Báo cáo hoa học: "Research Article Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in n-Dimensional Fuzzy Vector Space" potx

Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 734090, 16 pages doi:10.1155/2009/734090 Research Article Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in n-Dimensional Fuzzy Vector Space Young Chel Kwun,1 Jeong Soon Kim,1 Min Ji Park,1 and Jin Han Park2 Department of Mathematics, Dong-A University, Pusan 604-714, South Korea Division of Mathematics Sciences, Pukyong National University, Pusan 608-737, South Korea Correspondence should be addressed to Jin Han Park, jihpark@pknu.ac.kr Received 23 February 2009; Revised 20 June 2009; Accepted August 2009 Recommended by Tocka Diagana We study the existence and uniqueness of solutions and controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space EN n by using Banach fixed point theorem, that is, an extension of the result of J H Park et al to n-dimensional fuzzy vector space Copyright q 2009 Young Chel Kwun et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction Many authors have studied several concepts of fuzzy systems Diamond and Kloeden proved the fuzzy optimal control for the following system: x t ˙ a t x t u t , x x0 , 1.1 where x · and u · are nonempty compact interval-valued functions on E1 Kwun and Park proved the existence of fuzzy optimal control for the nonlinear fuzzy differential system with nonlocal initial condition in EN by using Kuhn-Tucker theorems Fuzzy integrodifferential equations are a field of interest, due to their applicability to the analysis of phenomena with memory where imprecision is inherent Balasubramaniam and Muralisankar proved the existence and uniqueness of fuzzy solutions for the semilinear fuzzy integrodifferential equation with nonlocal initial condition They considered the semilinear one-dimensional heat equation on a connected domain 0, for material with Advances in Difference Equations memory In one-dimensional fuzzy vector space EN , Park et al proved the existence and uniqueness of fuzzy solutions and presented the sufficient condition of nonlocal controllability for the following semilinear fuzzy integrodifferential equation with nonlocal initial condition: t dx t dt A x t G t − s x s ds f t, x t∈J u t , 0, T , x g t1 , t2 , , , x tm x0 ∈ EN , m 1.2 1, 2, , p, where T > 0, A : J → EN is a fuzzy coefficient, EN is the set of all upper semicontinuous convex normal fuzzy numbers with bounded α-level intervals, f : J ×EN → EN is a nonlinear continuous function, g : J p × EN → EN is a nonlinear continuous function, G t is an n × n continuous matrix such that dG t x/dt is continuous for x ∈ EN and t ∈ J with G t ≤ K, K > 0, with all nonnegative elements, u : J → EN is control function In , Kwun et al proved the existence and uniqueness of fuzzy solutions for the semilinear fuzzy integrodifferential equations by using successive iteration In , Kwun et al investigated the continuously initial observability for the semilinear fuzzy integrodifferential equations Bede and Gal studied almost periodic fuzzy-number-valued functions Gal and N’Gu´ r´ kata studied almost automorphic fuzzy-number-valued ee functions In this paper, we study the the existence and uniqueness of solutions and controllability for the following semilinear fuzzy integrodifferential equations: dxi t dt t Ai xi t G t − s xi s ds fi t, xi t xi gi xi i x0i ∈ EN i i ui t on EN , 1.3 1, 2, , n , i i where Ai : 0, T → EN is fuzzy coefficient, EN is the set of all upper semicontinuously j i i i convex fuzzy numbers on R with EN / EN i / j , fi : 0, T × EN → EN is a nonlinear regular i i fuzzy function, gi : EN → EN is a nonlinear continuous function, G t is n × n continuous i matrix such that dG t xi /dt is continuous for xi ∈ EN and t ∈ 0, T with G t ≤ k, k > 0, i i ui : 0, T → EN is control function and x0i ∈ EN is initial value Preliminaries A fuzzy set of Rn is a function u : Rn → 0, For each fuzzy set u, we denote by u α {x ∈ Rn : u x ≥ α} for any α ∈ 0, , its α-level set v α for each α ∈ 0, implies Let u, v be fuzzy sets of Rn It is well known that u α u v Let En denote the collection of all fuzzy sets of Rn that satisfies the following conditions: u is normal, that is, there exists an x0 ∈ Rn such that u xo u is fuzzy convex, that is, u λx ≤ λ ≤ 1; 1; − λ y ≥ min{u x , u y } for any x, y ∈ Rn , Advances in Difference Equations 3 u x is upper semicontinuous, that is, u x0 ≥ limk → ∞ u xk for any xk ∈ Rn k 0, 1, 2, , xk → x0 ; u is compact We call u ∈ En an n-dimension fuzzy number Wang et al defined n-dimensional fuzzy vector space and investigated its properties For any ui ∈ E, i 1, 2, , n, we call the ordered one-dimension fuzzy number class u1 , u2 , , un i.e., the Cartesian product of one-dimension fuzzy number u1 , u2 , , un an ndimension fuzzy vector, denote it as u1 , u2 , , un , and call the collection of all n-dimension fuzzy vectors i.e., the Cartesian product E × E × · · · × E n-dimensional fuzzy vector space, and denote it as E n Definition 2.1 see If u ∈ En , and u α is a hyperrectangle, that is, u α can be represented by n uα , uα , that is, uα , uα × uα , uα ×· · ·× uα , uα for every α ∈ 0, , where uα , uα ∈ R nr i 2r ir ir 1r il 1l 2l nl il with uα ≤ uα when α ∈ 0, , i 1, 2, , n, then we call u a fuzzy n-cell number We denote ir il the collection of all fuzzy n-cell numbers by L En n α α Theorem 2.2 see For any u ∈ L En with u α α ∈ 0, , there exists a i uil , uir n α α α unique u1 , u2 , , un ∈ E such that ui uil , uir (i 1, 2, , n and α ∈ 0, ) Conversely, for any u1 , u2 , , un ∈ E n with ui α uα , uα i 1, 2, , n and α ∈ ir il α n α n 0, , there exists a unique u ∈ L E such that u uil , uα α ∈ 0, i ir Note see Theorem 2.2 indicates that fuzzy n-cell numbers and n-dimension fuzzy vectors can represent each other, so L En and E n may be regarded as identity If u1 , u2 , , un ∈ E n is the unique n-dimension fuzzy vector determined by u ∈ L En , then we denote u u1 , u2 , , un n i i i Let EN n EN × EN × · · · × EN , EN i 1, 2, ×, n be fuzzy subset of R Then EN n ⊆ n E i Definition 2.3 see The complete metric DL on EN DL u, v sup dL u α , v n is defined by α 0 0, for all t ∈ I ij ij H2 c{h T cT kT cT } < In view of Definition 3.1 and H1 , 3.3 can be expressed as x t S t x0 − g x t S t − s f s, x s x 3.8 i g x 0, T u s ds, 3.9 x0 i Theorem 3.2 Let T > If hypotheses (H1)-(H2) are hold, then for every x0 ∈ EN n , 3.9 (u ≡ i n have a unique fuzzy solution x ∈ C 0, T : EN Advances in Difference Equations n i Proof For each x t ∈ EN i and t ∈ 0, T , define G0 x t ∈ EN t S t x0 − g x G0 x t n by S t − s f s, x s ds 3.10 i i Thus, G0 x : 0, T → EN n is continuous, so G0 is a mapping from C 0, T : EN n into itself By Definitions 2.3 and 2.4, some properties of dL , and inequalities 3.4 and 3.5 , we i have following inequalities For x, y ∈ C 0, T : EN n , dL G0 x t α , α G0 y t t S t x0 − g x dL α S t − s f s, x s ds , t S t x0 − g y α S t − s f s, y s ds t −S t g x dL α S t − s f s, x s ds , t −S t g y α S t − s f s, y s ds α ≤ dL S t g x t α , St g y dL S t − s f s, x s α , S t − s f s, y s α ds α α max Sα t gil x − gil y il α α , Sα t gir x − gir y ir 1≤i≤n t α max Sα t − s fil s, x s il 1≤i≤n ≤ c max 1≤i≤n t c α α gil x − gil y α max fil s, x s 1≤i≤n cdL g x α , g y α α − fil s, y s α , Sα t − s fir s, x s ir α − fir s, y s ds α α gir x − gir y , α − fil s, y s α , fir s, x s t dL f s, x s c α α − fir s, y s , f s, y s α ds ds ≤ chdL x · α , y · α t dL x s ck α , y s α ds 3.11 Advances in Difference Equations Therefore DL G0 x t , G0 y t sup dL α G0 x t , α G0 y t 0