Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in -Dimensional Fuzzy Vector Space

Abstract

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.

1. Introduction

Many authors have studied several concepts of fuzzy systems. Diamond and Kloeden [1] proved the fuzzy optimal control for the following system:

(1.1)

where and are nonempty compact interval-valued functions on . Kwun and Park [2] proved the existence of fuzzy optimal control for the nonlinear fuzzy differential system with nonlocal initial condition in 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 [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 for material with memory. In one-dimensional fuzzy vector space , Park et al. [4] 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:

(1.2)

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 [5], Kwun et al. proved the existence and uniqueness of fuzzy solutions for the semilinear fuzzy integrodifferential equations by using successive iteration. In [6], Kwun et al. investigated the continuously initial observability for the semilinear fuzzy integrodifferential equations. Bede and Gal [7] studied almost periodic fuzzy-number-valued functions. Gal and N'Guérékata [8] studied almost automorphic fuzzy-number-valued functions.

In this paper, we study the the existence and uniqueness of solutions and controllability for the following semilinear fuzzy integrodifferential equations:

(1.3)

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.

2. Preliminaries

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. [9] 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 [9]).

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 [9]).

For any with , there exists a unique such that ( and ).

Conversely, for any with and , there exists a unique such that .

Note (see [9]).

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 [9]).

The complete metric on is defined by

(2.1)

for any , which satisfies .

Definition 2.4.

Let , then

(2.2)

Definition 2.5 (see [9]).

The derivative of a fuzzy process is defined by

(2.3)

provided that the equation defines a fuzzy .

Definition 2.6 (see [9]).

The fuzzy integral , is defined by

(2.4)

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) ().

We define

(3.1)

Then

(3.2)

Instead of (1.3), we consider the following fuzzy integrodifferential equations in :

(3.3)

with fuzzy coefficient , initial value , and is a control function. Given nonlinear regular fuzzy function satisfies a global Lipschitz condition, that is, there exists a finite such that

(3.4)

for all . The nonlinear function is a continuous function and satisfies the Lipschitz condition

(3.5)

for all , is a finite positive constant.

Definition 3.1.

The fuzzy process with -level set is a fuzzy solution of (3.3) without nonhomogeneous term if and only if

(3.6)

For the sequel, we need the following assumptions.

(H1) is a fuzzy number satisfying, for , , the equation

(3.7)

where

(3.8)

and is continuous with , , for all .

(H2) .

In view of Definition 3.1 and (H1), (3.3) can be expressed as

(3.9)

Theorem 3.2.

Let . If hypotheses (H1)-(H2) are hold, then for every , (3.9) ( have a unique fuzzy solution .

Proof.

For each and , define by

(3.10)

Thus, is continuous, so is a mapping from into itself. By Definitions 2.3 and 2.4, some properties of , and inequalities (3.4) and (3.5), we have following inequalities. For ,

(3.11)

Therefore

(3.12)

Hence

(3.13)

By hypothesis (H2), is a contraction mapping.

Using the Banach fixed point theorem, (3.9) have a unique fixed point .

4. Controllability

In this section, we show the nonlocal controllability for the control system (1.3).

Definition 4.1.

Equation (1.3) is nonlocal controllable. Then there exists such that the fuzzy solution for (3.9) as where is target set.

Define the fuzzy mapping by

(4.1)

where is closed support of . Then there exists

(4.2)

such that

(4.3)

Then exists such that

(4.4)

We assume that are bijective mappings.

We can introduce -level set of of (3.4)-(3.5)

(4.5)

Then substituting this expression into (3.9) yields -level of .

For each ,

(4.6)

Therefore

(4.7)

We now set

(4.8)

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.

Theorem 4.2.

Suppose that hypotheses (H1)–(H3) are satisfied. Then (3.9) are nonlocal controllable.

Proof.

We can easily check that is continuous function from to itself. By Definitions 2.3 and 2.4, some properties of , and inequalities (3.4) and (3.5), we have the following inequalities. For any ,

(4.9)

Therefore

(4.10)

Hence

(4.11)

By hypothesis (H2), is a contraction mapping. Using the Banach fixed point theorem, (4.8) has a unique fixed point .

5. Example

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.

Let

(5.1)

then the balance equations become

(5.2)

The -level sets of fuzzy numbers are the following: , for all . Then -level set of is

(5.3)

Further, we have

(5.4)

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 .

From the definition of fuzzy solution,

(5.5)

where .

Thus the -level of is

(5.6)

Then -level of is

(5.7)

Similarly

(5.8)

Hence

(5.9)

Then all the conditions stated in Theorem 4.2 are satisfied, so system (5.2) is nonlocal controllable on .