Advances in Difference Equations Volume 2009, Article ID 734090,16pages doi:10.1155/2009/734090
Research Article
Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in n-Dimensional Fuzzy Vector Space
Young Chel Kwun,
1Jeong Soon Kim,
1Min Ji Park,
1and Jin Han Park
21Department of Mathematics, Dong-A University, Pusan 604-714, South Korea
2Division of Mathematics Sciences, Pukyong National University, Pusan 608-737, South Korea
Correspondence should be addressed to Jin Han Park,[email protected] 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 inn-dimensional fuzzy vector spaceENn by using Banach fixed point theorem, that is, an extension of the result of J. H. Park et al. ton-dimensional fuzzy vector space.
Copyrightq2009 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 Kloeden1 proved the fuzzy optimal control for the following system:
xt ˙ atxt ut, x0 x0, 1.1
where x· and u· are nonempty compact interval-valued functions on E1. Kwun and Park2proved the existence of fuzzy optimal control for the nonlinear fuzzy differential system with nonlocal initial condition in EN1 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 Muralisankar3proved 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
memory. In one-dimensional fuzzy vector space E1N, 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:
dxt
dt A
xt
t
0
Gt−sxsds
ft, x ut, t∈J 0, T, x0 g
t1, t2, . . . , tp, xtm
x0∈EN, m1,2, . . . , p,
1.2
whereT > 0,A : J → EN is a fuzzy coefficient,EN is the set of all upper semicontinuous convex normal fuzzy numbers with boundedα-level intervals,f:J×EN → ENis a nonlinear continuous function,g :Jp×EN → ENis a nonlinear continuous function,Gtis ann×n continuous matrix such thatdGtx/dtis continuous forx∈ENandt∈JwithGt ≤K, K >0, with all nonnegative elements,u:J → ENis 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 Gal7studied almost periodic fuzzy-number-valued functions. Gal and N’Gu´er´ekata 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:
dxit dt Ai
xit
t
0
Gt−sxisds
fit, xit uitonEiN, xi0 gixi x0i ∈EiN i1,2, . . . , n,
1.3
where Ai : 0, T → EiN is fuzzy coefficient, EiN is the set of all upper semicontinuously convex fuzzy numbers onRwithEiN/ENj i /j,fi:0, T×ENi → EiNis a nonlinear regular fuzzy function,gi :EiN → EiNis a nonlinear continuous function,Gtisn×ncontinuous matrix such thatdGtxi/dtis continuous forxi ∈ENi andt∈0, TwithGt ≤ k,k >0, ui:0, T → EiNis control function andx0i ∈ENi is initial value.
2. Preliminaries
A fuzzy set ofRnis a functionu:Rn → 0,1. For each fuzzy setu, we denote byuα{x∈ Rn :ux≥α}for anyα∈0,1, itsα-level set.
Letu, vbe fuzzy sets ofRn. It is well known thatuα vαfor eachα∈0,1implies uv.
Let En denote the collection of all fuzzy sets of Rn that satisfies the following conditions:
1uis normal, that is, there exists anx0∈Rnsuch thatuxo 1;
2uis fuzzy convex, that is, uλx 1−λy ≥ min{ux, uy}for anyx, y ∈ Rn, 0≤λ≤1;
3uxis upper semicontinuous, that is,ux0≥ limk→ ∞uxkfor anyxk ∈ Rn k 0,1,2, . . .,xk → x0;
4 u0is compact.
We callu∈Enann-dimension fuzzy number.
Wang et al. 9 defined n-dimensional fuzzy vector space and investigated its properties.
For anyui ∈ E,i1,2, . . . , n, we call the ordered one-dimension fuzzy number class u1, u2, . . . , uni.e., the Cartesian product of one-dimension fuzzy numberu1, u2, . . . , unann- dimension fuzzy vector, denote it asu1, u2, . . . , un, and call the collection of alln-dimension fuzzy vectorsi.e., the Cartesian product
E×E× · · · ×En-dimensional fuzzy vector space, and denote it asEn.
Definition 2.1see9. Ifu∈En, anduαis a hyperrectangle, that is,uαcan be represented byn
i1uαil, uαir, that is,uα1l, uα1r×uα2l, uα2r×· · ·×uαnl, uαnrfor everyα∈0,1, whereuαil, uαir∈R withuαil ≤uαirwhenα∈0,1,i1,2, . . . , n, then we callua fuzzyn-cell number. We denote the collection of all fuzzyn-cell numbers byLEn.
Theorem 2.2see9. For anyu∈LEnwithuα n
i1uαil, uαir α ∈0,1, there exists a uniqueu1, u2, . . . , un∈Ensuch thatuiα uαil, uαir(i1,2, . . . , nandα∈0,1).
Conversely, for anyu1, u2, . . . , un ∈ En with uiα uαil, uαiri 1,2, . . . , nand α ∈ 0,1, there exists a uniqueu∈LEnsuch thatuαn
i1uαil, uαir α∈0,1.
Note 1 see 9. Theorem 2.2indicates 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 uniquen-dimension fuzzy vector determined by u ∈ LEn, then we denoteu u1, u2, . . . , un.
LetEiNnEN1 ×E2N× · · · ×EnN,EiN i1,2,×, nbe fuzzy subset ofR. ThenEiNn ⊆ En.
Definition 2.3see9. The complete metricDLonEiNnis defined by
DLu, v sup
0<α≤1dL
uα,vα sup
0<α≤1
max
1≤i≤n
uαil−vilα,uαir−virα 2.1
for anyu, v∈EiNn, which satisfiesdLuw, vw dLu, v.
Definition 2.4. Letu, v∈C0, T:EiNn, then
H1u, v sup
0≤t≤TDLut, vt. 2.2
Definition 2.5see9. The derivativextof a fuzzy processx∈EiNnis defined by xtα
n
i1
xαil t,
xαir t
2.3
provided that the equation defines a fuzzyxt∈EiNn. Definition 2.6see9. The fuzzy integrala
bxtdt,a, b∈0,Tis defined by a
b
xtdt α
n
i1
a
b
xαiltdt, a
b
xαirtdt
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 for1.3 u≡0.
We define
A A1, A2, . . . , An, x x1, x2, . . . , xn, f
f1, f2, . . . , fn , u u1, u2, . . . , un, g
g1, g2, . . . , gn
, x0 x01, x02, . . . , x0n.
3.1
Then
A, x, f, x0, u, g∈ EiNn
. 3.2
Instead of 1.3, we consider the following fuzzy integrodifferential equations in EiNn:
dxt
dt A
xt
t
0
Gt−sxsds
ft, xt uton ENi n x0 gx x0∈
EiNn
3.3
with fuzzy coefficientA:0, T → EiNn, initial valuex0 ∈ENi n, andu:0, T → EiNn is a control function. Given nonlinear regular fuzzy functionf : 0, T×EiNn → EiNn satisfies a global Lipschitz condition, that is, there exists a finitek >0 such that
dL
fs, xsα ,
f
s, ysα
≤kdL
xsα, ysα
3.4
for allxs, ys∈EiNn. The nonlinear functiong :ENi n → ENi nis a continuous function and satisfies the Lipschitz condition
dL
gx·α
, g
y·α
≤hdL
x·α, y·α
3.5
for allx·, y·∈ENi n,his a finite positive constant.
Definition 3.1. The fuzzy processx:I 0, T → EiNnwithα-level setxtα Πni1xiα Πni1xαil, xαiris a fuzzy solution of3.3without nonhomogeneous term if and only if
xαil
t min
Aαijt
xαikt t
0
Gt−sxikαsds
:j, kl, r
, xαir
t max
Aαijt
xαikt t
0
Gt−sxikαsds
:j, kl, r
, xαil0 gilα
xαil xα0
il, xαir0 gαir
xirα xα0
ir, i1,2, . . . , n.
3.6
For the sequel, we need the following assumptions.
H1Stis a fuzzy number satisfying, fory∈EiNn,d/dtSty∈C1I :EiNn CI:EiNn, the equation
d
dtStyA
Sty
t
0
Gt−sSsy ds
StAy t
0
St−sAGsy ds, t∈I,
3.7
where
Stαn
i1
Sitαn
i1
Sαilt, Sαirt
, 3.8
andSαijt jl, ris continuous with|Sαijt| ≤c,c >0, for allt∈I 0, T. H2c{h1TcT kT1cT}<1.
In view ofDefinition 3.1andH1,3.3can be expressed as
xt St
x0−gx
t
0
St−s
fs, xs us ds, x0 gx x0.
3.9
Theorem 3.2. LetT >0. If hypotheses (H1)-(H2) are hold, then for everyx0 ∈EiNn,3.9(u≡0 have a unique fuzzy solutionx∈C0, T:EiNn.
Proof. For eachxt∈ENi nandt∈0, T, defineG0xt∈EiNnby
G0xt St
x0−gx
t
0
St−sfs, xsds. 3.10
Thus,G0x : 0, T → EiNnis continuous, soG0 is a mapping from C0, T : EiNninto itself. By Definitions2.3and2.4, some properties ofdL, and inequalities3.4and3.5, we have following inequalities. Forx, y∈C0, T:EiNn,
dL
G0xtα, G0y
tα dL
St
x0−gx
t
0
St−sfs, xsds α
,
St x0−g
y
t
0
St−sf s, ys
ds α
dL
−Stgx t
0
St−sfs, xsds α
,
−Stg y
t
0
St−sf s, ys
ds α
≤dL
Stgxα
, Stg
yα
t
0
dL
St−sfs, xsα
,
St−sf
s, ysα ds max
1≤i≤nSαilt
gilαx−gilα
y,Sαirt
girαx−girα
y
t
0
max1≤i≤nSαilt−s
filαs, xs−filα
s, ys,Sαirt−s
firαs, xs−firα
s, ysds
≤cmax
1≤i≤ngilαx−gilα
y,girαx−girα
y
c t
0
max1≤i≤nfilαs, xs−filα
s, ys,firαs, xs−firα
s, ysds cdL
gxα
, g
yα c
t
0
dL
fs, xsα
, f
s, ysα ds
≤chdL
x·α, y·α
ck t
0
dL
xsα, ysα
ds.
3.11
Therefore
DL
G0xt, G0y
t sup
0<α≤1dL
G0xtα, G0y
tα
≤chsup
0<α≤1dL
x·α, y·α
cksup
0<α≤1
t
0
dL
xsα, ysα
ds
≤chDL
x·, y·
ck t
0
DL
xs, ys ds.
3.12
Hence
H1
G0x, G0y sup
0≤t≤TDL
G0xt, G0y
t
≤chsup
0≤t≤TDL
x·, y·
cksup
0≤t≤T
t
0
DL
xs, ys ds
≤ch H1 x, y
ckT H1 x, y chkTH1
x, y .
3.13
By hypothesisH2,G0is a contraction mapping.
Using the Banach fixed point theorem,3.9have a unique fixed pointx ∈C0, T : EiNn.
4. Controllability
In this section, we show the nonlocal controllability for the control system1.3.
Definition 4.1. Equation 1.3 is nonlocal controllable. Then there exists ut such that the fuzzy solutionxtfor3.9asxT x1−gx i.e.,xTα x1−gxαwherex1∈EiNn is target set.
Define the fuzzy mappingβ:PR n → ENi nby
βαv
⎧⎪
⎪⎨
⎪⎪
⎩ T
0
SαT−svsds, v⊂Γu,
0, otherwise,
4.1
whereΓuis closed support ofu. Then there exists
βi:PR −→EiN i1,2, . . . , n 4.2
such that
βiαvi
⎧⎪
⎪⎨
⎪⎪
⎩ T
0
SαiT−svisds, vis⊂Γui,
0, otherwise.
4.3
Thenβijα jl, rexists such that
βαilvil
T
0
SαilT−svilsds, vils∈
uαils, u1i , βαirvir
T
0
SαirT−svirsds, virs∈
u1i, uαirs .
4.4
We assume thatβilα,βαirare bijective mappings.
We can introduceα-level set ofusof3.4-3.5
usαn
i1
uisα
n
i1
uαils, uαirs
n
i1
βαil−1 x1α
il−gαil xilα
−SαilT xα0
il−gilα
xαil
− T
0
SαilT−sfilα
s, xαils ds
, βirα−1
x1α
ir−girα xαir
−SαirT x0α
ir−girα
xαir
− T
0
SαirT−sfirα
s, xαirs ds
.
4.5
Then substituting this expression into3.9yieldsα-level ofxT.
For eachi1,2, . . . , n,
xiTα
SαilT xα0
il −gilα
xαil
T
0
SαilT−sfilα
s, xαils ds
T
0
SαilT−s
βαil−1 x1α
il−gαil xilα
−SαilT xα0
il−gilα
xαil
− T
0
SαilT−sfilα
s, xαils ds
ds,
SαirT xα0
ir −girα
xirα
T
0
SαirT−sfirα
s, xαirs ds
T
0
SαirT−s
βirα−1 x1α
ir−girα xαir
−SαirT
x0αir−girα xαir
− T
0
SαirT−sfirα
s, xαirs ds
ds
x1−gxα
il,
x1−gxα
ir
x1−gx
i
α .
4.6
Therefore
xTαn
i1
xiTαn
i1
x1−gx
i
α
x1−gxα
. 4.7
We now set Φxt St
x0−gx
t
0
St−sfs, xsds
t
0
St−sβ−1
x1−gx−ST
x0−gx
− T
0
ST−sfs, xsds
ds, 4.8 where the fuzzy mappingβ−1satisfies above statements.
Notice thatΦxT x1−gx, which means that the controlutsteers3.9from the origin tox1−gxin timeTprovided that we can obtain a fixed point of the operatorΦ.
H3Assume that the linear system of3.9 f≡0is controllable.
Theorem 4.2. Suppose that hypotheses (H1)–(H3) are satisfied. Then3.9are nonlocal controllable.
Proof. We can easily check thatΦis continuous function fromC0, T: EiNnto itself. By Definitions2.3and2.4, some properties ofdL, and inequalities3.4and3.5, we have the following inequalities. For anyx, y∈C0, T:ENi n,
dL
Φxtα,
Φytα dL
St
x0−gx
t
o
St−sfs, xsds t
0
St−sβ−1
×
x1−gx−ST
x0−gx
− T
0
ST−sfs, xsds
ds α
,
St x0−g
y
t
0
St−sf s, ys
ds t
0
St−sβ−1
×
x1−g y
−ST x0−g
y
− T
0
ST−sf s, ys
ds
ds α
≤dL
Stgxα
, Stg
yα
t
0
dL
St−sfs, xsα
,
St−sf
s, ysα ds
t
0
dL
St−sβ−1gxα ,
St−sβ−1g yα
ds
t
0
dL
St−sβ−1STgxα ,
St−sβ−1STg yα
ds
t
0
dL
St−sβ−1 T
0
ST−sfs, xsds α
,
St−sβ−1 T
0
ST−sf s, ys
ds α
ds max
1≤i≤nSαilt
gilαx−gαil
y,Sαirt
girαx−gαir
y
t
0
max1≤i≤nSαilt−s
filαs, xs−filα
s, ys,Sαirt−s
fαirs, xs−firα
s, ysds
t
0
max1≤i≤n
#Sαilt−s βαil−1
gilαx−gilα y,
Sαirt−s βαir−1
girαx−girα y
$ ds
t
0
max1≤i≤n
#Sαilt−s βαil−1
SαilT
gilαx−gilα y, Sαirt−s
βαir−1 SαirT
gαirx−girα y
$ ds
t
0
max1≤i≤n
Sαilt−s
βilα−1T 0
SαilT−sfilαs, xsds− T
0
SαilT−sfilα s, ys
ds
,
Sαirt−s
βαil−1T 0
SαirT−sfirαs, xsds− T
0
SαirT−sfirα s, ys
ds
ds
≤cmax
1≤i≤ngilαx−gαil
y,girαx−girα
y
c t
0
max1≤i≤nfilαs, xs−filα
s, ys,firαs, xs−firα
s, ysds c
t
0
max1≤i≤ngilαx−gαil
y,girαx−girα yds c2
t
0
max1≤i≤ngilαx−gαil
y,girαx−girα yds c2
t
0
T
0
max1≤i≤nfilαs, xs−filα
s, ys,firαs, xs−firα
s, ysds ds cdL
gxα
, g
yα c
t
0
dL
fs, xsα
, f
s, ysα ds c
t
0
dL gxα
, g
yα dsc2
t
0
dL gxα
, g
yα ds c2
t
0
T
0
dL
fs, xsα ,
f
s, ysα ds ds
≤ch
dL
x·α, y·α
1c t
0
dL
x·α, y·α
ds
ck t
0
dL
xsα, ysα
dsc t
0
T
0
dL
xsα, ysα
ds ds
.
4.9
Therefore DL
Φxt,Φyt sup
0<α≤1dL
Φxtα,
Φytα
≤ch
sup
0<α≤1dL
x·α, y·α
1c t
0
sup
0<α≤1dL
x·α, y·α
ds
ck t
0
sup
0<α≤1dL
xsα, ysα
dsc t
0
T
0
sup
0<α≤1dL
xsα, ysα
ds ds
ch
DL
x·, y·
1c t
0
DL
x·, y·
ds
ck t
0
DL
xs, ys dsc
t
0
T
0
DL
xs, ys ds ds
.
4.10
Hence H1
Φx,Φy sup
0≤t≤T
DL
Φxt,Φyt
≤ch
sup
0≤t≤TDL
x·, y·
1csup
0≤t≤T
t
0
DL
x·, y·
ds
ck
sup
0≤t≤T
t
0
DL
xs, ys
dscsup
0≤t≤T
t
0
T
0
DL
xs, ys ds ds
≤ch H1
x, y
1cT H1
x, y ck%
T H1 x, y
cT2H1 x, y&
c{h1TcT kT1cT}H1
x, y .
4.11
By hypothesisH2,Φis a contraction mapping. Using the Banach fixed point theorem,4.8 has a unique fixed pointx∈C0, T:ENi n.
5. Example
Consider the two semilinear one-dimensional heat equations on a connected domain0,1 for material with memory on EiN, i 1,2, boundary condition xit,0 xit,1 0, i 1,2 and with initial conditionsxi0, zi
'p
k1ckixitk, zi x0izi, wherex0izi ∈ EiN,'p
k1ckixitk, zi gixi,i 1,2. Let xit, zi,i 1,2, be the internal energy and let fit, xit, zi 2txit, zi2,i1,2, be the external heat.
Let
A A1, A2
2 ∂2
∂z21,2 ∂2
∂z22
, ft, xt
f1t, x1t, f2t, x2t
2tx1t, z12,2tx2t, z22 ,
gx
g1x1, g2x2
(p
k1
ck1x1tk, z1, (p k1
ck2x2tk, z2
,
x0 gx
x10 g1x, x20 g2x
, x0 x01, x02 0,0
, Gt−s
e−t−s, e−t−s ,
5.1
then the balance equations become dxt
dt A
xt
t
0
Gt−sxsds
ft, xton EiN2
, x0 gx x0∈
EiN2
.
5.2
Theα-level sets of fuzzy numbers are the following:0 α α−1,1−α,2α α 1,3−αfor allα∈0,1. Thenα-level set offt, xtis
ft, xtα
2tx1t2α
×
2tx2t2α
2α
·t x1t2α
× 2α
·t x2t2α
α1,3−α·t xα1lt2
,
xα1rt2
×α1,3−α·t x2lαt2
,
xα2rt2
α1t xα1lt2
,3−αt
xα1rt2
×
α1t x2lαt2
,3−αt
xα2rt2 .
5.3
Further, we have dL
ft, xtα , f
t, ytα dL
α1t xαilt2
,3−αt
xαirt2 ,
α1t yαilt2
,3−αt
yαirt2 tmax
1≤i≤2
%α1 xαilt2
−
yilαt2,3−α xαirt2
−
yirαt2&
≤T3−αmax
1≤i≤2xαilt−yαiltxαilt yαilt,xirαt−yαirtxαirt yirαt
≤3Txαirt yαirt×max
1≤i≤2xαilt−yilαt,xαirt−yirαt kdL
xtα, ytα
, dL
gx·α
, g
y·α dL
(p
k1
ckxtk α
, (p
k1
ck
ytkα
max
1≤i≤2
(p k1
cki xαiltk
− (p k1
cki
yαiltk ,
(p k1
cki xαirtk
− (p k1
cki
yirαtk
≤
(p k1
ck max
1≤i≤2xαiltk−yilαtk,xαirtk−yirαtk
(p k1
ck
dL
xtkα,
ytkα
≤
(p k1
ck max
k dL
xtkα,
ytkα hdL
x·α, y·α
,
5.4
