- Research Article
- Open Access
© Young Chel Kwun et al. 2009
- Received: 23 February 2009
- Accepted: 3 August 2009
- Published: 26 August 2009
- Fuzzy Number
- Lipschitz Condition
- Connected Domain
- Fuzzy Function
- Fuzzy Process
where , is a fuzzy coefficient, is the set of all upper semicontinuous convex normal fuzzy numbers with bounded -level intervals, is a nonlinear continuous function, is a nonlinear continuous function, is an continuous matrix such that is continuous for and with , , with all nonnegative elements, 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'Guérékata  studied almost automorphic fuzzy-number-valued functions.
where is fuzzy coefficient, is the set of all upper semicontinuously convex fuzzy numbers on with , is a nonlinear regular fuzzy function, is a nonlinear continuous function, is continuous matrix such that is continuous for and with , , is control function and is initial value.
Wang et al.  defined -dimensional fuzzy vector space and investigated its properties.
For any , , we call the ordered one-dimension fuzzy number class (i.e., the Cartesian product of one-dimension fuzzy number ) an -dimension fuzzy vector, denote it as , and call the collection of all -dimension fuzzy vectors (i.e., the Cartesian product ) -dimensional fuzzy vector space, and denote it as .
Definition 2.1 (see ).
Theorem 2.2 (see ).
Note (see ).
Theorem 2.2 indicates that fuzzy -cell numbers and -dimension fuzzy vectors can represent each other, so and may be regarded as identity. If is the unique -dimension fuzzy vector determined by , then we denote .
Definition 2.3 (see ).
Definition 2.5 (see ).
Definition 2.6 (see ).
provided that the Lebesgue integrals on the right-hand side exist.
For the sequel, we need the following assumptions.
In this section, we show the nonlocal controllability for the control system (1.3).
Suppose that hypotheses (H1)–(H3) are satisfied. Then (3.9) are nonlocal controllable.
Consider the two semilinear one-dimensional heat equations on a connected domain for material with memory on boundary condition , and with initial conditions , where , . Let , , be the internal energy and let , , be the external heat.
This study was supported by research funds from Dong-A University.
- Diamond P, Kloeden P: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, River Edge, NJ, USA; 1994:x+178.MATHView ArticleGoogle Scholar
- Kwun YC, Park DG: Optimal control problem for fuzzy differential equations. Proceedings of the Korea-Vietnam Joint Seminar, 1998 103–114.Google Scholar
- Balasubramaniam P, Muralisankar S: Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions. Computers & Mathematics with Applications 2004,47(6–7):1115–1122. 10.1016/S0898-1221(04)90091-0MATHMathSciNetView ArticleGoogle Scholar
- Park JH, Park JS, Kwun YC: Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions. In Fuzzy Systems and Knowledge Discovery, Lecture Notes in Computer Science, 2006. Volume 4223. Springer, Berlin, Germany; 221–230.Google Scholar
- Kwun YC, Kim MJ, Lee BY, Park JH: Existence of solutions for the semilinear fuzzy integrodifferential equations using by succesive iteration. Journal of Korean Institute of Intelligent Systems 2008, 18: 543–548. 10.5391/JKIIS.2008.18.4.543View ArticleGoogle Scholar
- Kwun YC, Kim MJ, Park JS, Park JH: Continuously initial observability for the semilinear fuzzy integrodifferential equations. Proceedings of the 5th International Conference on Fuzzy Systems and Knowledge Discovery, October 2008, Jinan, China 1: 225–229.Google Scholar
- Bede B, Gal SG: Almost periodic fuzzy-number-valued functions. Fuzzy Sets and Systems 2004,147(3):385–403. 10.1016/j.fss.2003.08.004MATHMathSciNetView ArticleGoogle Scholar
- Gal SG, N'Guérékata GM: Almost automorphic fuzzy-number-valued functions. Journal of Fuzzy Mathematics 2005,13(1):185–208.MATHMathSciNetGoogle Scholar
- Wang G, Li Y, Wen C: On fuzzy -cell numbers and -dimension fuzzy vectors. Fuzzy Sets and Systems 2007,158(1):71–84. 10.1016/j.fss.2006.09.006MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.