Báo cáo hoa học: "Research Article Nonlocal Controllability for the Semilinear Fuzzy Integrodifferential Equations in n-Dimensional Fuzzy Vector Space" potx

Volume 2009, Article ID 734090, 16 pagesdoi:10.1155/2009/734090 Research Article Nonlocal Controllability for the Semilinear Fuzzy Fuzzy Vector Space 1 Department of Mathematics, Dong-A

Volume 2009, Article ID 734090, 16 pages

doi:10.1155/2009/734090

Research Article

Nonlocal Controllability for the Semilinear Fuzzy

Fuzzy Vector Space

1 Department of Mathematics, Dong-A University, Pusan 604-714, South Korea

2 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 ENn 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

Copyrightq 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 Kloeden1 proved the fuzzy optimal control for the following system:

˙xt  atxt  ut, x 0  x0, 1.1

where x· and u· are nonempty compact interval-valued functions on E1 Kwun and Park2 proved the existence of fuzzy optimal control for the nonlinear fuzzy differential

system with nonlocal initial condition in E1

N 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 Muralisankar3 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

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:



x t 

t

0

G t − sxsds



 ft, x  ut, t ∈ J  0, T,

x 0  gt1, t2, , t p , x t m x0∈ E N , m  1, 2, , p,

1.2

where T > 0, A : J → E N is a fuzzy coefficient, EN is the set of all upper semicontinuous

convex normal fuzzy numbers with bounded α-level intervals, f : J×E N → E Nis a nonlinear

continuous function, g : J p × E N → E N is a nonlinear continuous function, Gt is an n × n continuous matrix such that dGtx/dt is continuous for x ∈ E N and t ∈ J with Gt ≤ K,

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’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:



x i t 

t

0

G t − sx i sds



 f i t, x i t  u i t on E i

N ,

x i 0  g i x i   x0i ∈ E i

N i  1, 2, , n,

1.3

where A i : 0, T → E i

N is fuzzy coefficient, Ei

N is the set of all upper semicontinuously

convex fuzzy numbers on R with E i N /  E N j i / j, f i:0, T × E i

N → E i

Nis a nonlinear regular

fuzzy function, g i : E i

N → E i

N is a nonlinear continuous function, Gt is n × n continuous matrix such that dGtx i /dt is continuous for x i ∈ E i

N and t ∈ 0, T with Gt ≤ k, k > 0,

u i:0, T → E i

N is control function and x0i ∈ E i

Nis initial value

2 Preliminaries

A fuzzy set of R n is a function u : R n → 0, 1 For each fuzzy set u, we denote by u α  {x ∈

R n : ux ≥ α} for any α ∈ 0, 1, its α-level set.

Let u, v be fuzzy sets of R n It is well known thatu α  v α for each α ∈ 0, 1 implies

Let E n denote the collection of all fuzzy sets of R n that satisfies the following conditions:

1 u is normal, that is, there exists an x0∈ R n such that ux o  1;

2 u is fuzzy convex, that is, uλx  1 − λy ≥ min{ux, uy} for any x, y ∈ R n,

0≤ λ ≤ 1;

3 ux is upper semicontinuous, that is, ux0 ≥ limk→ ∞u x k  for any x k ∈ R n k 

0, 1, 2, , x k → x0;

4 u0is compact

We call u ∈ E n an n-dimension fuzzy number.

Wang et al 9 defined n-dimensional fuzzy vector space and investigated its properties

For any u i ∈ E, i  1, 2, , n, we call the ordered one-dimension fuzzy number class

n-dimension fuzzy vector, denote it asu1, u2, , u n , and call the collection of all n-dimension

fuzzy vectorsi.e., the Cartesian productE × E × · · · × E n-dimensional fuzzy vector space,

and denote it asE n

byn

i1u α

il , u α

ir , that is, u α

1l , u α 1r ×u α 2l , u α 2r ×· · ·×u α

nl , u α

nr  for every α ∈ 0, 1, where u α

il , u α

ir ∈ R with u α il ≤ u α

ir 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 LE n

i1u α

il , u α

ir  α ∈ 0, 1, there exists a

il , u α ir  (i  1, 2, , n and α ∈ 0, 1).

il , u α

ir i  1, 2, , n and α ∈

0, 1, there exists a unique u ∈ LE n  such that u αn

i1u α

il , u α

ir  α ∈ 0, 1.

vectors can represent each other, so LE n  and E n may be regarded as identity If

then we denote u  u1, u2, , u n

LetE i

Nn  E1

N × E2

N × · · · × E n

N , E i

N i  1, 2, ×, n be fuzzy subset of R Then E i

Nn ⊆

E n

Nnis defined by

D L u, v  sup

0<α≤1 d L

u α , v α

 sup

0<α≤1

max

1≤i≤n

il − v α

il , u α

ir − v α ir

2.1

for any u, v ∈ E i

Nn , which satisfies d L u  w, v  w  d L u, v.

Nn, then

H1u, v  sup

Definition 2.5see 9 The derivative x t of a fuzzy process x ∈ E i

Nnis defined by



i1



x α il

t,x α ir

provided that the equation defines a fuzzy x t ∈ E i

Nn

b x tdt, a, b ∈ 0, T is defined by

a

b

x tdt

α

n

i1

a

b

x α il tdt,

a

b

x α ir 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 for1.3 u ≡ 0

We define

,



,

3.1

Then

Instead of 1.3, we consider the following fuzzy integrodifferential equations in

E i

Nn:



x t 

t

0

G t − sxsds



 ft, xt  ut onE N i n

x 0  gx  x0∈E i Nn

3.3

with fuzzy coefficient A : 0, T → Ei

Nn , initial value x0 ∈ E i

Nn , and u : 0, T → E i

Nn

is a control function Given nonlinear regular fuzzy function f : 0, T × E i

Nn → E i

Nn

satisfies a global Lipschitz condition, that is, there exists a finite k > 0 such that



f s, xsα

≤ kd L



xs α ,

3.4

for all xs, ys ∈ E i

Nn The nonlinear function g : E i

Nn → E i

Nnis a continuous function and satisfies the Lipschitz condition

g x·α

≤ hd L



x· α ,

3.5

for all x·, y· ∈ E i

Nn , h is a finite positive constant.

Nn with α-level set xt α Πn

i1x iα

Πn

i1x α

il , x α

ir is a fuzzy solution of 3.3 without nonhomogeneous term if and only if



x α il

t  min





x α ik t 

t

0

G t − sx α

ik sds



: j, k  l, r



,



ir



t  max



ij t



ik t 

t

0

G t − sx α

ik sds



: j, k  l, r



,

il 0  g α il



il



 x α

0il , x α

ir 0  g α

ir



ir



 x α

0ir , i  1, 2, , n.

3.6

For the sequel, we need the following assumptions

H1 St is a fuzzy number satisfying, for y ∈ E i

Nn,d/dt Sty ∈ C1I : E i

Nn

C I : E i

Nn, the equation

d

dt S ty  A



S ty 

t

0

G t − sSsy ds



 StAy 

t

0

S t − sAGsy ds, t ∈ I,

3.7

where

St αn

i1

S i t αn

i1



S α il t, S α

ir t, 3.8

and S α ij t j  l, r is continuous with |S α

ij t| ≤ c, c > 0, for all t ∈ I  0, T.

H2 c{h1  T  cT  kT1  cT} < 1.

In view ofDefinition 3.1andH1, 3.3 can be expressed as

x t  Stx0− gx

t

0

S t − sf s, xs  usds,

x 0  gx  x0.

3.9

Theorem 3.2 Let T > 0 If hypotheses (H1)-(H2) are hold, then for every x0 ∈ E i

Nn ,3.9 (u ≡ 0

Nn .

Proof For each x t ∈ E i

Nn and t ∈ 0, T, define G0x t ∈ E i

Nnby

G0x t  Stx0− gx

t

0

S t − sfs, xsds. 3.10

Thus, G0x : 0, T → E i

Nn is continuous, so G0 is a mapping from C0, T : E i

Nn into itself By Definitions2.3and2.4, some properties of dL, and inequalities3.4 and 3.5, we

have following inequalities For x, y ∈ C0, T : E i

Nn,



G0x t α ,

tα

 d L



S tx0− gx

t

0

S t − sfs, xsds

α

,





t

0

S t − sfs, y sds

α

 d L



−Stgx 

t

0

S t − sfs, xsds

α

,



−Stgy



t

0

S t − sfs, y sds

α

≤ d L



S tgxα



t

0

S t − sfs, xsα

S t − sfs, y sα

ds

 max

1≤i≤n S α

il tg α

il x − g α

il



ir tg α

ir x − g α

ir



y



t

0

max

1≤i≤n S α il t − sf il α s, xs − f α

il



s, y s , S α ir t − sf ir α s, xs − f α

ir



≤ c max

1≤i≤n g il α x − g α

il



y , g ir α x − g α

ir



y

 c

t

0

max

1≤i≤n f il α s, xs − f α

il



s, y s , f ir α s, xs − f α

ir



 cd L



 c

t

0

f s, xsα

ds

≤ chd L



x· α ,

 ck

t

0



xs α ,

ds.

3.11



G0x t,G0y

t

 sup

0<α≤1 d L



G0x t α ,

tα

≤ ch sup

0<α≤1 d L



x· α ,

 ck sup

0<α≤1

t

0



xs α ,

ds

≤ chD L



x ·, y· ck

t

0

x s, ysds.

3.12

Hence



 sup

0≤t≤TD L



G0x t,G0y

t

≤ chsup

0≤t≤TD L

x ·, y· ck sup

0≤t≤T

t

0

x s, ysds

≤ ch H1



 ckT H1



 ch  kT H1



.

3.13

By hypothesisH2, G0is a contraction mapping

Using the Banach fixed point theorem,3.9 have a unique fixed point x ∈ C0, T :

E i

Nn

4 Controllability

In this section, we show the nonlocal controllability for the control system1.3

fuzzy solution xt for 3.9 as xT  x1−gx i.e., xT α  x1−gx α  where x1∈ E i

Nn

is target set

Define the fuzzy mapping β :  P R n  → E i

Nnby

β α v 

⎧

⎪

⎪

T

0

S α T − svsds, v ⊂ Γ u ,

0, otherwise,

4.1

whereΓu is closed support of u Then there exists

β i: P R −→ E i

N i  1, 2, , n 4.2 such that

β α

i v i 

⎧

⎪

⎪

T

0

S α i T − sv i sds, v i s ⊂ Γ u i ,

0, otherwise.

4.3

Then β ij α j  l, r exists such that

β α

il v il 

T

0

S α il T − sv il sds, v il s ∈u α il s, u1

i



,

β α

ir v ir 

T

0

S α ir T − sv ir sds, v ir s ∈u1i , u α ir s.

4.4

We assume that β α

il ,  β α

irare bijective mappings

We can introduce α-level set of us of 3.4-3.5

us αn

i1

u i s α

n

i1



u α il s, u α

ir s

n

i1



β α il

−1

il − g α il



x il α

− S α

il Tx α0

il − g α il



−

T

0

S α il T − sf α

il



,



β α ir

−1

ir − g α ir



− S α

ir Tx0α

ir − g α ir



−

T

0

S α ir T − sf α

ir





.

4.5

Then substituting this expression into3.9 yields α-level of xT

For each i  1, 2, , n,

x i T α



S α il Tx α0

il − g α il





T

0

S α il T − sf α

il





T

0

S α il T − sβ α

il

−1

il − g α il



x il α

− S α

il Tx α0

il − g α il



−

T

0

S α il T − sf α

il



ds,

ir − g α ir





T

0

S α ir T − sf α

ir





T

0

S α ir T − sβ α

ir

−1

ir − g α ir



− S α

ir Tx0α ir − g α

ir



−

T

0

S α ir T − sf α

ir



ds



x1− gxα

il ,

x1− gxα

ir



x1− gx

i

α

.

4.6

Therefore

xT αn

i1

x i T αn

i1



x1− gx

i

α

x1− gxα 4.7

We now set

Φxt  Stx0− gx

t

0

S t − sfs, xsds



t

0

S t − sβ−1

x1− gx − STx0− gx−

T

0

S T − sfs, xsds

ds,

4.8

where the fuzzy mapping β−1satisfies above statements

Notice thatΦxT  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

Proof We can easily check that Φ is continuous function from C0, T : E i

Nn to itself By Definitions2.3and2.4, some properties of dL, and inequalities3.4 and 3.5, we have the

following inequalities For any x, y ∈ C0, T : E i

Nn,



Φxt α ,

Φytα

 d L



S tx0− gx

t

o

S t − sfs, xsds 

t

0

S t − sβ−1

×

x1− gx − STx0− gx−

T

0

S T − sfs, xsds

ds

α

,





t

0

S t − sfs, y sds

t

0

S t − sβ−1

×

− STx0− gy

−

T

0

S T − sfs, y sds

ds

α

≤ d L



S tgxα



t

0

S t − sfs, xsα

S t − sfs, y sα

ds



t

0

S t − sβ−1g xα ,

S t − sβ−1g

ds



t

0

S t − sβ−1S Tgxα ,

S t − sβ−1S Tgyα

ds



t

0



S t − sβ−1T

0

S T − sfs, xsds

α

,



S t − sβ−1T

0

S T − sfs, y sds

α

ds

 max

1≤i≤n S α il tg il α x − g α

il



y , S α ir tg ir α x − g α

ir



y



t

0

max

1≤i≤n S α il t − sf il α s, xs − f α

il



s, y s , S α ir t − sfα ir s, xs − f α

ir





t

0

max

1≤i≤n

α

il t − sβ α

il

−1

g il α x − g α

il



y α ir t − sβ α

ir

−1

g ir α x − g α

ir



y

$

ds



t

0

max

1≤i≤n

α

il t − sβ α

il

−1

S α il Tg il α x − g α

il



y

α

ir t − sβ α

ir

−1

S α ir Tg α ir x − g α

ir



y

$

ds



t

0

max

1≤i≤n S α il t − sβ α

il

−1T

0

S α il T − sf α

il s, xsds −

T

0

S α il T − sf α

il



S α ir t − sβ α

il

−1T

0

S α ir T −sf α

ir s, xsds −

T

0

S α ir T −sf α

ir





ds

≤ c max

1≤i≤n g il α x − g α

il



y , g ir α x − g α

ir



y

 c

t

0

max

1≤i≤n f α

il s, xs − f α

il



ir s, xs − f α

ir



 c

t

0

max

1≤i≤n g il α x − g α

il



y , g ir α x − g α

ir



 c2

t

0

max

1≤i≤n g il α x − g α

il



y , g ir α x − g α

ir



 c2

t

0

T

0

max

1≤i≤n f il α s, xs − f α

il



s, y s , f ir α s, xs − f α

ir



 cd L



 c

t

0

f s, xsα

ds

 c

t

0

t

0

ds

 c2

t

0

T

0

f s, xsα

ds ds

≤ ch





x· α ,

 1  c

t

0



x· α ,

ds



 ck

t

0



xs α ,

t

0

T

0



xs α ,

ds ds



.

4.9 Therefore

Φxt, Φyt

 sup

0<α≤1 d L



Φxt α ,

Φytα

≤ ch



sup

0<α≤1 d L



x· α ,

 1  c

t

0

sup

0<α≤1 d L



x· α ,

ds



 ck

t

0

sup

0<α≤1 d L



xs α ,

t

0

T

0

sup

0<α≤1 d L



xs α ,

ds ds



 ch





x ·, y· 1  c

t

0



x ·, y·ds



 ck

t

0



x s, ysds  c

t

0

T

0



x s, ysds ds



.

4.10

Φx, Φy sup

0≤t≤T

Φxt, Φyt

≤ ch

 sup

0≤t≤TD L



x ·, y· 1  csup

0≤t≤T

t

0



x ·, y·ds



 ck

 sup

0≤t≤T

t

0



x s, ysds  c sup

0≤t≤T

t

0

T

0



x s, ysds ds



≤ chH1

 1  cT H1



 ck%T H1

 cT2H1

x, y&

 c{h1  T  cT  kT1  cT}H1



.

4.11

By hypothesisH2, Φ is a contraction mapping Using the Banach fixed point theorem, 4.8

has a unique fixed point x ∈ C0, T : E i

Nn

5 Example

Consider the two semilinear one-dimensional heat equations on a connected domain0, 1 for material with memory on E i

N , i  1, 2, boundary condition x i t, 0  x i t, 1  0,

i  1, 2 and with initial conditions x i 0, z i 'p

k1c ki x i t k , z i   x0i z i , where x0i z i ∈

k1c ki x i t k , z i   g i x i , i  1, 2 Let x i t, z i , i  1, 2, be the internal energy and let

f i t, x i t, z i   2tx i t, z i2, i  1, 2, be the external heat.

Let

2 ∂2

1

2

2

,

f t, xt f1t, x1t, f2t, x2t2tx1t, z12, 2tx2t, z22

,

g x g1x1, g2x2

(p

k1

c k1x1t k , z1,

p

(

k1

c k2x2t k , z2

,

x 0  gx x10  g1x, x20  g2x, x0 x0 1, x02 0, 0,

G t − s e −t−s , e −t−s

,

5.1

then the balance equations become



x t 

t

0

G t − sxsds



 ft, xt onE i N2

,

x 0  gx  x0∈E i

N

2

.

5.2

.

4.11

By hypothesisH2, Φ is a contraction mapping Using the Banach fixed point theorem, 4.8

has a unique fixed point x ∈ C0, T : E i...

Nn

5 Example

Consider the two semilinear one-dimensional heat equations on a connected domain0, 1 for material with memory on E i

N... e −t−s , e −t−s

,

5.1

then the balance equations become



x t 

t

0