© Young Chel Kwun et al. 2009
Received: 23 February 2009
Accepted: 3 August 2009
Published: 26 August 2009
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.
3. Existence and Uniqueness
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.
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