Abdeldjalil Slama,　Ahmed Boudaoui

(1.Dept.of Math.and Computer Science,University of Adrar,National Road No.06,Adrar,Algeria; 2.Dept.of Probability and Statistics,USTHB,PO Box 32 EL Alia 16111 Bab Ezzouar,Algiers,Algeria, E-mail:slama dj@yahoo.fr(A.Slama))

APPROXIMATE CONTROLLABILITY OF FRACTIONAL IMPULSIVE NEUTRAL STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS AND INFINITE DELAY∗

Abdeldjalil Slama,1,2Ahmed Boudaoui1

(1.Dept.of Math.and Computer Science,University of Adrar,National Road No.06,Adrar,Algeria; 2.Dept.of Probability and Statistics,USTHB,PO Box 32 EL Alia 16111 Bab Ezzouar,Algiers,Algeria, E-mail:slama dj@yahoo.fr(A.Slama))

This paper is concerned with the approximate controllability of nonlinear fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable.By the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory,some sufficient conditions are given for the approximate controllability of the system.At the end,an example is given to illustrate the application of our result.

approximate controllability;fixed point principle;fractional impulsive neutral stochastic integro-differential equations;mild solution;nonlocal conditions

2000 Mathematics Subject Classification 65C30;93B05;34K40;34K45

Ann.of Appl.Math.

31:2(2015),127-139

1　Introduction

The controllability is one of the fundamental concept in linear and nonlinear control theory,and plays a crucial role in both deterministic and stochastic control systems(see e.g. Zabczyk,[19]).

From the mathematical point of view,the problems of exact and approximate controllability are to be distinguished.The concept of exact controllability is usually too strong and has limited applicability.Approximate controllability is a weaker concept than complete controllability and it is completely adequate in applications(see[5-7]).

The approximate controllability of stochastic or deterministic systems all have received extensive attention where a pioneering work has been reported by Bashirov and Mahmudov [2].An extensive list of these publications focused on the complete and approximate controllability of fractional stochastic differential systems can be found(see[6,11,12,14,16,17] and the references therein).

The problem with nonlocal condition,which is a generalization of the problem of classical condition,was motivated by physical problems.The pioneering work on nonlocal conditions is due to Byszewski and Lakshmikantham[3].They studied and obtained the existence anduniqueness of mild solutions to nonlocal differential equations.Since it is demonstrated that the nonlocal problems have better effects in applications than the classical Cauchy problems,stochastic differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained(see[1,8,18,20]and the references contained therein).On the other hand,the impulsive effects widely exist in fractional stochastic differential systems.So,it is important and necessary to discuss the qualitative properties for stochastic fractional equations with impulsive perturbations and infinite delay[18].

Recently,Zang and Li[20]studied the the approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions and infinite delay. Sufficient conditions were given for the approximate controllability of the system using the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory.

For the best of our knowledge,there is no work reported on approximate controllability of fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay.Motivated by this consideration,in this paper we study the approximate controllability of fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay in Hilbert space.Our approach is based on the fixed point theorem.The rest of this paper is organized as follows.In Section 2,we introduce some preliminaries such as definitions of fractional calculus and some useful lemmas.In Section 3,we prove our main results.Finally in Section 4,an example is given to demonstrate the application of our results.

2　Preliminaries and Basic Properties

In this section,we introduce some notations and preliminary results,which is useful for our results.Throughout this paper,let H,U be two separable Hilbert spaces and L(U,H) be the space of bounded linear operators from U into H.For convenience,we use the same notation‖·‖to denote the norms in H,U and L(U,H),and use〈·,·〉to denote the inner product of H and U without any confusion.Let(Ω,F,{Ft}t≥0,P)be a complete filtered probability space satisfying that usual conditions(that is,it is increasing and right continuous,while F0contains all P-null sets of F).Let W=(Wt)t≥0be a Q-Wiener process defined on (Ω,F,{Ft}t≥0,P)with the covariance operator Q such that TrQ＜∞.Let W=W(t)t≥0be a Q-Wiener process defined on(Ω,F,{Ft}t≥0,P)with the covariance operator Q,that is

E〈W(t),x〉〈W(s),y〉=(t∧s)〈Qx,y〉,for any x,y∈U and t,s∈[0,T],

where Q is a positive,self-adjoint,trace class operator on U.

We consider the following fractional stochastic impulsive neutral integro-differential systems with nonlocal conditions:

The control function u(·)is given in L2(J;U)and U is a Hilbert space;B is a bounded linear operator from U into H.The history xt:(-∞,0]→H,xt(θ)=x(t+θ),θ≤0 belongs to an abstract phase spaceand g:Bh→H are appropriate functions to be specified below;Ik:H→H,k= 1,2,···,m,are appropriate functions.The terms B1x(t)and B2x(t)are given byandrespectively,where K,P∈C(D,R+)are the sets of all positive continuous functions on D={(t,s)∈R2:0≤ s≤ t≤ T}.Here 0=t0≤ t1≤ ···≤tm≤tm+1=T,andrepresent the right and left limits of x(t)at t=tkrespectively. The initial data φ={φ(t);t∈(-∞,0]}is an F0-measurable,Bh-valued random variable independent of W(t)with finite second moments.

Now,we present an abstract space phase Bh.Assume that h:(-∞,0]→(0,+∞)withis a continuous function.We define an abstract phase space B byh

Bh:=is a bounded and measurable function on[-a,0]and

If Bhis endowed with the norm

then(Bh,‖·‖Bh)is a Banach space[13,15].

Now we consider the following space

where x|Jkis the restriction of x to Jk=(tk,tk+1],k=0,1,2,···,m.We endow a seminorm‖·‖Bbon Bbdefined by

We recall the following lemma.

Lemma 2.1[15]Assume that x∈Bb,then for t∈J,xt∈Bh.Moreover

Definition 2.1[4]The Caputo derivative of order α for a function f:[0,∞)→R,which is at least n-times differentiable can be defined as

for n-1≤α＜n,n∈N.If 0＜α≤1,then

Obviously,the Caputo derivative of a constant is equal to zero.The Laplace transform of the Caputo derivative of order α＞0 is given as

Definition 2.2 The fractional integral of order α with the lower limit 0 for a function f is defined as

provided that the right-hand side is pointwise defined on[0,∞),where the Γ is the gamma function.

Definition 2.3 A stochastic process x:J×Ω→H is called a mild solution to system (1)if:

(i)x(t)is measurable and Ft-adapted,for each t≥0;

(ii)x(t)∈H has c`adl`ag paths on t∈[0,T],and satisfies the following integral equation

(iii)x0=φ∈Bhon(-∞,0]satisfies‖φ‖Bh＜∞,where

with

and ξαbeing a probability density function defined on(0,∞),that is,

Lemma 2.2[21]The operators Tαand Sαhave the following properties:

(i)For any fixed t≥0,Tα(t)and Sα(t)are linear and bounded operators,that is,for any x∈X,

(ii){Tα(t),t≥0}and{Sα(t),t≥0}are strongly continuous,which means that for every x∈H and 0≤t′＜t′′≤T,we have

(iii)for every t≥0,Tα(t)and Sα(t)are also compact operators if T(t)is compact for every t＞0.

In order to study the approximate controllability for the fractional control system(1), we introduce the following linear fractional differential system

The controllability operator associated with(6)is defined by

where B∗anddenote the adjoints of B and Sα,respectively.

Let x(T;φ,u)be the state value of(1)at the terminal time T,corresponding to the control u and the initial value φ.Denote by R(T,φ)={x(T;φ,u):u∈L2(J,U)}the reachable set of system(1)at terminal time T,whose closure in H is denoted by

Definition 2.4 System(1)is said to be approximately controllable on J if= L2(Ω,H).

Lemma 2.3[9]The linear fractional control system(6)is approximately controllable on J if and only ifin the strong operator topology.

Lemma 2.4[21](Krasnoselskii’s fixed point theorem)Let E be a Banach space,be a bounded closed and convex subset of E,and F1,F2be maps ofinto E such that F1x+F2y∈for every pairIf F1is a contraction and F2is completely continuous,then the equation F1x+F2x=x has a solution on.

3　Main Results

In this section,we formulate sufficient conditions for the approximate controllability of system(1).

In order to establish the results,we impose the following conditions:

(H1)f:J×Bh×H→H is continuous and there exist ζ1,ζ2＞0 such that

and there exist two continuous functionsµ1,µ2:J→(0,∞)such that

(H2)There exist κ1,κ2＞0 such that

and there exist two continuous functions ν1,ν2:J→(0,∞)such that

(H3)A function h:J×Bh→H is continuous,and there exist an Mh＞0 such that

(H4)g is continuous,and there exists a positive constant δ1such that

(H5)A function Ik:H→ H is continuous and there exist continuous nondecreasing functions Lksuch that,for each x∈H,

(H6)The linear stochastic system(6)is approximately controllable on[0,T].

The following lemma is required to define the control function.

Lemma 3.1[10]For any,there exists an,such that

Theorem 3.1 Assume that conditions(H1)-(H5)hold.Then for each λ＞0,system (1)has a mild solution on[0,T],provided that

where

Proof For any λ＞0,define an operator Ψ:Bb→Bbby Ψx(t)=φ(t),t∈(-∞,0],

We shall show that the operator Ψ has a fixed point in the space Bb,which is the mild solution of(1).

For φ∈Bh,we defineby

For convenience,we divide the proof into several steps.

Step 1 We claim that there exists a positive number r such that Ψ(Br)⊂Br.If this isnot true,then,for each positive integer r,there exists a yr∈Brsuch that E‖Ψ(yr)(t)‖2＞r for t∈(-∞,T],t may depend upon r.However,on the other hand,we have

By(H1)-(H4),Lemma 2.1 and Hölder’s inequality,we obtain

Dividing both sides by r and taking the limit as r→∞,we obtain

which contradicts our assumption.Thus,for each λ＞0,there exists a positive number r such that Ψ(Br)⊂Br.

Next,we show that the operator Ψ is condensing,for convenience,we decompose Ψ as Ψ=Ψ1+Ψ2,where

Step 2 We prove that Ψ1is a contraction on Br.Let t∈J and y,y∗∈Br,then we have

hence Ψ1is a contraction.

Step 3Ψ2maps bounded sets into bounded sets in Br.

Let us prove that for r＞0 there exists ansuch that for each y∈Brwe haveNow we have

Step 4 The map Ψ2is equicontinuous.

Let u,v∈J,0≤u＜v≤T,y∈Br,then we obtain

Noting that for every∊＞0,there exists a δ＞0 such that,whenever|s1-s2|＜δ for every s1,s2∈J,‖Tα(s1)-Tα(s2)‖＜∊and‖Sα(s1)-Sα(s2)‖＜∊.Therefore,when|v-u|＜δ, we have

The right hand of the above inequality tends to 0 as v→ u and∊→ 0,hence the set {Ψ2y,y∈Br}is equicontinuous.

Step 5 The set V(t)={Ψ2y(t),y∈Br}is relatively compact in Br.Let 0＜t≤T be fixed and 0＜∊＜t.For δ＞0,y∈Br,we define

Then from the compactness ofis relatively compact in H for every∊,0＜∊＜t.Moreover,for y∈Br,we can easily prove thatis convergent to Ψ2y(t)in Bras∊→0 and δ→0,hence the set V(t)={Ψ2y(t):y∈Br}is also relatively compact in Br.Thus,by Arzela-Ascoli theorem Ψ2is completely continuous. Consequently,from Lemma 2.4,Ψ has a fixed point,which is a mild solution of(1).This completes the proof.

Theorem 3.2 Assume that(H1)-(H6)are satisfied,and the conditions of Theorem 3.1 hold.Further,if the functions f and σ are uniformly bounded,and T(t)is compact,then system(1)is approximately controllable on[0,T].

Proof Let xλbe a solution of(1),then we can easily get that

In view of the assumptions that f and σ are uniformly bounded on J,hence,there is a subsequence still denoted by,which converges weakly to f(s)in H,and σ(s)in L(U,H).On the other hand,by assumption(H5),the operatorstrongly as λ→0+for all 0≤s≤T,moreover,Thus,the Lebesgue dominated convergence theorem and the compactness of S yield

This gives the approximate controllability of(1),thus the proof is complete.

4　An Example

As an application,we consider an impulsive stochastic partial differential equation with the following form

Let U=H=L2([0,π])and h(t)=e2t,t＜0.To study the approximate controllability of(10),assume that H,Q,V and U are continuous and φ∈Bh.

Define operators h:J×Bh→H,f:J×Bh×L2([0,π])→,σ:J×Bh×L2([0,π]),g:Bh→L2([0,π])as

With the choices of A,h,f,σ and g,(10)can be rewritten as the abstract form of system(1).

Thus,under the appropriate conditions on the functions h,f,σ,g and Ikas those in (H1)-(H6),system(10)is approximately controllable.

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(edited by Liangwei Huang)

∗Manuscript January 9,2015