the rnaximul solution R ( t ) = R ( t ,to,wo) of

exists on [to,co) and r ( t ) =r R ( t ) , t

> to.

.

Assume that the function g(t, w:[) satisfies the conditions of Theorem 5.9.1. Then m E C [ R + R$] , and

Then for all [

< r ( t ) , it follows

that

We can now state the needed comparison results in terms of suitable Lyapunov-like functions which can easily be proved. For this purpose, we let

0 and a T-neighborhood of N of the perfect clock satisfying the condition that d o [ u o ,voj < S implies d o [ u ( t t, o , u o ) , v ( a ( t ) ,t o , v o ) ] < E , t 2 to where a E N; ( 2 ) 7 - u n i f o r m l y stable, if S in ( 1 ) i s independent of t o .

(3) r-asymptotically stable, if (1) holds and given E > 0: to E R+, there exist a So = S o ( t o ) > 0, a r-neighborhood N of the perfect clock, and a T = T ( t o E, ) > 0 such that

where a E N ;

( 4 ) r - u n i f o r m l y asymptotically stable, if So and T are independent of to. We note that a partial ordering of topologies induces a corresponding partial ordering of stability concept,^. Let us consider the following topologies of E:

( r l )the discrete topology, where every set of E is open; (r2)the chaotic topology, where the open sets are only the empty set, and the entire clock space E; (73)

the topology generated by the base

a E E : sup la(t) - ao(t)l< E t€[to,m)

( T ~ the ) topology defined by the base UCJO,€ -

[a E c1[R+; R+] : jo(t0) - oo(to)j< E and


5.9 A New Concept of Stability

129

It is easy to see that the topologies 7 3 , 7 4 lie between rl and 72. Also, an obvious conclusion is that if the unperturbed motion v ( t , to,v o ) is the trivial solution, then ( O S ) implies ( L S ) . In rl-topology, one can use the neighborhood consisting solely of the perfect clock a ( t ) = t and therefore Lyapunov stability follows immediately from the existing results. , to,v o ) and suppose that B is closed. Define B = B[to,vo] = v ( [ t ooo), Then the stability of the set B can be considered in the usual way in terins of Lyapunov functions since

p [ u ( t ,t o , uo),Bl =

inf s€[to@)I

dolu(t, t o , U O ) ,4% t o 97 / 0 1 ] ;

denoting the infimum for each t by st and defining a ( t ) = st for t > to, we see that a E E in the T~-topology.We therefore obtain orbital stability of the given solution v ( t ,to,v o ) in terins of the r2-topology. To investigate the results corresponding to the 7.3 and 7 4 topologies, we shall utilize the comparison Theorem 5.9.3 and suitably modify the proofs of standard stability results.

.4.Let condition ( i ) of Theorem 5.9.3 be satisfied. Suppose further that

where a ( t , a , .), b ( . ) and d ( . ) E K = [a E C[R+,R+],a ( 0 ) = 0 and a ( 7 ) i s increasing in 7 . T h e n the stability properties of the trivial solution of (5.9.2) imply the corresponding r3-stability properties of the fuzzy dzflerential system (5.9.1) relative to the given solution v ( t ,to,v o ) . Proof. Let v ( t ) = v ( t , to,v o ) be the given solution of (5.9.1) and let 0 < E and to E R+ be given. Suppose that the trivial solution of (5.9.2) is stable. Then given b ( ~ >) 0 and to E R+,there exists a 61 = d l (to,E ) > 0 such that

wio

< S1

w i ( t , t o ,w o ) < b(e): t

implies

2 to,

i=l

i= 1

where w ( t , to,w o ) is any solution of (5.9.2). We set wo = V ( t oa, ( t O )uo, , vo) and choose 6 = 6 ( t o ,E) 7 = T I ( € ) sat,isfying

a ( t o ,a ( t o )6, ) < dl © 2003 V. Lakshmikantham and R. N. Mohapatra

and

7 = d-'(b(~)).

(5.9.7)

130

Chapter 5. A4iscellaneous Topics

we have Using (b) and the fact that a E 0,

It then follows that It - a ( t )1

< q and hence a E N. We h i m that whenever

do [uo,vo] < S and

a E N,

it follows that

If this were not true, there would exist a solution u ( t ,t o , uo) and a t~ > to such that do[u(t1,t o , u o ) , v ( 4 1 ) ,t o , ~ 0 ) = l E and (5.9.8) d o [ u ( t ,t o , u o ) , v ( a ( t ) ,t o , vo>l 5 E for to

< t 5 t l . Then by Theorem 5.9.3, we get for to < t < t l ,

where r ( t , to,w o ) is the maximal solution of (5.9.2). It then follows from (5.9.2), (5.9.8), using (a); that

a contradictio~i,which proves r3-stability. © 2003 V. Lakshmikantham and R. N. Mohapatra

5.9 A New Concept of Stability

131

Suppose next that the trivial solution of (5.9.2) is asymptotically stable. Then it is stable and given b(c) > 0, to E R+, there exist hol = bol(to) > O and T = T ( t o c, ) > 0 satisfying d

d

woz < 610 implies The r3-stability yields, taking

~ ( to; t ,W O ) to + T .

> 0 and designating Go(to)= 6 ( t o ,p ) ,

do[uo,vo] < So implies d o [ u ( t ) v, ( a ( t ) ) ]< p, t for every o such that It

-

a1

to

This means that by Theorem 5.9.3

In view of (5.9.9), we find that

which in turn implies

Thus It - a(t)j < dP1b(e) = q ( e ) , t satisfying

> to + T . Hence there exists a a E N

which yields

do[zi(t),u ( a ( t ) ) ]< E .

t L to

+T,

whenever d o [ u o ,zlO] < 60 and o E AT. This proves -r3-asymptotic stability and the proof is complete. © 2003 V. Lakshmikantham and R. N. Mohapatra


zz 6.1

ifferen

Introduction

Recall that the theory of fuzzy differential equations (FDEs) considered so far utilizes the Hukuhara derivative (H-derivative) for the formulation. We have investigated, in the previous chapters, several basic results of fuzzy differential equations via the comparison principle in the metric space ( E n ,d) with no complete linear structure. This approach for fuzzy differential equations which employs the H-derivative suffers from a disadvantage because the solution z ( t ) of an FDE has the property that diarn[z(t)]D is nondecreasing in time, that is, the solution is irreversible in probabilistic terms. Consequently, it has been recently realized that this formulation of FDEs cannot really reflect any rich behavior of solutions of ODES such as stability, periodicity and bifurcation, and therefore is not well-suited for modeling. Alternative approaches have recently been introduced by Buckley and Feuring [8],Vorobiev and Seikkala [112], and Hullermeier [40]. A different and interesting framework suggested by Hullerrneier is more general than the others. It is based on a family of differential inclusions at each p-level, 0 p 1, namely

<
F(t,x;p)

{p,)

whenever

a 5 p.

is a nondecreasing sequence in I converging to (4 x) E 0 ~ ( tx;, = ~ ( tx;, B).

(ii) If

n

a)

P, then

(6.2.1) for all

(6.2.2)

n

The definition is adapted from the usual definition for real-valued functions dropping the convexity of level sets. Now consider the differential inclusion


6.2 Formulation of Fuzzy Differential Inclusions

137

where R is an open subset of Rn+l containing (0, xo), I is a compact interval and F : R x I ICE. Throughout, it is assumed that all maps are proper, that is have nonempty images of points in their domain. The boundedness assumption is said t o hold if there exist b, T, Ad > 0 such that the set Q = [0,T] x (xo+(b+MT)B) C R where B is the unit ball of R n , and F maps Q x I into the ball of radius Ad. Denote the set of all solutions of (6.2.3) on [0,T] by Sp(xo;T), the attainable set Ap(xo, T ) = {x(T) : x(.) E Sp(xo,T)) and write Z T ( R n ) {x E C([O,T];Rn) : x' E Lm([O,T];Rn)). It is known that for every XI E xo bintB, Sp(x1,T) exists, and is a compact subset of ZT(Rn), and each attainable section Ap(xl, r ) , 0 < T T, is a compact subset of Rn, see Aubin and Cellina [I]. In fact, although these sets are not in general convex, they are acyclic which is stronger than simply connected. See De Blasi and Myjak 1151.

-

+

0 such that

which is possible because G r ( F ) , the graph of F, is compact. By usc, there exists a neighborhood U of ( t ,v(t), P) such that for (s, x , a) E U , F ( s , x, a) c F ( t , v(t), P) +EB. Choosing n sufficiently large (t, xpn,Pn) E U and so


138

Chapter 6. Fuzzy Differential Inclusions

on,

+

w h i c h m e a n s ( t ,xa7,( t ) , xh7J E G r ( F ) N . T h e Convergence T h e o r e m implies v'(t) E F ( t , v ( t ) ,P ) a.e. and so v E S p as required. .I. Consider the F I V P in E l , x' = - A x , x ( 0 ) = X o , where X o i s a s y m m e t r i c triangular fuzzy n u m b e r with support [ - I , 11. W h e n this i s interpreted as a family of diflerential inclusions

regula,r quasi-concaz:ity i s especially evident. Since - A x p singleton set in K&, (6.2.4) becomes

=

{ - A x p ) is a

which has solution set S p ( X p ,t ) o n [ 0 ,t ] comprising the functions

Consequently, S p ( X p ,t ) = ( 1 - P)e-'t [- 1 , 1 ] . Obviously,

p

H

S p ( X p ,T )

-

i s regularly quasi-concave, as i s

( ( 1 - , ~ e - ' ~ [ - l 11 , : t E [0,TI}

P

++

A P ( X P ,T ) = ( 1 - P ) e X T[ - I , 11

Let V n d e n o t e t h e set o f u s c normal f u z z y sets on Rn w i t h compact s u p p o r t . Clearly, En c V n ,since elements o f V n have n o n e m p t y c o m p a c t , but n o t necessarily convex - l e v e l sets. The following characterization of elements of V n is required. eorern) Let { Y p C Rn : 0

< /3 5

1) be a

family of compact subsets satisfying

Yo E Kn for all O 5 !,3

Yp=

< 1;

Ya, for a n y nondecreasing sequence pi

4

P in [0,I ] .

T h e n there i s a fuzzy set u E V n such that [uIP= Y p . I n particular, if t h e Y2 are also convex, t h e n u I n . Conversely, the level sets of a n y u E En, [uIPare convex and satisfy these conditions. T h i s is T h e o r e m 1.5.1 i n a suitable f o r m . © 2003 V. Lakshmikantham and R. N. Mohapatra

6.2 Formulation of Fuzzy Differential Inclusions

139

. Let R be an open subset of R x Rn and suppose that G is a use mapping from R to E n . Define F(., .,P) : Rn+l -+ KE to be the mapping (t, x) H [G(t,x)]P. Then F ( . , ., P) is usc on R . Proof. Let B be the unit ball in En,that is B = {u E En : d ( ~ ( ~u)) 5 , 1). By definition, for each (to,xo) E R and each E > 0 there exists 6 > 0 such that lI(t, z) - (to,xo)l/ < 6 implies that G(t, x) 5 G(to,xo) $ - E M Here, . u5v for u, v E En means that u(J) 5 v(J) for all J E Rn and implies the level set containment [ujp E [v]P for all o L: p L: I. ~f w E B,dH([w]P,(01) 1 for each p, that is [w]P E B. Hence, [G(t,x)]' C [G(to,xo)jP E B and the result follows. We can now prove our main results concerning the level sets of solutions.

+

0, models dry friction. Here, sgn(y) = yllyl, y # 0. This can be replaced b y the inclusion

xtEG(t,x)=

(

O -1

-a

) X +

(

sin t

-

O pSgn(x2)

(6.3.3)

where the multivalued function Sgn(y) =sgn(y), y # 0: but is [ - I , 11 when y = 0. Clearly, G satisfies a linear growth condition and is 2~-periodic,so © 2003 V. Lakshmikantham and R. N. Mohapatra

6.3 Differential Incltlsions

143

by Theorem 6.3.1 all solutions exist o n [ O j m). If x , y are solutions, writing u ( t ) = $ j l x ( t ) - y(t)j12 gives

A function V : R+ x K (6.3.2) on K if

-+

R+ is

a Lyapunov f~mctionof the inclusion

>

( 1 ) V ( t , x ) p ( l / z i l ) on K n ( r B n ) , for some r > O and some continuous strictly increasing p : [0,r ) + RS.

for all t 2 to,all x E K and every u E G ( t ,x ) , where W : K continuous on K .

+Rf

is

If V is a Lyapunov function for (6.3.2) on K, define

E = { x : W ( x )= O , x E K ) . I11 particular, (6.3.4) implies that a Lyapunov function is nonincreasing on the solution set o f (6.3.2) as time evolves. So, if KO 2 K is compact and V ( t l , x ) a for all x E KO and some tl 2 to, then solutions which start in KO remain in KO and are thus bounded. If x ( t ) is a trajectory of the inclusion (6.3.2) and K1 c K, if p ( x ( t ) ) -+ K1 as t -+cc,write x ( t ) + K1, t +m .

0.

Let V(x) = x:

+ x:.

Then for every ro E r (t), checking (6.3.4) gives

so W ( x ) = yx;, E is the xl-axis and x2(t) -+0 as t --+ oo. But, since V is independent of t , x l ( t ) + a bounded subset of E as t 4 x.

Having the necessary results on differential inclusions in Sections 6.2 and 6.3, we are now ready to investigate fuzzy differential inclusions. Let N : Rx Rn -+ En and consider the fuzzy differential equation (FDE)

interpreted as a family of differential inclusions. Set [ H ( t ,x)]P = F ( t , x; 0) and identify the FDE with the family of differential inclusions

B E I := 10% 11 where R is an open subset of Rn" containing (0, [xojo), and F : fl x I + ICF The boundedness assumption now holds if the set Q is as above and F maps Q x I into the ball of radius ill. Denote the set of all solutions of (6.4.1) on [ O , r ] by S p ( x o , r ) and the attainable set by Ap(xo,T ) = {x(T) : x(.) E Sp(xO,7 ) ) . AS seen before, Sp(xl, T) exists and is a compact subset of Z T ( R n ) and . each attainable section Ap(xl, r), 0 < r 5 T, is a compact subset of Rn. The results on periodicity and stability for differential inclusions were in the space Rn. When extending these ideas t o FDEs, the definitions of stability have to be formulated in Dn, and notions of periodicity for solution sets are also in different spaces. If U E Dn is a fuzzy set and LI, W C Dn are closed subsets of Dn, define the distance from W and Hausdorff separation respectively by


6.4 F L Z Z Differential Z~ Inclnsions

145

pv(U, W ) = sup p, (U, W). UEU

The significance of these definitions is that, in the metric space (Dn,d) of fuzzy sets, p,(U, W ) is the distance of U E Vn from W c Dn and is the analog of p(x, A) in Rn. Correspondingly, pD(U, W ) is the Rausdorff separation between U, W C Dn with respect t o the metric d and is the analog of the Hausdorff separation p(A, B) in ICn. be the fuzzy singleton E Dn, write jlUll = d(U, the open unit ball in Dn by " = {U E D n : //U/I< 1). A set U c Dn is stable for the FDE (6.4.1) if for all E > 0 and to 2 0 there exists S = 6 ( ~to) , such that Xo E U 6Bn implies that A(&, t ) E LI + EB" on [to,a), where A(Xo,t ) is the fuzzy attainability set defined by the family (6.4.1). That is, p,(Xo,U) < S implies that pv(A(Xo,t ) , U ) 5 E on [to,m ) . If 6 = 6 ( ~is) independent of to and depends only on E , U for the FDE (6.4.1) is said to be uniformly stable. If p.o(A(Xo.t ) , U ) -+ 0 as t -a cc and IA is (uniformly) stable, the set U is said to be (uniformly) asymptotically stable. Most frequently, U will consist of a single fuzzy set U E Dn. Suppose that U c Dn is support bounded, that is

+

li = supp (U) :=

U supp(a) uEU

is a bounded set in Rn. A function V : Rt x K -+ R+ is a Lyapunov function for the FE (6.4.1) if V is a Lyapunov function for the differential inclusion @I E x; 01, 4 0 ) E supp(Xo), (6.4.2)

w,

for all Xo E LI. Recall that any crisp set X c Rn is also a fuzzy set, the menibership function being the chara,cteristic function xx .4.B. Let V be a Lyapunov function for (6.4.1) on K =supp(U), and suppose that the attainability set A(xo,t ) remains in U for all to 5 t < cc. If for each solution x(t) of (6.4.2), W ( x ( t ) )is absolutely continuous and its derivative is bounded above a.e. on [to,oo), then pv(A(xo,t),x E ) -+ 0 as t -+ w. In particular, if V = V(x) does not explicitly depend on t, then writing E(c) = E n { x : V(x) 5 c), pD(A(xo,t ) , x ~ ( ~-a) 0) for some c > 0 ast-+m.

Proof. From Theorem 6.3.2 as t -+ oo each solution x(t) of (6.4.2) approaches E, hence the attainability set A(xo;t ) satisfies p(A(xo,t), E) © 2003 V. Lakshmikantham and R. N. Mohapatra

146


i 0 as t i oo. B y T h e o r e m 6.2.3, t h e FDE (6.4.1) has fuzzy attainability set A(zo,t ) and A(zo,t ) = [ A ( x o .t)lo. Since [ A ( x o ,t)lP C. A ( z o ; t ) ; 0 < p 5 1, it follows t h a t p ( [ A ( x o , t ) l PE, ) -+ 0 as t -+ oo. Hence, p D ( A ( x o ,t ) ,x E ) i 0 as t + CQ. Again b y T h e o r e m 6.3.2, if V does not depend o n t , p ( A ( x o ,t ) ,E ( c ) ) i 0 for some c as -C i oo. T h e second part o f t h e result now proceeds b y t h e same reasoning as above.

.I. To illustrate the theorem, fuzzify the differential equation

by letting r ( t ) be a symmetric triangular valued fuzzy function with level sets [r( t ) ] P= ( 1 - P ) [rl( t ) r, 2 ( t ) ]:= ( 1 - P ) R ( t ) and with r l ( t ) 2 y > 0 . This gives the family of inclusions

+

Then, referring to (6.3.5), V ( x ) = 21 x ; is a Lyapunov function for the F D E (6.4.3) and p D ( A ( x o ,t ) ,x ~ ( ~ ) -+ ~ 0( as ~ t) + ) oo, where E ( c ) is the attractive set approached by x l ( t ) i n (6.3.5). I n Example 6.3.1, suppose that the frictional parameter p is known only vaguely and that the term F ( x l ) = p S g n ( x l ) , p 1, is modeled by the trapezoidal fuzzy number valued function whose p-levels are given by

>

Instead of (6.3.3), consider now the corresponding FDE represented by the family of in.clusions x' E [ G ( t ,z)lP=

( -4

-'a

<
l I

-

p(2 - P ) ,

B), 2 - PI. (6.4.7)

and consequently This last inequality (6.4.8) implies that p 2 1 / ( 2 - P ) . In particular, if x ( t ) 5 x* is a constant solution for each inclusion of the family, p 2 1. On the other hand, if x ( t ) is a 27i-periodic solution, then so also is y ( t ) = x ( t ) - x* for any x* = ( x ; ,o ) with ~ x; satisfying (6.4.8). So, yi = ~ 21 ~, ;= - 7 ~ 1 - 2; - Qy2

+ sin t - p w ( t ) ,

with w ( t ) E (2 - P ) Sgn(yz) C_ [- ( 2 - P, 2 - PI. From the first equation

and using th,is in the second equation gives

=

i2i7

2i7

yly2dt - a

+

y2(t)2dt

2.ir

(sint

- z;

- pw(t))y2dt

since I sin t - xr 1 < plw(t)1 and sgn(yz)y2= ly21 Thus, y2(t) = 0 a. e. and so y i ( t ) = 0. It follows that x 2 ( t ) = 0 and x l ( t ) = y l ( t ) x; is constant. Consequently, if p 2 1 the only 2~-periodicsolutions are x ( t ) = ( x ; ,0 ) , ,with xT satisfving 1 - p, 5 x; 5 -1 + p. Moreover, if p 2 1 / ( 2 - p), the only 27i-periodic solutions of the P-th inclusion are x ( t , /3) = ( x ; ( $ ) ,0 ) with x;(,o) satisfying (6.4.8). Now, using the Lyapunov function V = ( x l - ~ ; ) ~ / / 2 + x ; / 2 gives, for w ( t ) E [F(22)I0,

+

V'

= =

+

-

pw(t)

+ + + 1x2 I((sint - x ; ) sgn(x2)- pw ( t ) )

-ax; 5 --ax; = W ( x ) ,

=

+

( x l - x;)x2 x2(-x1 - ax2 sint - a x 22 x2(-x; sin t - p w ( t ) )


148


from (6.4.7). Hence, the attainability set A ( X o , t ) is attracted to U E D~ given by [ ~ ]=3 [I - p ( 2 - $),-I + p ( 2 - P ) ] x (0). Let us next consider the periodicity of solutions. Under appropriate conditions, such as those of Theorem 6.3.1, an inclusion x' E G(t, x) will have w-periodic solutions. Denote by Ap(W, t), 0 t w, the attainability set of all w-periodic solutions x(t) such that x(0) E W. Clearly, t i-t Ap(W, t ) is an w-periodic set-valued function on 0 t < w. An FDE of the form (6.4.1) is said t o have an w-periodic solution Ap (Xo,t) if there is a family of w-periodic set-valued functions t i+ U(t, P ) , 0 5 ,B 1, such that U ( t , p ) G A([XOlP,t)for t 2 0 and for each t 2 O the family {U(t,p ) } satisfies the conditions of the stacking theorem 6.2.2. That is, for all t 0 [Ap(Xo,t)lP = U(t, P).

<

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