# Controllability of Nonlinear Neutral Stochastic Differential Inclusions with Infinite Delay

Authors: Yong Li; Qiang Zou
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The paper is concerned with the controllability of nonlinear neutral stochastic differential inclusions
with infinite delay in a Hilbert space. Sufficient conditions for the controllability are obtained
by using a fixed-point theorem for condensing maps due to O'Regan.


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## Body

## 1. Introduction

LetK be a separable Hilbert space, and let (Ω,𝔉,𝔉t,P) be a complete probability space furnished with a complete family of right continuous increasing σ algebras {𝔉t} satisfying 𝔉t⊂𝔉 for t≥0. Suppose w(t) is a given K-valued, 𝔉t adapted Brownian motion with a finite trace nuclear covariance operator Q≥0. We are interested in the controllability of nonlinear stochastic differential inclusions with infinite delay
(1)d[x(t)-f(t,xt)]vvv∈Ax(t)dt+(Bu)(t)dtvvv+G(t,xt)dw(t),t∈J=[0,b],t>0,x0=ϕ∈𝔅h,t∈J0=(-∞,0],
where ϕ is 𝔉0 measurable and A is the infinitesimal generator of an analytic semigroup of bounded linear operator T(t), t≥0, the state x(·) takes values in Hilbert space H with the norm |·|, G:J×𝔅h→2L(K,H) is a bounded closed, convex-valued multivalued map, f:J×𝔅h→H is continuous, and the histories xt:(-∞,0]→H, xt(θ)=x(t+θ), and θ≤0 belong to the space 𝔅h. For σ1,σ2∈L(K,H), define 〈σ1,σ2〉=tr(σ1Qσ2*), where σ2* is the adjoint of the operator σ2, Q∈Ln+(K). L(K,H) furnished with the scalar product 〈·,·〉 is a pre-Hilbert space. The completion of L(K,H) with respect to the topology induced by the norm ∥·∥2, where ∥σ∥22=〈σ,σ〉2, is a Hilbert space.At first, we present the abstract phase space𝔅h. Assume that h:(-∞,0]→(0,+∞) is continuous function with l=∫-∞0‍h(s)ds<+∞. Define
(2)𝔅h={∫-∞0‍φ:(-∞,0]⟶H:foranya>0,(E|φ(θ)|2)1/2isaboundedandmeasurablefunctionon[-a,0]and∫-∞0‍h(s)sups≤θ≤0(E|φ(θ)|2)1/2ds<+∞}.
If 𝔅h is endowed with the norm
(3)∥ϕ∥𝔅h=∫-∞0‍h(s)sups≤θ≤0(E|ϕ(θ)|2)1/2ds,∀ϕ∈𝔅h,
then(𝔅h,∥·∥𝔅h) is a Banach space.Controllability is one of the fundamental concepts in mathematical control theory and plays an important role in control systems. The problem of controllability is to show the existence of a control function, which steers the solution of the system from its initial state to a final state, where the initial and final states may vary over the entire space. Controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. A standard approach is to transform the controllability problem into a fixed-point problem for an appropriate operator in a functional space. The problem of controllability for functional differential systems has been extensively studied in many papers [1–10]. For example, Sakthivel et al. [5] considered a class of fractional neutral control systems governed by abstract nonlinear fractional neutral differential equations and established a new set of sufficient conditions for the controllability of nonlinear fractional systems by using a fixed-point analysis approach. Using fixed-point technique, fractional calculations, stochastic analysis technique, and methods adopted directly from deterministic control problems, Sakthivel et al. [7] gave a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations. In [8], Sakthivel and Ren investigated the complete controllability property of a nonlinear stochastic control system with jumps in a separable Hilbert space. In [9], Sakthivel et al. studied the approximate controllability for nonlinear dispersion system under the assumption that the corresponding linear control system is approximately controllable.Benchohra et al. [11] discussed the controllability for one-order, second-order functional differential and integrodifferential inclusions in Banach space with finite delay. Park et al. [12] discussed the controllability for second-order neutral functional differential inclusions in Banach space with the help of some fixed-point theorems. In [13, 14], Liu investigated the controllability of neutral functional differential and integrodifferential inclusions with infinite delay.History about the research of stochastic differential equations is over fifty years; many important results on the stochastic differential equations have been reported in the literature; see [15, 16]. Recently, stochastic partial differential equations (SPDE) arise in the mathematical modeling of various fields in physics and engineering science cited by Sobczyk [16]. About neutral stochastic differential equations with delay, Mahmudov [17] discussed the existence and uniqueness of neutral stochastic differential equations by using Picard technique. For the controllability of stochastic functional differential inclusions, Balasubramaniam and Ntouyas [18] discussed the controllability of semilinear stochastic delay evolution equations in Hilbert spaces. Balasubramaniam and Dauer [19] obtained the controllability result of stochastic differential inclusions with infinite delay in abstract space. Since many systems arising from realistic models heavily depend on histories (i.e., there is the effect of infinite delay on state equations) [20], there is a real need to discuss partial functional differential systems with infinite delay. So in the present paper, we will concentrate on the case with infinite delay and establish sufficient conditions for the controllability of systems (1) by relying on a fixed-point theorem for condensing maps due to O’Regan [21].The structure of this paper is as follows. In Section2 we briefly present some basic notations and preliminaries. The controllability result of system (1) is investigated by means of fixed-point theorem and operator theory in Section 3. Conclusion is given in Section 4.


## 2. Preliminaries

Let(E,∥·∥) be a Banach space. A multivalued map 𝔍:E→2E is convex (closed)-valued, if 𝔍(x) is convex (closed) for all x∈E. 𝔍 is bounded on bounded set if 𝔍(B)=∪x∈B𝔍(x) is bounded in E for any bounded set B of E; that is,
(4)supx∈Bsup{∥y∥∈𝔍(x)}<∞.𝔍 is called upper semicontinuous (Usc) on E if for each x*∈E, the set 𝔍(x*) is nonempty, closed subset of E and if for each open set B of E containing 𝔍(x*), there exists an open neighborhood V of x* such that 𝔍(V)⊆B.𝔍 is said to be completely continuous if 𝔍(B) is relatively compact, for every bounded subset B⊆E.If the multivalued map𝔍 is completely continuous with nonempty compact values, then 𝔍 is Usc if and only if 𝔍 has a closed graph (i.e., xn=x*, yn=y*, yn∈𝔍xn imply y*∈𝔍x*).LetBCC(E) denote the sets of all the sets of all nonempty, bounded, closed, and convex subsets of E. For more details on multivalued maps see the books of Deimling [22] and Hu and Papageorgiou [23].An upper semicontinuous mapH:E→E is said to be condensing if for any subset B⊆E with α(B)≠0, we have α(H(B))<α(B), where α denotes the Kuratowski measure of noncompactness. It is easy to see that s completely continuous multivalued map is a condensing map.IfT is a uniformly bounded and analytic semigroup with infinitesimal generator A such that 0∈ρ(A) then it is possible to define the fractional power (-A)α, for 0<α≤1, as a closed linear operator on its domain D((-A)α). Furthermore, the subspace D((-A)α) is dense in X and the expression
(5)∥x∥α≔∥(-A)αx∥,x∈D((-A)α),
defines a norm on D((-A)α). Hereafter we represent by Xα the space D((-A)α) endowed with the norm ∥·∥α. Then the following properties are well known [24].Lemma 1 (see [24]).
Suppose that the preceding conditions are satisfied.(a)
Let0<α≤1. Then Xα is a Banach space.(b)
If0<γ<α≤1, then Xα↪Xγ and the imbedding is compact whenever the resolvent operator of A is compact.(c)
For everya>0, there exists a positive constant cα such that
(6)∥(-A)αT(t)∥≤cαtα,0<t≤a.
The key tool in our approach is following fixed-point theorem [21].Theorem 2 (see [21]).
LetX be a Hilbert space, and let U be an open, convex subset of X and u0∈U. If (a)
F:U¯→𝒫cd(X) has closed graph,(b)
F:U¯→𝒫cd(X) is condensing operator and F(U¯) is bounded set of X,then either(i)
F has fixed-point in U¯; or(ii)
there existu∈∂U and λ∈(0,1) such that u∈λF(u)+(1-λ){u0}.


## 3. Main Result

In the following, we will apply Theorem2 to study the existence of solutions for system (1).Definition 3.
A functionx:(-∞,b]→H is called a mild solution of system (1) if the following holds: let ϕ be 𝔉0 measurable H-valued stochastic processes x0=ϕ∈𝔅h on (-∞,0] and the integral equation
(7)x(t)=T(t)[ϕ(0)-f(0,ϕ)]+f(t,xt)+∫0tAT(t-s)f(s,xs)ds+∫0tT(t-s)(Bu)(s)ds+∫0tT(t-s)g(s)dw(s),fora.e.t∈J,x0=ϕ∈𝔅h,t∈J0
is satisfied, where
(8)g∈SG,x={g∈L2(J,H):g(t)∈G(t,xt),fora.e.t∈J}.Definition 4.
System (1) is said to be controllable on the interval J if for every continuous initial random process ϕ∈𝔅h, x1∈H, there exists a control u∈L2(J,U) such that the mild solution x(t) of (1) satisfies x(b)=x1.
To investigate the controllability of system (1), we use the following hypotheses.(H1)
A is the infinitesimal generator of an analytic semigroup of bounded linear operators T(t) in H, 0∈ρ(A), for t>0; there exist constants M1 such that |T(t)|≤M1.(H2)
The linear operatorW:L2(J,U)→H defined by
(9)Wu=∫0b‍T(b-s)Bu(s)ds
has an induced inverse operator W-1 which takes values in L2(J,U)/kerW and there exist positive constants M3,M4 such that |B|≤M3 and |W-1|≤M4.(H3)
There exist constants0≤c1<1,1/2<β<1, and c2≥0 such that l2c1∥(-A)-β∥2<1/5 and
(10)E|(-A)βf(t,u)|2≤c1∥u∥𝔅h2+c2,t∈J,u∈𝔅h,
where l=∫-∞0‍h(s)ds<+∞.(H4)
G:J×𝔅h→BCC(H); (t,ϕ)→G(t,ϕ) is measurable with respect to t for each ϕ∈𝔅h, Usc with respect to ϕ for each t∈J, and for each fixed ϕ∈𝔅h, the set
(11)SG,x={g∈L2(J,H):g(t)∈G(t,xt),fora.e.t∈J}
is nonempty.(H5)
E∥G(t,ϕ)∥22=sup{E∥v∥22:v∈G(t,ϕ)}≤p(t)ψ(∥ϕ∥𝔅h)2, t∈J, p∈L2(J,R+), ϕ∈𝔅h, and ψ:[0,+∞)→(0,+∞) is a continuous nondecreasing function. There exists a positive constant r such that
(12)rF*+M-r+k3ψ(3l2r+3q*)∫0b‍p(s)ds>1,
where
(13)F=5M12∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2)+5∥(-A)-β∥2c2+5b2βc2c1-β22β-1+25b2M12M32M42×[b2β-1c2c1-β22β-1E|x1|2+2M12E|ϕ(0)|2+2M12∥(-A)-β∥2vvvv×(c1∥ϕ∥𝔅h2+c2)+∥(-A)-β∥2c2+b2β-1c2c1-β22β-1],q*=∥ϕ∥𝔅h2+l2M12E|ϕ(0)|2,F*=F+k1q*+k2b2β-12β-1q*,M-=k1l2+k2b2β-1l22β-1,k1=15∥(-A)-β∥2c1+75b2M12M32M42∥(-A)-β∥2c1,k2=15bc1c1-β2+75b2M12M32M42,k3=5M12+25b2(M1M2M3M4)2.(H6)
The operatorF with values (F(x))(t)=f(t,xt), t∈J, F is completely continuous in C(J,H) and for any bounded set V⊆C(J,H), the set {t→f(t,xt):x∈V} is equicontinuous in C(J,H).(H7)
There exist constants1/2<β<1 and q>0 such that (-A)βf is continuous and for t∈J, ϕ1,  ϕ2∈𝔅h such that |(-A)βf(t,ϕ1)-(-A)βf(t,ϕ2)|≤q∥ϕ1-ϕ2∥𝔅h,  0<Q=2∥(-A)-β∥2q2+2c1-β2(1/((2β-1)))b2β-1q2l2<1.Lemma 5 (Lasota and Opial [25]).
LetI be a compact real interval and let E be a Banach space. Let G be a multivalued map satisfying (H4) and let Γ be a linear continuous mapping from L2(I,E) to C(I,E). Then the operator
(14)Γ∘SG:C(I,E)⟶BCC(C(I,E)),x⟶(Γ∘SG)(x)=Γ(SG,x)
is a closed graph operator in C(I,E)×C(I,E).Lemma 6 (see [26]).
Letv(·),r(·):[0,T]→[0,∞) be continuous functions. If r(·) is nondecreasing and there are constants θ>0, 0<τ<1 such that
(15)v(t)≤r(t)+θ∫0t‍v(s)(t-s)1-τds,t∈[0,T],
then
(16)v(t)≤eθnΓ(τ)ntnτ/Γ(nτ)∑j=0n-1‍(θTττ)jr(t),
for every t∈[0,T] and every n∈N such that nτ>1, and Γ(·) is the Gamma function.Now we consider the space𝔅b; let 𝔅b be the space of all H-valued stochastic processes x(t), -∞<t≤b, such that x0=ϕ∈𝔅h, let ∥·∥ be a seminorm in 𝔅b defined by
(17)∥x∥b=∥x0∥𝔅h+sups∈[0,b](E|x(s)|2)1/2,x∈𝔅b.Lemma 7 (see [27]).
Supposex∈𝔅b; then for t∈J, xt∈𝔅h. Moreover
(18)lE1/2|x(t)|2≤∥xt∥𝔅h≤lsups∈[0,t](E|x(s)|2)1/2+∥x0∥𝔅h,
where l=∫-∞0‍h(s)ds<+∞.Now, consider the multivalued map𝔏:𝔅b→2𝔅b defined by 𝔏x the set of ρ∈𝔅b such that
(19)ρ(t)={ϕ(t),t∈(-∞,0],T(t)[ϕ(0)-f(0,ϕ)]+f(t,xt)+∫0t‍AT(t-s)f(s,xs)ds+∫0t‍T(t-s)g(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,xb)-∫0b‍AT(b-s)f(s,xs)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη,t∈J,
where g∈SG,x.We will show that the operator𝔏 has fixed-points, which are then a solution of system (1). For ϕ∈𝔅h, we define ϕ- by
(20)ϕ-(t)={ϕ(t),-∞<t≤0,T(t)ϕ(0),0≤t≤b,
then ϕ-∈𝔅b. Set
(21)x(t)=y(t)+ϕ-(t),-∞<t≤b.
It is clear to see that x satisfies (7) if and only if y satisfies y0=0 and
(22)y(t)=-T(t)f(0,ϕ)+f(t,yt+ϕ-t)+∫0t‍AT(t-s)f(s,ys+ϕ-s)ds+∫0t‍T(t-s)g(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)vvv-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvvv-∫0b‍T(b-s)g(s)dw(s)}(η)dη,t∈J.
Let 𝔅b0={y∈𝔅b:y0=0∈𝔅h}. For any y∈𝔅b0,
(23)∥y∥b=∥y0∥𝔅h+sups∈[0,b]E1/2|y(s)|2=sups∈[0,b]E1/2|y(s)|2.
Thus (𝔅b0,∥·∥b) is a Banach space. Set 𝔅q={y∈𝔅b0:∥y∥b≤q} for some q≥0; then 𝔅q⊆𝔅b0 is uniformly bounded, for any y∈𝔅q; from Lemma 7, we have
(24)∥yt+ϕ-t∥𝔅h≤∥yt∥𝔅h+∥ϕ-t∥𝔅h≤lsup0≤s≤tE1/2|y(s)|2+∥y0∥𝔅h+lsup0≤s≤tE1/2|ϕ-(s)|2+∥ϕ-0∥𝔅h≤lq+∥ϕ∥𝔅h+lsup0≤s≤t|T(s)|E1/2|ϕ(0)|2≤l(q+M1E1/2|ϕ(0)|2)+∥ϕ∥𝔅h=q′.
Define the multivalued map 𝔏1:𝔅b0→2𝔅b0 defined by 𝔏1y the set of ρ-∈𝔅b0 such that
(25)ρ-(t)={0,t∈(-∞,0],-T(t)f(0,ϕ)+f(t,yt+ϕ-t)+∫0t‍AT(t-s)f(s,ys+ϕ-s)ds+∫0t‍T(t-s)g(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη,t∈J.Theorem 8.
Assume that hypotheses(H1)–(H7) hold; then system (1) is controllable on J.Proof.
We divide the proof into several steps.
Step   1. We show that there exists an open set U and there does not exist y∈∂U, for some λ∈(0,1), such that y∈λ𝔏1(y).
Letλ∈(0,1) and let y∈λ𝔏1y. Then there exists g∈SG,y such that
(26)y(t)=-λT(t)f(0,ϕ)+λf(t,yt+ϕ-t)+λ∫0t‍AT(t-s)f(s,ys+ϕ-s)ds+λ∫0t‍T(t-s)g(s)dw(s)+λ∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ¯b)vvv-∫0b‍AT(b-s)f(s,ys+ϕ¯s)dsvvv-∫0b‍T(b-s)g(s)dw(s)}(η)dη,t∈J.E|y(t)|2≤5M12E|f(0,ϕ)|2+5E|f(t,yt+ϕt)|2+5b∫0t‍[c1-β(t-s)1-β]2(c1∥ys+ϕ-s∥𝔅h2+c2)ds+5M12∫0t‍p(s)ψ(∥ys+ϕ-s∥𝔅h2)ds+25b2M12M32M42×{∫0b‍p(s)ψ(∥ys+ϕ-s∥𝔅h2)E|x1|2+2M12E|ϕ(0)|2vvv+2M12∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2)vvv+∥(-A)-β∥2(c1∥yt+ϕ-t∥𝔅h2+c2)vvv+∫0b‍[c1-β(b-s)1-β]2(c1∥ys+ϕ-s∥𝔅h2+c2)dsvvv+M12∫0b‍p(s)ψ(∥ys+ϕ-s∥𝔅h2)ds}≤5M12∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2)+5∥(-A)-β∥2(c1∥yt+ϕ-t∥𝔅h2+c2)+5b2βc2c1-β22β-1+5bc1c1-β2∫0t‍∥ys+ϕ-s∥𝔅h2(t-s)2-2βds+5M12∫0t‍p(s)ψ(∥ys+ϕ-s∥𝔅h2)ds+25b2M12M32M42×{∫0b‍p(s)ψ(∥ys+ϕ-s∥𝔅h2)E|x1|2+2M12E|ϕ(0)|2vvv+2M12∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2)vvv+∥(-A)-β∥2(c1∥yt+ϕ-t∥𝔅h2+c2)vvv+b2β-1c2c1-β22β-1+c1c1-β2∫0b‍∥ys+ϕ-s∥𝔅h2(b-s)2-2βdsvvv+M12∫0b‍p(s)ψ(∥ys+ϕ-s∥𝔅h2)ds}.
Since(27)∥yt+ϕ-t∥𝔅h≤lsup0≤s≤tE1/2|y(s)|2+∥ϕ∥𝔅h+lM1E1/2|ϕ(0)|2,∥y∥b=supt∈[0,b]{E|y(t)|2}1/2≤{l2sup0≤s≤b5M12∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2)+5∥(-A)-β∥2c2+5b2βc2c1-β22β-1+25b2M12M32M42×[b2β-1c2c1-β22β-1E|x1|2+2M12E|ϕ(0)|2+2M12∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2)+∥(-A)-β∥2c2+b2β-1c2c1-β22β-1]+[15∥(-A)-β∥2c1+75b2M12M32M42∥(-A)-β∥2c1]×[l2sup0≤s≤bE|y(s)|2+∥ϕ∥𝔅h2+l2M12E|ϕ(0)|2]+[15bc1c1-β2+75b2M12M32M42]×∫0b‍l2sup0≤s≤bE|y(s)|2+∥ϕ∥𝔅h2+l2M12E|ϕ(0)|2(b-s)2-2β+[5M12+25b2(M1M2M3M4)2]×∫0b‍p(s)ψ(3l2sup0≤s≤bE|y(s)|2+3∥ϕ∥𝔅h2+3l2M12E|ϕ(0)|2l2sup0≤s≤b∥(-A)-β∥2)ds}1/2={∫0b‍p(s)ψ(3l2sup0≤s≤bE|y(s)|2+3q*)F+k1(l2sup0≤s≤bE|y(s)|2+q*)vvv+k2∫0b‍l2sup0≤s≤bE|y(s)|2+q*(b-s)2-2βdsvvv+k3∫0b‍p(s)ψ(3l2sup0≤s≤bE|y(s)|2+3q*)ds}1/2,
where F,k1,k2,k3, and q* are defined in H5.
Consequently(28)∥y∥b2≤F+k1q*+k1l2∥y∥b2+k2b2β-12β-1(l2∥y∥b2+q*)+k3ψ(3l2∥y∥b2+3q*)∫0b‍p(s)ds=F*+M-∥y∥b2+k3ψ(3l2∥y∥b2+3q*)∫0b‍p(s)ds.
So(29)∥y∥b2F*+M-∥y∥b2+k3ψ(3l2∥y∥b2+3q*)∫0b‍p(s)ds≤1.
Then by H5 there exists r such that ∥y∥b2≠r; that is, ∥y∥b≠r1/2. Define the set: U={z∈ℬb0:∥z∥b<r1/2}. From the choice of U, we know that there does not exist y∈∂U, for some λ∈(0,1), such that y∈λ𝔏1(y).
Step   2.  𝔏1y is convex for each y∈𝔅b0.
In fact, ifρ-1, ρ-2 belong to 𝔏1y, then there exist g1, g2∈SG,y such that for each t∈J we have
(30)ρ-i(t)=-T(t)f(0,ϕ)+f(t,yt+ϕ-t)+∫0t‍AT(t-s)f(s,ys+ϕ-s)ds+∫0t‍T(t-s)gi(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)gi(s)dw(s)}(η)dη,t∈J.
Let λ∈[0,1]; we have
(31)(λρ-1+(1-λ)ρ-2)(t)=-T(t)f(0,ϕ)+f(t,yt+ϕ-t)vvvv+∫0t‍AT(t-s)f(s,ys+ϕ-s)dsvvvv+∫0t‍T(t-s)[λg1(s)+(1-λ)g2(s)]dw(s)vvvv+∫0t‍T(t-η)BW-1vvvv×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))vvvvvvv-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvvvvvvv-∫0b‍T(b-s)[λg1(s)+(1-λ)g2(s)]dw(s)}(η)dη.
Since SG,y is convex, we have λρ-1+(1-λ)ρ-2∈𝔏1y.
Step  3.
𝔏
1 maps bounded set into bounded set in 𝔅b0. Indeed, it is enough to show that there exists a positive constant Λ such that for each ρ-∈𝔏1y, y∈𝔅q={y∈𝔅b0:∥y∥b≤q} one has ∥ρ-∥b≤Λ. If ρ-∈𝔏1y, then there exist g∈SG,y, such that for each t∈J,
(32)ρ-(t)=-T(t)f(0,ϕ)+f(t,yt+ϕ-t)+∫0t‍AT(t-s)f(s,ys+ϕ-s)ds+∫0t‍T(t-s)g(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,xb)vvv-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvvv-∫0b‍T(b-s)g(s)dw(s)}(η)dη.
So, from Lemma 1(c) for t∈J(33)E|ρ¯(t)|2=E|-∫0b‍T(b-s)g(s)dw(s)}-T(t)f(0,ϕ)+f(t,yt+ϕ-t)+∫0t‍AT(t-s)f(s,ys+ϕ-s)ds+∫0t‍T(t-s)g(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,xb)-∫0b‍AT(b-s)f(s,y-s+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη|2≤5E|-T(t)f(0,ϕ)|2+5E|f(t,yt+ϕ-t)|2+5E|∫0t‍AT(t-s)f(s,ys+ϕ-s)ds|2+5E∥∫0t‍T(t-s)g(s)dw(s)∥22+5E|-∫0b‍T(b-s)g(s)dw(s)}(η)dη∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,xb)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη|2≤5M12∥(-A)-β∥2E|(-A)βf(0,ϕ)|2+5∥(-A)-β∥2E|(-A)βf(t,yt+ϕ-t)|2+5E(∫0t‍|(-A)1-βT(t-s)||(-A)βf(s,y-s+ϕ-s)|ds)2+5∫0t‍E∥T(t-s)g(s)∥22ds+5b∫0t‍E|-∫0b‍T(b-s)g(s)dw(s)}T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}|2(η)dη≤5M12∥(-A)-β∥2(c1∥ϕ∥2+c2)+5∥(-A)-β∥2(c1q′2+c2)+5∫0t‍|(-A)1-βT(t-s)|2ds∫0t‍E|(-A)βf(s,ys+ϕ-s)|2ds+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds+5bM12M32M42×∫0t‍E|∫0b‍T(b-s)g(s)dw(s)x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)|2(η)dη≤5M12∥(-A)-β∥2(c1∥ϕ∥2+c2)+5∥(-A)-β∥2(c1q′2+c2)+5∫0t‍(c1-β(t-s)1-β)2ds∫0t‍(c1∥ys+ϕ-s∥2+c2)ds+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds+5bM12M32M42×∫0t‍(∥∫0b‍T(b-s)g(s)dw(s)∥225E|x1|2+5M12E|ϕ(0)-f(0,ϕ)|2+5E|f(b,yb+ϕ-b)|2+5E|∫0b‍AT(b-s)f(s,ys+ϕ-s)ds|2+5E∥∫0b‍T(b-s)g(s)dw(s)∥22)(η)dη≤5M12∥(-A)-β∥2(c1∥ϕ∥2+c2)+5∥(-A)-β∥2(c1q′2+c2)+5c1-β212β-1b2β(c1q′2+c2)+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds+5bM12M32M42×∫0t‍{∫0t‍5|x1|2+5M12(2E|ϕ(0)|2+2∥(-A)-β∥2(c1∥ϕ∥2+c2))+5∥(-A)-β∥2(c1∥yb+ϕ-b∥2+c2)+5∫0b‍|(-A)1-βT(b-s)|2ds×∫0b‍E|(-A)βf(s,ys+ϕ-s)ds|2ds+5∫0b‍E∥T(b-s)g(s)∥22}(η)dη≤5M12∥(-A)-β∥2(c1∥ϕ∥2+c2)+5∥(-A)-β∥2(c1q′2+c2)+5c1-β2b2β2β-1(c1q′2+c2)+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds+5bM12M32M42×∫0t‍{+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds5|x1|2+5M12(2E|ϕ(0)|2+2∥(-A)-β∥2(c1∥ϕ∥2+c2))+5∥(-A)-β∥2(c1q′2+c2)+5∫0b‍(c1-β(b-s)1-β)2ds∫0b‍(c1∥ys+ϕ-s∥2+c2)ds+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds}(η)dη≤5M12∥(-A)-β∥2(c1∥ϕ∥2+c2)+5∥(-A)-β∥2(c1q′2+c2)+5c1-β2b2β2β-1(c1q′2+c2)+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds+5b2M12M32M42×{∫0b‍supx∈[0,q′]ψ(x2)∫0b‍p(s)ds5E|x1|2+5M12(2E|ϕ(0)|2+2∥(-A)-β∥2(c1∥ϕ∥2+c2))vvv+5∥(-A)-β∥2(c1q′2+c2)+5c1-β2b2β2β-1(c1q′2+c2)vvv+5M12supx∈[0,q′]ψ(x2)∫0b‍p(s)ds}=Λ2,
then for each ρ-∈L1(𝔅q), we have
(34)∥ρ-∥b≤Λ.
Step  4.𝔏1 has a closed graph.
Lety(n)→y*, ρ-n∈𝔏1y(n), and ρ¯n→ρ-*. We will prove that ρ-*∈𝔏1y*. Indeed, ρ-n∈𝔏1y(n) means that there exists gn∈SG,y(n), such that
(35)ρ-n(t)=-T(t)f(0,ϕ)+f(t,yt(n)+ϕ-t)+∫0t‍AT(t-s)f(s,ys(n)+ϕ-s)ds+∫0t‍T(t-s)gn(s)dw(s)+∫0t‍T(t-η)(Buy(n)(η))dη,t∈J,
where
(36)uy(n)(t)=W-1{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))vv-f(b,yb+ϕ-b)vv-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvv-∫0b‍T(b-s)gn(s)dw(s)}(t).
We must prove that there exists g*∈SG,y* such that
(37)ρ-*(t)=-T(t)f(0,ϕ)+f(t,yt*+ϕ-t)+∫0t‍AT(t-s)f(s,ys*+ϕ-s)ds+∫0t‍T(t-s)g*(s)dw(s)+∫0t‍T(t-η)(Buy*(η))dη,t∈J,
where
(38)uy*(t)=W-1{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb*+ϕ-b)-∫0b‍AT(b-s)f(s,ys*+ϕ-s)ds-∫0b‍T(b-s)g*(s)dw(s)}(t).
Set
(39)u-y(t)=W-1{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)vv-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds}(t).
Since f, W-1 are continuous, then u-y(n)(t)→u-y*(t) for t∈J,
(40)∥{∫0t‍ρ-n(t)+T(t)f(0,ϕ)-f(t,yt(n)+ϕ-t)-∫0t‍AT(t-s)f(s,ys(n)+ϕ-s)ds-∫0t‍T(t-η)(Buy(n)(η)dη}-{∫0t‍ρ-*(t)+T(t)f(0,ϕ)-f(t,yt*+ϕ-t)-∫0t‍AT(t-s)f(s,ys*+ϕ-s)ds-∫0t‍T(t-η)(Buy*(η)(η)dη}∥⟶0,asn⟶∞.
Consider the linear continuous operator
(41)Γ:L2(J,H)⟶C(J,H),g⟶Γ(g)(t)=∫0t‍T(t-s)g(s)dw(s)-∫0t‍T(t-s)BW-1[∫0b‍T(b-τ)g(τ)dw(τ)]ds.
From Lemma 5, it follows that T∘SG is a closed graph. Moreover, we have
(42)ρ-n(t)+T(t)f(0,ϕ)-f(t,yt(n)+ϕ-t)-∫0t‍AT(t-s)f(s,ys(n)+ϕ-s)ds-∫0t‍T(t-η)Buy(n)(η)dη∈Γ(SG,y(n)).
Since y(n)→y*, it follows from Lemma 5 that
(43)ρ-*(t)+T(t)f(0,ϕ)-f(t,yt*+ϕ-t)vvvv-∫0t‍AT(t-s)f(s,ys*+ϕ-s)dsvvvv-∫0t‍T(t-η)(Bu-y*)(η)dη=∫0t‍T(t-s)g*(s)dw(s)vvvv-∫0t‍T(t-s)BW-1[∫0b‍T(b-τ)g(τ)dw(τ)]ds
for some g*∈SG,y*.
Step  5. Multivalued function 𝔏1:U¯→𝒫cp,cv(𝔅b0) is a condensing operator.
From the decomposition of operator𝔏1, we define two operators T,S on space 𝔅b0. For operator T, T:U¯→𝒫cp,cv(𝔅b0), for each y, Ty defined by the set of h1(t)∈𝔅b0 such that
(44)h1(t)={0,t∈(-∞,0],-T(t)f(0,ϕ)+f(t,yt+ϕ-t)+∫0t‍AT(t-s)f(s,ys+ϕ-s)dst∈J.
For operatorS, S:U¯→𝒫cp,cv(𝔅b0), for each y, Sy defined by the set of h2(t)∈𝔅b0 such that
(45)h2(t)={0,t∈(-∞,0],∫0t‍T(t-s)g(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)vvv-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvvv-∫0b‍T(b-s)g(s)dw(s)}(η)dη,t∈J.
So, operator 𝔏1y=Ty+Sy. Next, we will prove that operator T is contraction operator and S is completely continuous operator.
First, we prove operatorT is contractive. For each y1,y2∈𝔅b0, we have
(46)E|h11(t)-h12(t)|2=E|∫0t‍AT(t-s)f(s,ys2+ϕ-s)dsf(t,yt1+ϕ-t)-f(t,yt2+ϕ-t)vvv+∫0t‍AT(t-s)f(s,ys1+ϕ-s)dsvvv-∫0t‍AT(t-s)f(s,ys2+ϕ-s)ds|2≤2E|f(t,yt1+ϕ-t)-f(t,yt2+ϕ-t)|2+2E|∫0t‍AT(t-s)f(s,ys1+ϕ-s)dsvvvvv-∫0t‍AT(t-s)f(s,ys2+ϕ-s)ds|2≤2∥(-A)-β∥2q2∥yt1-yt2∥2+2E|∫0t‍(-A)1-βT(t-s)×[(-A)βf(s,ys1+ϕ-s)-(-A)βf(s,ys2+ϕ-s)]ds∫0t‍|2≤2∥(-A)-β∥2q2∥yt1-yt2∥2+2∫0t‍|(-A)1-βT(t-s)|2ds×∫0t‍E|(-A)βf(s,ys1+ϕ-s)-(-A)βf(s,ys2+ϕ-s)|2ds≤2∥(-A)-β∥2q2∥yt1-yt2∥2+2∫0t‍(c1-β(t-s)1-β)2ds∫0t‍q2∥yt1-yt2∥2ds≤2∥(-A)-β∥2q2∥yt1-yt2∥2+2c1-β212β-1b2β-1q2sups∈[0,t]∥yt1-yt2∥2≤[2∥(-A)-β∥2q2+2c1-β212β-1b2β-1q2l2]∥y1-y2∥2=Q∥y1-y2∥2.
So operator T is contraction operator.
Next, we will prove that operatorS is completely continuous.
Fort1,t2∈J, 0<t1<t2<b, if y∈U¯, h2∈Sy, then there exist f∈SF,y such that for arbitrary t∈[0,b],
(47)h2(t)=∫0t‍T(t-s)g(s)dw(s)+∫0t‍T(t-η)BW-1×{∫0b‍T(b-s)g(s)dw(s)x1-T(b)(ϕ(0)-f(0,ϕ))vvv-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvvv-∫0b‍T(b-s)g(s)dw(s)}(η)dη,t∈J.
Thus(48)E|h2(t2)-h2(t1)|2=E|∫0t2‍(T(t2-s)-T(t1-s))g(s)dw(s)+∫t1t2‍T(t1-s)g(s)dw(s)+∫0t2‍(T(t2-η)-T(t1-η))×BW-1{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη+∫t1t2‍T(t1-η)BW-1×{∫0t‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη|2≤4E|∫0t2‍(T(t2-s)-T(t1-s))g(s)dw(s)|2+4E|∫t1t2‍T(t1-s)g(s)dw(s)|2+4E|-∫0b‍T(b-s)g(s)dw(s)}(η)dη∫0t2‍(T(t2-η)-T(t1-η))BW-1vvvvvv×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη|2+4E|∫t1t2‍T(t1-η)BW-1vvvvvv×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))vvvvvvvvvv-f(b,yb+ϕ-b)vvvvvvvvvv-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvvvvvvvvvv-∫0b‍T(b-s)g(s)dw(s)}(η)dη|2≤4E∥∫0t2‍(T(t2-s)-T(t1-s))g(s)dw(s)∥22+4E∥∫t1t2‍T(t1-s)g(s)dw(s)∥22+4E|∫0t2‍(T(t2-η)-T(t1-η))BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη|2+4E|∫t1t2‍T(t1-η)BW-1×{∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)}(η)dη|2≤4∫0t2‍|T(t2-s)-T(t1-s)|2E∥g(s)∥22ds+4∫t1t2‍|T(t1-s)2|E∥g(s)∥22dsvvvvvvv+4b∫0t2‍|T(t2-η)-T(t1-η)|2M32M42vvvvvvv×E|∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)-∫0b‍AT(b-s)f(s,ys+ϕ-s)ds-∫0b‍T(b-s)g(s)dw(s)|2(η)dη+4(t2-t1)∫t1t2‍|T(t1-η)|2M32M42×E|∫0b‍x1-T(b)(ϕ(0)-f(0,ϕ))-f(b,yb+ϕ-b)vvvv-∫0b‍AT(b-s)f(s,ys+ϕ-s)dsvvvv-∫0b‍T(b-s)g(s)dw(s)|2(η)dη≤4∫0t2‍|T(t2-s)-T(t1-s)|2E∥g(s)∥22ds+4∫t1t2‍|T(t1-s)|2E∥g(s)∥22ds+4b∫0t2‍|T(t2-η)-T(t1-η)|2M32M42×{∫0b‍5E|x1|2+5M12×(2E|ϕ(0)|2+2∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2))vvv+5∥(-A)-β∥2(c1∥yb+ϕ-b∥𝔅h2+c2)vvv+5b∫0b‍E|AT(b-s)f(s,ys+ϕ-s)|2dsvvv+5∫0b‍E∥T(b-s)g(s)∥22ds}(η)dη+4(t2-t1)M32M42∫t1t2‍|T(t1-η)|2×{∫0b‍5E|x1|2+5M12×(2E|ϕ(0)|2+2∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2))+5∥(-A)-β∥2(c1∥yb+ϕ-b∥𝔅h+c2)+5b∫0b‍E|AT(b-s)f(s,ys+ϕ-s)|2ds+5∫0b‍E∥T(b-s)g(s)∥22ds}(η)dη≤4∫0t2‍|T(t2-s)-T(t1-s)|2E∥g(s)∥22ds+4∫t1t2‍|T(t1-s)|2E∥g(s)∥22ds+4b∫0t2‍|T(t2-η)-T(t1-η)|2M32M42×{∫0b‍5E|x1|2+5M12×(2E|ϕ(0)|2+2∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2))+5∥(-A)-β∥2(c1∥yb+ϕ-b∥𝔅h2+c2)+5b∫0b‍|(-A)1-βT(b-s)|2×(c1∥ys+ϕ-s∥𝔅h2+c2)ds+5M12∫0b‍E∥g(s)∥22ds}(η)dη+4(t2-t1)M32M42∫t1t2‍|T(t1-η)|2×{∫0b‍5E|x1|2+5M12×(2E|ϕ(0)|2+2∥(-A)-β∥2(c1∥ϕ∥𝔅h2+c2))+5∥(-A)-β∥2(c1∥yb+ϕ-b∥𝔅h2+c2)+5b∫0b‍|(-A)1-βT(b-s)|2(c1∥ys+ϕ-s∥𝔅h2+c2)ds+5M12∫0b‍E∥g(s)∥22ds}(η)dη.
The right-hand side of the previous inequality is independent of y and tends to zero as t2→t1. It is easy to prove that S maps bounded set into bounded set, so operator S is completely continuous.
Hence, it follows from Theorem2 that the operator 𝔏1 has a fixed-point y*∈𝔅b0. Let x(t)=y*(t)+ϕ-(t), t∈(-∞,b]. Then x is a fixed-point of the operator 𝔏 which is a mild solution of problem (1); then system (1) is controllable on J.Remark 9.
Sakthivel et al. [28] considered approximate controllability for nonlinear stochastic systems with resolvent operators and unbounded delay. Approximate controllability results are being established for a class of nonlinear stochastic differential equations with resolvent operators in a real separable Hilbert spaces by using the resolvent operators and fixed-point technique.


## 4. Conclusion

In this paper, we have investigated the controllability of stochastic neutral functional differential inclusions in Hilbert spaces. Based on a fixed-point theorem with regard to condensing map, sufficient conditions for the controllability of the stochastic neutral functional differential inclusions have been derived.



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*Source: 419156-2013-05-28.xml*