# Approximate Controllability of Semilinear Neutral Stochastic Integrodifferential Inclusions with Infinite Delay

Authors: Meili Li; Man Liu
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

The approximate controllability of semilinear neutral stochastic integrodifferential inclusions with infinite delay in an abstract space is studied. Sufficient conditions are established for the approximate controllability. The results are obtained by using the theory of analytic resolvent operator, the fractional power theory, and the theorem of nonlinear alternative for Kakutani maps. Finally, an example is provided to illustrate the theory.


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## Body

## 1. Introduction

Controllability is an important concept which plays a vital role in many areas of applied mathematics. For the last decades, many authors established sufficient conditions for the controllability of nonlinear systems in Banach spaces; see [1–3]. Most of the controllability results for nonlinear systems concern the so-called semilinear control systems that consist of a linear part and a nonlinear part. Chang et al. [4] investigated the controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces.Integrodifferential equations have been recently proved to be strong tools in the modeling of many phenomena arising from many fields such as electronics. The theory of integrodifferential inclusions in deterministic cases may be found in several papers and monographs; for example, see [5–8]. In fact, many deterministic models often fluctuate because of noise, so we must move from deterministic control problems to stochastic control problems. The applications of stochastic differential equations and stochastic delay differential equations have attracted great interest; see [9–13]. In particular, Balasubramaniam and Ntouyas [14] have studied the controllability of the following neutral stochastic functional differential inclusions with infinite delay in abstract space:(1)dxt-ft,xt∈Axt+Butdt+Gt,xtdωt,t∈0,b,xt=ϕt∈L2Ω,B,for  a.et∈-∞,0,where f:[0,b]×B→H and G:[0,b]×B→PLQK,H are, respectively, measurable and multivalued measurable mapping. They achieved controllability for (1) by assuming the semigroup Tt is compact and uniformly bounded and applying the theory of fractional power operators.The concept of complete controllability is usually too strong which has limited applicability. Compared with complete controllability, approximate controllability is a weaker concept and it is completely adequate in applications; see [15–17]. Recently, Mokkedem and Fu [18] have studied the approximate controllability of the following semilinear neutral integrodifferential systems with finite delay:(2)ddtxt+ft,xt=-Axt+∫0tγt-sxsds+Gt,xt+But,t∈0,T,x0=ϕ,t∈-r,0,where f:[0,T]×Cα→Hα+β and G:0,T×Cα→H. They studied the problem by applying the theory of fractional power operators and α-norm. Meanwhile, they supposed that the operator -A,D-A generates a compact analytic semigroup on H and the corresponding linear system of (2) is approximately controllable. They established sufficient conditions of approximate controllability for the neutral integrodifferential control system in Hilbert space under a resolvent condition; in particular, they did not require that the resolvent operator Wt is compact for t>0.In fact, neutral stochastic integrodifferential inclusions arise in many areas for applied mathematics. The approximate controllability for neutral stochastic integrodifferential inclusions is also an important and interesting topic for mathematics, but it is also a difficult problem. Meanwhile, few researches have been done on the approximate controllability for semilinear neutral stochastic integrodifferential inclusions with infinite delay. Motivated by [14, 18], we will show the approximate controllability of the following semilinear neutral stochastic integrodifferential inclusions with infinite delay in a Hilbert space:(3)dxt+Ft,xt∈-Axt+But+∫0tγt-sxsdsdt+Gt,xtdωt,t∈J≔0,T,xt=ϕt∈L2Ω,Bα,for  a.et∈J0≔-∞,0,where -A is the infinitesimal generator of an analytic semigroup of bound linear operator Stt≥0, on a separable Hilbert space H with inner product ·,· and norm ·. The control function u· takes values in L2J,U of admissible control functions for a separable Hilbert space U and B is a bounded linear operator from U into H. γ· is a family of closed linear operators to be specified later. Let K be another separable Hilbert space with inner product ·,·K and norm ·K. Suppose ωtt≥0 is a given K-valued Wiener process in a finite trace nuclear covariance operator Q≥0. F:J×Bα→Hα+β and G:J×Bα→P(LQ(K,H)) are, respectively, measurable and multivalued measurable mapping, where P(LQ(K,H)) is the family of all nonempty subsets of LQ(K,H) and LQK,H denotes the space of all Q-Hilbert-Schmidt operators from K into H. The histories xt:-∞,0→Hα,xtθ=xt+θ,θ≤0, belong to an abstract phase space Bα.The aim of the present work is to investigate the approximate controllability for (3) by using the resolvent operator theory. Because the nonlinear terms involve frequently special derivatives in many practical models and the history variables of the functions F and G are only defined on C-r,0;H1/2, we cannot discuss the problem on the whole space H. We also suppose that there exists a semigroup which generated by -A,D-A is compact analytic on H so that the resolvent Wt is also analytic. So, we restrict this integrodifferential inclusion in the Hα and demonstrate the existence of mild solutions by applying ·α and the fractional power theory and then prove the approximate controllability for (3) in space H. In particular, we do not require that the resolvent operator W(t) be compact for t>0 which differs greatly from [14].The whole paper is arranged as follows: in Section2, we introduce some concepts, hypotheses, and basic results about resolvent operator and approximate controllability. Section 3 is devoted to studying the existence of mild solution of (3) and proving the approximate controllability. The application of our theoretical results is given in Section 4.


## 2. Preliminaries

The complete probability space, denoted byΩ,F,P, is furnished with a complete family of right continuous increasing sub-σ-algebras Ft,t∈J satisfying Ft∈F. Let the complete orthonormal basis {en}n≥1 in K be a bounded sequence of nonnegative real numbers λn such that Qen=λnenn=1,2,3,…,m and ω(t)=∑n=1∞λnωn(t)en, where ωntn=1∞ are independent one-dimensional standard Wiener processes. We assume that Ft=σωs:0≤s≤t is the sub-σ-algebra generated by ω and FT=F. Let LK,H be the space of all bounded linear operators from K into H with the usual operator norm ·. For ϕ∈LK,H define(4)ϕQ2=TrϕQϕ∗=∑n=1∞λnϕen2<∞.Then ϕ is called a Q-Hilbert-Schmidt operator. Let ρ(A) be the resolvent set of the operator A and 0∈ρA; then the fractional power operator Aα, 0<α≤1, is a closed linear operator on DAα in H and the expression(5)xα=Aαx,x∈DAαdefines a norm on DAα. Denoting the space DAα,·α by Hα, then Hα is a Banach space for each 0<α≤1. For 0<β<α≤1, the imbedding Hα→Hβ is compact. Meanwhile, Rλ,-A=λI+A-1, the resolvent operator of -A, is compact. Let L(Hα,Hβ) be the space of all bounded linear operators from Hα into Hβ with the norm ·α,β and H0=H. Let Cα≔C-∞,0;Hα be the Banach space of continuous functions from J0 to Hα with the norm(6)xCα=supθ∈-∞,0Aαxθ,x∈Cα≔C-∞,0;Hα,and the Banach space of continuous functions from -∞,T to Hα is defined by C-∞,T;Hα with the norm(7)xC=supθ∈-∞,TAαxθ,x∈C-∞,T;Hα.Keep it simple; letAαWtx=WtAαx, for any 0≤α≤1,x∈DAα. Let B be a F0-measurable function from J0 into H endowed with a seminorm ·B; we assume that B satisfies the following conditions:(a)
If:x∈-∞,b→H, b>0, is continuous on 0,b and x0 in B, then for every t∈0,b the following conditions hold:
(1)
xt is in B;(2)
xt≤H1xtB;(3)
xtB≤Ktsup⁡xs:0≤s≤t+Ntx0B, where H1 is a positive constant and Kt and Nt:0,∞→0,∞ are continuous and independent of x·.(b)
For the functionx· in (a), xt is a B-valued function on 0,b.(c)
The spaceB is complete.We define the set of all elements inB which takes values in space Hα by Bα. Since Hα is still a Banach space, we will assume that the subspace Bα also satisfies the following conditions:(a′)
If:x∈-∞,b→Hα,  b>0, is continuous on 0,b and x0 in Bα, then for every t∈0,b the following conditions hold:
(1)
xt is in Bα;(2)
xtα≤H1xtBα;(3)
xtBα≤K(t)sup⁡Aαxs:0≤s≤t+Ntx0Bα, where H1 is a positive constant and Kt and N(t):0,∞→0,∞ are continuous and independent of x·.(b′)
For the functionx· in a′, xt is a Bα-valued function on 0,b.(c′)
The spaceBα is complete.LetL2Ω,F,P;H=L2Ω,H be the collection of all strongly measurable, square-integrable K-valued stochastic variables with the norm ·L2=(Ex·,ωH2). The setting xt=xt+sω:s∈-∞,0 is the collection of all Bα-valued stochastic process and let L2Ω,F,P;Hα=L2Ω,Hα be the collection of all strongly measurable, square-integrable K-valued stochastic variables with the norm ·L2,α=(EAαx·,ωH2).The spaceC-∞,T,L2FΩ;Bα consists of measurable and Ft-adapted processes such that ϕ∈Bα and the restriction x:J→L2FΩ;Bα is continuous. The closed subspace of all continuous process x that belongs to C-∞,T,L2FΩ;Bα is defined by Z and the seminorm in Z is defined by(8)xZ2=supt∈JxtBα2,where(9)xtBα≤N¯EϕBα+K¯sup⁡EAαxs:0≤s≤T,N¯=supt∈JNt, and K¯=supt∈JKt. It is not hard to prove that Z is a Banach space.Now, we introduce some facts on multivalues analysis:(i)
If, for eachx∈H, the set Fx is a nonempty, closed subset of H and if, for each open set G of H containing Fx, there exists an open neighborhood c of x such that Fc⊆G, then F is upper semicontinuous (u.s.c.) on H.(ii)
If, for every bounded subsetB⊆H, FB is relatively compact, then F is completely continuous.(iii)
If the multivalued mapF is completely continuous with nonempty compact values, then F is upper semicontinuous (u.s.c.) if and only if F has a closed graph (i.e., xn→x,yn→y,   yn∈Fxn imply y∈Fx).For eachx∈L2LQK,H, define the set of selections of G by(10)g∈NG,x=g∈L2LQK,H:gt∈Gt,xtfor  a.e.t∈J.Definition 1 (see [19]).
A family of bounded linear operatorsWt∈LH for t∈J is called resolvent operator for(11)ddtxt=-Axt+∫0tγt-sxsds,t∈J,x0=x0∈H,if(i)
W0=I and Wt≤N1eςt for some N1≥0, ς∈R;(ii)
for allx∈H, Wt is strongly continuous in t on J;(iii)
Wt∈LY, for t∈J, where Y is the Banach space from D-A endowed with the graph norm. Moreover for x∈Y,W·x∈C1J;H∩CJ;Y and for t≥0, the following formula holds:(12)ddtWtx=-AWtx+∫0tγt-sWsxds=-WtAx+∫0tWt-sγsxds.
See from [18–20] that the resolvent operator Wt for the above linear system exists which is given by W0=I, and(13)Wtx=12πi∫ΓeλtλI+A-γ∗λ-1xdλ,t>0,where Γ is contour of the type used to obtain an analytic semigroup. W(t) is also analytic and there exists N, Nα>0, such that(14)Wt≤N,AαWt≤Nαtα,s∈J,0≤α≤1.Lemma 2 (see [21, Lemma  2.3]).
W
(
t
) is continuous for t>0 in the uniform operator topology of L(H).Lemma 3 (see [22, Lemma  2.3]).
A
W
(
t
) is continuous for t>0 in the uniform operator topology of L(H).Definition 4.
The multivalued mapG:J∗Bα→P(H) is said to be L2-Carathéodory if(i)
t⊢G(t,v) is measurable for each v∈Bα;(ii)
v⊢G(t,v) is u.s.c. for almost all t∈J;(iii)
for eachq>0, there exists hq∈L1(J,R+) such that(15)Gt,v2=sup⁡Eg2,g∈Gt,v≤hqt∀vBα2≤q,for  a.et∈J.Lemma 5 (see [23]).
LetI be a compact interval and Y a Hilbert space. Let G be an L2-Carathéodory multivalued map with NG,x≠Ø and let Θ be a linear continuous mapping from L2(I,Y) to C(I,Y). Then the operator(16)Θ∘NG:CI,Y⟶Pb,cl,cvCI,Y,x⟼Θ∘NGx=ΘNG,xis a closed graph operator in C(I,Y)×C(I,Y). Here Pb,cl,cv(C(I,Y)) denotes the family of nonempty, bound, close, compact convex subset of C(I,Y).Definition 6.
AFt-adapted stochastic process x(t)∈C((-∞,T];Hα) is the mild solution of (3) if x(·) is a continuous stochastic process on [0,T), the function AW(t-s)F(s,xs) is integrable for each s∈[0,t), and g∈NG,x is a selection of G(t,xt) such that(17)xt=ϕt∈L2Ω,Bα,t∈J0,Wtϕ0+F0,ϕ-Ft,xt+∫0tAWt-sFs,xsds+∫0tWt-sBusds+∫0tWt-sgsdωs-∫0tWt-s∫0sλs-vFv,xvdvds,t∈J.Theorem 7 (nonlinear alternative for Kakutani maps).
LetY be a Hilbert space, C a closed convex subset of Y, ℧ an open subset of C, and 0∈℧. Suppose that Φ:℧¯→Pc,cv(C) is an upper semicontinuous compact map; here Pc,cv(C) denotes the family of nonempty, compact convex subsets of C. Then either(i)
Φ has a fixed point in ℧¯, or(ii)
there arev∈∂℧ and λ∈(0,1) with v∈λΦ(v).Definition 8.
Equation (3) is said to be approximately controllable on the interval J if R(T,ϕ) is dense in H; that is,(18)RT,ϕ¯=H,where R(T,ϕ)={x(T,ϕ,u),u(·)∈L2(I;U)}.


## 3. Approximate Controllability

We introduce the following operators.(1)
The controllability operator Grammian is as follows:(19)ΓT=∫0TWT-tBB∗W∗T-tdt.(2)
The resolvent operator is as follows:(20)Rλ,ΓT=λI+ΓT-1,where B∗ and W∗(t) are the adjoints of the operators B and W(t), respectively. We also assume that the operator R(λ,ΓT) satisfies the following hypothesis:(21)H0:λRλ,ΓT⟶0asλ⟶0+in  the  strong  operator  topology.In this section, we discuss the approximate controllability of (3). Firstly, we compare approximate controllability of the semilinear equation (3) with approximate controllability of the associated linear system. Hypothesis (H0) is equivalent to the fact that the linear control system corresponding to (3)(22)xt=ϕt,t∈J0,ddtxt=-Axt+∫0tγt-sxsds+But,t∈J,is approximately controllable on J. For further details for the linear part of semilinear system, we can look up [24]. Secondly, we demonstrate that, for any given xT∈Hα and λ∈0,1, we choose a proper control uλ to prove that there exists a mild solution xλ(·,ϕ,u)∈C((-∞,T];Hα) of (3). At last, we prove that xλ(T)→xT in Hα which implies that (3) is approximately controllable on J. Assume α∈(0,1) is fixed:(H1)
B is a bounded linear operator from U to H and B≤M.(H2)
For eacht∈J, there exists a constant β∈(0,1) with 1/2≤α+β≤1, such that γ(t)∈L(Hα+β,H). And there exists a number M1>0, such that(23)γtα+β,0≤M1fort∈J.(H3)
The functionF:J×Bα→Hα+β is completely continuous. Moreover, there exists a positive, nondecreasing function f(·)∈L1(R) such that(24)Ft,ϕα+β2≤fϕBα2.(H4)
G:J×Bα→P(LQ(K,H)) is an L2-Carathéodory function.(H5)
There exists a continuous nondecreasing functionψ:R+→(0,∞),P∈L2(J,R+) such that(25)Gt,vQ2=sup⁡gQ2:g∈Gt,v≤PtψvBα2;for  almost  allt∈J,v∈Bα.For anyxT∈Hα, let x1=xT and λ∈(0,1], and we take the control function uλ(t,x), simply denoted by u(t), as follows:(26)ut=B∗W∗T-tRλ,ΓTx1-WTϕ0+F0,ϕ+FT,xT-∫0TAWT-sFs,xsds-∫0TWT-sgsdωs+∫0TWT-s∫0sγs-vFv,xvdvds,t∈J.We define the operatorPλ on C-∞,T;Hα by using this control as follows:(27)Pλxt=ϕt,t∈J0,Wtϕ0+WtF0,ϕ-Ft,xt+∫0tAWt-sFs,xsds+∫0tWt-sgsdωs-∫0tWt-s∫0sγs-vFv,xvdvds+∫0tWt-sBusds,t∈J.Theorem 9.
Letϕ∈Hα. If the hypotheses H0–H5 are satisfied, then, for each 0<λ≤1, the operator Pλ has a fixed point in C[(-∞,T],Hα] provided that there exists a constant M∗>0 so that(28)M∗-C2+C4T2β-1/2β-1+C3+C5T1-2α/1-2αfM∗C1+C6+C7ψvZ∫0TP2sds1/2>1,C2+C4T2β-12β-1+C3+C5T1-2α1-2αfpp<1,foranyp>0,where(29)C1=2N¯EϕBα2+K¯14H12N2ϕBα2+14A-β2N2fϕBα2+98M4N4T2λ2x1α2+H12N2ϕBα2+A-β2N2fϕBα2+14A-β2fxTBα2+98M4N4T2λ2A-β2fxTBα2,C2=14N1-β2T,C3=14M12T3Nα2,C4=98M4N4T2λ2N1-β2T,C5=98M4N4T2λ2M12T3Nα2,C6=14TrQNα2T1-4α1-4α,C7=98M4N4T2λ2TrQNα2T1-4α1-4α.Proof.
LetBα′ be the space of all function x:(-∞,T]→Hα, such that x0∈Bα′. The restriction x:J→Hα is continuous and ·T is a seminorm in Bα′ which is defined by(30)xT=x0Bα+sup⁡Aαxs:0≤s≤T.
LetZT=C((-∞,T],L2(Ω,Bα′)). We define the multivalued map Φ:ZT→P(ZT) by Φx and the set of Pλ∈ZT such that(31)Pλt=ϕt,t∈J0,Wtϕ0+WtF0,ϕ-Ft,xt+∫0tAWt-sFs,xsds+∫0tWt-sgsdωs-∫0tWt-s∫0sγs-vFv,xvdvds+∫0tWt-sBusds,t∈J.We will prove that the operator Φ has a fixed point, which then is a solution of (3). For ϕ∈Z, let y(·):(-∞,T]→ZT be the function defined by(32)yt=ϕt,t∈J0,Wt+ϕ0,t∈J.
Setx(t)=z(t)+y(t),-∞<t≤T. It is clear that x satisfies (26) if and only if z(t) satisfies z0=0 and(33)zt=WtF0,ϕ-Ft,zt+yt+∫0tAWt-sFs,zs+ysds-∫0tWt-s∫0sγs-vFv,zv+yvdvds+∫0tWt-sgsdωs+∫0tWt-sBusds,t∈J.Let Bα′0={z0∈Bα′,z0=0∈Bα}. For any z∈Bα′0, we have(34)zT=z0Bα+sup⁡Aαzs:s∈J=sup⁡Aαzs:s∈J.So, if ZT0=C(J1,L2(Ω,Bα′0)), then (ZT0,·T) is a Banach space. For any q≥0, set(35)Bq=z∈ZT0:zT2≤q;then Bq⊆ZT0 is uniformly bounded, and we have(36)ytBα2≤K¯sup⁡EAαys:0≤s≤T+N¯EϕBα2≤2K¯2sup⁡Eys0≤s≤t2+2N¯2EϕBα2≤2K¯2N2ϕ02+2N¯2EϕBα2.For z∈B(q), we can get(37)zt+ytBα2≤2ztBα2+ytBα2≤4K¯2q+N2Aαϕ02+N¯2EϕBα2≔ρ.
Let the operatorΦ^: ZT0→P(ZT0) be defined by Φ^z, the set of P^λ∈ZT0, such that(38)P^λxt=0,t∈J0,WtF0,ϕ-Ft,xt+∫0tAWt-sFs,xsds+∫0tWt-sgsdωs+∫0tWt-sBusds-∫0tWt-s∫0sγs-vFv,xvdvds,x∈J.
We divide the proof into five steps.
Step  1. Φ^z is convex for each z∈ZT0.
IfP^1λ and P^2λ belong to Φ^z, then there exists g1, g2∈NG,z such that(39)P^iλt=WtF0,ϕ-Ft,xt+∫0tAWt-sFs,xsds+∫0tWt-sgisdωs+∫0tWt-sBusds-∫0tWt-s∫0sγs-vFv,xvdvds,i=1,2,t∈J.Let r∈[0,1], because the operators B, B∗, W∗ are linear, t∈J; we have(40)rP^1λ+1-rP^2λt=WtF0,ϕ-Ft,zt+yt+∫0tAWt-sFs,xsds+∫0tWt-srg1s+1-rg2sdωs+∫0tWt-sB∗W∗T-sRλ,ΓTx1-WTϕ0+FT,xT-∫0TAWT-sFs,zs+ysds+∫0TWT-s∫0sγs-vFv,zv+yvdvds-∫0TWT-srg1s+1-rg2sdωs.Since G has convex values, NG,z is convex. Then rP^1λ+(1-r)P^2λ∈Φ^z.
Step  2.
Φ
^
(
B
q
) is bounded.
The proof can be found in AppendixA.
Step  3. We prove that Φ^(Bq) is equicontinuous.
The proof can be found in AppendixB.
As a consequence of Steps2 and 3, we can find that Φ^: ZT0→P(ZT0) is compact multivalued map.
Step  4.
Φ
^ has a closed graph.
The proof can be found in AppendixC.
We will prove thatΦ^ has a fixed point.
Step  5. We will prove that there exists an open set ℧∈Z and x∈∂℧, such that x∉λΦ^x, for λ∈(0,1).
Letx∈λΦ^(x) for λ∈(0,1); then there exists g∈NG,x such that(41)xt=λWtϕ0+F0,ϕ-λFt,xt+λ∫0tAWt-sFs,xsds+λ∫0tWt-sBusds-λ∫0tWt-s∫0sλs-vFv,xvdvds+λ∫0tWt-sgsdωs;by hypotheses (H1)–(H5), for each t∈J, we have(42)Extα2≤7H12N2ϕBα2+7A-β2N2fϕBα2+49M4N4T2λ2x1α2+H12N2ϕBα2+A-β2N2fϕBα2+7A-β2fxtBα2+7N1-β2T∫0tfxsBα2t-s2β-1ds+7M12T3Nα2∫0tfxsBα2t-s2αds+49M4N4T2λ2A-β2fxTBα2+N1-β2T∫0TfxsBα2t-s2β-1ds+M12T3Nα2∫0TfxsBα2t-s2αds+7Aα-12TrQ∫0tAWt-s4ds1/2∫0tP2sψ2xsBα21/2ds+Aα-1249M4N4T2λ2TrQ∫0TAWt-s4ds1/2∫0TP2sψ2xsBα21/2ds.Then, we define the function v(t)=sup⁡xsBα2:0≤s≤t, t∈J. We have(43)vt≤2N¯EϕBα2+2K¯Extα2≤2N¯EϕBα2+2K¯7H12N2ϕBα2+7A-β2N2fϕBα2+49M4N4T2λ2x1α2+H12N2ϕBα2+A-β2N2fϕBα2+7A-β2fxtBα2+7N1-β2T∫0tfxsBα2t-s2β-1ds+7M12T3Nα2∫0tfxsBα2t-s2αds+49M4N4T2λ2A-β2fxTBα2+N1-β2T∫0TfxsBα2t-s2β-1ds+M12T3Nα2∫0TfxsBα2t-s2αds+7Aα-12TrQ∫0tAWt-s4ds1/2∫0tP2sψ2xsBα21/2ds+Aα-1249M4N4T2λ2TrQ∫0TAWt-s4ds1/2∫0TP2sψ2xsBα21/2ds≤C1+C2∫0tfxsBα2t-s2-2βds+C3∫0tfxsBα2t-s2αds+C4∫0TfxsBα2t-s2-2βds+C5∫0TfxsBα2t-s2αds+C6∫0tP2sψ2xsBα2ds1/2+C7∫0TP2sψ2xsBα2ds1/2,t∈J.If t∗∈J0, then v(t)=ϕBα2 and the previous inequality holds. Consequently,(44)vZ≤C1+C2+C4T2β-12β-1+C3+C5T1-2α1-2αfvZ+C6+C7ψvZ∫0TP2sds1/2⟹vZ-C2+C4T2β-1/2β-1+C3+C5T1-2α/1-2αfvZC1+C6+C7ψvZ∫0TP2sds1/2≤1.
Then, according to the hypotheses, there existsM∗ such that xZ≠M∗ and set ℧={u∈:uZ<M∗}. It is clear that there is no v∈∂℧ for λ∈(0,1) such that v∈λΦ^(v). So we deduce that Φ^ has a fixed point by Theorem 7 and (3) has a mild solution.
In the end, we prove that (3) is approximately controllable on J.Theorem 10.
Assume that the hypotheses (H0)–(H5) are satisfied, and the functions F and G are uniformly bounded in the space H; then (3) is approximately controllable on J.Proof.
Letx1=xT∈H, and xλ(·):(-∞,T]→H is a mild solution of (3) which is got from Theorem 9. From the definition of ΓT,xλ(·) satisfies(45)xλT=WTϕ0+F0,ϕ-FT,xTλ+∫0TAWT-sFs,xsλds-∫0TWT-s∫0sγs-vFv,xvλdvds+∫0TWT-sgsdωs+∫0TWT-sBB∗W∗T-sRλ,ΓTx1-WTϕ0+F0,ϕ-FT,xTλ+∫0TAWT-sFs,xsλds+∫0TWT-sgsdωs-∫0TWT-s∫0sγs-vFv,xvλdvdsds=WTϕ0+F0,ϕ-FT,xTλ+∫0TAWT-sFs,xsλds-∫0TWT-s∫0sγs-vFv,xvλdvds+∫0TWT-sgsdωs+ΓTRλ,ΓTx1-WTϕ0+F0,ϕ-FT,xTλ+∫0TAWT-sFs,xsλds+∫0TWT-sgsdωs-∫0TWT-s∫0sγs-vFv,xvλdvds=x1+ΓTRλ,ΓT-Ix1-R,where(46)R=WTϕ0+F0,ϕ-FT,xTλ+∫0TAWT-sFs,xsλds-∫0TWT-s∫0sγs-vFv,xvλdvds+∫0TWT-sgsdωs.Or we obtain that(47)ExλT-x12≤EΓTRλ,ΓT-Ix1-R2=EλRλ,ΓTx1-R2.Then, we have(48)Ex1-R2≤Ex1-WTϕ0+F0,ϕ+FT,xTλ-∫0TAWT-sFs,xsλds+∫0TWT-s∫0sγs-vFv,xvλdvds-∫0TWT-sgsdωs2≤6Ex12+WTϕ0+F0,ϕ2+FT,xTλ2+∫0TAWT-sFs,xsλds2+∫0TWT-sgsdωs2+∫0TWT-s∫0sγs-vFv,xvλdvds2.Now, for ϵ>0 small and α+ϵ<1, (49)Aα+ϵ∫0TAWT-sFs,xsλds≤∫0TN1+ϵ-βt-s1+ϵ-βdsFs,xsλα+β.The operator A-(α+ϵ):H→Hα is compactness and the function F is uniformly bounded. So(50)∫0TAWT-sFs,xsλds:λ∈0,1=A-α+ϵAα+ϵ∫0TAWT-sFs,xsλds:λ∈0,1is relatively compact in Hα and H. Then, there exists FT1∈H such that(51)∫0TAWT-sFs,xsλds2⟶FT1,asλ⟶0+.So, we can get FT2, FT3, and FT4 by the same way, such that(52)FT,xTλ2⟶FT2,∫0TWT-s∫0sγs-vFv,xvλdvds2⟶FT3,∫0TWT-sgsdωs2⟶FT4,respectively, as λ→0+ in space H. Clearly, x1-R2 is bounded, and there exists a positive constant M′′ such that E(x1-R2)≤M′′. According to (H0), we have(53)ExλT-x12≤EλRλ,ΓT2x1-R2⟶0asλ⟶0+.So xλ(T)→xT holds in H, and we obtain the approximate controllability of (3).
The proof is finished.


## 4. Example

In this section, we study the following neutral inclusion which arises in the study of heat flow in materials of the so-called retarded type [25, 26] as an application to Theorem 10: (54)∂∂tvt,x+∫-∞tas-t,zs,xds∈∂2∂x2vt,s+∫0tbt-s∂2∂x2vs,xds+Gt,vt-s,xdωt+ut,x,0≤x≤π,t∈J,vt,0=vt,π=0,t≥0,vt,x=φt,x,t∈J0,0≤x≤π,where v(t,x) represents the temperature of the point x at time t, and a(·,·), G(·,·), and b(·) will be described below.In the spaceH=L2[0,π], ω(t) stands for cylindrical Wiener process in H defined on a probability space (Ω,F,P). Define the operator A as Aξ=-∂2/∂x2ξ, where D(A)={v∈H:v,dv/dx are absolutely continuous, and (-d2/dx2)v∈H, v(0)=v(π)}.-
A generates a strongly continuous semigroup (S(t))t≥0 which is analytic, compact, and self-adjoint. Moreover, the resolvent operator R(λ,-A)=(λI+A)-1 is compact when it exists. The complete orthonormal set {zn}(n=1,2,3,…) of eigenvectors of A with zn=2/πsin⁡nx also exists. Then, the operator A1/2 is given by A1/2z=∑n=1∞nz,znzn on the space D[A1/2]=z·∈H:∑n=1∞nζ,znzn∈H and the following properties hold:(i)
Ifζ∈D(A), then Aζ=∑n=1∞n2〈ζ,zn〉zn.(ii)
Ifζ∈H, then A1/2ζ=∑n=1∞1/n〈ζ,zn〉zn.(iii)
The operatorA1/2 is given by A1/2ζ=∑n=1∞n〈ζ,zn〉zn.Then, rewrite the above system as follows:(55)dxt+Ft,xt∈-Axt+But+∫0tγt-sxsdsdt+Gt,xtdωt.Here we take α=β=1/2 and ε<0 and define the phase space(56)B=ϕ∈CJ0,H:limθ→-reδθϕθexists  inH,and let ϕB=supθ∈J0eδθϕθL2. Then (B,ϕB) is a Banach space which satisfies (a)–(c). Meanwhile, the subspace B1/2 is defined by(57)B1/2=ϕ∈CJ0,H1/2:limθ→-reδθϕθexists  inH1/2,endowed with the norm ϕB1/2=supθ∈J0eδθA1/2ϕθL2. Thus for (t,ϕ)∈J×B1/2, where ϕ(θ)(x)=ϕ(θ,x), (θ,x)∈J0×[0,π]. Clearly, it satisfies the axioms (a′)–c′. Let v(t)(x)=v(t,x) and define the functions F:J×B1/2→H,G:J×B1/2→P(LQ(K,H)) and γ(t) for the infinite delay as follows:(58)γt=btA,Ft,ϕx=∫-∞0∫0πaθ,x,ϕθydydθ,Gt,vx=∫-∞0u1t,x,θG1v,θxdθ.We assume that the following conditions hold:(i)
b(t)∈L1(R+)⋂C1(R+) with primitive B(t)∈Lloc1(R+). B(t) is nonpositive, nondecreasing and B(0)=1.(ii)
The functiona(·,·)∈C2 and there exists a function a1(·,·)∈L1((-∞,0]×R,R+) such that(59)∂2∂x2aθ,zx<a1θzx,c1=-12ε∫-∞0a12θdθ.(iii)
The functionu1(t,ξ,θ)≥0 is continuous in J×[0,π]×(-∞,0),(60)pt,x=∫-∞0u1θdθ<∞,Qt=∫0πp2t,xdx1/2<∞.(iv)
The functionG1(·) is continuous, 0≤G1≤ψeθvθ,·L2, for (θ,x)∈J0×[0,π], where ψ(·):[0,∞)→(0,∞) is continuous and nondecreasing.Moreover, it is clear thatB=I is a bounded and linear operator which satisfies (H1), and γ(t) satisfies condition (H2). By condition (ii), we have(61)Fϕx,zn=1n∫-∞0∫0π∂∂xax,ϕθydydθ,z¯nx=-1n2∫-∞0∫0π∂2∂x2ax,ϕθydydθ,znx,so we get R(F)⊂D(A) and(62)ϕθ2=∑n=1∞φ,zn2≤∑n=1∞n2φ,zn2≤φθ1/22.Then, we have(63)Fϕx12≤∫0π∫-∞0∂2∂x2aθ,ϕθxdθ2dx≤∫0π∫-∞0a1θ,xϕθxdydθ2dx≤∫0π∫-∞0a1θϕθx1/2dθ2dx≤π∫-∞0a12θdθ∫0π∫-∞0ϕθx1/22dθdx≤π∫-∞0a12θdθ∫-∞0e-2ξθdθe2ξθ∫0πϕθx1/22dx≤-12ξ∫-∞0a12θdθsup⁡eξθϕθL2,1/22=-12ξ∫-∞0a12θdθϕB1/22=c1ϕB1/22,which implies that Ft,ϕ2≤fϕB1/22 and satisfies the assumption (H3). Clearly, according to the definition of G, it satisfies the assumption (H4) and(64)Gt,ϕL2=∫0π∫-∞0u1t,x,θG1ϕθxdθ2dx1/2≤∫0π∫-∞0u1t,x,θψϕθ·L2dθ2dx1/2≤∫0π∫-∞0u1t,x,θψϕθL2,1/2dθ2dx1/2≤∫0π∫-∞0u1t,x,θψeξθϕθL2,1/2dθ2dx1/2=∫0πp2t,xds1/2ψϕB1/2=QtψϕB1/2,which satisfies the assumption (H5).To sum up, the conditions stated in Theorem10 have been satisfied in the above case; therefore the inclusion is approximately controllable.


## 5. Conclusion

In this paper, the approximate controllability of semilinear neutral stochastic integrodifferential inclusions with infinite delay has been studied. A new set of sufficient conditions are formulated for the approximately controllable (3). The family λ(t) are closed linear operators and the standard semigroup approaches cannot be used here. The sufficient conditions are established in Hilbert space by applying the theory of analytic resolvent operator, the fractional power theory, and the theorem of nonlinear alternative for Kakutani maps under the assumption that the corresponding linear inclusion is approximately controllable. Moreover, an application is provided to illustrate the obtained theoretical results.Finally, we have to point out here that, like most related work in the literature of distributed parameter control systems, the definition of controllability for inputs in an infinitely dimensional space is not so practical and there is no indication on how to find the controller to achieve the desired state for the system. We expect some important improvement on these issues to appear in the future work. As the impulsive perturbations and state-dependent delay are very common in many real life phenomena, in future we can study the approximate controllability of integrodifferential systems with impulsive effects and state-dependent delay.



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*Source: 420826-2015-12-20.xml*