Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 734090, 16 pages
doi:10.1155/2009/734090
Research Article
Nonlocal Controllability for the Semilinear Fuzzy
Integrodifferential Equations in n-Dimensional
Fuzzy Vector Space
Young Chel Kwun,1 Jeong Soon Kim,1 Min Ji Park,1
and Jin Han Park2
1
2
Department of Mathematics, Dong-A University, Pusan 604-714, South Korea
Division of Mathematics Sciences, Pukyong National University, Pusan 608-737, South Korea
Correspondence should be addressed to Jin Han Park, jihpark@pknu.ac.kr
Received 23 February 2009; Revised 20 June 2009; Accepted 3 August 2009
Recommended by Tocka Diagana
We study the existence and uniqueness of solutions and controllability for the semilinear fuzzy
integrodifferential equations in n-dimensional fuzzy vector space EN n by using Banach fixed
point theorem, that is, an extension of the result of J. H. Park et al. to n-dimensional fuzzy vector
space.
Copyright q 2009 Young Chel Kwun et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Many authors have studied several concepts of fuzzy systems. Diamond and Kloeden 1
proved the fuzzy optimal control for the following system:
ẋt atxt ut,
x0 x0 ,
1.1
where x· and u· are nonempty compact interval-valued functions on E1 . Kwun and
Park 2 proved the existence of fuzzy optimal control for the nonlinear fuzzy differential
1
by using Kuhn-Tucker theorems. Fuzzy
system with nonlocal initial condition in EN
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 0, 1 for material with
2
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1
, Park et al. 4 proved the existence
memory. In one-dimensional fuzzy vector space EN
and uniqueness of fuzzy solutions and presented the sufficient condition of nonlocal
controllability for the following semilinear fuzzy integrodifferential equation with nonlocal
initial condition:
t
dxt
A xt Gt − sxsds ft, x ut,
dt
0
x0 g t1 , t2 , . . . , tp , xtm x0 ∈ EN ,
t ∈ J 0, T ,
1.2
m 1, 2, . . . , p,
where T > 0, A : J → EN is a fuzzy coefficient, EN is the set of all upper semicontinuous
convex normal fuzzy numbers with bounded α-level intervals, f : J ×EN → EN is a nonlinear
continuous function, g : J p × EN → EN is a nonlinear continuous function, Gt is an n × n
continuous matrix such that dGtx/dt is continuous for x ∈ EN and t ∈ J with Gt ≤ K,
K > 0, with all nonnegative elements, u : J → EN 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:
t
dxi t
i
Ai xi t Gt − sxi sds fi t, xi t ui t on EN
,
dt
0
i
xi 0 gi xi x0i ∈ EN
1.3
i 1, 2, . . . , n,
i
i
is fuzzy coefficient, EN
is the set of all upper semicontinuously
where Ai : 0, T → EN
j
i
i
i
convex fuzzy numbers on R with EN i
j,
f
E
/ N /
i : 0, T × EN → EN is a nonlinear regular
i
i
fuzzy function, gi : EN → EN is a nonlinear continuous function, Gt is n × n continuous
i
and t ∈ 0, T with Gt ≤ k, k > 0,
matrix such that dGtxi /dt is continuous for xi ∈ EN
i
i
ui : 0, T → EN is control function and x0i ∈ EN is initial value.
2. Preliminaries
A fuzzy set of Rn is a function u : Rn → 0, 1. For each fuzzy set u, we denote by uα {x ∈
Rn : ux ≥ α} for any α ∈ 0, 1, its α-level set.
Let u, v be fuzzy sets of Rn . It is well known that uα vα for each α ∈ 0, 1 implies
u v.
Let En denote the collection of all fuzzy sets of Rn that satisfies the following
conditions:
1 u is normal, that is, there exists an x0 ∈ Rn such that uxo 1;
2 u is fuzzy convex, that is, uλx 1 − λy ≥ min{ux, uy} for any x, y ∈ Rn ,
0 ≤ λ ≤ 1;
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3
3 ux is upper semicontinuous, that is, ux0 ≥ limk → ∞ uxk for any xk ∈ Rn k 0, 1, 2, . . ., xk → x0 ;
4 u0 is compact.
We call u ∈ En an n-dimension fuzzy number.
Wang et al. 9 defined n-dimensional fuzzy vector space and investigated its
properties.
For any ui ∈ E, i 1, 2, . . . , n, we call the ordered one-dimension fuzzy number class
u1 , u2 , . . . , un i.e., the Cartesian product of one-dimension fuzzy number u1 , u2 , . . . , un an ndimension fuzzy vector, denote it as u1 , u2 , . . . , un , and call the collection of all n-dimension
fuzzy vectors i.e., the Cartesian product E × E × · · · × E n-dimensional fuzzy vector space,
and denote it as En .
Definition 2.1 see 9. If u ∈ En , and uα is a hyperrectangle, that is, uα can be represented
by ni1 uαil , uαir , that is, uα1l , uα1r ×uα2l , uα2r ×· · ·×uαnl , uαnr for every α ∈ 0, 1, where uαil , uαir ∈ R
with uαil ≤ uαir when α ∈ 0, 1, i 1, 2, . . . , n, then we call u a fuzzy n-cell number. We denote
the collection of all fuzzy n-cell numbers by LEn .
Theorem 2.2 see 9. For any u ∈ LEn with uα ni1 uαil , uαir α ∈ 0, 1, there exists a
unique u1 , u2 , . . . , un ∈ En such that ui α uαil , uαir (i 1, 2, . . . , n and α ∈ 0, 1).
Conversely, for any u1 , u2 , . . . , un ∈ En with ui α uαil , uαir i 1, 2, . . . , n and α ∈
0, 1, there exists a unique u ∈ LEn such that uα ni1 uαil , uαir α ∈ 0, 1.
Note 1 see 9. Theorem 2.2 indicates that fuzzy n-cell numbers and n-dimension fuzzy
vectors can represent each other, so LEn and En may be regarded as identity. If
u1 , u2 , . . . , un ∈ En is the unique n-dimension fuzzy vector determined by u ∈ LEn ,
then we denote u u1 , u2 , . . . , un .
n
i n
i
i n
1
2
Let EN
EN
× EN
× · · · × EN
, EN
i 1, 2, ×, n be fuzzy subset of R. Then EN
⊆
n
E .
i n
Definition 2.3 see 9. The complete metric DL on EN
is defined by
DL u, v sup dL uα , vα
0<α≤1
sup max uαil − vilα , uαir − virα 2.1
0<α≤1 1≤i≤n
i n
for any u, v ∈ EN
, which satisfies dL u w, v w dL u, v.
i n
Definition 2.4. Let u, v ∈ C0, T : EN
, then
H1 u, v sup DL ut, vt.
0≤t≤T
2.2
4
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i n
is defined by
Definition 2.5 see 9. The derivative x t of a fuzzy process x ∈ EN
x t
α
xilα t, xirα t
n
2.3
i1
i n
provided that the equation defines a fuzzy x t ∈ EN
.
a
Definition 2.6 see 9. The fuzzy integral b xtdt, a, b ∈ 0, T is defined by
a
α
xtdt
b
n a
i1
b
xilα tdt,
a
b
xirα tdt
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 u ≡ 0.
We define
A A1 , A2 , . . . , An ,
x x1 , x2 , . . . , xn ,
f f1 , f2 , . . . , fn ,
3.1
u u1 , u2 , . . . , un ,
g g1 , g2 , . . . , gn ,
x0 x01 , x02 , . . . , x0n .
Then
n
i
A, x, f, x0 , u, g ∈ EN
.
i n
:
EN
3.2
Instead of 1.3, we consider the following fuzzy integrodifferential equations in
t
n
dxt
i
A xt Gt − sxsds ft, xt ut on EN
dt
0
i
x0 gx x0 ∈ EN
n
3.3
i n
i n
i n
with fuzzy coefficient A : 0, T → EN
, initial value x0 ∈ EN
, and u : 0, T → EN
i n
i n
is a control function. Given nonlinear regular fuzzy function f : 0, T × EN → EN
satisfies a global Lipschitz condition, that is, there exists a finite k > 0 such that
α α α dL fs, xs , f s, ys
≤ kdL xsα , ys
3.4
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5
i n
i n
i n
. The nonlinear function g : EN
→ EN
is a continuous function
for all xs, ys ∈ EN
and satisfies the Lipschitz condition
α α α dL gx· , g y·
≤ hdL x·α , y·
3.5
i n
for all x·, y· ∈ EN
, h is a finite positive constant.
i n
Definition 3.1. The fuzzy process x : I 0, T → EN
with α-level set xtα Πni1 xi α n
α
α
Πi1 xil , xir is a fuzzy solution of 3.3 without nonhomogeneous term if and only if
xilα t
xirα t
min
Aαij t
max
α
t
α
xik
t
α
xik
t
α
Gt −
Gt −
α
sxik
sds
0
Aαij t
t
: j, k l, r ,
0
xirα 0 girα xirα x0αir ,
xilα 0 gil xil x0αil ,
α
sxik
sds
: j, k l, r ,
3.6
i 1, 2, . . . , n.
For the sequel, we need the following assumptions.
i n
i n , d/dt Sty ∈ C1 I : EN
H1 St is a fuzzy number satisfying, for y ∈ EN
i n
, the equation
CI : EN
t
d
Sty A Sty Gt − sSsy ds
dt
0
StAy t
St − sAGsy ds,
3.7
t ∈ I,
0
where
Stα n
Si tα i1
n
Sαil t, Sαir t ,
3.8
i1
and Sαij t j l, r is continuous with |Sαij t| ≤ c, c > 0, for all t ∈ I 0, T .
H2 c{h1 T cT kT 1 cT } < 1.
In view of Definition 3.1 and H1, 3.3 can be expressed as
xt St x0 − gx t
St − s fs, xs us ds,
0
3.9
x0 gx x0 .
i n
Theorem 3.2. Let T > 0. If hypotheses (H1)-(H2) are hold, then for every x0 ∈ EN
, 3.9 (u ≡ 0
i n
have a unique fuzzy solution x ∈ C0, T : EN .
6
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i n
i n
and t ∈ 0, T , define G0 xt ∈ EN
by
Proof. For each xt ∈ EN
G0 xt St x0 − gx t
St − sfs, xsds.
3.10
0
i n
i n
is continuous, so G0 is a mapping from C0, T : EN
into
Thus, G0 x : 0, T → EN
itself. By Definitions 2.3 and 2.4, some properties of dL , and inequalities 3.4 and 3.5, we
i n
,
have following inequalities. For x, y ∈ C0, T : EN
α dL G0 xtα , G0 y t
St x0 − gx dL
t
α
St − sfs, xsds
,
0
St x0 − g y t
α St − sf s, ys ds
0
dL
−Stgx t
α
St − sfs, xsds
,
0
−Stg y t
α St − sf s, ys ds
0
α α ≤ dL Stgx , Stg y
t
α α dL St − sfs, xs , St − sf s, ys
ds
0
max Sαil t gilα x − gilα y , Sαir t girα x − girα y 1≤i≤n
t
max Sαil t − s filα s, xs − filα s, ys , Sαir t − s firα s, xs − firα s, ys ds
0 1≤i≤n
≤ c max gilα x − gilα y , girα x − girα y 1≤i≤n
c
t
max filα s, xs − filα s, ys , firα s, xs − firα s, ys ds
0 1≤i≤n
α α cdL gx , g y
c
t
α α dL fs, xs , f s, ys
ds
0
α ≤ chdL x·α , y·
ck
t
0
α dL xsα , ys ds.
3.11
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Therefore
DL G0 xt, G0 y t
α sup dL G0 xtα , G0 y t
0<α≤1
α
≤ ch sup dL x· , y·
α ck sup
t
α dL xsα , ys ds
3.12
0<α≤1 0
0<α≤1
≤ chDL x·, y· ck
t
DL xs, ys ds.
0
Hence
H1 G0 x, G0 y sup DL G0 xt, G0 y t
0≤t≤T
≤ ch sup DL x·, y· ck sup
0≤t≤T
0≤t≤T
≤ ch H1 x, y ckT H1 x, y
t
DL xs, ys ds
0
3.13
ch kT H1 x, y .
By hypothesis H2, G0 is a contraction mapping.
Using the Banach fixed point theorem, 3.9 have a unique fixed point x ∈ C0, T :
i n
.
EN
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 ut such that the
i n
fuzzy solution xt for 3.9 as xT x1 − gx i.e., xT α x1 − gxα where x1 ∈ EN
is target set.
i n
Define the fuzzy mapping β : P Rn → EN
by
βα v ⎧ T
⎪
⎪
⎨ Sα T − svsds, v ⊂ Γu ,
⎪
⎪
⎩
0
0,
otherwise,
4.1
8
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where Γu is closed support of u. Then there exists
i
βi : P R −→ EN
4.2
i 1, 2, . . . , n
such that
⎧ T
⎪
⎪
⎨ Sαi T − svi sds, vi s ⊂ Γui ,
βiα vi ⎪
⎪
⎩
4.3
0
0,
otherwise.
Then βijα j l, r exists such that
βilα vil βirα vir T
0
T
0
Sαil T − svil sds,
vil s ∈ uαil s, u1i ,
Sαir T − svir sds,
vir s ∈ u1i , uαir s .
4.4
We assume that βilα , βirα are bijective mappings.
We can introduce α-level set of us of 3.4-3.5
usα n
ui sα
i1
n
α
uil s, uαir s
i1
n
βilα
−1 α
x1 − gilα xilα − Sαil T x0αil − gilα xilα
il
i1
−
T
0
βirα
Sαil T
−
sfilα
s, xilα s
4.5
ds ,
−1 α
x1 − girα xirα − Sαir T x0αir − girα xirα
ir
−
T
0
Sαir T
−
sfirα
s, xirα s
ds
Then substituting this expression into 3.9 yields α-level of xT .
.
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For each i 1, 2, . . . , n,
T
xi T Sαil T x0αil − gilα xilα Sαil T − sfilα s, xilα s ds
α
0
T
0
Sαil T
−1 α
α
x1 − gilα xilα − Sαil T x0αil − gilα xilα
− s βil
il
−
T
0
Sαil T
−
sfilα
s, xilα s
ds ds,
T
Sαir T x0αir − girα xirα Sαir T − sfirα s, xirα s ds
4.6
0
T
0
Sαir T
− s βirα
−1 α
x1 − girα xirα − Sαir T x0αir − girα xirα
ir
−
T
0
Sαir T
−
sfirα
s, xirα s
ds ds
α α α
x1 − gx
.
x1 − gx , x1 − gx
il
i
ir
Therefore
xT α n
xi T α i1
n x1 − gx
i1
α
i
α
x1 − gx .
4.7
We now set
Φxt St x0 − gx t
St − sfs, xsds
0
t
0
−1
St − sβ
x − gx − ST x0 − gx −
1
T
ST − sfs, xsds ds,
0
4.8
where the fuzzy mapping β−1 satisfies above statements.
Notice that ΦxT x1 − gx, which means that the control ut steers 3.9 from the
origin to x1 − gx in time T provided that we can obtain a fixed point of the operator Φ.
H3 Assume that the linear system of 3.9 f ≡ 0 is controllable.
Theorem 4.2. Suppose that hypotheses (H1)–(H3) are satisfied. Then 3.9 are nonlocal controllable.
10
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i n
to itself. By
Proof. We can easily check that Φ is continuous function from C0, T : EN
Definitions 2.3 and 2.4, some properties of dL , and inequalities 3.4 and 3.5, we have the
i n
,
following inequalities. For any x, y ∈ C0, T : EN
α dL Φxtα , Φyt
t
t
St x0 − gx St − sfs, xsds St − sβ−1
dL
o
0
× x − gx − ST x0 − gx −
1
T
α
ST − sfs, xsds ds
,
0
St x0 − g y t
St − sf s, ys ds t
0
St − sβ−1
0
× x − g y − ST x0 − g y −
1
T
α ST − sf s, ys ds ds
0
α α ≤ dL Stgx , Stg y
t
α α dL St − sfs, xs , St − sf s, ys
ds
0
t
dL
α α St − sβ−1 gx , St − sβ−1 g y
ds
α α St − sβ−1 ST gx , St − sβ−1 ST g y
ds
0
t
dL
0
t
dL
0
−1
St − sβ
T
α ST − sfs, xsds
−1
, St − sβ
0
T
α ds
ST − sf s, ys ds
0
max Sαil t gilα x − gilα y , Sαir t girα x − girα y 1≤i≤n
t
0 1≤i≤n
t
0
max Sαil t − s filα s, xs − filα s, ys , Sαir t − s fαir s, xs − firα s, ys ds
t
0
$
#
−1 −1 max Sαil t − s βilα
gilα x − gilα y , Sαir t − s βirα
girα x − girα y ds
1≤i≤n
#
−1
max Sαil t − s βilα Sαil T gilα x − gilα y ,
1≤i≤n
$
−1
α
α
α
α
S t − s βα
ds
S
g
−
g
y
T
x
ir
ir
ir
ir
ir
t
T
−1 T
α
max Sil t − s βilα
Sαil T − sfilα s, xsds − Sαil T − sfilα s, ys ds ,
1≤i≤n
0
0
0
T
−1 T
α
α
α
α
α
α
Sir T −sfir s, xsds − Sir T −sfir s, ys ds ds
Sir t − s βil
0
0
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≤ c max gilα x − gilα y , girα x − girα y 1≤i≤n
c
c
t
max filα s, xs − filα s, ys , firα s, xs − firα s, ys ds
0 1≤i≤n
t
max gilα x − gilα y , girα x − girα y ds
0 1≤i≤n
c2
c2
t
max gilα x − gilα y , girα x − girα y ds
0 1≤i≤n
t T
max filα s, xs − filα s, ys , firα s, xs − firα s, ys ds ds
0 1≤i≤n
0
α α c
cdL gx , g y
c
t
t
α α dL fs, xs , f s, ys
ds
0
α α dL gx , g y
ds c2
t
0
c2
t T
0
α α dL gx , g y
ds
0
α α dL fs, xs , f s, ys
ds ds
0
α
≤ ch dL x· , y·
α t
α α 1 c dL x· , y· ds
0
ck
t
α dL xs , ys ds c
α
t T
0
0
α
dL xs , ys
α ds ds .
0
4.9
Therefore
DL Φxt, Φyt
α sup dL Φxtα , Φyt
0<α≤1
t
α α α ≤ ch sup dL x· , y·
1 c sup dL x· , y· ds
α
0 0<α≤1
0<α≤1
ck
t
α
sup dL xs , ys
α ds c
0 0<α≤1
t T
0
α
sup dL xs , ys
0 0<α≤1
t
ch DL x·, y· 1 c DL x·, y· ds
0
ck
t
0
DL xs, ys ds c
t T
DL xs, ys ds ds .
0
0
α ds ds
4.10
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Hence
H1 Φx, Φy sup DL Φxt, Φyt
0≤t≤T
≤ ch sup DL x·, y· 1 c sup
0≤t≤T
ck sup
0≤t≤T
0≤t≤T
t
t
DL x·, y· ds
0
DL xs, ys ds c sup
0
DL xs, ys ds ds
t T
0≤t≤T
0
4.11
0
%
&
≤ ch H1 x, y 1 cT H1 x, y ck T H1 x, y cT 2 H1 x, y
c{h1 T cT kT 1 cT }H1 x, y .
By hypothesis H2, Φ is a contraction mapping. Using the Banach fixed point theorem, 4.8
i n
.
has a unique fixed point x ∈ C0, T : EN
5. Example
Consider the two semilinear one-dimensional heat equations on a connected domain 0, 1
i
, i 1, 2, boundary condition xi t, 0 xi t, 1 0,
for material with memory on EN
'p
i 1, 2 and with initial conditions xi 0, zi k1 ck i xi tk , zi x0i zi , where x0i zi ∈
i 'p
EN
, k1 ck i xi tk , zi gi xi , i 1, 2. Let xi t, zi , i 1, 2, be the internal energy and let
i t, zi 2 , i 1, 2, be the external heat.
fi t, xi t, zi 2tx
Let
2
2
∂
∂
A A1 , A2 2 2 , 2 2 ,
∂z1 ∂z2
2 t, z2 2 ,
1 t, z1 2 , 2tx
ft, xt f1 t, x1 t, f2 t, x2 t 2tx
p
p
(
(
gx g1 x1 , g2 x2 ck 1 x1 tk , z1 , ck 2 x2 tk , z2 ,
k1
5.1
k1
0 ,
x0 x01 , x02 0,
x0 gx x1 0 g1 x, x2 0 g2 x ,
Gt − s e−t−s , e−t−s ,
then the balance equations become
t
2
dxt
i
A xt Gt − sxsds ft, xt on EN
,
dt
0
i
x0 gx x0 ∈ EN
2
.
5.2
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α α α α − 1, 1 − α, 2
The α-level sets of fuzzy numbers are the following: 0
1, 3 − α for all α ∈ 0, 1. Then α-level set of ft, xt is
α
ft, xt
α α
1 t2 × 2tx
2 t2
2tx
α α α
α 2 · t x1 t2 × 2 · t x2 t2
2 α
α
α 1, 3 − α · t x1l
t , x1r
t
5.3
2 2 α
α
× α 1, 3 − α · t x2l
t , x2r
t
2 α 2 α 2 α 2
α 2
× α 1t x2l
α 1t x1l
t , 3 − αt x1r
t
t , 3 − αt x2r
t .
Further, we have
α α dL ft, xt , f t, yt
2
2 2
2 dL α 1t xilα t , 3 − αt xirα t , α 1t yilα t , 3 − αt yirα t
%
2 2 2 2 &
t max α 1 xilα t − yilα t , 3 − α xirα t − yirα t 1≤i≤2
≤ T 3 − α max xilα t − yilα txilα t yilα t, xirα t − yirα txirα t yirα t
1≤i≤2
≤ 3T xα t yα t × max xα t − yα t, xα t − yα t
ir
ir
1≤i≤2
il
il
ir
ir
α kdL xtα , yt ,
α α dL gx· , g y·
p
α p
α (
( dL
ck xtk ,
ck ytk k1
k1
p
p
p
p
(
(
(
(
α
α
α
α
max ck i xil tk − ck i yil tk , ck i xir tk − ck i yir tk k1
1≤i≤2 k1
k1
k1
p ( ≤ ck max xilα tk − yilα tk , xirα tk − yirα tk k1 1≤i≤2
p ( α ck dL xtk α , ytk k1 p ( α ≤ ck max dL xtk α , ytk k1 k
α hdL x·α , y· ,
5.4
14
Advances in Difference Equations
where k and h satisfy the inequality 3.4 and 3.5, respectively. Choose T such that T <
1 − ch/ck. Then all conditions stated in Theorem 3.2 are satisfied, so problem 5.2 has a
unique fuzzy solution.
3.
The α-level set of fuzzy numbers is 3
3
α Let target set be x1 x11 , x21 2,
α 2, 4 − α.
From the definition of fuzzy solution,
xilα t
x0 αil
Sαil t
−
p
(
ck i xilα tk k1
t
0
t
2
Sαil t − sα 1s xilα s ds xirα t Sαir t x0 αir −
p
(
0
ck i xirα tk Sαil t − suαil sds,
5.5
k1
t
0
2
Sαir t − s3 − αs xirα s ds t
0
Sαir t − suαir sds,
where i 1, 2.
Thus the α-level of us is
uα1l s
α
β1l
−1
k1
−
uα1r s
α
β1r
−1
α
β2l
−1
3 − α −
α
β2r
T
x0 α1r −
p
(
α
ck 1 x1r
tk x0 α2l
−
p
(
α
ck 2 x2l
tk Sα2r T 3 −
αSα1r T
p
(
T
α 1Sα2l T
3 −
αSα2r T
0
k1
4 − α −
T
0
k1
1Sα1l T
2
α
x1l
s ds
− ss
,
2
α
x1r
s ds
− ss
,
p
(
ck 2 xilα tk Sα2l T α 0
k1
α 2 −
−
α
ck 1 x1l
tk ck 1 xirα tk −1
k1
−
p
(
Sα1r T uα2r s
−
p
(
k1
−
x0 α1l
Sα1l T uα2l s
p
(
ck 1 xilα tk α 1 −
− ss
2
α
x2l
s ds
,
ck 2 xirα tk k1
x0 α2r
p
(
α
− ck x2r
tk k1
T
0
− ss
2
α
x2r
s ds
.
5.6
Advances in Difference Equations
15
Then α-level of xT x1 T , x2 T is
x1 T α
α
α
x1l
T , x1r
T Sα1l T x0 α1l
−
p
(
α
ck 1 x1l
tk T
0
k1
α 2
α 1Sα1l T − ss x1l
s ds
p
−1
(
α α
β1l
β1l
α 1 − ck 1 xilα tk k1
−
x0 α1l
Sα1l T −
p
(
α
ck 1 x1l
tk k1
T
α 0
1Sα1l T
2
α
x1l
s ds
− ss
T
ds,
α
2
3 − αSα1r T − ss x1r
s ds
0
k1
p
(
α
Sα1r T x0 α1r − ck 1 x1r
tk 5.7
p
−1
(
α
α α
β1r
β1r
tk 3 − α − ck 1 x1r
k1
−
Sα1r T x0 α1r
p
(
α
− ck 1 x1r
tk k1
T
3 −
0
α 1 −
p
(
α
ck 1 x1l
tk k1
αSα1r T
− ss
2
α
x1r
s ds
ds
α
p
p
(
(
α
, 3 − α − ck 1 x1r tk 2 − ck 1 x1 tk .
k1
k1
Similarly
α
p
(
α
α
x2 T x2l T , x2r T 3 − ck 2 x2 tk .
α
5.8
k1
Hence
xT x1 T , x2 T p
p
(
(
2 − ck 1 x1 tk , 3 − ck 2 x2 tk x1 − gx.
k1
5.9
k1
Then all the conditions stated in Theorem 4.2 are satisfied, so system 5.2 is nonlocal
controllable on 0, T .
16
Advances in Difference Equations
Acknowledgment
This study was supported by research funds from Dong-A University.
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