- Research Article
- Open Access
Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in
-Dimensional Fuzzy Vector Space
- YoungChel Kwun1,
- JeongSoon Kim1,
- MinJi Park1 and
- JinHan Park2Email author
https://doi.org/10.1155/2009/734090
© Young Chel Kwun et al. 2009
- Received: 23 February 2009
- Accepted: 3 August 2009
- Published: 26 August 2009
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.
Keywords
- Fuzzy Number
- Lipschitz Condition
- Connected Domain
- Fuzzy Function
- Fuzzy Process
1. Introduction






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.
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]).
for any
, which satisfies
.
Definition 2.4.
Definition 2.5 (see [9]).
provided that the equation defines a fuzzy
.
Definition 2.6 (see [9]).
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.
Definition 3.1.



For the sequel, we need the following assumptions.
and
is continuous with
,
, for all
.
(H2)
.
Theorem 3.2.
Let
. If hypotheses (H1)-(H2) are hold, then for every
, (3.9) (
have a unique fuzzy solution
.
Proof.





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.
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.
Theorem 4.2.
Suppose that hypotheses (H1)–(H3) are satisfied. Then (3.9) are nonlocal controllable.
Proof.




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.
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
.
where
.
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
.
Declarations
Acknowledgment
This study was supported by research funds from Dong-A University.
Authors’ Affiliations
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