Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in -Dimensional Fuzzy Vector Space
© Young Chel Kwun et al. 2009
Received: 23 February 2009
Accepted: 3 August 2009
Published: 26 August 2009
We study the existence and uniqueness of solutions and controllability for the semilinear fuzzy integrodifferential equations in -dimensional fuzzy vector space by using Banach fixed point theorem, that is, an extension of the result of J. H. Park et al. to -dimensional fuzzy vector space.
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.
A fuzzy set of is a function . For each fuzzy set , we denote by for any , its -level set.
Let be fuzzy sets of . It is well known that for each implies .
Let denote the collection of all fuzzy sets of that satisfies the following conditions:
(1) is normal, that is, there exists an such that ;
(2) is fuzzy convex, that is, for any , ;
(3) is upper semicontinuous, that is, for any , ;
(4) is compact.
We call an -dimension fuzzy number.
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 ).
If , and is a hyperrectangle, that is, can be represented by , that is, for every , where with when , , then we call a fuzzy -cell number. We denote the collection of all fuzzy -cell numbers by .
Theorem 2.2 (see ).
For any with , there exists a unique such that ( and ).
Conversely, for any with and , there exists a unique such that .
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 .
Let , be fuzzy subset of . Then .
Definition 2.3 (see ).
for any , which satisfies .
Definition 2.5 (see ).
provided that the equation defines a fuzzy .
Definition 2.6 (see ).
provided that the Lebesgue integrals on the right-hand side exist.
3. Existence and Uniqueness
In this section we consider the existence and uniqueness of the fuzzy solution for (1.3) ( ).
for all , is a finite positive constant.
For the sequel, we need the following assumptions.
and is continuous with , , for all .
Let . If hypotheses (H1)-(H2) are hold, then for every , (3.9) ( have a unique fuzzy solution .
By hypothesis (H2), is a contraction mapping.
Using the Banach fixed point theorem, (3.9) have a unique fixed point .
In this section, we show the nonlocal controllability for the control system (1.3).
Equation (1.3) is nonlocal controllable. Then there exists such that the fuzzy solution for (3.9) as where is target set.
We assume that are bijective mappings.
Then substituting this expression into (3.9) yields -level of .
where the fuzzy mapping satisfies above statements.
Notice that , which means that the control steers (3.9) from the origin to in time provided that we can obtain a fixed point of the operator .
(H3) Assume that the linear system of (3.9) is controllable.
Suppose that hypotheses (H1)–(H3) are satisfied. Then (3.9) are nonlocal controllable.
By hypothesis (H2), is a contraction mapping. Using the Banach fixed point theorem, (4.8) has a unique fixed point .
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.
where and satisfy the inequality (3.4) and (3.5), respectively. Choose such that . Then all conditions stated in Theorem 3.2 are satisfied, so problem (5.2) has a unique fuzzy solution.
Let target set be . The -level set of fuzzy numbers is .
Thus the -level of is
Then -level of is
Then all the conditions stated in Theorem 4.2 are satisfied, so system (5.2) is nonlocal controllable on .
This study was supported by research funds from Dong-A University.
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