Controllability of impulsive differential systems with nonlocal conditions

Applied Mathematics and Computation 217 (2011) 6981–6989

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Controllability of impulsive differential systems with nonlocal conditions Shaochun Ji a,b,⇑, Gang Li a, Min Wang c a b c

School of Mathematical Science,Yangzhou University, Yangzhou, Jiangsu 225002, PR China Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, PR China Library, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, PR China

a r t i c l e

i n f o

Keywords: Controllability Impulsive functional differential systems Nonlocal conditions Measure of noncompactness Mild solutions

a b s t r a c t The paper is concerned with the controllability of impulsive functional differential equations with nonlocal conditions. Using the measure of noncompactness and Mönch fixedpoint theorem, we establish some sufficient conditions for controllability. Firstly, we require the equicontinuity of evolution system, and next we only suppose that the evolution system is strongly continuous. Since we do not assume that the evolution system generates a compact semigroup, our theorems extend some analogous results of (impulsive) control systems. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we consider the following impulsive functional differential systems:

x0 ðtÞ ¼ AðtÞxðtÞ þ f ðt; xðtÞÞ þ ðBuÞðtÞ; a:e: on ½0; b;

ð1:1Þ

Dxðt i Þ ¼ xðtþi Þ  xðti Þ ¼ Ii ðxðti ÞÞ; i ¼ 1; . . . ; s; xð0Þ þ MðxÞ ¼ x0 ;

ð1:2Þ ð1:3Þ

where A(t) is a family of linear operators which generates an evolution operator

U : M ¼ fðt; sÞ 2 ½0; b  ½0; b : 0 6 s 6 t 6 bg ! LðXÞ; here, X is a Banach space, L(X) is the space of all bounded linear operators in X; f : [0, b]  X ? X; 0 < t1 <    < ts < ts+1 = b; Ii : X ? X, i = 1, . . . , s are impulsive functions; M : PC([0, b]; X) ? X; B is a bounded linear operator from a Banach space V to X and the control function u( ) is given in L2([0, b],V). Controllability for differential systems in Banach spaces has been studied by many authors [2,4,9] and the references therein. Benchohra and Ntouyas [4], using the Martelli fixed-point theorem, studied the controllability of second-order differential inclusions in Banach spaces. Guo et al. [9] proved the controllability of impulsive evolution inclusions with nonlocal conditions. The impulsive differential systems can be used to model processes which are subjected to abrupt changes. The study of dynamical systems with impulsive effects has been an object of intensive investigations [8,14,15]. The semilinear nonlocal initial problem was first discussed by Byszewski [5,6] and the importance of the problem consists in the fact that it is more general and has better effect than the classical initial conditions. Therefore it has been studied extensively under various conditions on A(or A(t)) and f by several authors [1,11,13,17].

⇑ Corresponding author at: School of Mathematical Science,Yangzhou University, Yangzhou, Jiangsu 225002, PR China. E-mail addresses: [email protected] (S. Ji), [email protected] (G. Li), [email protected] (M. Wang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.01.107

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Recently, Hernández and O’Regan [10] point out that some papers on exact controllability of abstract control system contain a similar technical error when the compactness of semigroup and other hypotheses are satisfied, that is, in this case the application of controllability results are restricted to finite dimensional space. The goal of this paper is to find conditions guaranteeing the controllability of impulsive differential systems when the Banach space is nonseparable and evolution systems U(t, s) is not compact, by means of Mönch fixed-point theorem and the measure of noncompactness. Since the method used in this paper is also available for evolution inclusions in Banach space, we can improve the corresponding results in [2,4]. 2. Preliminaries Let (X,k  k) be a real Banach space. We denote by C([0, b]; X) the space of X-valued continuous functions on [0, b] with the norm kxk = sup{kx(t)k, t 2 [0, b]} and by L1([0, b]; X) the space of X-valued Bochner integrable functions on [0, b] with the norm Rb kf kL1 ¼ 0 kf ðtÞk dt. For the sake of simplicity, we put J = [0, b]; J0 = [0, t1]; Ji = (ti, ti+1], i = 1, . . . , s. In order to define the mild solution of problem (1.1)–(1.3), we introduce the set PCð½0; b; XÞ ¼ fu : ½0; b ! X : u is continuous on J i ; i ¼ 0; 1; . . . ; s and the right limit uðt þ i Þ exists; i ¼ 1; . . . ; sg. It is easy to verify that PC([0, b]; X) is a Banach space with the norm kukPC = sup{ku(t)k, t 2 [0, b]}. Let us recall the following definitions. Definition 2.1. Let E+ be the positive cone of an order Banach space (E, 6). A function U defined on the set of all bounded subsets of the Banach space X with values in E+ is called a measure of noncompactness (MNC) on X if UðcoXÞ ¼ UðXÞ for all bounded subsets X  X, where coX stands for the closed convex hull of X. The MNC U is said: (1) monotone if for all bounded subsets X1, X2 of X we have: (X1 # X2) ) (U(X1) 6 U(X2)); (2) nonsingular if U({a} [ X) = U(X) for every a 2 X, X  X; (3) regular if U(X) = 0 if and only if X is relatively compact in X. One of the most important examples of MNC is the noncompactness measure of Hausdorff b defined on each bounded subset X of X by

bðXÞ ¼ inf fe > 0; X has a finite

e  net in X g:

It is well known that MNC b enjoys the above properties and other properties (see [3,12]): for all bounded subset X, X1, X2 of X, (4) b(X1 + X2) 6 b(X1) + b(X2), where X1 + X2 = {x + y : x 2 X1, y 2 X2}; (5) b(X1 [ X2) 6 max{b(X1), b(X2)}; (6) b(kX) 6 jkjb(X) for any k 2 R; (7) If the map Q : D(Q) # X ? Z is Lipschitz continuous with constant k, then bZ(QX) 6 kb(X) for any bounded subset X # D(Q), where Z is a Banach space. Definition 2.2. A function x() 2 PC([0, b]; X) is a mild solution of (1.1)–(1.3) if

xðtÞ ¼ Uðt; 0Þxð0Þ þ

Z

t

Uðt; sÞðf þ BuÞðsÞ ds þ

0

X

Uðt; ti ÞIi ðxðti ÞÞ;

0
for all t 2 [0, b], where x(0) + M(x) = x0. Definition 2.3. The system (1.1)–(1.3) is said to be nonlocally controllable on J if, for every x0, x1 2 X, there exists a control u 2 L2(J, V) such that the mild solution x() of (1.1)–(1.3) satisfies x(b) + M(x) = x1. A two parameter family of bounded linear operators U(t, s), 0 6 s 6 t 6 b on X is called an evolution system if the following two conditions are satisfied: (i) U(s, s) = I, U(t, r)U(r, s) = U(t, s) for 0 6 s 6 r 6 t 6 b; (ii) (t, s) ? U(t, s) is strongly continuous for 0 6 s 6 t 6 b. Since the evolution system U(t, s) is strongly continuous on the compact set M = J  J, then there exists LU > 0 such that kU(t, s)k 6 LU for any (t, s) 2 M. More details about evolution system can be found in [18]. 1 Definition 2.4. A countable set ffn gþ1 n¼1  L ð½0; b; XÞ is said to be semicompact if:

 the sequence ffn ðtÞgþ1 n¼1 is relatively compact in X for a.a. t 2 [0, b];  there is a function l 2 L1([0, b]; R+) satisfying supnP1kfn(t)k 6 l(t) for a.e. t 2 [0, b].

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The following interchange results about b-estimation are shown in [12] Theorems 4.2.2 and 5.1.1, respectively. 1 + 1 + Lemma 2.1 [12]. Let ffn gþ1 n¼1 be a sequence of functions in L ([0, b]; R ). Assume that there exist l,g 2 L ([0, b]; R ) satisfying

sup kfn ðtÞk 6 lðtÞ and bðffn ðtÞgþ1 n¼1 Þ 6 gðtÞ a:e: t 2 ½0; b: nP1

Then for all t 2 [0, b], we have

Z t  Z t b Uðt; sÞfn ðsÞds : n P 1 6 2LU gðsÞds: 0

0

Rt þ1 1 Lemma 2.2 [12]. Let ðGf ÞðtÞ ¼ 0 Uðt; sÞf ðsÞds. If ffn gþ1 n¼1  L ð½0; b; XÞ is semicompact, then the set fGfn gn¼1 is relatively compact in C([0, b]; X) and moreover, if fn N f0, then for all t 2 [0, b], (Gfn)(t) ? (Gf0)(t) as n ? + 1. The following fixed-point theorem, a nonlinear alternative of Mönch fixed-point theorem, plays a key role in our proof of controllability (see Theorem 2.2 in [16]). Lemma 2.3. Let D be a closed convex subset of a Banach space X and 0 2 D. Assume that F : D ? X is a continuous map which satisfies Mönch’s condition, that is,

M # D is countable; M # coðf0g

[

FðMÞÞ ) M is compact:

Then, there exists x 2 D with x = F(x). 3. Main results We first give the following hypotheses: (H1) A(t) is a family of linear operators, A(t) : D(A) ? X, D(A) not depending on t and dense subset of X, generating an equicontinuous evolution system {U(t, s) : (t, s) 2 M}, i.e., (t, s) ? {U(t, s)x : x 2 B} is equicontinuous for t > 0 and for all bounded subsets B. (H2) The function f : [0, b]  X ? X satisfies: (i) for a.e. t 2 [0, b], the function f(t, ) : X ? X is continuous and for all x 2 X, the function f(, x) : [0, b] ? X is measurable; (ii) there exists a function m 2 L1([0, b]; R+) and a nondecreasing continuous function X : R+ ? R+ such that

kf ðt; xÞk 6 mðtÞXðkxkÞ;

x 2 X; t 2 ½0; b

and

lim inf

XðnÞ n

n!þ1

¼ 0;

(iii) there exists h 2 L1([0, b]; R+) such that , for any bounded subset D  X,

bðf ðt; DÞÞ 6 hðtÞbðDÞ for a.e. t 2 [0, b], where b is the Hausdorff MNC. (H3) M : PC(J, X) ? X is a continuous compact operator such that

lim

kykPC !þ1

kMðyÞk ¼ 0: kykPC

(H4) The linear operator W : L2(J, V) ? X is defined by

Wu ¼

Z

b

Uðb; sÞBuðsÞds

0

such that: (i) W has an invertible operator W1 which take values in L2(J, V)/kerW and there exist positive constants LB and LW such that kBk 6 LB and kW1k 6 LW; (ii) there is KW 2 L1(J, R+) such that, for any bounded set Q  X,

bððW 1 Q ÞðtÞÞ 6 K W ðtÞbðQ Þ: (H5) Let Ii : X ? X, i = 1, . . . , s be a continuous operator such that: (i) there are nondecreasing functions li : R+ ? R+, i = 1, . . . , s, such that

kIi ðxÞk 6 li ðkxkÞ and lim inf n!þ1

li ðnÞ ¼ 0; n

i ¼ 1; . . . ; s;

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(ii) there exist constants Ki P 0, such that

bðIi ðDÞÞ 6 K i bðDÞ;

i ¼ 1; . . . ; s;

for every bounded subset D  X. (H6) The following estimation holds true:

L ¼ ðLU þ 2L2U LB kK W kL1 Þ

s X

K i þ ð2LU þ 4L2U LB kK W kL1 ÞkhkL1 < 1;

i¼1

where LU = sup{kU(t, s)k, (t, s) 2 M}. Theorem 3.1. Assume that the hypotheses (H1)–(H6) are satisfied, then the impulsive differential system 1.1, 1.2, 1.3 is nonlocally controllable on J. Proof. Using hypothesis (H4)(i), for every x 2 PC(J, X), define the control

ux ðtÞ ¼ W 1 ½x1  MðxÞ  Uðb; 0Þðx0  MðxÞÞ 

Z

b

Uðb; sÞf ðs; xðsÞÞds 

0

s X

Uðt; t i ÞIi ðxðti ÞÞðtÞ:

i¼1

We shall show that, when using this control, the operator, defined by

ðGxÞðtÞ ¼ Uðt; 0Þðx0  MðxÞÞ þ cðf þ Bux ÞðtÞ þ

X

Uðt; ti ÞIi ðxðti ÞÞ;

ð3:1Þ

0
where c(f + Bux)(t) 2 C(J, X) is defined by

cðf þ Bux ÞðtÞ ¼

Z

t

Uðt; sÞðf þ Bux ÞðsÞds; 0

has a fixed point. This fixed point is then a solution of the system (1.1)–(1.3). Clearly, x1  M(x) = G(x)(b), which implies that the system (1.1)–(1.3) is controllable. We define G = G1 + G2, where

ðG1 xÞðtÞ ¼ Uðt; 0Þðx0  MðxÞÞ þ

X

Uðt; ti ÞIi ðxðti ÞÞ; ðG2 xÞðtÞ ¼ cðf þ Bux ÞðtÞ;

0
for all t 2 [0, b]. Subsequently, we will prove that G has a fixed point by using Lemma 2.3. Step 1. The operator G is continuous on PC([0, b]; X). For this purpose, we assume that xn ? x in PC([0, b]; X). Then by hypothesis (H3) and (H5), we have that

kG1 xn  G1 xkPC 6 LU kMðxn Þ  MðxÞk þ LU

s X

kIi ðxn ðt i ÞÞ  Ii ðxðti ÞÞk

ð3:2Þ

i¼1

Note that

kG2 xn  G2 xkC 6 LU 6 LU

Z

b

kf ðs; xn ðsÞÞ  f ðs; xðsÞÞkds þ LU LB

0

Z

0

Z

b

kuxn ðsÞ  ux ðsÞkds

0 b

1

kf ðs; xn ðsÞÞ  f ðs; xðsÞÞkds þ LU LB b2 kuxn  ux kL2 ;

kuxn  ux kL2 6 LW ½kMðxn Þ  MðxÞk þ LU ðkMðxn Þ  MðxÞkÞ þ LU

Z

b

kf ðs; xn ðsÞ  f ðs; xðsÞÞÞkds þ LU

0

ð3:3Þ s X

kIi ðxn ðti ÞÞ

i¼1

 Ii ðxðti ÞÞk:

ð3:4Þ

Observing (3.2)–(3.4), by hypotheses (H2), (H3), (H5) and domination convergence theorem, we have that

kGxn  GxkPC 6 kG1 xn  G1 xkPC þ kG2 xn  G2 xkC ! 0;

as n ! þ1;

i.e., G is continuous. Step 2. There exists a positive integer n0 P 1 such that GðBn0 Þ # Bn0 , where Bn0 ¼ fx 2 PCðJ; XÞ : kxk 6 n0 g. Suppose the contrary. Then we can find xn 2 PC(J, X), yn = Gxn 2 PC(J, X), such that

kxn kPC 6 n and kyn kPC > n; for every n P 1. Now we have that

yn ðtÞ ¼ Uðt; 0Þðx0  Mðxn ÞÞ þ cðfn þ Buxn ÞðtÞ þ

X 0
Uðt; t i ÞIi ðxn ðt i ÞÞ:

S. Ji et al. / Applied Mathematics and Computation 217 (2011) 6981–6989

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So we get that

n < kyn kPC 6 LU ðkx0 k þ kMðxn ÞkÞ þ kcðfn ÞkC þ kcðBuxn ÞkC þ LU

s X

li ðkxn kPC Þ:

ð3:5Þ

i¼1

Note that, by (H2)(ii), (H4)(i) and (H5)(i),

kcðfn ÞkC 6 sup

Z

t2J

Z

t

kUðt; sÞkkf ðs; xn ðsÞÞkds 6 LU

0

mðsÞXðkxkPC Þds ¼ LU XðkxkPC ÞkmkL1 ;

0

Z

kcðBuxn ÞkC 6 sup

b

t2J

t

kUðt; sÞBuxn ðsÞkds 6 LU LB

Z

0

0

b

1

kuxn ðsÞkds 6 LU LB b2 kuxn kL2 ;

kuxn kL2 ¼ kW 1 ½x1  Mðxn Þ  Uðb; 0Þðx0  Mðxn ÞÞ  6 LW ½kx1 k þ LU kx0 k þ ð1 þ LU ÞkMðxn Þk þ LU

Z

b

Uðb; sÞf ðs; xn ðsÞds  0

Z

ð3:7Þ

s X

Uðt; t i ÞIi ðxn ðti ÞÞkL2

i¼1

b

0

ð3:6Þ

mðsÞXðkxn kPC Þds þ LU

s X

li ðkxn kPC Þ:

ð3:8Þ

i¼1

Hence, by (3.5)–(3.8), we have that 1

1

n <ðLU þ L2U LB b2 LW Þkx0 k þ LU LB b2 LW kx1 k 1

1

þ ðLU þ LU LB b2 LW þ L2U LB b2 LW ÞkMðxn Þk 1

þ ðLU kmkL1 þ L2U LB b2 LW kmkL1 ÞXðkxn kPC Þ 1

þ ðLU þ L2U LB b2 LW Þ

s X

li ðkxn kPC Þ;

i¼1

which implies that

1<

s X 1 li ðnÞ; ½C 1 þ C 2 kMðxn Þk þ C 3 XðnÞ þ C 4 n i¼1

ð3:9Þ

where 1

1

C 1 ¼ ðLU þ L2U LB b2 LW Þkx0 k þ LU LB b2 LW kx1 k; 1

1

C 2 ¼ LU þ LU LB b2 LW þ L2U LB b2 LW ; 1

C 3 ¼ LU kmkL1 þ L2U LB b2 LW kmkL1 ; 1

C 4 ¼ LU þ L2U LB b2 LW : Observing (H2)(ii), (H3) and (H5)(i), by passing to the limit as n ? + 1 in (3.9), we get 1 6 0, which is a contradiction. Thus we deduce that there is n0 P 1 such that GðBn0 Þ # Bn0 . Step 3. The Mönch’s condition holds. S Suppose D # Bn0 is countable and D # coðf0g GðDÞÞ, we shall show that b(D) = 0, where b is the Hausdorff MNC. 1 Without loss of generality, we may suppose that D ¼ fxn g1 n¼1 . If we can show that fGxn gn¼1 is equicontinuous on Ji, i = 0, . . . , s, S 0 00 0 00 then D # coðf0g GðDÞÞ is also equicontinuous on every Ji. To this end, let y 2 G(D), and t , t 2 Ji, t 6 t , there is x 2 D such that

 Z 00 Z t0   t   00 0 kyðt Þ  yðt Þk ¼ k½Uðt ; 0Þ  Uðt ; 0Þðx0  MðxÞÞk þ  Uðt ; sÞðf þ Bux ÞðsÞds  Uðt ; sÞðf þ Bux ÞðsÞds   0 0 Z t0 6 k½Uðt 00 ; 0Þ  Uðt0 ; 0Þðx0  MðxÞÞk þ k½Uðt 00 ; sÞ  Uðt 0 ; sÞðf þ Bux ÞðsÞkds 00

0

00

þ

0

0

Z

t 00 00

t0

kUðt ; sÞk  kðf þ Bux ÞðsÞkds:

ð3:10Þ

By the equicontinuity property of U(, s) and the absolute continuity of the Lebesgue integral, we can see that the right-hand 00 0 side of the inequality (3.10) tends to zero independent of y as t ? t . Therefore, G(D) is equicontinuous on every Ji. Now we shall show that (GD)(t) is relatively compact in X for each t 2 J. From the compactness of M() and (H5)(ii), we have that

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bðfðG1 xn ÞðtÞg1 n¼1 Þ

6 bðfUðt; 0Þðx0 

Mðxn ÞÞg1 n¼1 Þ

0( )1 1 s X X A 6 LU þ b@ Uðt; t i ÞIi ðxn ðt i ÞÞ bðIi ðDðt i ÞÞÞ 0
6 LU

s X

i¼1

n¼1

K i bðDðti ÞÞ

ð3:11Þ

i¼1

for t 2 [0, b]. By Lemma 2.1, from (H2)(ii), (H4)(ii) and (H5)(ii), we have that

" bV ðfuxn ðsÞg1 n¼1 Þ

(Z

6 K W ðsÞ b

)1 !

b

Uðb; sÞf ðs; xn ðsÞÞds

þb

0

" 6 K W ðsÞ 2LU

Z

hðsÞbðDðsÞÞds þ LU

0

s X

)1 !# Uðt; t i ÞIi ðxn ðt i ÞÞ

i¼1

n¼1

b

( s X

n¼1

#

K i bðDðti ÞÞ :

ð3:12Þ

i¼1

Then this implied that

  b fðG2 xn ÞðtÞg1 n¼1 6 b

Z

6 2LU

Z

1 

t

0

6 2LU

0

n¼1

b

hðsÞbðDðsÞÞds þ 2LU LB

0

Z

Z t 1  þb Uðt; sÞBuxn ðsÞds

Uðt; sÞf ðs; xn ðsÞÞds Z

b 0

b

hðsÞbðDðsÞÞds þ

0

þ 2L2U LB

Z

b

K W ðgÞdg

0

4L2U LB

s X

Z 0

n¼1

bV ðfuxn ðsÞg1 n¼1 Þds b

! Z K W ðsÞds

!

b

hðsÞbðDðsÞÞds

0

!

K i bðDðti ÞÞ ;

ð3:13Þ

i¼1

for each t 2 [0, b]. From (3.11) and (3.13), we obtain that

bððGDÞðtÞÞ 6 bððG1 DÞðtÞÞ þ bððG2 DÞðtÞÞ 6 LU

s X

K i bðDðt i ÞÞ þ ð2LU þ 4L2U LB

i¼1



Z

b

K W ðgÞdg

0

s X

Z

!

b

K W ðsÞdsÞ

0

Z 0

b

hðsÞbðDðsÞÞds þ 2L2U LB

K i bðDðt i ÞÞ ;

ð3:14Þ

i¼1

for each t 2 [0, b]. Since GD is equicontinuous on every Ji, by Proposition 7.3 of [7], we find that

bðGDÞ ¼ max bððGDÞðJ i ÞÞ ¼ max max bððGDÞðtÞÞ: 06i6s

06i6s

t2J i

Then, according to inequality (3.14), we have that

" bðGDÞ 6 LU "

s X

# K i þ ð2LU þ 4L2U LB kK W kL1 ÞkhkL1 bðDÞ þ ð2L2U LB kK W kL1

i¼1

¼ ðLU þ 2L2U LB kK W kL1 Þ

s X

#

s X

K i ÞbðDÞ

i¼1

K i þ ð2LU þ 4L2U LB kK W kL1 ÞkhkL1 bðDÞ

i¼1

¼ LbðDÞ; where L is defined in (H6). Thus, from the Mönch’s condition, we get that

bðDÞ 6 bðcoðf0g

[

GðDÞÞ ¼ bðGðDÞÞ 6 LbðDÞ;

which implies that b(D) = 0, since hypothesis (H6) holds. So we have that D is relatively compact. h Finally, due to Lemma 2.3, G has at least a fixed point and thus the system (1.1)–(1.3) is nonlocally controllable on [0, b]. Remark 3.1. In Theorem 3.1 we require f to satisfy a compactness condition (H2)(iii), but not require the compactness of evolution system U(t, s). Note that if f is compact or Lipschitz continuous, then condition (H2)(iii) is satisfied. Therefore, our work extends some previous results, where the compactness of T(t) and f, or the Lipschitz continuity of f are needed.

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In the following, by using another MNC, we will prove the result of Theorem 3.1 in the case there is no equicontinuity of the evolution system U(t, s) and hypothesis (H6). Then the result we get is more general than most of the previous control0 lability results and it is interesting. Instead of (H5), we give the hypothesis (H5 ): 0 (H5 ) Let Ii : X ? X, i = 1, . . . , s be a continuous compact operator such that there are nondecreasing functions li : R+ ? R+, i = 1, . . . , s, satisfying

kIi ðxÞ 6 li ðkxkÞ and

lim inf n!þ1

li ðnÞ ¼ 0; n

i ¼ 1; . . . ; s:

Theorem 3.2. Let fAðtÞgt 2 ½0; b be a family of linear operators that generates a strongly continuous evolution system 0 fUðt; sÞ : ðt; sÞ 2 Dg. Assume the hypotheses (H2)–(H4), (H5 ) are satisfied. Then the impulsive differential system 1.1, 1.2, 1.3 is nonlocally controllable on J. Proof. On account of Theorem 3.1, we should only prove that the function G : PC([0, b]; X) ? PC([0, b]; X) given by formula (3.1) satisfies the Mönch’s condition. S For this purpose, let D # Bn0 be countable and D # coðf0g GðDÞÞ. We shall prove that D is relatively compact. In the sequel we will denote by U the following measure of noncompactness in PC([0, b]; X) defined by (see [12])

UðXÞ ¼ max ðaðEÞ; modC ðEÞÞ

ð3:15Þ

E2DðXÞ

for all bounded subset X of PC([0, b]; X), where D(X) stands for the set of countable subsets of X; a is the real MNC defined by

aðEÞ ¼ sup eLt bðEðtÞÞ t2½0;b

with E(t) = {x(t),x 2 E}, L is a constant that we shall appropriately choose. modC(E) is the modulus of equicontinuity of the function set E given by the formula

modC ðEÞ ¼ lim sup max d!0

x2E

max

06i6s t1 ;t 2 2J i ;jt 1 t 2 j6d

kxðt 1 Þ  xðt 2 Þk:

It was proved in [12] that U is well defined ( i.e., there is E0 2 D(X) which achieves the maximum in (3.15)) and is a monotone, nonsingular, regular MNC. Let us choose a constant L > 0 such that

q ¼ ð2LU þ 4L2U LB kK W kL1 Þ sup

t2½0;b

Z

t

eLðtsÞ hðsÞds < 1;

ð3:16Þ

0

where LU = {kU(t, s)k: (t, s) 2 D} and h is the integrable function of hypothesis (H2). Put Gx = G1x + G2x, as defined in Theorem 3.1. From the regularity of U, it is enough to prove that U(D) = (0,0). Since U(G(D)) is a maximum, let fyn g1 n¼1 # GðDÞ be the denumerable set which achieves its maximum. Then there exists a set fxn g1 n¼1 # D such that

yn ðtÞ ¼ ðGxn ÞðtÞ ¼ ðG1 xn ÞðtÞ þ ðG2 xn ÞðtÞ

ð3:17Þ

for all n P 1, t 2 [0, b]. Now we give an estimation for aðfyn g1 n¼1 Þ. From (3.12) and (3.13), noticing that Ki = 0 as Ii is compact, we have that

bðfðG2 xn ÞðtÞg1 n¼1 Þ 6 2LU

Z 0

t

2 hðsÞbðfxn ðsÞg1 n¼1 Þds þ 4LU LB kK W kL1

6 ð2LU þ 4L2U LB kK W kL1 Þ ¼ ð2LU þ 4L2U LB kK W kL1 Þ

Z

t

0

t 0

hðsÞbðfxn ðsÞg1 n¼1 Þds

hðsÞeLs sup ðeLt bðfxn ðtÞg1 n¼1 ÞÞds t2½0;b

0

Z

Z

t

hðsÞeLs ds  aðfxn g1 n¼1 Þ;

ð3:18Þ

for t 2 [0, b]. Since M() and Ii() are compact operators, we get that

bðfðG1 xn ÞðtÞg1 n¼1 Þ ¼ 0

ð3:19Þ

for t 2 [0, b]. From (3.18) and (3.19), it follows that Lt Lt aðfyn g1 bðfðG1 xn ÞðtÞ þ ðG2 xn ÞðtÞg1 ð2LU þ 4L2U LB kK W kL1 Þ n¼1 Þ ¼ sup e n¼1 Þ 6 sup e t2½0;b

t2½0;b

2 ¼ aðfxn g1 n¼1 Þð2LU þ 4LU LB kK W kL1 Þ sup

t2½0;b

Z 0

t

hðsÞeLðtsÞ ds ¼ aðfxn g1 n¼1 Þq:

Z 0

t

hðsÞeLs ds  aðfxn g1 n¼1 Þ

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S. Ji et al. / Applied Mathematics and Computation 217 (2011) 6981–6989

Therefore, we have that

aðfxn g1 n¼1 Þ 6 aðDÞ 6 aðcoðf0g

[

1 GðDÞÞÞ ¼ aðfyn g1 n¼1 Þ 6 aðfxn gn¼1 Þq:

From (3.16), we obtain that 1 aðfxn g1 n¼1 Þ ¼ aðDÞ ¼ aðfyn gn¼1 Þ ¼ 0:

Coming back to the definition of a, we can see that 1 bðfxn ðtÞg1 n¼1 Þ ¼ bðfyn ðtÞgn¼1 Þ ¼ 0;

ð3:20Þ

for every t 2 [0, b]. From (3.12) and (3.20), noticing that ki = 0 in (3.12), we get that 1

bðff ðt; xn ðtÞÞ þ ðBuxn ÞðtÞgn¼1 Þ 6 hðtÞbðfxn ðtÞg1 n¼1 Þ þ 2LU LB K W ðsÞ

Z 0

b

hðsÞbðfxn ðsÞg1 n¼1 Þds ¼ 0;

1 i.e., ff ðt; xn ðtÞÞ þ ðBuxn ÞðtÞg1 n¼1 is relatively compact for a.a. t 2 [0, b] in X. Moreover, from the fact fxn gn¼1 # Bn0 , by (H2)(ii) and 1 (3.8), it is easy to see that ff ðt; xn ðtÞÞ þ ðBuxn ÞðtÞgn¼1 is uniformly integrable for a.e. t 2 [0, b]. So ff ð; xn ðÞÞ þ Buxn g1 n¼1 is semicompact according to Definition 2.4. By applying Lemma 2.2, we have that set fcðfn þ Buxn Þg1 n¼1 is relatively compact in C([0, b]; X). So is the set G2 ðfxn g1 n¼1 Þ. On the other hand, by the strong continuity of T(t) and the compactness of g, Ii, we can easily verify that the set G1 ðfxn g1 n¼1 Þ is relatively compact. Then the representation of yn given by (3.17) yields that the set fyn g1 n¼1 is also relatively compact in PC([0, b]; X). Since U is a monotone, nonsingular, regular MNC, we, from Möch’s condition, have that

UðDÞ 6 Uðcoðf0g

[

GðDÞÞÞ ¼ Uðfyn g1 n¼1 Þ ¼ ð0; 0Þ:

Therefore, D is relatively compact in PC([0, b]; X). This completes the proof. Finally, we give an example to illustrate our abstract results above.

h

Example 3.1. Consider the partial differential system of the form

8 xt ðt; hÞ ¼ xh ðt; hÞ þ mðhÞuðt; hÞ þ Fðt; xðt; hÞÞ; for a:e: ðt; hÞ 2 ½0; b  X; > > > > < xðt; hÞ ¼ 0; for ðt; hÞ 2 ½0; b  @ X; xðt þi ; hÞ  xðt i ; hÞ ¼ Ii ðxðt i ; hÞÞ; i ¼ 1; 2; . . . ; s; > > > > Rb : xð0; hÞ þ 0 hðsÞ logð1 þ jxðs; hÞjÞds ¼ x0 ðhÞ; where X is a bounded domain in Rn with smooth boundary oX, m is the characteristic function of an open subset D  X, and h() 2 L1([0, b],R). 0 0 Take X ¼ CðXÞ, and x0 2 X. Define A(t)  A : D(A)  X ? X by Ax = x with the domain D(A) = {x 2 X : x 2 X, x(h) = 0, h 2 oX}. It is well known that A is an infinitesimal generator of a semigroup T(t) defined by T(t)x(s) = x(t + s) for each x 2 X. T(t) is not a compact semigroup on X and b(T(t)D) 6 b(D), where b is the Hausdorff MNC. We also define the bounded linear control operator B : X ? X by (Bu)(h) = m(h)u(h) for almost every h 2 X. We assume that (1) f : [0, b]  X ? X is a continuous function defined by

f ðt; xÞðhÞ ¼ Fðt; xðhÞÞ;

t 2 ½0; b; h 2 X:

We take F(t, x(h)) = c0 sin(x(h)), c0 is a constant. F is Lipschitz continuous for the second variable. Then f satisfies hypothesis (H2) of Section 3. (2) Ii : X ? X is a continuous function for each i = 1, 2, . . . , s, defined by

Ii ðxÞðhÞ ¼ Ii ðxðhÞÞ: R We take Ii ðxðhÞÞ ¼ X qi ðh; yÞ cos2 ðxðyÞÞdy, x 2 X, qi 2 CðX  X; RÞ; for each i = 1, 2, . . . , s. Then Ii is compact and satisfies hypoth0 esis (H5 ). (3) M : PC([0, b]; X) ? X is a continuous function defined by

MðuÞðhÞ ¼

Z

b

hðsÞ logð1 þ juðsÞ ðhÞjÞds;

u 2 PCð½0; b; XÞ; with uðSÞðhÞ ¼ xðS; hÞ:

0

Then M is a compact operator and satisfies hypothesis (H3). Under these assumptions, the above partial differential system can be reformulated as the abstract problem (1.1)–(1.3). And due to Theorem 3.2, this partial differential system is controllable on [0, b].

S. Ji et al. / Applied Mathematics and Computation 217 (2011) 6981–6989

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Acknowledgements The authors are grateful to the referee for his/her valuable comments and suggestions. Research is supported by the National Natural Science Foundation of China (10971182) and the first author is also supported by Youth Teachers Foundation of Huaiyin Institute of Technology (HGC0929). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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