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Controllability for the Impulsive Semilinear Nonlocal Fuzzy Integrodifferential Equations in Dimensional Fuzzy Vector Space
Advances in Difference Equations volume 2010, Article number: 983483 (2010)
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 shortterm 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 shortterm 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 onedimensional fuzzy vector space . RodríguezLópez [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 onedimensional heat equation on a connected domain for material with memory. In onedimensional 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 fuzzynumbervalued functions. Gal and N'Guerekata [13] studied almost automorphic fuzzynumbervalued 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 shortterm perturbations techniques and Banach fixed point theorem:
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 onedimension fuzzy number class (i.e., the Cartesian product of onedimension 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
for any , which satisfies .
Definition 2.4.
Let ,
Definition 2.5 (see [15]).
The derivative of a fuzzy process is defined by
provided that equation defines a fuzzy .
Definition 2.6 (see [15]).
The fuzzy integral , is defined by
provided that the Lebesgue integrals on the righthand side exist.
3. Existence and Uniqueness
In this section we consider the existence and uniqueness of the fuzzy solution for (1.1) ().
We define
Then
Instead of (1.1), we consider the following fuzzy integrodifferential equations in .
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
for all ,the nonlinear function is continuous and satisfies the Lipschitz condition
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
For the sequel, we need the following assumption:
(H1) is a fuzzy number satisfying, for , , the equation
where
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
Proof.
Let be a solution of (3.4)–(3.6). Define . Then we have that
Consider . Then integrating the previous equation, we have
For ,
or
Now for we have that
Then
if and only if
Hence
which proves the lemma.
Assume the following:
(H2)there exists such that
where ;
(H3)
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
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 ,
Therefore
Hence
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:
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
where is closed support of . Then there exists
such that
Then there exists such that
We assume that are bijective mappings.
We can introduce level set of of (4.1):
Then substituting this expression into (4.1) yields level of .
For each ,
Therefore
We now set
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 ,
Therefore
Hence
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 onedimensional heat equations on a connected domain for material with memory on boundary condition
and with initial conditions
where ,
Let be the internal energy and
be the external heat withmemory.
is impulsive effect at .
Let
then the balance equations become
The level sets of fuzzy numbers are the following
, for all . Then level set of is
Further, we have
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,
where .
Thus the levels of
Then level of is
Similarly,
Hence
Then all the conditions stated in Theorem 4.2 are satisfied, so the system (5.7) is nonlocal controllable on .
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Acknowledgment
This study was supported by research funds from DongA University.
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Kwun, Y., Kim, J., Park, M. et al. Controllability for the Impulsive Semilinear Nonlocal Fuzzy Integrodifferential Equations in Dimensional Fuzzy Vector Space. Adv Differ Equ 2010, 983483 (2010). https://doi.org/10.1155/2010/983483
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Keywords
 Fuzzy Number
 Lipschitz Condition
 Connected Domain
 Impulsive Differential Equation
 Fuzzy Function