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Applied Mathematics Letters
Applied Mathematics Letters 13 (2000) 31-35
PERGAMON
www.elsevier.nl/locate/aml
R e g u l a r i t y of S o l u t i o n S e t s for Differential I n c l u s i o n s Q u a s i - C o n c a v e in a P a r a m e t e r P.
DIAMOND AND P. WATSON Department of Mathematics The University of Queensland Brisbane, Queensland 4072, Australia ~maths. uq. edu. au
(Received April 1999; accepted May 1999) Communicated by H. T. Banks Abstract--The
concept of quasi-concavity is extended to multifunctions. It is then shown that if
the velocity of a differential inclusion is regularly quasi-concave in a parameter, the solution set and
attainability set are also dependent upon the parameter in like manner. The result is applied to give a vastly improved notion of fuzzy differential equations. @ 1999 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - D i f f e r e n t i a l inclusion, Quasi-concave, Fuzzy differential equation.
1.
INTRODUCTION
For v a r i o u s a p p r o p r i a t e c o n d i t i o n s on f : ]~n+k+l __~ R, t h e initial value p r o b l e m w i t h p a r a m e t e r p E R k,
x'(t) = f(t,x;p),
X(to) = X0,
has s o l u t i o n s x(t; to,xo,p) which a r e c o n t i n u o u s or differentiable w i t h r e s p e c t to x0 a n d p. T h e s i t u a t i o n w i t h a differential inclusion
z'(t) E F(t,z;p),
z(to) = xo,
is s o m e w h a t different. If F is u p p e r s e m i c o n t i n u o u s (usc) convex c o m p a c t valued, t h e set of s o l u t i o n s on [t0,T], Sp(xo, [t0,T]) is c o m p a c t , c o n n e c t e d a n d usc in x0 a n d p [1,2], b u t n o t g e n e r a l l y convex. If F is o n l y lower semicontinuous, Sp(xo, [t0,T]) need n o t even be c l o s e d T h i s p a p e r c o n s i d e r s a quasi-concave t y p e of r e g u l a r i t y w i t h r e s p e c t to a single p a r a m e t e r / 3 . D e n o t e b y / C ~ ( r e s p e c t i v e l y , / C ~ ) t h e n o n e m p t y c o m p a c t (respectively, convex c o m p a c t ) s u b s e t s of R ~, let ft C 1R x R n be o p e n a n d let I be a real c o m p a c t interval. A m a p p i n g F : 90 × I --+ K:~ is s a i d t o b e regularly quasi-concave on I if • for all ( t , x ) E f~ a n d a , ~ C I ,
F ( t , x ; a ) D_ F(t,x;~),
w h e n e v e r a 0 such that the set Q = [0, T] × (xo + (b + M T ) B ) C ~, where B is the unit ball of R n, and F maps Q × I into the ball of radius M . Denote the set of all solutions of (3) on [0, T] by S~(xo, r), the attainable set .Af~(x0,~') = {X(T) : X(.) E S~(XO,T)} and write ZT(R n) = {x E C([O,T];R n) : x' E L°°([0, T];~n)}. It is known that for every Xl E x0 + b i n t B , SZ(Xl,T) exists and is a compact subset of ZT(I~n), and each attainable section .Af~(Xl,7), 0 ( T _< T, is a compact subset of R n, see [1]. In fact, although these sets are not in general convex, they are acyclic which is stronger than simply connected [4]. THEOREM 1. Let F : f~ x I --~ IC~ be usc on ~, regularly quasi-concave on I, and suppose that the boundedness assumption holds. Then the mapping/3 ~ S~(x0, T) is a regularly quasi-concave map from I to ICn. PROOF. Abbreviate S~ = S~(xo,T). It is clear from (1) that for a _< /3, SZ c_ Sa. Let/3n be a nondecreasing sequence converging to/3. Then SZ. is a decreasing sequence of compact sets and so Nn S ~ = S is nonempty and compact. Furthermore, d~(S~., S) ~ 0, where o~H is the Hausdorff metric in ZT(Rn). To see that S~ = S, it suffices to show S C S~ since S~ C S is clear. For each n let x ~ E SZ.. Since F is bounded by the ball of radius M, x}. is bounded, so { x ~ } is an equicontinuous family. By Theorem 0.3.4 of [1] a subsequence x~o(~) converges to some v E C([0, T], R n) and x'~(i) converges in the weak topology of LI([0,T], R n) to v' and thus weakly* in L~([0, T], R n) by Alaoglu's Theorem. Observe that v E S. That v E SZ is a consequence of the convergence theorem (Theorem 1.4.1) in [1] and the usc of F. Let N be an arbitrary neighbourhood of the origin in ~ x [0, 1] x R n and choose e > 0 such that < inf{lla - bll : a E Gr(F), b E (Gr(F) + N)C}, which is possible because Gr(F), the graph of F, is compact. By usc, there exists a neighbourhood U of (t, v(t),/3) such that for (s,x,a) E U, F(s, x,a) C F(t,v(t),/3) + ¢B. Choosing n sufficiently large, (t, xf~,/3n) E U and so
F(t, x ~ , ~n) C F(t, v(t), ~) + sB, which means (t, x ~ (t), 13n,x}. ) E G r ( F ) + N . The convergence theorem implies v'(t) E F(t, v(t),/3) a.e., and so v E Sf~ as required. I
Regularity of Solution Sets
33
3. A P P L I C A T I O N S 3.1. F u z z y D E s F u z z y differential equations have been suggested as a way of modelling uncertain and incompletely specified systems. Let g ~ be the space of all usc normal fuzzy convex fuzzy sets on R *~, with c o m p a c t s u p p o r t and with metric doo(u, v) = suP0_ 0 such t h a t II(t,z) - (t0,zo)ll < 6 implies t h a t a(t, z)
Applied Mathematics Letters 13 (2000) 31-35
PERGAMON
www.elsevier.nl/locate/aml
R e g u l a r i t y of S o l u t i o n S e t s for Differential I n c l u s i o n s Q u a s i - C o n c a v e in a P a r a m e t e r P.
DIAMOND AND P. WATSON Department of Mathematics The University of Queensland Brisbane, Queensland 4072, Australia ~maths. uq. edu. au
(Received April 1999; accepted May 1999) Communicated by H. T. Banks Abstract--The
concept of quasi-concavity is extended to multifunctions. It is then shown that if
the velocity of a differential inclusion is regularly quasi-concave in a parameter, the solution set and
attainability set are also dependent upon the parameter in like manner. The result is applied to give a vastly improved notion of fuzzy differential equations. @ 1999 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - D i f f e r e n t i a l inclusion, Quasi-concave, Fuzzy differential equation.
1.
INTRODUCTION
For v a r i o u s a p p r o p r i a t e c o n d i t i o n s on f : ]~n+k+l __~ R, t h e initial value p r o b l e m w i t h p a r a m e t e r p E R k,
x'(t) = f(t,x;p),
X(to) = X0,
has s o l u t i o n s x(t; to,xo,p) which a r e c o n t i n u o u s or differentiable w i t h r e s p e c t to x0 a n d p. T h e s i t u a t i o n w i t h a differential inclusion
z'(t) E F(t,z;p),
z(to) = xo,
is s o m e w h a t different. If F is u p p e r s e m i c o n t i n u o u s (usc) convex c o m p a c t valued, t h e set of s o l u t i o n s on [t0,T], Sp(xo, [t0,T]) is c o m p a c t , c o n n e c t e d a n d usc in x0 a n d p [1,2], b u t n o t g e n e r a l l y convex. If F is o n l y lower semicontinuous, Sp(xo, [t0,T]) need n o t even be c l o s e d T h i s p a p e r c o n s i d e r s a quasi-concave t y p e of r e g u l a r i t y w i t h r e s p e c t to a single p a r a m e t e r / 3 . D e n o t e b y / C ~ ( r e s p e c t i v e l y , / C ~ ) t h e n o n e m p t y c o m p a c t (respectively, convex c o m p a c t ) s u b s e t s of R ~, let ft C 1R x R n be o p e n a n d let I be a real c o m p a c t interval. A m a p p i n g F : 90 × I --+ K:~ is s a i d t o b e regularly quasi-concave on I if • for all ( t , x ) E f~ a n d a , ~ C I ,
F ( t , x ; a ) D_ F(t,x;~),
w h e n e v e r a 0 such that the set Q = [0, T] × (xo + (b + M T ) B ) C ~, where B is the unit ball of R n, and F maps Q × I into the ball of radius M . Denote the set of all solutions of (3) on [0, T] by S~(xo, r), the attainable set .Af~(x0,~') = {X(T) : X(.) E S~(XO,T)} and write ZT(R n) = {x E C([O,T];R n) : x' E L°°([0, T];~n)}. It is known that for every Xl E x0 + b i n t B , SZ(Xl,T) exists and is a compact subset of ZT(I~n), and each attainable section .Af~(Xl,7), 0 ( T _< T, is a compact subset of R n, see [1]. In fact, although these sets are not in general convex, they are acyclic which is stronger than simply connected [4]. THEOREM 1. Let F : f~ x I --~ IC~ be usc on ~, regularly quasi-concave on I, and suppose that the boundedness assumption holds. Then the mapping/3 ~ S~(x0, T) is a regularly quasi-concave map from I to ICn. PROOF. Abbreviate S~ = S~(xo,T). It is clear from (1) that for a _< /3, SZ c_ Sa. Let/3n be a nondecreasing sequence converging to/3. Then SZ. is a decreasing sequence of compact sets and so Nn S ~ = S is nonempty and compact. Furthermore, d~(S~., S) ~ 0, where o~H is the Hausdorff metric in ZT(Rn). To see that S~ = S, it suffices to show S C S~ since S~ C S is clear. For each n let x ~ E SZ.. Since F is bounded by the ball of radius M, x}. is bounded, so { x ~ } is an equicontinuous family. By Theorem 0.3.4 of [1] a subsequence x~o(~) converges to some v E C([0, T], R n) and x'~(i) converges in the weak topology of LI([0,T], R n) to v' and thus weakly* in L~([0, T], R n) by Alaoglu's Theorem. Observe that v E S. That v E SZ is a consequence of the convergence theorem (Theorem 1.4.1) in [1] and the usc of F. Let N be an arbitrary neighbourhood of the origin in ~ x [0, 1] x R n and choose e > 0 such that < inf{lla - bll : a E Gr(F), b E (Gr(F) + N)C}, which is possible because Gr(F), the graph of F, is compact. By usc, there exists a neighbourhood U of (t, v(t),/3) such that for (s,x,a) E U, F(s, x,a) C F(t,v(t),/3) + ¢B. Choosing n sufficiently large, (t, xf~,/3n) E U and so
F(t, x ~ , ~n) C F(t, v(t), ~) + sB, which means (t, x ~ (t), 13n,x}. ) E G r ( F ) + N . The convergence theorem implies v'(t) E F(t, v(t),/3) a.e., and so v E Sf~ as required. I
Regularity of Solution Sets
33
3. A P P L I C A T I O N S 3.1. F u z z y D E s F u z z y differential equations have been suggested as a way of modelling uncertain and incompletely specified systems. Let g ~ be the space of all usc normal fuzzy convex fuzzy sets on R *~, with c o m p a c t s u p p o r t and with metric doo(u, v) = suP0_ 0 such t h a t II(t,z) - (t0,zo)ll < 6 implies t h a t a(t, z)
