\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 53, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/53\hfil Approximate controllability]
{Approximate controllability of neutral stochastic integro-differential
 systems with impulsive effects}

\author[M. Li, X. Li \hfil EJDE-2016/53\hfilneg]
{Meili Li, Xiang Li}

\address{Meili Li \newline
School of Science, Donghua University,
Shanghai 201620, China}
\email{stylml@dhu.edu.cn}

\address{Xiang Li \newline
School of Science, Donghua University,
Shanghai 201620, China}
\email{lixiangdhu@163.com}

\thanks{Submitted July 24, 2015. Published February 18, 2016.}
\subjclass[2010]{93B05, 34K50, 34A37}
\keywords{Approximate controllability; resolvent operator; impulsive effects;
\hfill\break\indent neutral stochastic integro-differential system;
  fractional power operator}

\begin{abstract}
 This paper studies the approximate controllability of neutral stochastic
 integro-differential systems with impulsive effects. Sufficient conditions
 are formulated and proved for the approximate controllability.
 The results are obtained by using the Nussbaum fixed point theorem and the
 theory of analytic resolvent operator. An example is given to show the
 applications of the proposed results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

 In this article, we study the approximate controllability of the following
 neutral stochastic integro-differential systems with impulsive effects
\begin{equation} \label{e1.1}
\begin{gathered}
 d[x(t)+F(t,x_t)]= [-Ax(t)+\int _0^{t}\gamma(t-s)x(s)\,ds+Bu(t)]dt
+G(t,x_t)dw(t),\\
 t\in J,\; t \neq t_k,\\
\Delta x(t)=I_k(x(t^-)), \quad t=t_k,\; k=1,2,3,\ldots, m , \\
x_0=\phi \in L_2(\Omega, C_{\alpha}),\quad t\in [-r,0] ,
\end{gathered}
\end{equation}
where $J=[0,T]$, $ \phi $ is $ \mathfrak{F_0} $-measurable and the linear
operator $-A$ generates an analytic semigroup on a separable Hilbert space $H$
with inner product $\langle\cdot,\cdot \rangle$ and norm $\|\cdot\|$.
$u(\cdot) \in L_2^{\mathfrak{F_t}}(J,U)$ is the control function where $U$
is a Hilbert space. $\gamma(\cdot)$ is a family of closed linear operators
to be specified later. $B$ is a bounded linear operator from $U$ into $H$.
Define the Banach space $D(A^{\alpha})$ with the norm
$\|x\|_{\alpha}=\|A^{\alpha}x\|$ for $x\in D(A^{\alpha})$, where $ D(A^{\alpha})$
denotes the domain of the fractional power operator $A^{\alpha}:H\to H$.
Let $ H_{\alpha}:=D(A^{\alpha})$ and $ C_{\alpha}=C([-r,0],H_{\alpha}) $
be the space of all continuous functions from $ [-r,0] $ into $H_{\alpha} $.
Define $K$ be an another separable Hilbert space. Suppose $w(t)$ is a given
$K$-valued wiener process with a finite trace nuclear covariance operator
$Q\geq 0 $. $F: J\times C_{\alpha}\to H_\alpha$,
$G: J\times C_{\alpha}\to L_2^{0}(K,H)$ and $I_{k}:H\to H $,
where $L_2^{0}(K,H)$ is the space of all $Q $-Hilbert-Schmidt operators
from $K$ into $H$. The collection of all strongly measurable, square integrable,
$C_{\alpha}$-valued random variables denoted by $L_2(\Omega, C_{\alpha})$.
The histories $x_t:\Omega \to C_{\alpha}$,
$t\in J$, which are defined by setting
$ x_t(\theta)=x(t+\theta)$, $\theta \in [-r,0]$. $\Delta x(t) $ denotes
the jump of $x$ at $t$, $\Delta x(t)=x(t^{+})-x(t^{-})=x(t^{+})-x(t)$.

 The concept of controllability is an important part of mathematical
control theory. Generally speaking, controllability 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.
Controllability problems for different kinds of dynamical systems have
been studied by several authors, see
 \cite{Dauer, Liu, Li, Naito, Sakthivel, Shen-1}. 

Dauer and Mahmudov \cite{Dauer}
established sufficient conditions for the controllability of stochastic
semi-linear functional differential equations in Hilbert spaces under the
assumption that the associated linear part of systems is approximately
controllable. They obtained the results by using the Banach fixed point
theorem and the fractional power theory. Sakthivel et al \cite{Sakthivel}
considered the approximate controllability issue for nonlinear impulsive
differential and neutral functional differential equations in Hilbert spaces.
Finally, they applied their results to a control system governed by a heat
equation with impulses. 

In \cite{Shen-1}, the authors studied the approximate
 controllability of stochastic impulsive functional system with infinite delay
in abstract space. They obtained some sufficient conditions with no compactness
 requirement imposed on the semigroup generated by the linear part of the system
by using the contraction mapping principle. Then they dropped the restriction
of the combination of system parameters with the help of the Nussbaum fixed
point theorem.

The theory of integro-differential systems has recently become an important
area of investigation, stimulated by their numerous applications to problems
from electronics, fluid dynamics, biological models.
In many cases, deterministic models often fluctuate due to noise,
which is random or at least to be so.
So, we have to move from deterministic problems to stochastic ones.

 Balachandran et al \cite{Balachandran} derived sufficient conditions for
the controllability of stochastic integro-differential systems in finite
dimensional spaces. Muthukumar and Balasubramaniam \cite{Muthukumar}
investigated the appromimate controllability of mixed stochastic
Volterra-Fredholm type integro-differential in Hilbert space by employing
the Banach fixed point theorem.

In recent years, the study of impulsive integro-differential systems has
received increasing interest, since dynamical systems involving impulsive
effects occur in numerous applications: the radiation of electromagnetic waves,
population dynamics, biological systems, etc. Subalakshmi and Balachandran
\cite{Subalakshmi} studied the approximate controllability properties of
nonlinear stochastic impulsive integro-differential and neutral stochastic
impulsive integro-differential equations in Hilbert spaces under the assumption
that the associated linear part of system is approximately controllable.
Moreover, Shen et al. \cite{Shen-2} obtained the complete controllability of
impulsive stochastic integro-differential systems by using Schaefer's fixed
point theorem.

Recently, Mokkedem and Fu \cite{Mokkedem} studied the approximate controllability
 of the following semi-linear neutral integro-differential equations with finite
delay
\begin{gather}
\frac{d}{dt}[x(t)+F(t,x_t)]= -Ax(t)+\int _0^{t}\gamma(t-s)x(s)\,ds+Bu(t)
+G(t,x_t),\quad t\in J, \nonumber \\
 x_0=\phi,\quad t\in [-r,0]. \label{e1.2}
\end{gather}
They assumed that the linear control system corresponding to system \eqref{e1.2}
\begin{equation} \label{e1.3}
\begin{gathered}
 \frac{d}{dt}x(t)= -Ax(t)+\int _0^{t}\gamma(t-s)x(s)\,ds+Bu(t), \quad t\in J, \\
 x(t)=\phi(t), \quad t\in [-r,0],
 \end{gathered}\
\end{equation}
is approximately controllable on $J$.
With the help of the theory of analytic resolvent operator, the authors defined
the mild solution of system \eqref{e1.2}, then they used the Sadovskii fixed
point theorem and the fractional power operator theorem
to prove the existence of solution. Then the authors obtained the approximate
controllability of semi-linear neutral integro-differential systems with
finite delay in Hilbert space. However, the authors did not consider the
stochastic and impulsive effects. Very recently, Yan and Lu \cite{Yan} studied
the approximate controllability of a class of impulsive partial stochastic
functional integro-differential inclusions with infinite delay in Hilbert
spaces of the form
\begin{gather}
 d[x(t)-G(t,x_t)]\in A[x(t)+\int_0^{t}h(t-s)x(s)ds]dt+Bu(t)dt+F(t,x_t)dw(t),
\nonumber \\
 t\in J=[0,b],\; t\neq t_k,\; k=1,2,\dots,  m, \nonumber\\
 x_0=\varphi \in\mathcal{B},  \label{e1.4}\\
 \Delta x(t_k)=I_k(x_{t_k}),\; k=1,2,\dots, m. \nonumber
 \end{gather}
They achieved the approximate controllability result for \eqref{e1.4}
 by imposing compactness assumption on the resolvent operator $\Phi(t)$,
they also assumed that the corresponding linear system of \eqref{e1.4} is
approximately controllable.

 The aim of the present work is to study the approximate controllability
for the system \eqref{e1.1} with the aid of the resolvent operator theory
and the fractional power theory. The resolvent operator is similar to the
semigroup operator for abstract differential equations in Banach spaces.
However, the resolvent operator does not satisfy semigroup properties.
In many practical models the nonlinear terms involve frequently spacial
derivatives, in this case, we can not discuss the problem on the whole space
$H$ because the history variables of the functions $F$ and $G$ are only
defined on $C([-r,0];H_\alpha)$. In order to study the controllability
for system \eqref{e1.1}, we first apply the theory of fractional power
operator and $\alpha$-norm. We also suppose that $(-A, D(-A))$ generates
a compact analytic semigroup on $H$ so that the resolvent operator $\Phi(t)$
is analytic. We point out here that we do not require that the resolvent
operator be compact which differs greatly from that in \cite{Yan}.
Then with the help of the Nussbaum fixed theorem, some sufficient conditions
will be obtained.

 This article is organized as follows. In section 2, we give the preliminaries
for the paper. In section 3, we consider the existence of mild solutions of
system  \eqref{e1.1} and provide the main result.
In section 4, an example is given to illustrate the applications
of the approximate controllability results.

\section{Preliminaries}


 In this article, the operator $-A$ is the infinitesimal generator of a compact
analytic semigroup $(S(t))_{t\geq0}$. $H_\alpha$ is the space
$(D(A^\alpha), \|\cdot\|_\alpha)$, $H_\alpha \subset H$. For each
$0< \alpha \leq 1$, $H_\alpha$ is a Banach space, $H_\alpha\to H_\beta$ for
$0<\beta<\alpha \leq 1$ and the embedding is compact whenever the resolvent
 operator of $A$ is compact. $\pounds(H_\alpha;H_\beta)$ is the space of bounded
linear operators from $H_\alpha$ into $H_\beta$ with norm
$\|\cdot\|_{\alpha, \beta}$ and $H_0=H$.

Let $(\Omega,\mathfrak F,P)$ be a probability space on which an increasing and
right continuous family \{ $\mathfrak F_t:t\geq 0 $\} of complete
sub-$\sigma$-algebras of $\mathfrak F $ is defined. The collection of all
square integrable and $\mathfrak F_t$-adapted processes is denoted by
$L_2^{\mathfrak F_t}(J,H)$. Let $\beta_n(t) (n=1, 2, \cdots)$ be a sequence
of real valued one dimensional standard Brownian motions mutually independent
over $(\Omega,\mathfrak F,P)$. We assume there exists a complete orthonormal
basis $\{e_n\}$ in K and a bounded sequence of nonnegative real numbers
$\lambda_n$ such that $w(t)=\sum_{n=1}^{\infty}\sqrt{\lambda_n}\beta_n(t)e_n$,
$t\geq 0$. Let $Q\in L(K,K)$ be an operator defined by
 $Qe_n=\lambda_ne_n, (n=1,2,3\ldots ) $ with finite trace
$\operatorname{tr} Q=\sum_{n=1}^{\infty}\lambda_n<\infty$.
Then the above $K$-valued stochastic process $w(t)$ is called a $Q$-Wiener process.
We assume that $\mathfrak F_t=\sigma(w(s):0\leq s \leq t)$ is the $\sigma$-algebra
generated by $w$ and $\mathfrak F_t=\mathfrak F$.
Let $\Psi \in L_2^{0}(K,H)$ with the norm
 \[
 \|\Psi\|_{Q}^2=tr(\Psi Q\Psi^{*})=\sum_{n=1}^{\infty}
\|\sqrt{\lambda_n} \Psi e_n\|^2.
\]
If $\|\Psi\|_{Q}< \infty$, then $\Psi$ is called a $Q$-Hilbert-Schmidt operator.
 Define the space of all $\mathfrak{F_0}$-measurable $C_{\alpha}$-valued
function $\psi: \Omega\to C_{\alpha}$ with the norm
 \[
 \mathbb{E}\|\psi\|^2_{C_{\alpha}}=\mathbb{E}\{\sup _{\theta\in [-r,0]}
\|A^{\alpha}\psi (\theta)\|^2\}< \infty.
 \]
 Let $L_2(\Omega, \mathfrak F,P; H)$ be the space of all
$\mathfrak F_t$-measurable square integrable random variables with value in $H$.
 We assume that: $PC(J_0,L_2(\Omega, \mathfrak F,P; H))=\{x(t): J_0=[-r,T]
\to L_2(\Omega, \mathfrak F,P; H)$ is continuous everywhere except some
$t_{k}$ at which $x(t_{k}^{+})$ and $x(t_{k}^{-})$ exist with
$x(t_{k})=x(t_{k}^{-})$ satisfying $sup_{s\in J_0}E\|x(s)\|^2< \infty $\}.
Let $PC(J_0,L_2)$ be the closed subspace of
$PC(J_0,L_2(\Omega, \mathfrak F,P; H))$ consisting of measurable and
$\mathfrak F_t $-adapted processes and $ \mathfrak F_0$-adapted processes
$y \in L_2(\Omega, \mathfrak F_0,P; C_{\alpha})$. Let $\|\cdot\|_*$ be
a seminorm in $PC(J_0,L_2)$ defined by
$\|y\|_{*} = (\sup_{t \in J} \mathbb{E}\|y_t\|_{C_{\alpha}}^2)^{1/2}<\infty$.


\begin{definition}[\cite{Ezzinbi}] \label{def2.1} \rm
 A family of bounded linear operators $\Phi(t)\in \pounds(H)$ for $t\in J$
is called resolvent operator for
\begin{equation} \label{e2.1}
\begin{gathered}
 \frac{d}{dt} x(t)= -Ax(t)+\int _0^{t}\gamma(t-s)x(s)\,ds ,\quad t\in J, \\
 x(0)=x_0\in H,
 \end{gathered}
\end{equation}
if
\begin{itemize}
\item[(i)] $\Phi(0)=I$ and $\|\Phi(t)\|\leq N_1e^{\omega t}$ for some
$N_1>0,\omega \in R$.

\item[(ii)] For all $x\in H$, $\Phi(t)x$ is strongly continuous in $t$ on $J$.

\item[(iii)] $\Phi(t) \in \pounds(Y)$, for $t\in J$, where $Y$ is the Banach
space formed from $D(-A)$ endowed with the graph norm. Moreover for
$x\in Y, \Phi(\cdot)x \in C^{1}(J;H)\cap C(J;Y) $ and, for $t\geq 0$,
the following equality holds
\[
 \frac{d}{dt}\Phi(t)x=-A\Phi(t)x+ \int _0^{t}\gamma(t-s)\Phi(s)x\,ds
=-\Phi(t)Ax+\int _0^{t}\Phi(t-s)\gamma(s)x\,ds.
\]
\end{itemize}
\end{definition}

The hypotheses on the operator $A$ and $\gamma(\cdot)$ follow
from \cite[Hypotheses $(V_1)-(V_3)$]{Mokkedem}.
Then, $\Phi(t)$ is also analytic and there exist $N, N_{\alpha}>0$ such that
$\|\Phi(t)\|\leq N$ and
\[
 \|A^{\alpha}\Phi(t)\|\leq \frac{N_{\alpha}}{t^\alpha},0<t\leq T, \quad
0\leq \alpha \leq 1.
\]

\begin{lemma}[\cite{Fu-1}] \label{lem2.1}
 $\Phi(t)$ is continuous for $t>0$ in the uniform operator topology
of $\pounds(H)$.
\end{lemma}

\begin{lemma}[\cite{Fu-2}] \label{lem2.2}
 $A\Phi(t)$ is continuous for $t>0$ in the uniform operator topology
 of $\pounds(H)$.
\end{lemma}

To simplify notation, let $A^{\alpha}\Phi(t)x=\Phi(t)A^{\alpha}x$,
for any $ 0 \leq \alpha \leq 1,x\in D(A^{\alpha}) $.
Now we define the mild solution of \eqref{e1.1} expressed by
the resolvent operator $\Phi(t)$.

\begin{definition} \label{def2.2}\rm
 A stochastic process $x(\cdot)\in PC(J_0, L_2(\Omega, \mathfrak F,P; H))$
is called a mild solution of  \eqref{e1.1} if the following condition are satisfied:
\begin{itemize}
\item[(1)] the initial value $\phi \in L_2(\Omega,C_{\alpha})$ and the control
$u(\cdot) \in L_2^{\mathfrak{F_t}}(J,U)$.

\item[(2)] the function $A\Phi(t-s)F(s,x_{s}), s\in J$ is integrable and on
$J_0$ it satisfies
\begin{equation} \label{e2.2}
x(t)=\begin{cases}
 \phi(t), & t\in [-r,0], \\
 \Phi(t)(\phi(0)+F(0,\phi))-F(t,x_t)+\int _0^{t}A \Phi(t-s)F(s,x_{s})\,ds\\
+\int _0^{t}\Phi(t-s)Bu(s)\,ds -\int _0^{t}\Phi(t-s)\int _0^{s}\gamma(s-v)
F(v,x_{v})\,dv\,ds \\
+\int _0^{t}\Phi(t-s)G(s, x_{s})\,dw(s)
+\Sigma_{0<t_{k}<t}\Phi(t-t_{k})I_{k}(x(t_{k}), & t\in J.
 \end{cases}
\end{equation}
\end{itemize}
\end{definition}

\begin{definition} \label{def2.3} \rm
 System \eqref{e1.1} is said to be approximately controllable on $J$ if
\[
 \overline{R(T; \phi, u)}=L_2(\Omega, \mathfrak F,P; H),
\]
where $R(T; \phi, u)=\{x(T; \phi, u), u(\cdot) \in L_2^{\mathfrak{F_t}}(J,U)\}$.
\end{definition}

 To discuss the approximate controllability of system \eqref{e1.1},
 we introduce the following operators.

 (1) The controllability Grammian $\Gamma_t^{T}$ is defined by
\[
 \Gamma_t^{T}=\int _t^{T}\Phi(T-s)BB^{*}\Phi^{*}(T-s)\,ds,
\]
where $\Phi^{*}$ denotes the adjoint operator of $\Phi$.

 (2) The resolvent operator
\[
 R(\lambda,\Gamma_t^{T})=(\lambda I+\Gamma_t^{T})^{-1}.
\]

 At first, we assume
\begin{itemize}
\item[(H0)] $\lambda R(\lambda,\Gamma_t^{T}) \to 0 $, as
 $ \lambda \to 0^{+}$ in the strong operator topology.
\end{itemize}

Note  that the deterministic linear system corresponding to
\eqref{e1.1} is approximately controllable on $[t,T]$ if and only if
 (H0) holds , see \cite{Mahmudov-1}.

We need the Nussbaum fixed-point theorem to prove the existence of mild
solutions of system \eqref{e1.1}.

\begin{lemma}[\cite{Nussbaum}] \label{lem2.3}
 Let $S$ be a closed, bounded convex subset of a Banach space $H$, and
$P_1, P_2$ be continuous mappings from $S$ into $H$ such that
$(P_1+P_2)S\subset S, \|P_1x-P_1y\|\leq k\|x-y\|$ for all $x, y \in S$,
 where $0\leq k <1$ is a constant, and $P_2S$ is compact, then the
operator $ P_1+P_2$ has a fixed point in $S$.
\end{lemma}

\begin{lemma}[\cite{Mahmudov-2}] \label{lem2.4}
 For any $h \in L_2(\Omega, \mathfrak{F}, P; H)$, there exist
$\varphi \in L_2^{\mathfrak{F_t}}(J, L_2^{0}(K, H))$ such that
\[
 h=\mathbb{E}h+\int _0^{T}\varphi(s)\,dw(s).
\]
\end{lemma}


\section{Approximate controllability}

To study the approximate controllability of system \eqref{e1.1}, we introduce
 the following hypotheses:
\begin{itemize}
\item[(H1)] There exist positive constants $M_1$, $d$ and $d_{k}$,
$k=1,2,\cdots m$, such that
\[
 \|B\|\leq M_1, \|I_{k}(x)\|\leq d_{k}, d=\sum_{k=1}^{m} d_{k}.
\]

\item[(H2)] For arbitrary $\beta, \xi \in C_{\alpha}, x, y \in H$ and
 $t\in J$, suppose that there exists constants $d_{1k}, N_1 > 0$, such that
\begin{gather*}
 \|F(t, \beta)-F(t, \xi) \|_{\alpha}^2 + \|G(t, \beta)-G(t, \xi)\|_{Q}^2
\leq N_1\|\beta-\xi \|_{C_{\alpha}}^2,\\
 \|F(t, \xi)\|_{\alpha}^2+\|G(t, \xi)\|_{Q}^2
\leq N_1(1+\|\xi\|_{C_{\alpha}}^2),\\
 \|I_{k}(x)-I_{k}(y)\|^2\leq d_{1k}\|x-y\|^2.
 \end{gather*}

\item[(H3)]  The function $\gamma(t) \in L(H_{\alpha}, H)$ for each
$t\in J$ suppose that there exist a positive constant $M_2$, such that
$\|\gamma(t)\|_{\alpha,0} \leq M_2$.

\item[(H4)] The function $F: J\times C_{\alpha}\to H_\alpha$ and
$G:J\times C_{\alpha}\to L_2^{0}(K,H)$ are uniformly bounded for
$t\in J$, $x_t\in C_{\alpha}$, there exist a positive constant $M_{3}$, such that
\[
 \|F(t, x_t)\|_\alpha + \|G(t, x_t)\|_{Q} \leq M_{3}.
\]
\end{itemize}
For any $\lambda \in (0, 1]$, we define the control function for system
\eqref{e1.1} as:
\begin{align*}
u^{\lambda}(t, x)
&= B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})[\mathbb{E}h
 -\Phi(T)(\phi(0)+F(0, \phi))+F(T, x_{T})]\\
&\quad -B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})A\Phi(T-s)
 F(s,x_{s})\,ds\\
&\quad +B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})\Phi(T-s)
 \int_0^{s}\gamma(s-v)F(v,x_{v})\,dvds \\
&\quad -B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 [\Phi(T-s)G(s,x_{s})-\varphi(s)]\, dw(s)\\
&\quad -B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})\sum_{0<t_{k}<t}
 \Phi(T-t_{k})I_{k}(x(t_{k})).
\end{align*}
and the operator $P^{\lambda}$ on $PC(J_0, L_2(\Omega, \mathfrak{F}, P; H)$
 as follows
\begin{equation} \label{e3.1}
\begin{aligned}
&(P^{\lambda}x)(t) \\
&=\begin{cases}
 \phi(t), & t\in [-r,0], \\
 \Phi(t)(\phi(0)+F(0,\phi))-F(t,x_t)+\int _0^{t}A \Phi(t-s)F(s,x_{s})\,ds \\
 +\int _0^{t}\Phi(t-s)G(s,x_{s})\,dw(s)\\
 -\int _0^{t}\Phi(t-s)\int _0^{s}\gamma(s-v)F(v,x_{v})\,dv\,ds \\
 +\int _0^{t}\Phi(t-s)Bu^{\lambda}(s, x)\,ds+\sum_{0<t_{k}<t}
 \Phi(t-t_{k})I_{k}(x(t_{k})), & t\in J.
 \end{cases}
\end{aligned}
\end{equation}
We can see the fixed point of $P^{\lambda}$ is a mild solution of
system \eqref{e1.1}. Now, we prove the following existence theorem.

\begin{theorem} \label{thm3.1}
Let $\phi\in L_2(\Omega, C_{\alpha})$. If the assumptions {\rm (H0)--(H3)}
 are satisfied then, the operator $P^{\lambda}$ has a fixed point provided that
\[
 K:=2N_1\Big(\|A^{-\alpha}\|^2+\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2}\Big)<1.
\]
\end{theorem}

\begin{proof}
 We prove this theorem by using Lemma \ref{lem2.3}. Put
\[
 Y_{r}=\{ x \in PC(J_0, L_2), \|x\|_{*}\leq r \}.
\]
It is obvious that $Y_{r}$ is a bounded, closed and convex set.
We first prove that for arbitrary $0<\lambda\leq 1$, there is a positive
constant $r_0=r_0(\lambda)$ such that $P^{\lambda}(Y_{r_0})\subset Y_{r_0}$.
For any $x\in Y_{r_0}$, we have
\begin{align}
&\mathbb{E}\|u^{\lambda}(t, x)\|^2 \nonumber \\
&= \mathbb{E}\|B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})[\mathbb{E}h-\Phi(T)
 (\phi(0)+F(0, \phi))+F(T, x_{T})] \nonumber \\
&\quad -B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})A\Phi(T-s)
 F(s,x_{s})\,ds \nonumber \\
&\quad +B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 \Phi(T-s)\int_0^{s}\gamma(s-v)F(v,x_{v})\,dv\,ds \nonumber \\
&\quad -B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 [\Phi(T-s)G(s,x_{s})-\varphi(s)]\, dw(s) \nonumber \\
&\quad -B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})\sum_{0<t_{k}<t}
 \Phi(T-t_{k})I_{k}(x(t_{k}))\|^2 \label{e3.2} \\
&\leq  6\mathbb{E}\|B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})
 [\mathbb{E}h-\Phi(T)(\phi(0)+F(0, \phi))]\|^2 \nonumber \\
&\quad +6\mathbb{E}\|B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})
 A^{-\alpha}A^{\alpha}F(T, x_{T})\|^2 \nonumber \\
&\quad +6\mathbb{E}\|B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 A^{1 - \alpha}A^{\alpha}\Phi(t-s)F(s,x_{s})\,ds\|^2 \nonumber \\
&\quad +6\mathbb{E}\|B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 \Phi(T-s)\int_0^{s}\gamma(s-v)F(v,x_{v})\,dvds\|^2 \nonumber \\
&\quad +6\mathbb{E}\|B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 [\Phi(T-s)G(s,x_{s})-\varphi(s)]\, dw(s)\|^2 \nonumber  \\
&\quad +6\mathbb{E}\|B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})
 \sum_{0<t_{k}<t}\Phi(T-t_{k})I_{k}(x(t_{k}))\|^2 \nonumber \\
&\leq  \frac{6M_1^2N^2}{\lambda^2}\mathbb{E}
 \|\mathbb{E}h-N(\phi(0)+F(0, \phi))\|^2
 +\frac{6M_1^2N^2}{\lambda^2}\|A^{-\alpha}\|^2\mathbb{E}
 \|F(T,x_{T})\|_{\alpha}^2\nonumber \\
&\quad +\frac{6M_1^2N^2}{\lambda^2}
\frac{N_{1-\alpha}^2T ^{2\alpha}}{\alpha^2}\mathbb{E}\|F(s, x_{s})\|_{\alpha}^2
 +\frac{6M_1^2N^4M_2^2T^2}{\lambda^2}\mathbb{E}
 \int_0^{T}\|F(v, x_{v})\|_{\alpha}^2\,dv \nonumber \\
&+\frac{12M_1^2N^4}{\lambda^2}\mathbb{E}\int_0^{T}\|G(s, x_{s})\|_{Q}^2\,ds
 + \frac{12M_1^2N^2}{\lambda^2}\mathbb{E}\int_0^{T}\|\varphi(s)\|_{Q}^2\,ds
 +\frac{6M_1^2N^4d^2}{\lambda^2}\nonumber \\
&\leq \frac{6M_1^2N^2} {\lambda^2}\mathbb{E}\{\|\mathbb{E}h
 -N(\phi(0)+F(0,\phi))\|^2+\|A^{-\alpha}\|^2N_1(1+\|x_{s}\|_{C_{\alpha}}^2)
 \nonumber \\
&\quad +\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2}N_1(1+\|x_{s}\|_{C_{\alpha}}^2)
 +N^2(2+M_2^2T^2)\int_0^{T}N_1(1+\|x_{s}\|_{C_{\alpha}}^2)\,ds \nonumber \\
&\quad +2\int_0^{T}\|\varphi(s)\|_{Q}^2\,ds+N^2d^2\}. \nonumber
\end{align}
Here we employ the assumption (H0) that
\[
 \|R(\lambda,\Gamma_{s}^{T})\|\leq \frac{1}{\lambda}, \lambda\in (0, 1].
\]
Similarly, for $x, y\in PC(J_0, L_2)$, we can also obtain
\begin{align}
&\mathbb{E}\|u^{\lambda}(t,x)-u^{\lambda}(t,y)\|^2 \nonumber \\
&\leq 5\mathbb{E}\|B^{*}\Phi^{*}
(T-t)R(\lambda,\Gamma_0^{T})A^{-\alpha}A^{\alpha}[F(T, x_{T})-F(T, y_{T})]\|^2
 \nonumber\\
&\quad +5\mathbb{E}\|B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 A^{1 - \alpha}A^{\alpha}\Phi(T-s)[F(s, x_{s})-F(s, y_{s})]\,ds\|^2 \nonumber\\
&\quad +5\mathbb{E}\|B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 \Phi(T-s) \nonumber\\
&\quad\times \int_0^{s}\gamma(s-v)[F(v,x_{v})-F(v,y_{v})]\,dvds\|^2\nonumber \\
&\quad +5\mathbb{E}\|B^{*}\Phi^{*}(T-t)\int _0^{t}R(\lambda,\Gamma_{s}^{T})
 \Phi(T-s)[G(s,x_{s})-G(s,y_{s})]\,dw(s)\|^2\nonumber \\
&\quad +5\mathbb{E}\|B^{*}\Phi^{*}(T-t)R(\lambda,\Gamma_0^{T})
 \sum_{0<t_{k}<t}\Phi(T-t_{k})[I_{k}(x(t_{k}))-I_{k}(y(t_{k}))]\|^2 \nonumber\\
&\leq \frac{5M_1^2N^2}{\lambda^2}\|A^{-\alpha}\|^2\mathbb{E}N_1
 \|x_{T}-y_{T}\|_{C_{\alpha}}^2 \label{e3.3} \\
&\quad +\frac{5M_1^2N^2}{\lambda^2}\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2}
 \mathbb{E}N_1\|x_{s}-y_{s}\|_{C_{\alpha}}^2 \nonumber\\
&\quad +\frac{5M_1^2N^4M_2^2T^2}{\lambda^2}\int_0^{T}\mathbb{E}
 N_1\|x_{s}-y_{s}\|_{C_{\alpha}}^2\,ds \nonumber\\
&\quad +\frac{5M_1^2N^4}{\lambda^2}
 \int_0^{T}\mathbb{E}N_1\|x_{s}-y_{s}\|_{C_{\alpha}}^2\,ds \nonumber\\
&\quad +\frac{5M_1^2N^4}{\lambda^2}\sum_{k=1}^{m}d_{1k}^2
 \mathbb{E}\|x_{t_{k}}-y_{t_{k}}\|^2 \nonumber\\
&\leq \frac{5M_1^2N^2}{\lambda^2}\{N_1[N^2T(M_2^2T^2+1)
 +\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2} \nonumber\\
&\quad +\|A^{-\alpha}\|^2]+N^2\sum_{k=1}^{m}d_{1k}^2\}\|x-y\|_{*}^2. \nonumber
\end{align}
Then we have
\begin{align*}
&E\|(P^{\lambda}x)(t)\|^2\\
&=E\|\Phi(t)(\phi(0)+F(0,\phi))-F(t,x_t)+\int _0^{t}A \Phi(t-s)F(s,x_{s})\,ds \\
&\quad +\int _0^{t}\Phi(t-s)G(s,x_{s})\,dw(s)
 - \int _0^{t}\Phi(t-s)\int _0^{s}\gamma(s-v)F(v,x_{v})\,dv\,ds \\
&\quad +\int _0^{t}\Phi(t-s)Bu^{\lambda}(s)\,ds+\sum_{0<t_{k}<t}
 \Phi(t-t_{k})I_{k}(x(t_{k}))\|^2\\
&\leq 7E\|\Phi(t)(\phi(0)+F(0,\phi))\|^2+7E\|A^{-\alpha}A^{\alpha}F(t, x_t)\|^2\\
&\quad +7E\|\int _0^{t}A^{1 - \alpha}A^{\alpha}\Phi(t-s)F(s,x_{s})\,ds\|^2
 +7E\|\int _0^{t}\Phi(t-s)Bu^{\lambda}(s)\,ds\|^2\\
&\quad +7E\|\int _0^{t}\Phi(t-s)\int_0^{s}\gamma(s-v)F(v,x_{v})\,dvds\|^2\\
&\quad  +7E\|\int _0^{t}\Phi(t-s)G(s,x_{s})\,dw(s)\|^2
 +7E\|\sum_{0<t_{k}<t}\Phi(t-t_{k})I_{k}(x(t_{k}))\|^2\\
&\leq 7E\|N(\phi(0)+F(0,\phi))\|^2+7N_1\|A^{-\alpha}\|^2(1+\|x\|_{*}^2)\\
&\quad +7\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2}
 N_1(1+\|x\|_{*}^2)+7T^4N^2M_2^2N_1(1+\|x\|_{*}^2)\\
&\quad  +7N^2TN_1(1+\|x\|_{*}^2)+7N^2M_1^2TE\|u^{\lambda}(t, s)\|^2
 +7N^2d^2\\
&\leq 7N_1(1+\|x\|_{*}^2)(\|A^{-\alpha}\|^2
 +\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2}+T^4N^2M_2^2+N^2T)\\
&\quad  +7E\|N(\phi(0)+F(0,\phi))\|^2+7N^2M_1^2TE\|u^{\lambda}(t, s)\|^2
 +7N^2d^2.
\end{align*} %\label{e3.4}
From \eqref{e3.2} we can imply that $E\|(P^{\lambda}x)(t)\|^2< \infty$.
So there exist a constant $r_0$ such that $P^{\lambda}(Y_{r_0})\in Y_{r_0}$.

Next we prove that $P^{\lambda}$ has a fixed point on $Y_{r_0}$.
To begin, we rewrite $P^{\lambda}$ as $P^{\lambda}=P_1^{\lambda}+P_2^{\lambda}$,
where
\begin{gather*}
(P^\lambda _1x)(t)=\begin{cases}
 0, & t\in [-r,0], \\
 \Phi(t)F(0,\phi)-F(t,x_t)+\int _0^{t}A \Phi(t-s)F(s,x_{s})\,ds, & t\in J.
 \end{cases}
\\
\begin{aligned}
&(P^\lambda _2x)(t)\\
&=\begin{cases}
 \phi(t), & t\in [-r,0], \\
 \Phi(t)\phi(0)+\int _0^{t}\Phi(t-s)G(s,x_{s})\,dw(s)\\
- \int _0^{t}\Phi(t-s)\int _0^{s}\gamma(s-v)F(v,x_{v})\,dv\,ds \\
+\int _0^{t}\Phi(t-s)Bu^{\lambda}(s, x)\,ds
+ \sum_{0<t_{k}<t}\Phi(t-t_{k})I_{k}(x(t_{k})), & t\in J.
 \end{cases}
\end{aligned}
\end{gather*}
To prove that $P^\lambda _1$ is a contraction, we take $x, y\in Y_{r_0}$, then,
for each $t \in J$, we can verify
\begin{align*}
&\mathbb{E}\|(P^\lambda _1x)(t)-(P^\lambda _1y)(t)\|^2\\
&=\mathbb{E}\|A^{-\alpha}A^{\alpha}(F(t,x_t)-F(t,y_t))
+\int _0^{t}A^{1 - \alpha}A^{\alpha}\Phi(t-s)(F(t,x_{s})-F(t,y_{s}))\,ds\|^2\\
&\leq 2\|A^{-\alpha}\|^2N_1\|x-y\|_{*}^2
 +2\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2}N_1\|x-y\|_{*}^2\\
&=2N_1(\|A^{-\alpha}\|^2+\frac{N_{1-\alpha}^2T^{2\alpha}}{\alpha^2})
\|x-y\|_{*}^2.
\end{align*}
So we have
\[
 \|P^\lambda _1x-P^\lambda _1y\|_{*}^2\leq K\|x-y\|_{*}^2.
\]
So $P^\lambda _1$ is a contraction mapping on $Y_{r_0}$.

Next we prove that $P^\lambda _2$ is continuous on $Y_{r_0}$.
Let $\{x^m(\cdot)\}\subseteq Y_{r_0}$ with $x^m(\cdot) \to x(\cdot), (m\to \infty)$.
Then we have
\begin{align*}
&\mathbb{E}\|(P^\lambda _2x^m)(t)-(P^\lambda _2x)(t)\|^2\\
&\leq 4\mathbb{E}\|\int _0^{t}\Phi(t-s)[G(s,x_{s}^m)-G(s,x_{s})\,dw(s)]\|^2\\
&\quad + 4\mathbb{E}\|\int _0^{t}\Phi(t-s)\int _0^{s}\gamma(s-v)[F(v,x_{v}^m)
 -F(v,x_{v})]\,dvds\|^2\\
&\quad + 4\mathbb{E}\|\int _0^{t}\Phi(t-s)B[u^{\lambda}(s, x^m)
 -u^{\lambda}(s, x)]\,ds\|^2\\
&\quad +4\mathbb{E}\|\sum_{0<t_{k}<t}\Phi(t-t_{k})[I_{k}(x^m(t_{k}))
 -I_{k}(x(t_{k}))]\|^2\\
&\leq  4N^2\mathbb{E}\int _0^{T}N_1\|x^{m}_{s}-x_{s}\|_{C_{\alpha}}^2)\,ds\\
&\quad +4N^2M_2^2\mathbb{E}\int _0^{T}\int _0^{T}N_1
 \|x^{m}_{v}-x_{v}\|_{C_{\alpha}}^2\,dv\,ds\\
&\quad +4N^2M_1^2\int _0^{T}\mathbb{E}\|u^{\lambda}(s, x^m)
 -u^{\lambda}(s,x)\|^2\,ds\\
&\quad +4N^2\sum_{k=1}^{m}d_{1k}^2\mathbb{E}\|x^{m}_{t_{k}}-x_{t_{k}}\|^2.
\end{align*}
From \eqref{e3.3} and the Lebesgue-dominated convergence theorem, we obtain
\[
 \mathbb{E}\|(P^\lambda _2x^m)(t)-(P^\lambda _2x)(t)\|^2\to 0 ,
\]
as $m\to \infty$. So $P_2^{\lambda}$ is continuous.

We finally prove that the operator $P^\lambda _2$ maps $Y_{r_0}$
into a relatively compact subset of $Y_{r_0}$. Denote the set
\[
 V(t)=\{(P^\lambda _2x)(t): x\in Y_{r_0}\}.
\]

\noindent\textbf{Step 1.} $P^\lambda _2(Y_{r_0})$ is clearly bounded.
\smallskip

\noindent\textbf{Step 2.} we have to show that $V(t)$ is equicontinuous on $J_0$.
Let $x\in Y_{r_0}, t_1, t_2\in (0, T]$. Then
\begin{align*}
&\mathbb{E}\|(P^\lambda _2x)(t_2)-(P^\lambda _2x)(t_1)\|^2\\
&\leq 5\mathbb{E}\|\phi(0)(\Phi(t_2)-\Phi(t_1))\|^2 \\
&\quad +5\mathbb{E}\|[\int _0^{t_2}\Phi(t_2-s)G(s,x_{s})
 -\int _0^{t_1}\Phi(t_1-s)G(s,x_{s})]\,dw(s)\|^2\\
&\quad +5\mathbb{E}\|\int _0^{s}\gamma(s-v)F(v,x_{v})
\Big[\int _0^{t_2}\Phi(t_2-s)-\int _0^{t_1}\Phi(t_1-s)\Big]\,dvds\|^2
 \\
&\quad +5\mathbb{E}\|[\int _0^{t_2}\Phi(t_2-s)
 -\int _0^{t_1}\Phi(t_1-s)]Bu^{\lambda}(s, x)\,ds\|^2\\
&\quad +5\mathbb{E}\|\sum_{0<t_{k}<t}\big[\Phi(t_2-t_k)-\Phi(t_2-t_k)\big]
 I_{k}(x(t_{k})\|^2\\
&= I_1+I_2+I_3+I_4+I_5.
\end{align*}
Thus we have
\begin{align*}
I_2&\leq  10\mathbb{E}\|\int _0^{t_1}(\Phi(t_2-s)-\Phi(t_1-s))G(s,x_{s})\,dw(s)\|^2\\
&\quad +10\mathbb{E}\|\int _{t_1}^{t_2}\Phi(t_2-s)G(s,x_{s})\,dw(s)\|^2\\
&\leq  10N^2N_1T(1+\|x\|_{*}^2)\mathbb{E}\|\Phi(t_2-t_1)-I\|^2
 +10N^2N_1(t_2-t_1)(1+\|x\|_{*}^2),
\end{align*}
\begin{align*}
I_3&\leq  10\mathbb{E}\|\int _0^{t_1}\Phi(t_1-s)(\Phi(t_2-t_1)-I)
 \int _0^{s}\gamma(s-v)F(v,x_{v})\,dvds\|^2\\
&\quad +10\mathbb{E}\|\int _{t_1}^{t_2}\Phi(t_2-s)
 \int _0^{s}\gamma(s-v)F(v,x_{v})\,dvds\|^2\\
&\leq  10N^2M_2^2T^2N_1(1+\|x\|_{*}^2)\mathbb{E}
 \|\Phi(t_2-t_1)-I\|^2\\
&\quad +10N^2M_2^2TN_1(1+\|x\|_{*}^2)(t_2-t_1),
\end{align*}
\begin{align*}
I_4&\leq 10\mathbb{E}\|\int _0^{t_1}\Phi(t_1-s)(\Phi(t_2-t_1)-I)
 Bu^{\lambda}(s, x)\,ds\|^2\\
&\quad +10\mathbb{E}
\|\int _{t_1}^{t_2}\Phi(t_2-s)Bu^{\lambda}(s, x)\,ds\|^2\\
&\leq 10N^2M_1^2T\mathbb{E}\|u^{\lambda}(s, x)\|^2\|\Phi(t_2-t_1)-I\|^2
 +10N^2M_1^2(t_2-t_1)\mathbb{E}\|u^{\lambda}(s, x)\|^2.
\end{align*}
In a similar way, we have
\[
 I_5\leq10N^2d^2\mathbb{E}\|\Phi(t_2-t_1)-I\|^2+10N^2d^2(t_2-t_1).
\]
It is easy to see that, as $t_2 \to t_1$ the right-hand side of the above
inequality tends to zero, since $\Phi(t)$ is continuous in $t$ in the uniform
operator topology by Lemma \ref{lem2.1}. So we obtain the equicontinuity of $V$.
\smallskip

\noindent\textbf{Step3.} We show that for fixed $t$, the set $V(t)$
is relatively compact.
Obviously, $V(t)=\phi(t), t\in [-r,0]$ which is trivially relatively compact.
So let $t\in (0,T]$ be fixed, then
\[
 V(t)=\Phi(t)\phi(0)+\sum_{k=1}^{m}\Phi(t-t_{k})I_{k}(x(t_{k}))+V_1(t),
\]
where $V_1(t)$ is defined by
\begin{align*}
V_1(t):=\Big\{&\nu(t)=\int _0^{t}\Phi(t-s)G(s,x_{s})\,dw(s)
 -\! \int _0^{t}\Phi(t-s)\int _0^{s}\gamma(s-v)F(v,x_{v})\,dv\,ds \\
&+\int _0^{t}\Phi(t-s)Bu^{\lambda}(s, x)\,ds , x\in Y_{r_0} \Big\}.
\end{align*}
 Since $\sum_{k=1}^{m}\Phi(t-t_{k})I_{k}(x(t_{k}))$ is uniform bounded and
equicontinuous. By the Ascoli-Arzela theorem
$\sum_{k=1}^{m}\Phi(t-t_{k})I_{k}(x(t_{k}))$ is relatively compact.
$\Phi(t)\phi(0)$ is a single point in $H$. So we just have to show that
$V_1(t)$ is relatively compact, let $0<\alpha<\alpha_1<1$, we have
\begin{align*}
\mathbb{E}\|A^{\alpha_1}\nu(t)\|^2
&\leq 3\mathbb{E}\|\int _0^{t}A^{\alpha_1}\Phi(t-s)G(s,x_{s})\,dw(s)\|^2\\
&\quad +3\mathbb{E}\|\int _0^{t}A^{\alpha_1}\Phi(t-s)
 \int _0^{s}\gamma(s-v)F(v,x_{v})\,dvds\|^2\\
&\quad +3\mathbb{E}\|\int _0^{t}A^{\alpha_1}\Phi(t-s)
 Bu^{\lambda}(s, x)\,ds\|^2\\
&\leq 3\frac{N_{\alpha_1}^2T^{2-2\alpha_1}}{{(1-\alpha_1)}^2}
 (1+\|x\|_{*}^2)+3\frac{N_{\alpha_1}^2T^{4-2\alpha_1}M_2^2}{{(1-\alpha_1)}^2}
 (1+\|x\|_{*}^2)\\
&\quad +3\frac{N_{\alpha_1}^2T^{2-2\alpha_1}M_1^2}{{(1-\alpha_1)}^2}
\mathbb{E}\|u^{\lambda}(s, x)\|^2
< \infty,
\end{align*}
which implies that $A^{\alpha_1}V_1(t)$ is bounded in $H$. Hence we obtain
that $V_1(t)$ is relatively compact in $H_\alpha$ by the compactness of the
operator $A^{-\alpha_1}:H\to H_{\alpha_1}$, (noting that the embedding
$H_{\alpha_1}\to H_\alpha$ is compact). Therefore, from the Ascoli-Arzela theorem,
$P_2^\lambda(Y_{r_0})$ is compact.
So, the operator $P^{\lambda}=P_1^{\lambda}+P_2^{\lambda}$ has a fixed point
from Lemma \ref{lem2.3}. The proof is complete.
\end{proof}

\begin{theorem} \label{thm3.2}
Assume that {\rm (H0)--(H4)} are satisfied, then system \eqref{e1.1}
 is approximately controllable on $J$.
\end{theorem}

\begin{proof}
Because the hypotheses of Theorem \ref{thm3.1} are fulfilled, there is a solution
$x^{\lambda}(\cdot)$ of \eqref{e1.1} under the control $u^{\lambda}(t,x)$.
Using the stochastic Fubini theorem we can obtain
\begin{align*}
x^{\lambda}(T)
&=  \Phi(T)(\phi(0)+F(0,\phi))-F(T,x_{T}^\lambda)
+\int _0^{T}A \Phi(T-s)F(s,x_{s}^\lambda)\,ds \\
&\quad +\int _0^{T}\Phi(T-s)G(s,x_{s}^\lambda)\,dw(s)
 - \int _0^{T}\Phi(T-s)\int _0^{s}\gamma(s-v)F(v,x_{v}^\lambda)\,dv\,ds \\
&\quad +\sum_{k=1}^{m}\Phi(T-t_{k})I_{k}(x^\lambda(t_{k}))
 +\Gamma_0^TR(\lambda,\Gamma_0^{T})
\big[Eh-\Phi(T)(\phi(0)+F(0, \phi))\\
&\quad +F(T, x_{T})\big]\\
&\quad -\int _0^{T}\int _{r}^{T}\Phi(T-s)BB^{*}\Phi^{*}(T-s)
 R(\lambda,\Gamma_{r}^{T})A\Phi(T-r)F(r,x_{r}^\lambda)\,ds\,dr \\
&\quad +\int _0^{T}\int _{r}^{T}\Phi(T-s)BB^{*}\Phi^{*}(T-s)
 R(\lambda,\Gamma_{r}^{T})\Phi(T-r)\\
&\quad\times \int _0^{r}\gamma(r-v)F(v,x_{v}^\lambda) \,dv\,ds\,dr\\
&\quad -\int _0^{T}\int _{r}^{T}\Phi(T-s)BB^{*}\Phi^{*}(T-s)
 R(\lambda,\Gamma_{r}^{T})\\
&\quad\times [\Phi(T-r)G(r,x_{r}^\lambda)-\varphi(r)]\,ds\,dw(r)\\
&\quad -\Gamma_0^TR(\lambda,\Gamma_0^{T})
 \sum_{k=1}^{m}\Phi(T-t_{k})I_{k}(x^\lambda(t_{k}))\\
&= h-\lambda R(\lambda,\Gamma_0^{T})[Eh-\Phi(T)(\phi(0)+F(0, \phi))+F(T, x_{T})]\\
&\quad +\int _0^{T}\lambda R(\lambda,\Gamma_{s}^{T})A \Phi(T-s)F(s,x_{s}^\lambda)\,ds\\
&\quad -\int _0^{T}\lambda R(\lambda,\Gamma_{s}^{T})\Phi(T-s)
 \int _0^{s}\gamma(s-v)F(v,x_{v}^\lambda)\,dv\,ds\\
&\quad +\int _0^{T}\lambda R(\lambda,\Gamma_{s}^{T})[\Phi(T-s)
 G(s,x_{s}^\lambda)-\varphi(s)]\,dw(s)\\
&\quad +\lambda R(\lambda,\Gamma_0^{T})\sum_{k=1}^{m}
 \Phi(T-t_{k})I_{k}(x^\lambda(t_{k})).
\end{align*}
By (H4),
\[
 \|F(t, x_t)\|_\alpha + \|G(t, x_t)\|_{Q} \leq M_{3}.
\]
Then there is a subsequence, still denoted by
$\{F(s,x_{s}^\lambda), G(s,x_{s}^\lambda)\}$, weakly converging to say,
$\{F(s, w), G(s, w)\}$ in $H\times L_2^0(K, H)$.
 By (H0), $\lambda R(\lambda,\Gamma_{s}^{T}) \to 0 $ as
$ \lambda \to 0^{+}$ and $\|\lambda R(\lambda,\Gamma_{s}^{T})\|\leq 1$,
from which, together with Lebesgue dominated convergence theorem, we have
\begin{align*}
&\mathbb{E}\|x^{\lambda}(T)-h\|^2\\
&\leq 10\mathbb{E}\|\lambda R(\lambda,\Gamma_0^{T})
 [Eh-\Phi(T)(\phi(0)+F(0, \phi))\|^2\\
&\quad +10\mathbb{E}\|\lambda R(\lambda,\Gamma_0^{T})
 \|^2\|A^{-\alpha}\|^2\|F(T, x_{T}^\lambda)]\|_{\alpha}^2\\
&\quad +10\mathbb{E}\Big(\int _0^{T}\|\lambda R(\lambda,\Gamma_{s}^{T})
 \| \|A\Phi(T-s)\|\|A^{-\alpha}\| \|F(s,x_{s}^\lambda)-F(s)\|_\alpha\,ds\Big)^2\\
&\quad +10\mathbb{E}\Big(\int _0^{T}\|\lambda R(\lambda,\Gamma_{s}^{T})
 \| \|A\Phi(T-s)\|\|A^{-\alpha}\| \|F(s)\|_\alpha\,ds\Big)^2\\
&\quad +10\mathbb{E}\Big(\int _0^{T}\|\lambda R(\lambda,\Gamma_{s}^{T})\Phi(T-s)
 \int _0^{s}\gamma(s-v)A^{-\alpha}\| \|F(v,x_{v}^\lambda)
 -F(v)\|_\alpha\,dv\,ds\Big)^2\\
&\quad +10\mathbb{E}\Big(\int _0^{T}\|\lambda R(\lambda,\Gamma_{s}^{T})
 \Phi(T-s)\int _0^{s}\gamma(s-v)F(v)\|\,dvds\Big)^2\\
&\quad +10\mathbb{E}\int _0^{T} \|\lambda R(\lambda,\Gamma_{s}^{T})\|^2
 \|\Phi(T-s)\|^2 \|G(s,x_{s}^{\lambda})-G(s)\|_{Q}^2 \,ds\\
&\quad +10\mathbb{E}\int _0^{T} \|\lambda R(\lambda,\Gamma_{s}^{T})\|^2
 \|\Phi(T-s)\|^2\|G(s)\|_{Q}^2 \,ds\\
&\quad +10\mathbb{E}\int _0^{T} \|\lambda R(\lambda,\Gamma_{s}^{T})\|^2
 \|\varphi(s)\|_{Q}^2 \,ds\\
&\quad +10\mathbb{E}\big\|\lambda R(\lambda,\Gamma_0^{T})\sum_{k=1}^{m}
 \Phi(T-t_{k})I_{k}(x^\lambda(t_{k}))\big\|^2\to 0,
\quad \text{as } \lambda \to 0^+.
\end{align*}
This gives the approximate controllability. The proof is complete.
\end{proof}

\section{Example}

As an application, we consider the  neutral stochastic integro-differential
 system with impulses,
\begin{gather}
\begin{aligned}
& d[z(t, x)+f(t, z(t-r_1(t), x))]\\
&=\big[\frac{\partial^2} {\partial x^2}z(t,x)+u(t, x)
 +\int_0^{t}b(t-s)\frac{\partial^2} {\partial x^2}z(s,x)ds\big]dt
 +g(t, z(t-r_2(t),x))d\beta(t),
\end{aligned} \nonumber \\
  0<t\leq T,\quad  t\neq t_k,\quad  0\leq x\leq \pi,
\nonumber \\
 z(t,0)=z(t, \pi)=0,\quad  0\leq t \leq T, \label{e4.1}\\
 z(t, x)=\phi(t, x), \quad -r\leq t \leq 0,\quad  0\leq x\leq \pi, \nonumber\\
 \Delta z(t_k, x)=\int_0^{\pi}K(t_{k},x,y)z(t_{k},y)\,dy,\quad 
 k=1, 2, 3, \dots, m, \nonumber
 \end{gather}
where $r_1, r_2$ are continuous functions with $0<r_1(t)\leq r, 0<r_2(t)\leq r$
for all $t\in J$, $\beta(t)$ denotes a one-dimensional standard Brownian motion.
The functions $f, g, \phi, K$ and $b$ will be described below.

System \eqref{e4.1} arises in the study of heat flow in materials of the
so-called retarded type \cite{Lunardi,Nunziato}. Here, $z(t,x)$
represents the temperature of the point $x$ at time $t$.
As stated in \cite{Caraballo}, these problems also arise in systems
related to couple oscillators in a noisy environment or in viscoelastic
materials under random or stochastic influences. Meanwhile, an
impulsive perturbation occurs very often in many practical models.
For instance, the system of rigid heat conduction with impulsive effect
can be modeled in the form of \eqref{e4.1}.

Let $H=L_2[0, \pi]$, and let $A: H\to H$ be the operator defined by
\[
 A\xi=-\frac{\partial^2}{\partial x^2}\xi,
\]
with domain
\[
 D(A)=\big\{\xi\in H: \xi, \frac{\partial}{\partial x}\xi
 \text{ are absolutely continuous},\,
 \frac{\partial^2}{\partial x^2}\xi\in H, \xi(0)=\xi(\pi)=0\big\}.
\]
Then $-A$ is self-adjoint, negative definite and the resolvent operator
$R(\lambda, -A)=(\lambda I+A)^{-1}$ is compact when it exist.
Moreover, $-A$ generates a strongly semigroup $\{S(t)\}_{t\geq 0}$ which is
analytic, compact and self-adjoint. There exists a complete orthonormal
set $\{e_n\}$ of eigenvectors of $-A $ with
$e_n(x)=\sqrt{\frac{2}{\Pi}}\sin(nx), n=1,2,3,\dots$.
 Then the following properties hold:
\begin{gather*}
 A\xi= \sum_{n=1}^{\infty}n^2\langle \xi, e_n\rangle e_n, \quad  \xi\in D(A),\\
 S(t)\xi= \sum_{n=1}^{\infty}\mbox{exp}(-n^2t)\langle\xi, e_n\rangle e_n, \quad
 \xi\in H.
\end{gather*}
Set $H_\alpha=D(A^\alpha)$ and $C_\alpha=C([-r, 0], H_\alpha)$.
By the classical spectral theorem, we deduce that
\[
 A^{\alpha}S(t)\xi=\sum_{n=1}^{\infty}(n^2)^{\alpha}\mbox{exp}(-n^2t)\langle \xi, e_n\rangle e_n.
\]

We assume that the following conditions hold:
\begin{itemize}
\item[(i)] The functions $f: J\times \mathbb{R}\to \mathbb{R}$ and
$g: J\times \mathbb{R}\to \mathbb{R}$ are continuous and global
Lipschitz continuous and uniformly bounded.

\item[(ii)] The function $\phi$ is defined by
$\phi(\theta)(x)=\phi(\theta, x)$ belongs to $C_\alpha$.

\item[(iii)] $b(t)\in L^1(R^+)\cap C^1(R^+)$ with primitive
$B(t)\in L_{loc}^1(R^+)$, $B(t)$ is non-positive, non-decreasing and $B(0)=-1$.

\item[(iiii)] $K(t, x, y): J\to L^2([0, \pi]\times [0, \pi])$ is
measurable and continuous, thus bounded.
 $l_{k}:=\int_0^{\pi}\int_0^{\pi}|K(t_{k},x,y)|^2 \, dxdy, k=1, 2, \dots, m$.
\end{itemize}
Now define the abstract functions $F$, $G$, $I_k$ and the operator $\gamma(t)$
respectively by
\begin{gather*}
F(t, \psi)(x)=f(t, \psi(-r_1(t))(x),\quad  \psi\in C_\alpha,\;  x\in [0, \pi],\\
G(t, \psi)(x)=g(t, \psi(-r_1(t))(x),\quad  \psi\in C_\alpha,\;  x\in [0, \pi],\\
I_{k}(\varphi)(x)=\int_0^{\pi}K(t_{k}, x, y)\varphi(y)\,dy, \;  \varphi \in H,\\
\gamma(t)=b(t)A, \quad t\in J.
\end{gather*}
Then system \eqref{e4.1} is rewritten into the form of \eqref{e1.1}.
 Moreover, $B=I$, satisfy (H1), and $\gamma(t)$ satisfies condition (H3).
From \cite{Mokkedem}, the linear system of \eqref{e4.1} has an analytic
resolvent operator $(W(t))_{t\geq 0}$ which is given by $W(0)=I$.

By  condition (i) we have
 \begin{align*}
 \|F(t, \psi)-F(t, \phi)\|_{\alpha}^2
&=\int_0^\pi|f(t, \psi(-r_1(t))(x))-f(t, \phi(-r_1(t))(x))|^2\,dx\\
 &\leq N^2\|\psi(-r_1(t))(x)-\phi(-r_1(t))(x)\|^2\\
 &=N^2\sum_{n=1}^{\infty}\langle\psi(-r_1(t))-\phi(-r_1(t)), e_n\rangle^2\\
 &\leq N^2\sum_{n=1}^{\infty}n^{4\alpha}\langle\psi(-r_1(t))-\phi(-r_1(t)), e_n\rangle^2\\
 &=N^2\|A^\alpha(\psi(-r_1(t))-\phi(-r_1(t))\|^2\\
 &\leq N^2\|\psi-\phi\|_{C_\alpha}^2,
 \end{align*}
where $N$ is appropriate constant.

By  condition (iiii) we have
\begin{align*}
\|I_{k}(\varphi_1)-I_{k}(\varphi_2)\|^2
&=\int_0^{\pi}\Big[\int_0^{\pi}K(t_{k}, x, y)\varphi_1(y)\,dy
 -\int_0^{\pi}K(t_{k}, x, y)\varphi_2(y)\,dy\Big]^2 \,dx,\\
&=\int_0^{\pi}\Big[\int_0^{\pi}K(t_{k}, x, y)(\varphi_1(y)-\varphi_2(y))\Big]^2 \,dx,\\
&\leq \int_0^{\pi}\Big[\int_0^{\pi}|K(t_{k}, x, y)|^2
 \int_0^{\pi}\|\varphi_1(y)-\varphi_2(y)\|^2 \,dy\Big] \,dx,\\
&\leq \pi l_{k}\|\varphi_1-\varphi_2\|^2.
\end{align*}

Similarly we can show that $F$, $G$ and $I_k$ satisfy the assumptions
(H2), (H4). From \cite[Lemma 4.1]{Mokkedem}, the resolvent operator $W(t)$
of \eqref{e4.1} is self adjoint. Thus
\[
 B^{*}W^{*}(t)\xi=W(t)\xi,\quad  \xi\in H.
\]
Let $B^{*}W^{*}(t)\xi=0$, for all $t\in J$, thus
\[
 B^{*}W^{*}(t)\xi=W(t)\xi=0,\quad t\in J.
\]
It follows from the fact $W(0)=I$ that $\xi=0$, so by
\cite[Theorem 4.1.7]{Cyrtain}, the deterministic linear system corresponding
to \eqref{e4.1} is approximately controllable on $J$. Hence, (H0) holds.
Therefore, by Theorem \ref{thm3.2}, system \eqref{e4.1} is approximately
controllable on $J$.

\subsection*{Acknowledgments}
We are very grateful to the referees for their important comments
and suggestions. This work was supported by the NNSF
of China (No. 11371087).

\begin{thebibliography}{00}

\bibitem{Balachandran} K. Balachandran, J. H. Kim, S. Karthikeyan;
\emph{Controllability of semilinear stochastic integro-differential equations},
Kybernetika, 43 (2007), 31-44.

\bibitem{Cyrtain} R. Cyrtain, H. J. Zwart;
\emph{An introduction to infinite dimensional linear systems theory}.
New York: Springer, 1995.

\bibitem{Caraballo} T. Caraballo, J. Real, T. Taniguchi;
\emph{The exponential stability of neutral stochastic delay partial
differential equations}, Discrete and Continuous Dynamical Systems.
Series A, 18 (2007), 295-313.

\bibitem{Dauer} J. P. Dauer, N. I. Mahmudov;
\emph{Controllability of stochastic semilinear functional differential equations
in Hilbert spaces}, Journal of
Mathematical Analysis and Applications, 290 (2004), 373-394.

\bibitem{Ezzinbi} K. Ezzinbi, H. Toure, I. zabsonre;
\emph{Local existence and regularity of solutions for some partial functional
integro-differential equations with infinite delay
in Banach spaces}, Nonlinear Analysis, 70 (2009), 3378-3389.

\bibitem{Fu-1} X. Fu, R. Huang;
\emph{Existence of solutions for neutral integro-differential equations
with state-dependent delay}, Applied Mathematics and Computation, 224 (2013), 743-759.

\bibitem{Fu-2} X. Fu, Y. Gao, Y. Zhang;
\emph{Existence of solutions for neutral integro-differential equations
 with nonlocal conditions}, Taiwanese Journal Mathematics, 16 (2012), 1879-1909.

\bibitem{Liu} B. Liu;
\emph{Controllability of neutral functional differential and integro-differential
inclusions with infinite delay}, Journal of Optimization Theory and Applications,
123 (2004), 573-593.

\bibitem{Li} M. L. Li, F. Q. Zhang, M. S. Wang;
\emph{Conttrollability of impulsive functional differential systems in Banach
spaces}, Chaos, Solitons and Fractals, 29 (2006), 175-181.

\bibitem{Lunardi} A. Lunardi;
\emph{On the linear heat equation with fading memory},
SIAM Journal of Mathematical Analysis, 21 (1990), 1213-1224.

\bibitem{Muthukumar} P. Muthukumar, P. Balasubramaniam;
\emph{Approximate controllability of mixed stochastic Volterra-Fredholm
type integro-differential systems in Hilbert space},
Journal of the Franklin Institute, 348 (2011), 2911-2922.

\bibitem{Mokkedem} F. Z. Mokkedem, X. Fu;
\emph{Approximate controllability of semilinear neutral integro-differential
 systems with finite delay}, Applied Mathematics and Computation, 242 (2014), 202-215.

\bibitem{Mahmudov-1} N. I. Mahmudov;
\emph{Controllability of linear stochastic systems in Hilbert spaces}, Journal of
Mathematical Analysis and Applications, 259 (2001), 64-82.

\bibitem{Mahmudov-2} N. I. Mahmudov;
\emph{Controllability of linear stochastic systems},
IEEE Transactions on Automatic Control, 46 (2001), 724-731.

\bibitem{Naito} K. Naito;
\emph{Controllability of semilinear control systems dominated by the linear part},
SIAM Journal on Control and Optimization, 25 (1987), 715-722.

\bibitem{Nussbaum} B. P. Nussbaum;
\emph{The fixed point index and asymptotic fixed point theorems for k-set
contractions}, American Mathematical Society Bulletin, 75 (1969), 490-495.

\bibitem{Nunziato} J. W. Nunziato;
\emph{On heat conduction in materials with memory},
 Quarterly Journal of Mechanics and Applied Mathematics, 29 (1971), 187-304.

\bibitem{Sakthivel} R. Sakthivel, N. I. Mahmudov, J. H. Kim;
\emph{Approximate controllability of nonlinear impulsive differential systems},
Reports on Mathematical Physics, 60 (2007), 85-96.

\bibitem{Shen-1} L. J. Shen, J. T. Sun;
\emph{Approximate controllability of stochastic impulsive functional
systems with infinite delay}, Automatica, 48 (2012), 2705-2709.

\bibitem{Shen-2} L. J. Shen, J. P. Shi, J. T. Sun;
\emph{Complete controllability of impulsive stochastic integro-differential systems},
Automatica, 46 (2010), 1068-1073.

\bibitem{Subalakshmi} R. Subalakshmi, K. Balachandran;
\emph{Approximate controllability of nonlinear stochastic impulsive
integro-differential systems in Hilbert spaces},
 Chaos, Solitons and Fractals, 42 (2009), 2035-2046.

\bibitem{Yan} Z. Yan, F. Lu;
\emph{Approximate controllability of impulsive partial neutral stochastic
functional integro-differential inclusions with infinite delay},
Advances in Difference Equations, (2015) 2015:106, DOI: 10.1186/s13662-015-0368-z.

\end{thebibliography}

\end{document}