Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in -Dimensional Fuzzy Vector Space
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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 1 2
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, [email protected] Received 23 February 2009; Revised 20 June 2009; Accepted 3 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.
1. Introduction Many authors have studied several concepts of fuzzy systems. Diamond and Kloeden 1 proved the fuzzy optimal control for the following system: xt ˙ atxt ut,
x0 x0 ,
1.1
where x· and u· are nonempty compact interval-valued functions on E1 . Kwun and Park 2 proved the existence of fuzzy optimal control for the nonlinear fuzzy differential 1 by using Kuhn-Tucker theorems. Fuzzy system with nonlocal initial condition in EN 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 3 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, 1 for material with
2
Advances in Difference Equations
1 , Park et al. 4 proved the existence memory. In one-dimensional fuzzy vector space EN 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 dxt A xt Gt − sxsds ft, x ut, dt 0
x0 g t1 , t2 , . . . , tp , xtm x0 ∈ EN ,
t ∈ J 0, T ,
1.2
m 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, Gt is an n × n continuous matrix such that dGtx/dt is continuous for x ∈ EN and t ∈ J with Gt ≤ K, K > 0, with all nonnegative elements, u : J → EN is control function. In 5, Kwun et al. proved the existence and uniqueness of fuzzy solutions for the semilinear fuzzy integrodifferential equations by using successive i
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