Vol 9, No 1 (2018)

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Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 9(1), Jan. – 2018 191

APPROXIMATE CONTROLLABILITY OF SECOND-ORDER

NEUTRAL IMPULSIVE EVOLUTION DIFFERENTIAL INCLUSIONS

OF SOBOLEV-TYPE WITH NONLOCAL CONDITIONS

M. TAMIL SELVAN*

1

_{& R. MURUGESU}

2

1,2

_{Department of Mathematics,}

SRMV College of Arts and Science, Coimbatore - 641 020, Tamil Nadu, India.

(Received On: 06-10-17; Revised & Accepted On: 05-01-18)

ABSTRACT

I

n this paper, we consider a class of second-order neutral impulsive evolution differential inclusions of Sobolev- type in Hilbert spaces. This paper deals with the approximate controllability for a class of second-order control systems. We establish a set of sufficient conditions for the approximate controllability for a class of second- order neutral impulsive evolution differential inclusions of Sobolev-type in Hilbert spaces by using the Bohnenblust- Karlin’s fixed point theorem. Finally, an example is given to illustrate our main results.

Keywords: Approximate controllability, Second- order differential inclusions of Sobolev-type, Cosine function of

operators, Impulsive systems, Evolution equations, Nonlocal conditions.

2010 Mathematics Subject Classification: 26A33, 34B10, 34K09, 47H10, 93B05.

1. INTRODUCTION

In this paper, we establish a set of sufficient conditions for the approximate controllability for a class of second- order neutral impulsive evolution differential inclusions with nonlocal conditions in Hilbert spaces of the form

[

(

( )) '

( , ( ))

]

( ) ( )

( , ( ))

( ),

[0, ],

d

Ex t

g t x t

A t x t

F t x t

Bu t

t

I

b

dt

−

∈

+

∈ =

(1.1)

0 0

(0)

( ), '(0)

( ),

x

=

x

+

p x

x

=

y

+

q x

(1.2)

( ) ( ( )),

_i _i _i

x t

I x t

∆

=

(1.3)

'( )

_i _i

( ( )),

_i

1, 2,..., .

x t

J x t

i

n

∆

=

(1.4)

In this equation, A(t):D(A(t))⊆X →XandE:D(E)⊆X →X are closed linear operators on a Hilbert space

X

and the control function u(⋅)∈L2(I,U), a Hilbert space of admissible control functions. Further,B is a bounded linear operator from U toX,g:I×X →X,p,q :PC(I,X)→XandF:I×X →2X \{0/}is a nonempty, bounded, closed and convex multivalued map. Similarly, the impulsive moments

{ }

t_i are given that 0=t₁<t₂ <⋅ ⋅⋅<t_n=bare fixed numbers and the functions I_i(⋅):X →X,J_i(⋅):X →Xdefined later in the preliminaries section. The symbol ∆ξ(t) represents the jump of the function ξ(⋅) at ,t which is defined by ∆ξ(t)=ξ(t+)−ξ(t−).

Various evolutionary processes from fields such as physics, population dynamics, aeronautics, economics and engineering are characterized by the fact that they undergo abrupt changes of state at certain moments of time between intervals of continuous evolution. The duration of the changes made here are often negligible compared to the total period of time but such changes can be reasonably well approximated in the form of impulses. These process tend to more suitably modeled by impulsive differential equations. For more details on this theory, we refer the monographs of Samoilenko and Perestyuk [32] , Lakshmikantham et al. [23], Bainov et al. [7] and the references cited therein. The literature on this type of problrms is vast, and different topics on the existence and qualitative properties of solutions are considered. Concerning the general motivations, relevant developments and the current status of the theory, we refer the reader to papers [2, 16, 34, 35, 38, 42, 43].

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There exists an extensive literature of differential equations with nonlocal conditions. Since it is demonstrated that the nonlocal problems have better effects in applications than the classical ones, differential equations with nonlocal problems have been studied extensively in the literature. The result concerning the existence and uniqueness of mild solutions to abstract Cauchy problems with nonlocal initial conditions was first formulated and proved by Byszewski, see [10, 11]. Since the appearance of these papers, several papers have addressed the issue of existence and uniqueness of nonlocal differential equations. Existence and controllability results of nonlinear differential equations and fractional differential equations with nonlocal conditions has been studied by several authors for different kinds of problems [3, 13, 16, 26]. Sobolev-type evolution equations often arise in various applications such as in the flow of fluid through fissured rocks, thermodynamics, and shear in second order fluids. For Sobolev- type differential equations, readers may refer [1, 5, 8, 14]. It is helpful to study the controllability of the second- order abstract differential system directly rather than to convert that into first- order system since the conversion does not give the desired results due to the behavior of the semi group generated by the linear part of the converted first order system.

Since the introduction of controllability concept by Kalman in 1960, it has been one of the fundamental concepts of the mathematical control theory. This concept led to some very important conclusions regarding the linear and nonlinear dynamical systems. Many scientific and engineering problems are nonlinear in nature and so it is necessary to study the controllability results in infinite dimensional spaces. Two basic concepts of controllability can be distinguished which are given as exact and approximate controllability. Exact controllability enables to drive the system to arbitrary final state with a set of admissible control function while approximate controllability means that system can be steered to arbitrary small neighborhood of final state. In other words, approximate controllability gives the possibility of steering the system to states which form the dense subspace in the state space. Various authors extended the controllability concept to infinite dimensional systems and established sufficient conditions for the controllability of nonlinear systems in abstract spaces, see the references [2, 18, 26-28, 31, 39-41, 43, 45]. In the work [26], Vijayakumar et al. established a set of sufficient conditions for the approximate controllability for a class of second- order evolution differential inclusions in Hilbert spaces by using Bohnenblust- Karlin’s fixed point theorem. In this paper, we extended the work to a class of second-order evolution impulsive neutral differential inclusions of Sobolev-type with nonlocal conditions in Hilbert spaces.

This paper is organized as follows. In Section 2, some necessary definitions and preliminary results to be used in the article are given. In Section 3, a set of sufficient conditions are established for the approximate controllability for a class of Sobolev type second-order neutral impulsive evolution differential inclusions with nonlocal conditions in Hilbert spaces. An example is presented in Section 4 to illustrate the theory of the obtained results.

2. PRELIMINARIES

In this section, we introduce few results, notations and lemmas which are required to establish our main results. The notations which are mentioned here will be used throughout the article without any further references. Let

) , ( and ) , (

Y Y X

X ⋅ ⋅ be Hilbert spaces and L(Y,X)be the Hilbert space of bounded linear operators from YintoX equipped with its natural topology; in particular, when Y = X we use the notation L(X). The resolvent set of a linear operator Ais given by ρ(A).Throughout this paper, B_r(x,X)will denote the closed ball with center at xand radius

0

>

r in a Hilbert space X.We denote by PC,the Hilbert space endowed with supnorm given by )

( sup_t _I xt PC

x ≡ _∈ , for x∈PC.

The operators A(t):D(A(t))⊆X →X andE:D(E)⊆X →X satisfy the following hypothesis. (S1) A(t)and E are linear operators, and A(t)is closed;

(S2) D(E)⊂D(A(t))andEis bijective; (S3)E−1 :X →D(E)iscompact..

The hypothesis (S1) - (S3) and the closed graph theorem imply that the boundedness of the linear operator .

:

1 _X _X

AE− →

In recent times there has been an increasing interest in studying the abstract non-autonomous second order initial value problem

''( )

( ) ( )

( ), 0

,

x t

=

A t x t

+

f t

≤

s t

≤

a

(2.1)

0 1

( )

, '( )

(3)

© 2018, IJMA. All Rights Reserved 193 references mentioned in these works. In the most of works, the existence of solutions to the problem (2.1) – (2.2) is related to the existence of an evolution operator S(t,s)for the homogenous equation

''( )

( ) ( ), 0

,

x t

=

A t x t

≤

s t

≤

a

(2.3)

Let us assume that the domain of A(t)is a subspace Ddense in X and independent of t, and for each x∈Dthe function t→A(t)xis continuous.

Following Kozak [22], in this work, we will use the following concept of evolution operator.

Definition 2.1: A family Sof bounded linear operators S(t,s) :[0,a]×[0,a]→L(X) is called an evolution operator for (2.3) if the following conditions are satisfied:

(Z1) For each x∈X, the mapping [0,a]×[0,a]∋(t,s)→S(t,s)x∈Xisofclass C1 and

, ],

, 0 [ , )

(

, 0 ) , ( ], , 0 [ )

(

X x each for and a s t all for ii

t t S a t each for i

∈ ∈

= ∈

( , ) |

_{t s}

,

( , ) |

_{t s}

.

S t s x

x

S t s x

x

t

=

s

=

∂

₌

∂

_{= −}

∂

(Z2) For all t,s∈[0,a], if x∈D(A),then S(t,s)x∈D(A),the mapping [0,a]×[0,a]∋(t,s)→S(t,s)x∈Xis of class C2 and

. 0 | ) , ( )

(

, ) ( ) , ( ) , ( ) (

, ) , ( ) ( ) , ( ) (

2 2

= ∂

∂ ∂

∂

= ∂

∂

= ∂

∂

=s t

x s t S t s iii

x s A s t S x s t S s ii

x s t S t A x s t S t i

(Z3) For all t,s∈[0,a], if x∈D(A),then S(t,s)x∈D(A),then (, ) 2 _sS t s x

t ∂

∂ ∂

∂

, (, )

2 _tS t s x

s ∂

∂ ∂

∂

and

)

(i (, )

2 _sS t s x

t ∂

∂ ∂

∂

= () S(t,s)x, s

t A

∂ ∂

)

(ii (, )

2 _t S t s x

s ∂

∂ ∂

∂

= S(t,s)A(s)x, s

∂ ∂

and the mapping S t s x s t A s t a

a] [0, ] (, ) () (, ) ,

0 [

∂ ∂ → ∋

× is continuous.

Throughout the work, we assume that there exists an evolution operator S(t,s)associated to the operator A(t).To abbreviate the text, we introduce the operator (, ) (, ).

s s t S s t C

∂ ∂ −

= In addition, we set NandN~for positive constants

such that sup₀_≤_t_,_s_≤_a S(t,s) ≤Nand sup₀_≤_t_,_s_≤_a C(t,s) ≤N~. Furthermore, we denote by N₁a positive constant such that

, )

, ( ) ,

(t h s S t s N₁h

S + − ≤ (2.4)

for all s,t,t+h∈[0,a]. Assuming that f :[0,a]→X is an integrable function, the mild solution x:[0,a]→Xof the problem (2.1) – (2.2) is given by

∫

+ +

=

t

d f t S x s t S x s t C t x

0 1

0 (, ) (, ) ( ) .

) , ( )

( τ τ τ

In the literature several techniques have been discussed to establish the existence of the evolution operator S(t,s). In particular, a much studied situation is that A(t)is the perturbation of an operator Athat generates a cosine operator function. For this reason, below we briefly review some essential properties of the theory of cosine functions. Let

X X A D

A: ( )⊆ → be the infinitesimal generator of a strongly continuous cosine family of bounded linear operators R

t

t C ())_∈

(4)

0 0 0

( )

,

.

t

S t x

=

∫

C s xds

x

∈

X t

∈

R

We refer the reader to [15, 36, 37] for the necessary concepts about cosine functions. Next we only mention a few results and notations about this matter needed to establish our results. The function S₀(⋅)satisfies

0

(

)

0

( )

0

( )

0

( ), ,

.

S t

+ =

s

S t C t S s

s t

∈

R

(2.5)

The notation

[

D(A)

]

stands for the domain of the operator Aendowed with the graph norm x _A = x + Ax ,x∈D(A). Moreover, in this paper the notation Estands for the space formed by the vectors x∈X for which C₀(⋅)x is a function of class C1. It was proved by Kisynski [21] that the space Eendowed with the norm

, , ) ( sup ₀

1 0

X x x t AS x

x

t

E = + _≤_≤ ∈

is a Hilbert space. The operator valued function

   

  =

) ( ) (

) ( ) ( )

(

0 0

t C t AS

t S t C t G

is a strongly continuous group of linear operators on the space E×Xgenerated by the operator _

     =

0 0 A

I

A defined on

E A

D( )× . It follows from that AS₀(t)x:E→X is abounded linear operator and that AS₀(t)x→0,t→0, for each

. E

x∈ Furthermore, if x:[0,∞)→Xis a locally integrable function, then =

∫

−

t

ds s t S t z

0

0( )

)

( defines a E−valued continuous function.

The existence of solutions for the second- order abstract Cauchy problem

''( )

( )

( ), 0

,

x t

=

Ax t

+

h t

≤ ≤

t

a

(2.6)

0 1,

( )

, '( )

x s

=

x

x s

=

x

(2.7) whereh:[0,a]→X , is an integrable function, has been discussed in [36]. Similarly, the existence of solutions of the semi linear second order Cauchy problem it has been treated in [37]. We only mention here that the function x(⋅)given by

0 0 0 1 0

( )

(

)

(

)

(

) ( )

, 0

,

t s

x t

=

C t

−

s x

+

S t

−

s x

+

∫

S t

−

τ τ τ

h

d

≤ ≤

t

a

(2.8) is called the mild solution of (2.6) - (2.7), and that when x₀∈E,x(⋅)is continuously differentiable and

0 0 0 1 0

'( )

(

)

(

)

(

) ( )

, 0

,

t s

x t

=

AS t

−

s x

+

C t

−

s x

+

∫

C t

−

τ τ τ

h

d

≤ ≤

t

a

In addition, if x₀∈D(A),x₁∈E and h is a continuously differentiable function, then the function x(⋅)is a solution of the initial value problem (2.6) - (2.7).

Assume now that A(t)=A+B~(t) where B~(⋅):R→L(E,X)is a map such that the function t→B~(t)xis continuously differentiable in X for each x∈X. It has been established by Serizawa [33] that for each(x0,x1)∈D(A)×E, the non

autonomous abstract Cauchy problem

''( )

(

( )) ( ), ,

x t

=

A B t x t

+



t

∈

R

(2.9)

0 1,

(0)

, '(0)

x

=

x

=

x

(2.10) has a unique solution x(⋅) such that the function t→x(t) is continuously differentiable in .E It is clear that the same argument allows us to conclude that equation (2.9) with the initial condition (2.7) has a unique solution x(⋅,s) such that the function t→x(t)is continuously differentiable in E. It follows from (2.8) that

∫

−

+ − + − =

t

s

d s x B t S x s t S x s t C t

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∫

−

+ − =

t

s

d s x B t S x s t S t

x(,s) ₀( ) ₁ ₀( τ)~(τ) (τ, ) τ. (2.12) Consequently,

∫

−

+ −

≤

t

s

E E

X L E

X L

E S t s x S t s B x s d

s t

x(, ) ( ) ( ) ~(τ) ( ,τ) τ

) , ( )

, ( 0

1 ) , ( 0

and applying the Gronwall-Bellman lemma, we infer that

1

( , )

_E

, ,

[0, ].

x t s

≤

M x



s t

∈

a

We define the operator S(t,s)x1=x(t,s).It follows from the previous estimate that S(t,s):(E, ⋅)→(E, ⋅_E) is a bounded linear map onE. Since (E, ⋅)is dense in Xand (E, ⋅)is complete, we can extend S(t,s) toX. We keep the notation S(t,s) for this extension. Combining this assertion with (2.12), we obtain that

0 0

( , )

(

)

(

) ( ) ( , )

,

.

t s

S t s x

=

S t

−

s x

+

∫

S t

−

τ

B



τ

S

τ

s xd

τ

x

∈

X

(2.13) In similar way, from (2.11) we have

∫

−

+ − =

t

s

d x s C B t S x s t C x s t

C(, ) ₀ ₀( ) ₀ ₀( τ)~(τ) (τ, ) ₀ τ.

forx₀∈D(A)..

Assume that the following condition holds: (S4) The operator ( )

~ ) ( 0 t B

τ

S has an extension H₀(t,τ)∈L(X)for all 0≤τ,t≤a, and the operator valued map )

( ] , 0 [ ] , 0 [ :

0 a a L X

H × → is strongly continuous.

In this case, proceeding as above, we can consider C(t,s) defined onX, and

0 0

( , )

(

)

(

, ) ( , )

, .

t s

C t s x

=

C t

−

s x

+

∫

H t

−

τ τ

C

τ

s xd

τ

x

∈

X

(2.14)

We denote

0

( , )

(

, ) ( , )

,

.

t s

R t s x

=

∫

H t

−

τ τ

C

τ

s xd

τ

x

∈

X

Theorem 2.2: [18, Theorem 1.2]. Under the preceding conditions, and assuming that condition S₄holds, then S(⋅,⋅) is an evolution operator for (2.9) - (2.10). Moreover, if S0(t) is compact for allt∈R, then S(t,s)is also compact for all

. t s≤

On the other hand, it is well known that, except in the casedim(X)<∞, the cosine function C(t) cannot be compact for all t∈R. By contrast, for the cosine functions that arise in specific applications, the sine function S(t) is very often a compact operator for allt→R..

Concerning the impulsive conditions in system (1.1)-(1.4), it is convenient to introduce some additional concepts and notations.

A function u:[

σ

,

τ

]→X is said to be a normalized piecewise continuous function on [

σ

,

τ

] if u is piecewise continuous and left continuous on (σ,τ]. We denote by PC([σ,τ];X) the space of normalized piecewise continuous functions from [

σ

,

τ

] intoX. In particular, for fixed a>0 and t_i∈(0,a),i=1,2,...,n,, we introduce the space PC formed by all normalized piecewise continuous functions u:[0,a]→X such that u is continuous at t=t_i,i=1,2,...,n. It is clear that PC endowed with the norm u _PC =sup_s_∈_I u(s) is a Hilbert space. In what follows, we set

, ,

0 ₁

0 t a

(6)

   

= ∈

= ₊ +

. for , (

], , ( for ), ( ) (

~ 1

i i

t t t

u

t t t t

u t u

Moreover, for a setB⊆PC, we represented by B~_i,i=0,1,2,...n, the sets B~_i ={u~_i :u∈B}.

Lemma 2.3: A set B⊆PC is relatively compact in PC if and only if, the set B~_i, is relatively compact in the space .

,... 2 , 1 , 0 all for ) ]; ,

([t t ₁ X i n

C _i _i₊ =

Throughout this paper, B_r(x,X)will denote the closed ball with center at xand radius r>0 in a Hilbert space X. Some of our results are proved using the next well-known results.

We also introduce some basic definitions and results of multivalued maps. For more details on multivalued maps, see the books of Deimling [12] and Hu and Papageorgious [20].

A multivalued map G:X →2X \{0/}is convex (closed) valued if G(x) is convex (closed) for all x∈X. G is bounded on bounded sets if G(C)=∪x∈CG(x) is bounded in X for any bounded set C of X, i.e.,

. )}} ( : {sup{

sup_x_∈_C y y∈G x <∞

Definition 2.4: G is called upper semicontinuous (u.s.c. for short) on X if for each x0∈X,the set G(x0)is a

nonempty closed subset of X,and if for each open set C of X containing G(x₀), there exists an open neighborhood V of x₀ such that G(V)⊆C..

Definition 2.5: G is called completely continuous if G(C) is relatively compact for every bounded subset C of X.

If the multivalued map G is completely continuous with nonempty values, then G is u.s.c., if and only if G has a closed graph, i.e., x_n →x_*, y_n →y_*,y_n∈Gx_nimply y_*∈Gx_*.G has a fixed point if there is a x∈X such that

). (x G x∈

Definition 2.6: A function x∈PC is said to be a mild solution of system (1.1)-(1.4) if

0 0, '(0) ( ) )

( ) 0

( p x x x q x y

x + = + = and there exists f ∈L1(I,X) such that f(t)∈F(t,x(t)) on t∈Iand the integral equation

∫

−

∫

−

− ₊ ₊ ₊ ₋ ₊ ₊

=

t t

ds s f s t S E ds s x s g s t C E x

g x Eq Ey t S E x Ep Ex t C E t x

0 0

1 1

0 1

0

1 ₍_,₀_)[ ₍ _)] ₍_,₀_)[ ₍ ₎ ₍₀_, ₍₀_)] ₍_, ₎ ₍ _, ₍ ₎₎ ₍_, ₎ ₍ ₎

) (

+

∫

∑

< <

−

< <

−

− ₊ ₊ _∈

t i t

i i i t

i t

i i i t

I t t

x J t t S E t

x I t t C E ds

s Bu s t S E

0 1

. )), ( ( ) , ( ))

( ( ) , ( )

( ) ,

(

is satisfied.

In order to address the problem, it is convenient at this point to introduce two relevant operators and basic assumptions on these operators:

∫

→

=

Γ − −

b

b _E _S_b _s _BB _E _S _b _s_ds _X _X

0

* 1 * 1

0 ( , ) ( , ) : ,

X X I

R(α,Γ₀b)=(α +Γ₀b)−1: →

whereB*denotes the adjoint of B and S*(t) is the adjoint of S(t). It is straightforward that the operator Γ₀b is a linear bounded operator.

To investigate the approximate controllability of system (1.1)-(1.2), we impose the following condition: +

→ →

Γ ) 0as 0

, ( )

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(

Ex t

( )) ''

=

Ax t

( ) (

+

Bu t

)( ),

t

∈

[0, ],

b

(2.15)

0 0

(0)

, '(0)

,

x

=

x

=

y

(2.16) is approximately controllable on [0,b].

Some of our results are proved using the next well-known results.

Lemma 2.7: [24, Lasota and Opial] Let Ibe a compact real interval, BCC(X) be the set of all nonempty, bounded, closed and convex subset of X and F be a multivalued map satisfying F:I×X →BCC(X) is measurable to t for each fixed x∈X , u.s.c. to x for each t∈I, and for each x∈Cthe set

,

{

'( ,

) : ( )

( , ( ), }

F x

S

=

f

∈

L I X

f t

∈

F t x t

t

∈

I

is nonempty. Let F be a linear continuous from L'(I,X)to ,C then the operator

,

:

( ),

(

)( )

(

),

F F F x

F S



C

→

BCC X

x

→

F S



x

=

F S

is a closed graph operator in C×C.

Lemma 2.8: [9, Bohnenblust and Karlin] Let Dbe a nonempty subset of X, which is bounded, closed and convex. Suppose G:D→2X \{0/}is u.s.c. with closed, convex values and such that G(D)⊆Dand G(D)is compact. Then G has a fixed point.

3. APPROXIMATE CONTROLLABILITY RESULTS

In this section, we need the following hypotheses to establish a set of sufficient conditions for the approximate controllability for a class of second-order evolution differential inclusions of the form (1.1)-(1.4) in Hilbert spaces by using Bohnenblust-Karlin's fixed point theorem.

(H1)S0(t),t>0 is compact.

(H2) For each positive number r and x∈PC with x _PC ≤r, there exists Lf,r(⋅)∈L1(I,R+)such that

{

: () (, ())

}

(),

sup f f t ∈F t xt ≤L_f_,_r t for a.e. t∈I.

(H3) The function s→Lf,r(s)∈L1([0,t],R+)and there exists a

γ

>0such that

. )

( lim 0

,

+∞ < =

∫

∞

→ r γ

ds s L

t

r f

r

(H4) The function g :I×X →Xsatisfies the following conditions: (i) The function g(t,⋅):X →X is continuous a.e. t∈I,

(ii) The function g(t,⋅):I →Xis strongly measurable for each x∈X, (iii)There exists positive constants L_g >0,L~_g >0,such that

, , , e a. for ))

( , ( )) ( ,

(t xt g t yt L x y x y X t I

g − ≤ g − ∈ ∈

g(t,x(t)) ≤L_g x +L~_g

where L~_g =max_t_∈_I g(t,0) .

(H5) The function p,q :PC(I,X)→X are continuous and there exists constants Lp >0 ,Lq >0such that

. , for , )

( ) (

, , for , )

( ) (

PC y x y

x L y q x q

PC y x y x L y p x p

q p

∈ −

≤ −

∈ −

≤ −

(H6) (i) There are positive constants LI_i ,LJ_isuch that

.. ,... 2 , 1 , ,

, )

( ) (

, )

( ) (

2 1 2 1 2

1

2 1 2

1

n i

X L

J J

L I

I

i J i

i

i I i

i

= ∈ −

≤ −

− ≤

−

ψ ψ ψ ψ ψ

ψ

ψ ψ ψ

(8)

(ii) The maps I_i,J_i :X →X,i=1,2,...nare completely continuous and there exists continuous non-decreasing functions M_i ,M~_i :[0,∞)→(0,∞), i=1,2,...n,such that

( )

(

),

( )

(

),

.

i i

I

M

J

M

X

ψ

≤



∈

It will be shown that the system (1.1-1.4) is approximately controllable, if for all α >0,there exists a continuous function x(⋅)such that

1 1 1

0 0

0

1 1 1 1

,

0 0

( )

( , 0)[

( )]

( , 0)[

( )

(0, (0))]

( , ) ( , ( ))

( , ) ( )

( , )

( , ) ( ( ))

( , ) ( ( )),

i i

t

t t

i i i i i i F x t t t t

x t

E C t

Ex

Ep x

E S t

Ey

Eq x

g

x

E C t s g s x s ds

E S t s f s d s

E S t s Bu s x d s

E C t t I x t

E S t t J x t

f

S

− − −

− − − −

< < < <

=

+

−

+

∈

∫

∑

∫

)), ( ( ) , ( ) , ( ) ,

(t x =B*E−1S* bt R Γ₀ p x⋅

u α b

. )) ( ( ) , ( )) ( ( ) , ( ) ( ) , ( )) ( , ( ) , ( ))] 0 ( , 0 ( ) ( )[ 0 , ( )] ( )[ 0 , ( )) ( ( 0 1 0 1 0 1 0 1 0 1 0 1

∑

∫

< < − < < − − − − − − − − − − + − + − = ⋅ b i t i i i b i t i i i b b b t x J t b S E t x I t b C E ds s f s b S E ds s x s g s b C E x g x Eq Ey b S E x Ep Ex b C E x x p

Theorem 3.1: Suppose that the hypotheses (H0)-(H6) are satisfied. Assume also

, 1 ) ~ ( ~ ) ( ~ ~ ) ( ~ ~ ~ 1 1 1 2 2 2 _<         + + + + + +     +

_∑

= n i i J i I g q g p

B G b GN GL bL GN GL L G NL NL

M

N γ

α (3.1)

where M_B = B,then the system (1.1) – (1.4) has a solution on .I

Proof: The main aim in this section is to find conditions for solvability of system (1.1)-(1.4) for α> 0. We show that, using the control u(t,x),the operator Υ:PC→2PC,defined by

, )) ( ( ) , ( )) ( ( ) , ( ) , ( ) , ( ) ( ) , ( )) ( , ( ) , ( ))] 0 ( , 0 ( ) ( )[ 0 , ( )] ( )[ 0 , ( ) ( : ) ( 0 1 0 1 0 1 0 1 0 1 0 1 0 1                   + + + + + − + + + = ∈ = Υ

∑

∫

< < − < < − − − − − − t i t i i i t i t i i i t t t t x J t t S E t x I t t C E ds x s Bu s t S E ds s f s t S E ds s x s g s t C E x g x Eq Ey t S E x Ep Ex t C E t PC x ϕ ϕ

has a fixed point x,which is a mild solution of system (1.1)-(1.4).

We now show that Υsatisfies all the conditions of Lemma 2.8. We subdivide the proof into five steps for our convenience.

Step-1: Υ is convex for each x∈PC.

In fact, if ϕ₁,ϕ₂ belong to Υ(x),then there exist f₁,f₂∈S_F_,_xsuch that for each t∈I,we have

1 1 1

0 0

0

1 1 * 1 1

0 0

1

0

( )

( , 0)[

( )]

( , 0)[

( )

(0, (0))]

( , ) ( , ( ))

( , )

( )

( , )

( , ) ( ,

)[

( , 0)[

( )]

( , 0)[

t

i

t t

b

i b

t

E C t

Ex

Ep x

E S t

Ey

Eq x

g

x

E C t s g s x s ds

E S t s f s d s

E S t s BB E S b t R

x

E C b

x

Ep x

E S b

Ey

E

φ

α

− − − − − − − −

=

+

−

+

Γ

−

+

−

+

∫

1 1

0 0

1 1 1

0 0 0

1 0

( )

(0, (0))]

( , ) ( , ( ))

( , )

( )

( , ) ( ( ))

( , )

( ( ))]( )

( , ) ( ( ))

( , )

( ( )).

i i i

i

b b

i

i i i i i i i i i

t b t b t t

i i i

t t

q x

g

x

E C b s g s x s d s

E S b s f

d

E C b t I x t

E S b t J x t

s d s

E C t t I x t

E S t t J x t

η η

− −

− − −

< < < < < <

(9)

1 1 1

1 2 0 0

0

1 1 * 1

1 2

0

Let [0,1]. Then for each , we get

( ) (1 ) ( ) ( , 0)[ ( )] ( , 0)[ ( ) (0, (0))] ( , ) ( , ( ))

( , )[ ( ) (1 ) ( )] ( , ) (

t

t I

t t E C t Ex Ep x E S t Ey Eq x g x E C t s g s x s ds

E S t s f s f s d s E S t s BB E S

λ

λφ

λ φ

λ

− − − − − − ∈ ∈ + − = + + + − + + + − +

∫

1 0 0 0

1 1 1

0 1 2

0 0

1 1

0 0

, ) ( , )[ ( , 0)[ ( )]

( , 0)[ ( ) (0, (0))] ( , ) ( , ( )) ( , )[ ( ) (1 ) ( )]

( , ) ( ( )) ( , ) ( ( )) i t b b b b

i i i i i i t b

b t R x E C b x Ep x

E S b Ey Eq x g x E C b s g s x s ds E S b s f f d

E C b t I x t E S b t J x t

α

λ η

λ

η η

−

− − −

− −

< <

Γ − +

− + − − − + − − −

∫

∑

1 0 1 0 ]( ) ( , ) ( ( )) ( , ) ( ( )). i i i

i i i

t b t t

i i i t t

s ds E C t t I x t

E S t t J x t

−

< < < <

− < < + +

∑

It is easy to see that S_F_,_xis convex since Fhas convex values. So, λf₁+(1−λ)f₂∈S_F_,_x.Thus, ).

( ) 1

( ₂

1+ −λϕ ∈Υ x

λϕ

Step-2: For each positive number r>0,let B_r =

{

x∈PC: x _PC ≤r

}

. Obviously, B_ris a bounded, closed and convex set of PC.We claim that there exists a positive number r such that Υ(B_r)⊆B_r.

If this is not true, then for each positive number r,there exists a function xr∈B_r,but Υ(xr)does not belong to B_r, i.e.,

r x

x r r

PC r PC r _>     

 _∈_Υ

≡

Υ( ) sup ϕ :ϕ ( )

and , )) ( ( ) , ( )) ( ( ) , ( ) , ( ) , ( ) ( ) , ( )) ( , ( ) , ( ))] 0 ( , 0 ( ) ( )[ 0 , ( )] ( )[ 0 , ( ) ( 1 1 1 1 0 1 0 1 0 1 0 1 0 1

∑

∫

= = − − − − − − − + + + + + − + + + = n i n i i r i i i r i i t r t r t r r t x J t t S E t x I t t C E ds x s Bu s t S E ds s f s t S E ds s x s g s t C E x g x Eq Ey t S E x Ep Ex t C E t ϕ

for some .

,xr F r _S

f ∈ Furthermore we take G= E and G~= E−1 and by using (H1)-(H6), we have )

)( (x t r< Υ r

(

)

1 1 1

0 0

0

1 1 1 1

1 1

0 0

( , 0)[

( )]

( , 0)[

( )

(0, (0))]

( , ) ( , ( ))

( , )

( )

( , )

( , ) (

( ))

( , ) (

( ))

(0)

t

r

t t n n

r r r r

i i i i i i

i i

p

E C t

Ex

Ep x

E S t

Ey

Eq x

g

x

E C t s g s x s

ds

E S t s f

s d s

E S t s Bu s x d s

E C t t I x t

E S t t J x t

GNG

x

L r

p

GN G y

− − − − − − − = =

≤

+

−

+

≤

+

∫

∑

∫



 

(

)

(

)

(

)

(

)

(

)

[

]

, 0 0

2 2 2

0 0

0

,

1 0

(0)

( )

1 (0)

(0)

( )

( ( ))

(0)

( ( ))

(0)

t t

q g g g g f r

t

B p q g g g g

t n

f r i i i i i i i i

GL r

G q

L r

L

GN

L r

L ds

GN L

s ds

N M G b GNG

x

L r

p

GN G y

GL r

G q

L r

L

GN

L r

L ds

GN L

s ds

GN

I x t

I

GN

J x t

J

α

=

+

−

+

−



















∫

∑

∫



_[

_]

[

]

[

]

1 1 1

(0)

( ( ))

(0)

( ( ))

(0)

(0) .

n

i i

n n

i i i i i i i i

i i

J

GN

I x t

I

GN

J x t

J

(10)

Dividing both sides of the above inequality by rand taking the limit as r→∞,using (H3), we get , 1 ) ~ (

~ ) (

~ ~ ) (

~ ~ ~

1 1

1 2

2

2 _≥

    

  

+ +

+ + +

+ 

 

 +

_∑

=

n

i

i J i I g

q g

p

B G b GN GL bL GN GL L G NL NL

M

N γ

α

This contradicts with the condition (3.1). Hence, for some r>0,Υ(B_r)⊆B_r.

Step-3: Υ sends bounded sets into equicontinuous sets of PC. For each x∈B_r,ϕ∈Υ(x), there exists a f ∈S_F_,_x such that

1 1 1

0 0

0

1 1 1 1

0 0

( )

( , 0)[

( )]

( , 0)[

( )

(0, (0))]

( , ) ( , ( ))

( , ) ( )

( , )

( , ) ( ( ))

( , )

( ( )),

i i

t

t t

i i i i i i t t t t

t

E C t

Ex

Ep x

E S t

Ey

Eq x

g

x

E C t s g s x s ds

E S t s f s d s

E S t s Bu s x d s

E C t t I x t

E S t t J x t

φ

− − −

− − − −

< < < <

=

+

−

+

∫

∑

∫

Let ε >0and0<t₁<t₂ ≤b,then

1 1

1 2 1 2 0 1 2 0

( )

t

( )

t

E

C t

( , 0)

C t

( , 0)

Ex

Ep x

( )

E

S t

( , 0)

S t

( , 0)

Ey

Eq x

( )

g

(0, (0))

x

φ

₋

φ

₌

−

₋

₊

−

₋

₊

₋

[

]

[

]

[

]

[

]

1 1

1

2 1

1

1 2

1 1

1 2 1 2

0

1 1

2 1 2

0

1 1

1 2 2

( , )

( , ( ))

( , )

( , ( ))

( , ) ( , ( ))

( , )

( )

( , )

( )

( , ) ( )

t t

t

t t

t

t t

E

C t s

g s x s d s

E

C t s

g s x s d s

E C t s g s x s ds

E

S t s

f s ds

E

S t s

f s d s

E S t s f s d s

ε

ε ε

ε

−

− −

−

− −

−

+

−

+

−

+

−

+

−

+

∫

[

]

[

]

[

]

[

]

1 1

1 2

1 2 1

1 1

1 2 1 2

0

1 1 1

2 2 1 2

0

1 1

2 1

0

( , )

( , ) ( , )

( , )

( ( ))

( , ) ( ( ))

( , )

( ( ))

i i

i

t t

i i i i i i i

t t t t t

i i i i t t

E

S t

Bu

x d

E

S t

Bu

x d

E S t

B u

d

E

C t t

I x t

E C t t I x t

E

S t t

J x t

E

ε

η

η η

−

− −

−

− − −

< < < <

− −

< <

+

−

+

−

+

−

+

−

+

∫

∑

∫

∑

1 2

2

( , )

( ( ))

i

i i i t t t

S t t J x t

< <

∑

(

)

(

)

(

)

(

)

(

)

1 1 2

1 1

1

1 2 0 1 2 0 0

1 2 1 2

0

1 2 ,

0

( , 0)

(0)

( , 0)

( , )

( )

(

p q g g

t t t

g g g g g g

t t

t

f r

G C t

C t

G x

GL r

G p

G S t

S t

G y

GL r

G q

L r

L

G

C t s

L r

L

d s G

C t s

L r

L

d s GN

L r

L

d s

G

S t s

L

s d s G

S t

ε

ε ε

−

≤

−

+

−

+

−

+

−

+

−

+

∫



1 2

1 1

1 2

1 2 1

1 2 , ,

1 2 1 2

0

2 1

0

, )

( , ) ( )

( )

( , )

i i

t t

f r f r

t t

B B

t t

B i i i t i t

t t t t t

t

s

S t s

L

s ds

GN L

s ds

GM

S t

u

x d

GM

S t

u

x d

GNM

u

x d

G

C t t

M

x

GN

M

x

ε ε

ε

η

−

< < < <

−

+

−

+

−

+

−

+

∫

∑

∫

 



1 2

2 1

0

( , )

i i

i i i t i t

t t t t t

G

S t t

M

x

GN

M

x

< < < <

−

+

(11)

© 2018, IJMA. All Rights Reserved 201 The right-hand side of the above inequality tends to zero independently of x∈B_ras (t₁−t₂)→0and ε sufficiently small, since the compactness of the evolution operator S(t,s)implies the continuity in the uniform operator topology. Thus Υ(xr)sends B_rinto equicontinuous family of functions.

Step-4: The set Π(t)=

{

ϕ(t):ϕ∈Υ(B_r)

}

is relatively compact in X. Let t∈(0,b]be fixed and εa real number satisfying 0<ε<t._forx∈B_r,we define

1 1 1

0 0

0

1 1 1 1

0 0

( )

( , 0)[

( )]

( , 0)[

( )

(0, (0))]

( , ) ( , ( ))

( , ) ( )

( , )

( , ) ( ( ))

i i

t

t t

i i i i i i

t t t t

t

E C t

Ex

Ep x

E S t

Ey

Eq x

g

x

E C t s g s x s ds

E S t s f s d s

E S t s Bu s x d s

E C t t I x t

E S t t J x t

ε

ε ε

φ

− − − −

− −

− − − −

< < − < < −

=

+

−

+

∫

∑

∫

Since S(t)is a compact operator, the set Πε(t)=

{

ϕε(t):ϕε ∈Υ(B_r)

}

is relatively compact in Xfor each ε,0<ε<t. Moreover, for each 0<ε<t,we have

(

)

,

( )

( , )

(

)

(

)

i i

t t t

g g f r B i t

t t t

i t t t t

t

GN

L r

L

ds

GN

L

s ds

GNM

u

x d

GN

M

x

GN

M

x

ε

ε ε ε

ε

φ

η

− < <

− − −

− < <

−

≤

+

∑

∫

∑



Hence there exist relatively compact sets arbitrarily close to the set Π(t)=

{

ϕ(t):ϕ∈Υ(B_r)

}

and the set Π(t) is relatively compact in Xfor all t∈(0,b]. Since it is compact at =0, hence Π(t)is relatively compact in X for all

]. , 0

[ b

t∈

Step-5: Υhas a closed graph. Let x_n→x_* as n→∞,ϕ_n∈Υ(x_n),andϕ_n →ϕ_*asn→∞ .we will show that ).