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

Abstract

We study the existence and uniqueness of solutions and nonlocal controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in -dimensional fuzzy vector space by using short-term perturbations techniques and Banach fixed point theorem. This is an extension of the result of Kwun et al. (Kwun et al., 2009) to impulsive system.

1. Introduction

The theory of differential equations with discontinuous trajectories during the last twenty years has been to a great extent stimulated by their numerous applications to problem arising in mechanics, electrical engineering, the theory of automatic control, medicine and biology. For the monographs of the theory of impulsive differential equations, see the papers of Bainov and Simenov [1], Lakshmikantham et al. [2] and Samoileuko and Perestyuk [3], where numerous properties of their solutions are studied and detailed bibliographies are given. Rogovchenko [4] followed the ideas of the theory of impulsive differential equations which treats the changes of the state of the evolution process due to a short-term perturbations whose duration can be negligible in comparison with the duration of the process as an instant impulses. In 2001, Lakshmikantham and McRae [5] studied basic results for fuzzy impulsive differential equations. Park et al. [6] studied the existence and uniqueness of fuzzy solutions and controllability for the impulsive semilinear fuzzy integrodifferential equations in one-dimensional fuzzy vector space . Rodríguez-López [7] studied periodic boundary value problems for impulsive fuzzy differential equations. 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 [8] 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. [9] 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.

In [10], Kwun et al. proved the existence and uniqueness of fuzzy solutions for the semilinear fuzzy integrodifferential equations by using successive iteration. In [11], Kwun et al. investigated the continuously initial observability for the semilinear fuzzy integrodifferential equations. Bede and Gal [12] studied almost periodic fuzzy-number-valued functions. Gal and N'Guerekata [13] studied almost automorphic fuzzy-number-valued functions. More recently, Kwun et al. [14] studied the existence and uniqueness of solutions and nonlocal controllability for the semilinear fuzzy integrodifferential equations in -dimensional fuzzy vector space.

In this paper, we study the existence and uniqueness of solutions and nonlocal controllability for the following impulsive semilinear nonlocal fuzzy integrodifferential equations in -dimensional fuzzy vector space by using short-term perturbations techniques and Banach fixed point theorem:

(1.1)

where is fuzzy coefficient, is the set of all upper semicontinuously convex fuzzy numbers on with , and are nonlinear regular fuzzy functions, is a nonlinear continuous function, is an continuous matrix such that is continuous for and with , , is a control function, is an initial value and are bounded functions, , where and represent the left and right limits of at , respectively.

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. [15] 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 [15]).

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 [15]).

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

Note (see [15]).

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 , where is a fuzzy subset of . Then .

Definition 2.3 (see [15]).

The complete metric on is defined by

(2.1)

for any , which satisfies .

Definition 2.4.

Let ,

(2.2)

Definition 2.5 (see [15]).

The derivative of a fuzzy process is defined by

(2.3)

provided that equation defines a fuzzy .

Definition 2.6 (see [15]).

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 (1.1) ().

We define

(3.1)

(3.2)

Then

(3.3)

Instead of (1.1), we consider the following fuzzy integrodifferential equations in .

(3.4)

(3.5)

(3.6)

with fuzzy coefficient , initial value , and being a control function. Given nonlinear regular fuzzy functions and satisfy global Lipschitz conditions, that is, there exist finite constants such that

(3.7)

(3.8)

for all ,the nonlinear function is continuous and satisfies the Lipschitz condition

(3.9)

for all , is a finite positive constant.

Definition 3.1.

The fuzzy process with -level set is a fuzzy solution of (3.4) and (3.5) without nonhomogeneous term if and only if

(3.10)

For the sequel, we need the following assumption:

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

(3.11)

where

(3.12)

and is continuous with , , for all .

In order to define the solution of (3.4)–(3.6), we will consider the space = and there exist and with

Let .

Lemma 3.2.

If is an integral solution of (3.4)–(3.6) , then is given by

(3.13)

Proof.

Let be a solution of (3.4)–(3.6). Define . Then we have that

(3.14)

Consider . Then integrating the previous equation, we have

(3.15)

For ,

(3.16)

or

(3.17)

Now for we have that

(3.18)

Then

(3.19)

if and only if

(3.20)

Hence

(3.21)

which proves the lemma.

Assume the following:

(H2)there exists such that

(3.22)

where ;

(H3)

(3.23)

Theorem 3.3.

Let . If hypotheses (H1)–(H3) are hold, then, for every , (3.13) has a unique fuzzy solution .

Proof.

For each and , define by

(3.24)

Thus, is continuous, so is a mapping from into itself. By Definitions 2.3 and 2.4, some properties of and inequalities (3.7), (3.8), and (3.9), we have the following inequalities. For ,

(3.25)

Therefore

(3.26)

Hence

(3.27)

By hypothesis (H3), is a contraction mapping. Using the Banach fixed point theorem, (3.13) has a unique fixed point .

4. Nonlocal Controllability

In this section, we show the nonlocal controllability for the control system (1.1).

The control system (1.1) is related to the following fuzzy integral system:

(4.1)

Definition 4.1.

Equations (1.1)–(3) are nonlocal controllable. Then there exists such that the fuzzy solution for (4.1) as , where , is target set.

Define the fuzzy mapping by

(4.2)

where is closed support of . Then there exists

(4.3)

such that

(4.4)

Then there exists such that

(4.5)

We assume that are bijective mappings.

We can introduce -level set of of (4.1):

(4.6)

Then substituting this expression into (4.1) yields -level of .

For each ,

 

(4.7)

Therefore

(4.8)

We now set

(4.9)

where the fuzzy mapping satisfies the previous statements.

Notice that , which means that the control steers (4.9) from the origin to in time provided we can obtain a fixed point of the operator .

(H4)Assume that the linear system of (4.9) is controllable.

Theorem 4.2.

Suppose that hypotheses (H1)–(H4) are satisfied. Then (4.9) is 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 (3.7), (3.8), and (3.9), we have following inequalities. For any ,

(4.10)

Therefore

(4.11)

Hence

(4.12)

By hypothesis (H3), is a contraction mapping. Using the Banach fixed point theorem, (4.9) 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

(5.1)

and with initial conditions

(5.2)

where ,

(5.3)

Let be the internal energy and

(5.4)

be the external heat withmemory.

(5.5)

is impulsive effect at .

Let

(5.6)

then the balance equations become

(5.7)

The -level sets of fuzzy numbers are the following

, for all . Then -level set of is

(5.8)

Further, we have

(5.9)

where , , and satisfy inequalities (3.7), (3.8), and (3.9), respectively. Choose such that . Then all conditions stated in Theorem 3.3 are satisfied, so the problem (5.7) has a unique fuzzy solution.

Let target set be . The -level set of fuzzy numbersis.

From the definition of fuzzy solution,

(5.10)

where .

Thus the -levels of

 

(5.11)

Then -level of is

(5.12)

Similarly,

(5.13)

Hence

(5.14)

Then all the conditions stated in Theorem 4.2 are satisfied, so the system (5.7) is nonlocal controllable on .