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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 162, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/162\hfil
Controllability of stochastic  systems]
{Approximate controllability of neutral stochastic
integrodifferential systems \\ in Hilbert spaces}

\author[R. Subalakshmi, K. Balachandran  \hfil EJDE-2008/162\hfilneg]
{Ravikumar Subalakshmi, Krishnan Balachandran}

\address{Ravikumar Subalakshmi \newline
Department of Mathematics\\
Bharathiar University\\
Coimbatore 641 046, India}
\email{suba.ab.bu@gmail.com}

\address{Krishnan Balachandran \newline
Department of Mathematics\\
Bharathiar University\\
Coimbatore 641 046, India}
\email{balachandran\_k@lycos.com}

\thanks{Submitted June 1, 2008. Published December 12, 2008.}
\subjclass[2000]{93B05}
\keywords{Compact semigroup; approximate  controllability; \hfill\break\indent
   neutral stochastic  integrodifferential system;
  Nussbaum fixed point theorem}

\begin{abstract}
 In this paper sufficient conditions are established for the
 controllability of a class of neutral stochastic
 integrodifferential equations with nonlocal conditions in
 abstract space. The Nussbaum fixed point theorem is used to
 obtain the controllability results, which extends the linear
 system to the stochastic settings with the help of compact
 semigroup. An example is provided to illustrate the theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Controllability is one of the fundamental concepts in mathematical
control theory and plays an important role in both deterministic
and stochastic control systems. It is well known that
controllability  of deterministic systems are widely used in many
fields  of science and technology. The controllability of
nonlinear deterministic systems represented by evolution equations
in abstract spaces has been extensively studied by several authors
\cite{b1, b2}. Stochastic control theory is a stochastic
generalization of classical control theory.

However, in many cases, the accurate analysis, design and
assessment  of systems subjected to realistic environments must
take into account the potential of random loads and randomness in
the system properties. Randomness is intrinsic to the mathematical
formulation of many phenomena such as fluctuations in the stock
market, or noise in communication networks. Mathematical modelling
of such systems often leads to differential equations with random
parameters. The use of deterministic equations that ignore the
randomness of the parameter or replace them by their mean values
can result in gross errors. All such problems are mathematically
modelled and described by various stochastic systems described by
stochastic differential equations, stochastic delay equations and
in some cases stochastic integrodifferential equations which are
mathematical models for phenomena with irregular fluctuations.

The problem of controllability of the linear stochastic system of the form
\begin{gather*}
dx(t)=[Ax(t)+Bu(t)]dt+\tilde{\sigma}(t)dw(t)\\
x(0) =x_0,\quad t \in I=[0,T]
\end{gather*}
in Hilbert spaces has been studied by Dubov and Mordukhovich
\cite{d3}, Mahmudov \cite{m1}.

The problem of controllability of nonlinear stochastic system in
infinite  dimensional spaces have been studied by many authors.
Sirbu and Tessitore \cite{s1} studied null controllability of an
infinite dimensional stochastic differential equations with state
and control dependent noise using Riccati equation approach.
Mahmudov \cite{m2} investigated the sufficient conditions for
approximate controllability of nonlinear systems in Hilbert spaces
by using the Nussbaum fixed point theorem. Dauer and Mahmudov
\cite{d2},  Mahmudov \cite{m3} studied  controllability of
semilinear stochastic system by using the Banach fixed point
technique. Sunahara et al \cite{s2} introduced the concept of
stochastic $\epsilon$-controllability and controllability with
probability and established sufficient conditions for stochastic
controllability of a class of nonlinear systems.  Sufficient
conditions for stochastic $\epsilon$-controllability  have been
established by  Klamka and Socha \cite{k1} using a stochastic
Lyapunov-like approach.

Balachandran  et al \cite{b5} discussed the
controllability  of neutral functional integrodifferential systems
in Banach spaces by using semigroup theory and the Nussbaum
fixed point theorem. Recently, Balachandran and Karthikeyan
\cite{b3}, Balachandran et al \cite{b4} derived  sufficient
conditions for the controllability of stochastic
integrodifferential systems in finite dimensional spaces. This
paper is different from previous works in which dependence of the
nonlinear map contain integrodifferential term with nonlocal
condition. Here we are interested to establish a set of sufficient
conditions for the approximate controllability of the following
nonlinear neutral stochastic integrodifferential systems with
non-local condition
\begin{equation}
\begin{gathered}
\begin{aligned}
&d[x(t)-q(t,x)]\\
&=[Ax(t)+Bu(t)+f(t,x(t)) +\int_0^t g(t,s,x(s))ds]dt
 +\sigma(t,x(t))dw(t)
\end{aligned}\\
  x(0)+h(x) =x_0,\quad  t \in I=[0,T].
\end{gathered} \label{e1.1}
\end{equation}
in a Hilbert space $H$ by using the Nussbaum fixed point
theorem. Here $(\Omega,\mathcal{F},P)$ is a probability space
with a normal  filtration
$$
\{\mathcal{F}_t = \sigma(w(s): s\leq t),\; 0 \leq t \leq T\}
$$
generated by $w$; $H, E, U $ are three separable Hilbert spaces,
and $w$ is a $Q$-Wiener process on $(\Omega,\mathcal{F},P)$, with
the covariance operator $Q\in \mathcal{L}(E)$. We assume that
there exists a complete orthonormal system $\{e_k\}$ in $E$, a
bounded sequence of non-negative real numbers ${\lambda_k}$ such
that $Q e_k =\lambda_k e_k$ and  a sequence of real independent
Brownian motions such that $w(t)= \sum_{k=1}^\infty
\sqrt{\lambda_k }\beta_k (t)e_k$.  Let
$L_2^0=L_2(Q^{\frac{1}{2}}E,H)$ be the space of all
Hilbert-Schmidt operators. The space $L_2^0$ is a separable
Hilbert space, equipped with the norm
$\|\Psi\|^2_Q =\mathop{\rm tr}[\Psi Q \Psi^*]$.
$L_2^\mathcal{F}(I, H)$ is the space of all
$\mathcal{F}_t$-adapted, $H$-valued measurable square integrable
processes on $I\times \Omega$. $\mathcal{C}(I,L_2(\Omega,F,P,H))$
is the Banach space of continuous maps from $I$ into
$L_2(\Omega,F,P,H)$ satisfying the condition that
$\sup_{t \in I}\mathbf{E}\|x(t)\|^2 < \infty$. $\mathcal{C}(I,L_2)$ is the
closed subspace of $\mathcal{C}(I,L_2(\Omega,F,P,H))$  consisting
of measurable and $\mathcal{F}_t$-adapted processes $x(t)$ with
norm $\|x\|_*^2=\sup_{t \in I}\mathbf{E}\|x(t)\|^2$.

Concerning the operators $A$, $B$, $f$, $q$, $g$, $\sigma$, $h$
we assume the following hypotheses:

\begin{itemize}
\item[(H1)] The operator $A$ generates a compact semigroup $S(\cdot)$
 and $B$ is a bounded linear operator from a Hilbert space $U$ into $H$.

\item[(H2)] The functions $f:I \times H \to H$,
 $q:I \times H \to H$, $g:I \times I\times H \to H$,
 $\sigma:I \times H \to L_2^0$ and $h:C(I,H) \to H$, satisfy the
 Lipschitz condition and there exist constants $L_1, L_2, L_3, l>0$ for
$x_1,x_2 \in H$ and $0\leq s < t \leq T$ such that
\begin{gather*}
\|f(t,x_1)-f(t,x_2)\|^2+\|\sigma(t,x_1)-\sigma(t,x_2)\|^2_Q
 \leq L_1\|x_1-x_2\|^2\\
\|g(t,s,x_1(s))-g(t,s,x_2(s))\|^2\leq L_2\|x_1-x_2\|^2\\
\|q(t,x_1)-q(t,x_2)\|^2\leq L_3\|x_1-x_2\|^2\\
\|h(x_1)-h(x_2)\|^2\leq l\|x_1-x_2\|^2
\end{gather*}

\item[(H3)] The functions $f, q, g, h $ and $\sigma$ are continuous
and  there exist constants $L_4, L_5, L_6, l_1>0$ for
$x \in H$ and $0\leq t \leq T$ such that
\begin{gather*}
\|f(t,x)\|^2+\|\sigma(t,x)\|^2_Q \leq L_4\\
\|g(t,s,x(s))\|^2 \leq L_5\\
\|q(t,x)\|^2 \leq L_6\\
\|h(x)\|^2 \leq l_1
\end{gather*}
\end{itemize}

It is clear that under these conditions the system \eqref{e1.1}
admits a mild solution $x(\cdot) \in \mathcal{C}(I,L_2)$ for any
$x_0\in H$, $u(\cdot)\in L_2^\mathcal{F}(I,U)$ in the following
form (see \cite{d1}).
\begin{equation} \label{e1.2}
\begin{aligned}
x(t)&=S(t)[x_0-h(x)-q(0,x(0))]+q(t,x(t))+\int_0^t AS(t-s)q(s,x(s))ds \\
&\quad +\int_0^tS(t-s)Bu(s)ds
 +\int_0^tS(t-s)f(s,x(s))ds \\
&\quad +\int_0^tS(t-s)\sigma(s,x(s))dw(s)
 +\int_0^tS(t-s)\Big[\int_0^sg(s,\tau,x(\tau))d\tau\Big]ds.
\end{aligned}
\end{equation}

To study the approximate controllability of the system
\eqref{e1.2}, we consider the approximate controllability of
its corresponding linear part
\begin{equation} \label{e1.3}
\begin{gathered}
d[x(t)-q(t)]=\Big[Ax(t)+Bu(t)+f(t)+\int_0^t
g(t,s)ds\Big]dt+\tilde\sigma(t)dw(t)\\
  x(0)+h(x)=x_0,\quad t \in I=[0,T].
\end{gathered}
\end{equation}
where $\tilde{\sigma}\in L_2^0$ and assume the
approximate controllability of the system \eqref{e1.3}

We need the Nussbaum fixed-point theorem (see \cite{n1}) to establish our results.

\begin{theorem} \label{thm1.1}
 Suppose that $Y$ is a  closed, bounded convex subset of a Banach space
$\mathcal{H}$. Suppose that $\mathbb{P}_1,\mathbb{P}_2$ are
continuous mappings from $ Y$ into $\mathcal{H}$ such that
\begin {itemize}
\item [(i)] $(\mathbb{P}_1+\mathbb{P}_2) Y \subset Y$,
\item [(ii)] $\|\mathbb{P}_1x-\mathbb{P}_2 y\| \leq  k\|x-y\|$
  for all $x, y \in Y$ where $0\leq k<1$ is a constant,
\item [(iii)] $\overline{\mathbb{P}}_2[Y]$ is compact.
\end{itemize}
Then the operator $\mathbb{P}_1+\mathbb{P}_2$ has a fixed point in
$Y$.
\end{theorem}

\section{Controllability Results for  Linear Systems}

In this section we find an optimal control for solving the
stochastic  linear regulator problem in terms of stochastic
controllability operator which drives a point $x_0 \in H$ to a
small neighbourhood of an arbitrary point $b \in
L_2(\mathcal{F}_T,H)$. Further, we study the relation between
controllability operator $\Gamma_0^T$ and its stochastic analogue
$\Pi_0^T$.

 Define the linear regulator problem: minimize
\begin{equation} \label{e2.1}
 J(u)=\mathbf{E}\| x_\alpha(T)-b\|^2+\alpha
\mathbf{E}\int_0^T \|u(t)\|^2dt
\end{equation}
over all $u(\cdot)\in L_2^\mathcal{F}(I,U)$, where the solution
$x(\cdot)$ of  \eqref{e1.3}
is given by
\begin{equation} \label{e2.2}
\begin{aligned}
x(t)&= S(t)[x_0-h(x)-q(0)]+q(t)+\int_0^t AS(t-s)q(s)ds\\
&\quad +\int_0^tS(t-s)Bu(s)ds  +\int_0^tS(t-s)f(s)ds\\
&\quad +\int_0^tS(t-s)\tilde\sigma(s)dw(s)
+\int_0^tS(t-s)\Big[\int_0^sg(s,\tau)d\tau\Big]ds,
\end{aligned}
\end{equation}
Here $b \in L_2(\mathcal{F}_T,H)$ and $\alpha >0$ are
parameters and $f(\cdot)\in L_2^\mathcal{F}(I,H)$, $q(\cdot)\in
L_2^\mathcal{F}(I,H)$, $g(\cdot)\in L_2^\mathcal{F}(I,I,H)$,
$\tilde{\sigma}(\cdot)\in L_2^\mathcal{F}(I,L_0^2)$ and
$h(\cdot)\in C(I,H)$.

It is convenient  to introduce the relevant operators and the basic
controllability condition
\begin{itemize}
\item [(i)] The operator $L_0^T \in \mathcal{L}(L_2^\mathcal{
F}(I,{H}),L_2(\Omega,\mathcal{F}_T,{H}))$ is defined by
$$
L_0^T u = \int_0^T S(T-s)Bu(s) ds.
$$
Clearly the adjoint
$(L_0^T)^*:L_2(\Omega,\mathcal{F}_T,{H})\to L_2^\mathcal{F}(I,{H})$
is defined by
$$
[(L_0^T)^*z](t)=B^*S^*(T-t) \mathbf{E}\{z \mid \mathcal{F}_t\}.
$$

\item [(ii)]
The controllability operator $\Pi_0^T$ associated with \eqref{e1.3}
is defined by
$$
\Pi_0^T\{\cdot\}=L_0^T(L_0^T)^* \{\cdot\}=\int_0^T
S(T-t)BB^*S^*(T-t)\mathbf{E}\{\cdot \mid \mathcal{F}_t \}dt.
$$
which belongs to
$\mathcal{L}(L_2(\Omega,\mathcal{F}_T,{H}),L_2(\Omega,\mathcal{F}_T,{H}))$
and the controllability
operator $\Gamma_s^T \in \mathcal{L}({H},{H})$ is
$$
\Gamma_s^T=\int_s^T S(T-t)BB^*S^*(T-t)dt,\quad 0  \leq s < t.
$$

\item [(iii)] The resolvent operator
$$
\mathcal{R}(\alpha,\Gamma_0^T) := (\alpha I+\Gamma_0^T)^{-1},
\quad \mathcal{R}(\alpha,\Pi_0^T) := (\alpha I+\Pi_0^T)^{-1}.
$$

\item[(AC)] $\alpha \mathcal{R}(\alpha,\Pi_0^T)
:= (\alpha I+\Pi_0^T)^{-1}\to 0$  as $\alpha \to 0^+ $ in the strong
topology.
\end{itemize}

It is known that the assumption (AC) holds if and only if  the linear
stochastic system\eqref{e1.3} is approximately controllable on $[0,T]$
(see \cite{m1}).
The following lemmas whose proof can be found in \cite{m2} and
lemma 2.2 give a formula for a control  which steers the
system \eqref{e2.2} from a point $x_0 \in H$ to a small neighbourhood
of an arbitrary point $b \in L_2(\mathcal{F}_T,H)$.

\begin{lemma} \label{lem2.1}
(a) For arbitrary  $z \in
L_2(\mathcal{F}_T,H)$ there exists $k_z(\cdot)\in
L_2^\mathcal{F}(I,L_2^0)$ such that
\begin{gather}
\mathbf{E}\{z| \mathcal{F}_t\}
 = \mathbf{E}z+\int_0^tk_z(s)dw(s),\label{e2.3}\\
{\Pi}_0^T z= {\Gamma}_0^T\mathbf{E}z+\int_0^t\Gamma_s^T k_z(s)dw(s),\label{e2.4}\\
R(\alpha,{\Pi}_0^T) z = R(\alpha,{\Gamma}_0^T)\mathbf{E}z+\int_0^T
R(\alpha,\Gamma_s^T) k_z(s)dw(s), \label{e2.5}\\
\begin{aligned}
&{\Pi}_0^t S^*(T-t)R(\alpha,{\Pi}_0^T) z\\
&= {\Gamma}_0^t S^*(T-t)R(\alpha,{\Gamma}_0^T)\mathbf{E}z
+\int_0^t\Gamma_r^t S^*(T-t) R(\alpha,\Gamma_r^T) k_z(r)dw(r).
\end{aligned} \label{e2.6}
\end{gather}

(b) If $f:I \times H \to H$, $q:I \times H \to H$, $g:I\times
I\times H\to H$,  satisfies the condition (H2) and \!$x(\cdot)\in
L_2^F(I,H)$, then there exist $k_f(\cdot,x(s))\in L_2^F(I,L_2^0)$,
$k_q(\cdot,x(s))\in L_2^F(I,L_2^0)$ and $k_g(\cdot,\cdot,x(s))\in
L_2^F(I,I,L_2^0)$ such that
\begin{gather}
\begin{aligned}
&\mathbf{E}\{\int_0^T S(T-s)f(s,x(s))ds| \mathcal{F}_t\}\\
& =\mathbf{E}\int_0^T S(T-s)f(s,x(s))ds
 +\int_0^T k_f(s,x(s))dw(s) 
 \end{aligned}\label{e2.7}\\
\begin{aligned}
&\mathbf{E}\{\int_0^T AS(T-s)q(s,x(s))ds| \mathcal{F}_t\}\\
& =\mathbf{E}\int_0^T AS(T-s)q(s,x(s))ds
 +\int_0^T k_q(s,x(s))dw(s) 
 \end{aligned} \label{e2.8}\\
\begin{aligned}
&\mathbf{E}\{\int_0^T S(T-s)\Big[\int_0^sg(s,\tau,x(\tau))d\tau\Big]ds| \mathcal{F}_t\}\\
&=\mathbf{E}\int_0^T S(T-s)\Big[\int_0^sg(s,\tau,x(\tau))d\tau\Big]ds
+\int_0^T\Big[\int_0^s k_g(s,\tau,x(\tau))d\tau\Big]dw(s)
\end{aligned} \label{e2.9}
\end{gather}
and for all $x(\cdot), y(\cdot)\in L_2^F(I,H)$
\begin{gather}
\mathbf{E}\int_0^T\|k_f(s,x(s))-k_f(s,y(s))\|^2ds
\leq K^2TL_1\Big(\mathbf{E}\int_0^T\|x(s)-y(s)\|^2ds\Big) \label{e2.10}\\
\mathbf{E}\int_0^T\|k_f(s,x(s))\|^2ds
\leq  K^2T^2L_4  \label{e2.11}\\
\mathbf{E}\int_0^T\|k_q(s,x(s))-k_q(s,y(s))\|^2ds
\leq K^2l_0^2TL_3\Big(\mathbf{E}\int_0^T\|x(s)-y(s)\|^2ds\Big) \label{e2.12}\\
\mathbf{E}\int_0^T\|k_q(s,x(s))\|^2ds \leq K^2l_0^2T^2L_6, \label{e2.13}\\
\begin{aligned}
& \mathbf{E}\int_0^T\big\|\int_0^s\Big[k_g(s,\tau,x(\tau))
  -k_g(s,\tau,y(\tau))\Big]d\tau\big\|^2ds \\
&\leq K^2T^2L_2\Big(\mathbf{E}\int_0^s\|x(\tau)-y(\tau)\|^2d\tau\Big)
\end{aligned} \label{e2.14}\\
\mathbf{E}\int_0^T\big\|\int_0^s\Big[
k_g(s,\tau,x(\tau))d\tau\Big]\big\|^2ds \leq K^2T^3L_5 \label{e2.15}
\end{gather}
where $K = \max\{\|S(t)\|:0\leq t\leq T\}$ and $l_0=\|AS(t)q\|$.
\end{lemma}

\begin{lemma} \label{lem2.2}
There exists a unique control
$u_{\alpha}(\cdot)\in L_2^F(I,U)$ such that
 \begin{equation}
\begin{aligned} \label{e2.16}
u_\alpha(t)
 &= B^*S^*(T-t)\mathbf{E}\Big\{R(\alpha,\Pi_0^T)
\Big(b-S(T)[x_0-h(x)-q(0)]-q(T)\\
&\quad -\int_0^TAS(T-s)q(s)ds-\int_0^T S(T-s)f(s)ds \\
&\quad -\int_0^TS(T-s)\Big[\int_0^s g(s,\tau)d\tau\Big]ds
-\int_0^TS(T-s)\tilde{\sigma}(s)dw(s)\Big) |\mathcal{F}_t\Big\}
\end{aligned}
\end{equation}
and
\begin{equation} \label{e2.17}
\begin{aligned}
&x_{\alpha}(T)\\
&= b -{\alpha}R(\alpha,\Gamma_0^T)
 \Big(\mathbf{E}b-S(T)[x_0-h(x)-q(0)]-q(T)\\
&\quad -\mathbf{E}\int_0^TAS(T-s)q(s)ds-\mathbf{E}
 \int_0^T S(T-s)\Big[f(s)ds+\int_0^s g(s,\tau)d\tau\Big]ds\Big)\\
&\quad -\alpha\int_0^T R(\alpha,\Gamma_s^T)\Big(k_b(s)-S(T-s)
 \tilde\sigma(s)-k_f(s)-k_q(s)-\int_0^s k_g(s,\tau)d\tau\Big)dw(s)
\end{aligned}
\end{equation}
where
\begin{gather*}
\mathbf{E}\{\int_0^T S(T-s)f(s)ds| \mathcal{F}_t\}
 = \mathbf{E}\int_0^T S(T-s)f(s)ds+\int_0^T k_f(s)dw(s),\\
\mathbf{E}\{\int_0^T AS(T-s)q(s)ds| \mathcal{F}_t\}
 = \mathbf{E}\int_0^T AS(T-s)q(s)ds+\int_0^T Ak_q(s)dw(s),\\
\begin{aligned}
&\mathbf{E}\{\int_0^T S(T-s)\Big[\int_0^s g(s,\tau)d\tau\Big]ds| \mathcal{F}_t\}\\
&= \mathbf{E}\int_0^T S(T-s)\Big[\int_0^sg(s,\tau)d\tau\Big]ds
 +\int_0^T\Big[\int_0^s k_g(s,\tau)d\tau\Big]dw(s).
\end{aligned}
\end{gather*}
\end{lemma}

\begin{proof}
 The problem of minimizing the functional
\eqref{e2.1} has a unique solution
$u_\alpha(\cdot)\in L_2^\mathcal{F}(I,U)$ which is completely
characterized by the
stochastic maximum principle (see \cite{a1}) and has the following
form:
$$
u_\alpha (t)=-\alpha^{-1}B^*S^*(T-t)\mathbf{E}\{x_\alpha(T)-b|\mathcal{F}_t\}.
$$
Formula \eqref{e2.17} shows that the linear system \eqref{e2.2} is
approximately
controllable on $[0,T]$ if and only if $ \alpha R(\alpha,\Pi_0^T)$
converges to zero operator as $\alpha\to 0^+$ in the
strong topology \cite{m1}.
\end{proof}

\section{Approximate Controllability}

In this section sufficient conditions are established for the
approximate controllability of the stochastic control system
\eqref{e1.2} under the assumption that the associated linear
system is approximately controllable.

\noindent \textbf{Definition.}
The stochastic system \eqref{e1.2} is approximately controllable on
the interval $I$ if
$$
\overline{\mathcal{R}}_T(x_0) = L_2(\mathcal{F}_T,H),
$$
where
$\mathcal{R}_T(x_0)=\{x(T;x_0,u):u(\cdot) \in L_2^\mathcal{F}(I,U)\}$.
\medskip

 Define the control
\begin{equation} \label{e3.1}
\begin{aligned}
u_\alpha(t)
&= B^*S^*(T-t)\mathbf{E}\Big\{R(\alpha,\Pi_0^T)
  \Big(b-S(T)[x_0-h(x)-q(0,x(0))] -q(T,x(T)) \\
&\quad   -\int_0^TAS(T-s)q(s,x(s))ds-\int_0^T
S(T-s)f(s,x(s))ds\\
&\quad -\int_0^TS(T-s)\Big[\int_0^s g(s,\tau,x(\tau))d\tau\Big]ds\\
&\quad   -\int_0^TS(T-s)\sigma(s,x(s))dw(s)\Big)\Big|\mathcal{F}_t\Big\}.
\end{aligned}
\end{equation}
To formulate the controllability problem in the form
suitable for application of the Nussbaum fixed-point theorem, we
put the control $u_\alpha(\cdot)$ into the stochastic control
system \eqref{e1.2} and obtain a nonlinear operator
$\mathbb{P}^\alpha:C(I,L_2)\to C(I,L_2)$
\begin{align*}
(\mathbb{P}^\alpha x)(t)
&= S(t)[x_0-h(x)-q(0,x(0))]+q(t,x(t))+\int_0^TAS(T-s)q(s,x(s))ds\\
&\quad +\int_0^tS(t-s)f(s,x(s))ds+\int_0^tS(t-s)\sigma(s,x(s))dw(s)\\
&\quad +\int_0^tS(t-s)\Big[\int_0^s g(s,\tau,x(\tau))d\tau\Big]ds
  +\Pi_0^tS^*(T-t)R(\alpha,\Pi_0^T)\\
&\quad \times\Big(b-S(T)[x_0-h(x)-q(0,x(0))]-q(t,x(t))\\
&\quad -\int_0^TAS(T-s)q(s,x(s))ds-\int_0^T S(T-s)f(s,x(s))ds\\
&\quad -\int_0^T S(T-s)\Big[\int_0^s g(s,\tau,x(\tau))d\tau\Big]ds
  -\int_0^T S(T-s)\sigma(s,x(s))dw(s)\Big).
\end{align*}
It will be shown that the stochastic control system
\eqref{e1.2} is approximately controllable if for all
$\alpha>0$ there exists a fixed point of the operator
$\mathbb{P}^\alpha$. To show that $\mathbb{P}^\alpha$ has a fixed
point we employ the Nussbaum fixed-point theorem in $C(I,L_2)$.We
now define the operators $\mathbb{P}_1^\alpha:C(I,L_2)\to
C(I,L_2)$ and $\mathbb{P}_2^\alpha:C(I,L_2)\to C(I,H)$ as
follows:
\begin{equation} \label{e3.2}
\begin{aligned}
&(\mathbb{P}_1^\alpha x)(t)\\
&= S(t)[x_0-h(x)-q(0,x(0))]+q(t,x(t))+\int_0^TAS(T-s)q(s,x(s))ds\\
&\quad +\int_0^t S(t-s)f(s,x(s))ds+\int_0^tS(t-s)\sigma(s,x(s))dw(s)\\
&\quad +\int_0^tS(t-s)\Big[\int_0^s g(s,\tau,x(\tau))d\tau\Big]ds
  +\int_0^t\Gamma_s^tS^*(T-t)R(\alpha,\Gamma_s^T)\Big[k_b(s)\\
&\quad -S(T-s)\sigma(s,x(s))-k_f(s,x(s))-Ak_q(s,x(s))
  -\int_0^s k_g(s,\tau,x(\tau))d\tau\Big]dw(s),
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.3}
\begin{aligned}
(\mathbb{P}_2^\alpha x)(t)
&= \Gamma_0^tS^*(T-t)R(\alpha,\Gamma_0^T)\Big(\mathbf{E}b-S(T)
   [x_0-h(x)-q(0,x(0))]-q(T,x(T))\\
&\quad -\mathbf{E}\int_0^TAS(T-s)q(s,x(s))ds
   -\mathbf{E}\int_0^T S(T-s)f(s,x(s))ds\\
&\quad -\mathbf{E}\int_0^T S(T-s)\Big[\int_0^s
   g(s,\tau,x(\tau))d\tau\Big]\Big),
\end{aligned}
\end{equation}
where $k_b(s)$,
$k_f(s,x)$, $k_q(s,x)$ and $k_g(s,\tau,x)$ are defined by \eqref{e2.3},
\eqref{e2.7}, \eqref{e2.8} and \eqref{e2.9} respectively.
By using \eqref{e2.6} along with
\begin{align*}
z&\equiv b-S(t)[x_0-h(x)-q(0,x(0))]-q(T,x(T))+\int_0^T
AS(T-s)q(s,x(s))ds\\
&\quad -\int_0^T S(T-s)f(s,x(s)ds-\int_0^T
 S(T-s)\int_0^s g(s,\tau,x(\tau))d\tau\Big]ds\\
&\quad -\int_0^T S(T-s)\sigma(s,x(s))dw(s),
\end{align*}
it is easy to observe that
$\mathbb{P}^\alpha x=(\mathbb{P}_1^\alpha+\mathbb{P}_2^\alpha)x$.
Define the set
$$
Y_r = \{x(\cdot) \in C(I,L_2):  \mathbf{E}\|x(t)\|^2\leq r\},
$$
where $r$ is a positive constant. Let us take
$$
M=\|B\|, \quad  N=  T \max  \{\|S(t)BB^*S^*(t)\|: 0\leq t<T\}.
$$

\begin{theorem} \label{thm3.1}
Assume that {\rm (H1)-(H2), (AC)} hold.
Then the system \eqref{e1.2} is approximately controllable
on $[0,T]$.
\end{theorem}

\begin{proof} The proof is done  by the several steps.

\textbf{Step 1.} For arbitrary $\alpha>0$ there is a
positive constant $r_0=r_0(\alpha)$ such that
$\mathbb{P}:Y_{r_0}\to Y_{r_0}$. From the definition of
$\mathbb{P}_1^\alpha$ and $\mathbb{P}_2^\alpha$,  for any
$x(\cdot)\in Y_{r_0}$, we have
\begin{align*}
&\|\mathbb{P}_1^\alpha x\|_*\\
&\leq K[\|x_0\|+l_1+\sqrt{L_6}]+\sqrt{L_6}+l_0KT\sqrt{L_6}+KT\sqrt{L_4}
 +K\sqrt{TL_4}\\
&\quad +KT\sqrt{TL_5}+ \frac{1}{\alpha}N
 K\Big(\int_0^T\mathbf{E}\|k_b(s)\|^2ds
 +\int_0^T\mathbf{E}\|A\|\|k_q(s,x(s))\|^2ds\\
&\quad +\int_0^T\mathbf{E}\|k_f(s,x(s))\|^2ds
 +\int_0^T\Big[\int_0^s\mathbf{E}\|k_g(s,\tau,x(\tau))\|^2d\tau\Big]ds
 + K^2TL_3\Big)^{1/2}\\
&\leq  K\|x_0\|+Kl_1+(K+1+KTl_0)\sqrt{L_6}+K(\sqrt{T}+1)\sqrt{TL_4}
 +KT\sqrt{TL_5}\\
&\quad +\frac{1}{\alpha}NK\Big(\int_0^T
 \mathbf{E}\|k_b(s)\|^2ds+K^2T^2l_0^2L_6+K^2T^3L_5+K^2T
 \Big(T+1\Big)L_4\Big)^{1/2},
\end{align*}
$$
\|\mathbb{P}_2^\alpha x\|\leq
\frac{1}{\alpha}NK\Big(\|\mathbf{E}b\|+K(\|x_0\|+l_1)+(K+1+KTl_0)
\sqrt{L_6}+KT(\sqrt{L_4}+\sqrt{TL_5})\Big),
$$
which implies for sufficiently large $r_0=r_0(\alpha)$
$$
\|\mathbb{P}^\alpha x\|_*\leq\|\mathbb{P}_1^\alpha x\|_*
+\|\mathbb{P}_2^\alpha x\|\leq r_0(\alpha).
$$
Hence, $\mathbb{P}^\alpha$ maps $Y_{r_0}$ into itself for some
$r_0$.

\textbf{Step 2.} For arbitrary $\alpha>0$ the operator
$\mathbb{P}_2^\alpha$ maps $Y_{r_0}$ into a relatively compact
subset of $Y_{r_0}$.
According to the infinite-dimensional version of the
 Ascoli-Arzela theorem we have to show that
\begin{enumerate}
\item for arbitrary $t\in [0,T]$ the set
$$
V(t)=\{(\mathbb{P}_2^\alpha x)(t) :x\in Y_{r_0}\}\subset X
$$
is relatively compact.

\item for arbitrary $\epsilon>0$ there exists $\delta>0$ such that
$$
\|(\mathbb{P}_2^\alpha x)(t+\tau)-(\mathbb{P}_2^\alpha x)(t)\|<\epsilon,
$$
if $\|x\|\leq r$, $|\tau|\leq \delta$, $t, t+\tau \in [0,T]$.
\end{enumerate}
Notice that the uniform boundedness is proved in step 1.

 Let us prove (1). In fact, the case where $t=0$ is trivial, since
$V(0)=\{x_0\}$, so let $t$, $0<t\leq T$, be fixed and let $\eta$ be
a given real number satisfying $0<\eta<t$. Define
\begin{align*}
&(\mathbb{P}_2^{\alpha,\eta} x)(t)\\
&= \int_0^{t-\eta} S(t-r)BB^*S^*(T-r)dr
   R(\alpha,\Gamma_0^T)\Big(\mathbf{E}b-S(T)[x_0-h(x)-q(0,x(0))]\\
&\quad -q(T,x(T))-\mathbf{E}
 \int_0^T A S(T-s)q(s,x(s))ds-\mathbf{E}
 \int_0^T S(T-s)f(s,x(s))ds\\
&\quad -\mathbf{E}\int_0^T S(T-s)\Big[\int_0^s
g(s,\tau,x(\tau))d\tau\Big]\Big)ds\\
&= S(\eta)(\mathbb{P}_2^{\alpha} x)(t-\eta).
\end{align*}
Since $S(\eta)$ is compact and $(\mathbb{P}_2^\alpha x)(t-\eta)$
is bounded on $Y_{r_0}$ the set
$$
V_{\eta}(t)=\{(\mathbb{P}_2^{\alpha,\eta} x)(t) :x(\cdot)\in Y_{r}\}
$$
is relatively compact set in $H$, that is, we can find a finite set
 $\{y_i, 1\leq i\leq m\}$ in $H$ such that
$$
V_{\eta}(t)\subset \bigcup_{i=1}^m N(y_i,\frac{\epsilon}{2}).
$$
On the other hand, there exists $\eta >0$ such that
\begin{align*}
&\|(\mathbb{P}_2^\alpha x)(t)-(\mathbb{P}_2^{\alpha,\eta} x)(t)\| \\
&= \big\|\int_{t-\eta}^t S(t-r)BB^*S^*(T-r)dr
 R(\alpha,\Gamma_0^T)\Big(\mathbf{E}b-S(T)[x_0-h(x)-q(0,x(0))]\\
&\quad -q(T,x(T))-\mathbf{E}\int_0^TA
 S(T-s)q(s,x(s))ds-\mathbf{E}\int_0^T S(T-s)f(s,x(s)ds\\
&\quad -\mathbf{E}\int_0^T S(T-s)\Big(\int_0^s
 g(s,\tau,x(\tau))d\tau\Big)ds\Big)\big\|\\
&\leq\! \frac{1}{\alpha}K^2M^2\Big(\|\mathbf{E}b\|+K(\|x_0\|+l_1)
 +(K+1+KTl_0)\sqrt{L_6}+KT(\sqrt{L_4}+\sqrt{TL_5})\Big)\eta\\
&\leq \frac{\epsilon}{2}.
\end{align*}
Consequently,
$$
V(t)\subset \bigcup_{i=1}^m N(y_i,{\epsilon}).
$$
Hence, for each $t\in [0,T]$, $V(t)$ is relatively compact in $X$.

Next, we prove (2). We have to show that $V=\{\mathbb{P}_2^\alpha
x)(\cdot) :x \in Y_{r_0}\}$ is equicontinuous on $[0,T]$. In fact,
for $0<t<t+\tau\leq T$ and $0<\eta\leq t$.
\begin{align*} %\label{e3.4}
&\|(\mathbb{P}_2^\alpha x)(t+\tau)-(\mathbb{P}_2^\alpha x)(t)\| \\
&\leq \big\|\Gamma_0^{t+\tau}S^*(T-t-\tau)-\Gamma_0^t
S^*(T-t)\big\|\big\|R(\alpha,\Gamma_0^T) \\
&\times \Big(\mathbf{E}b-S(T)[x_0-h(x)-q(0,x(0))]
 -q(T,x(T))-\mathbf{E}\int_0^TA S(T-s)q(s,x(s))ds \\
&\quad -\mathbf{E}\int_0^T S(T-s)f(s,x(s))
-\mathbf{E}\int_0^T S(T-s)\Big[\int_0^s
g(s,\tau,x(\tau))d\tau\Big]ds\Big)\big\|
\\
&\leq \big\|\int_t^{t+\tau}S(t+\tau-s)BB^*S^*(T-s)ds-[S(\tau)-I]
 \int_0^tS(t-s)BB^*S^*(T-s)ds\big\|\\
&\quad \times\frac{1}{\alpha}\Big(\|\mathbf{E}b\|+K[\|x_0\|+l_1]
 +(K+1+KTl_0)\sqrt{L_6}+KT\sqrt{L_4}+KT\sqrt{TL_5}\Big)\\
&\leq \frac{1}{\alpha}\Big(\tau+\|S(\tau)-I\|\Big)K^2M^2
 \Big(\|\mathbf{E}b\|+K[\|x_0+l_1]\|+(K+1+KTl_0)\sqrt{L_6}\\
&\quad +KT(\sqrt{L_4}+\sqrt{TL_5})\Big).
\end{align*}
The right-hand side of the above inequality does not depend on particular
choice of $x(\cdot)$ and approaches zero as $\tau\to 0^+$.
The case $0<t+\tau<t\leq T$ can be considered in a similar manner.
So, we obtain the equicontinuity of $V$. Thus,
$\mathbb{P}_2^\alpha$ maps $Y_{r_0}$ into an equicontinuous family
of deterministic functions which are also bounded. By the
Ascoli-Arzela theorem $\mathbb{P}_2^\alpha[y_{r_0}]$ is relatively
compact in $C(I,L_2)$.

\textbf{Step 3.} Here we prove $\mathbb{P}_1^\alpha$ is a
contraction mapping. In fact
\begin{align*}
&\|\mathbb{P}_1^\alpha x-\mathbb{P}_1^\alpha y\|_* \\
&\leq \big\|\int_0^t S(t-s)\Big[h(x(s))-h(y(s))\Big]
 ds\big\|_*+\big\|q(t,x(t))-q(t,y(t)\big\|_*\\
&\quad +\big\|\int_0^tA S(t-s)\Big[q(s,x(s))-q(s,y(s))\Big]ds\big\|_*\\
&\quad +\big\|\int_0^t S(t-s)\Big[f(s,x(s))-f(s,y(s))\Big]ds\big\|_*\\
&\quad +\big\|\int_0^t S(t-s)\Big[\sigma(s,x(s))-\sigma(s,y(s))\Big]dw(s)\big\|_*\\
&\quad +\big\|\int_0^t S(t-s)\Big[\int_0^s\Big[g(s,\tau,x(\tau))-g(s,\tau,y(\tau))\Big]d(\tau)\Big]ds\big\|_*\\
&\quad +NK\big\|\int_0^tR(\alpha,\Gamma_s^T)S(T-s)\Big[\sigma(s,x(s))-\sigma(s,x(s))\Big]dw(s)\big\|_*\\
&\quad +NK\big\|\int_0^t R(\alpha,\Gamma_s^T)S(T-s)\Big[k_f(s,x(s))- k_f(s,y(s))\Big]dw(s)\big\|_*\\
&\quad +NK\big\|\int_0^t R(\alpha,\Gamma_s^T)AS(T-s)\Big[k_q(s,x(s))- k_q(s,y(s))\Big]dw(s)\big\|_*\\
&\quad +NK\big\|\int_0^tR(\alpha,\Gamma_s^T)S(T-s)\Big[\int_0^s\Big[k_g(s,\tau,x(\tau))-k_g(s,\tau,y(\tau))\Big]d\tau\Big]dw(s)\big\|_*\\
&\leq  Kl+(1+KTl_0)\sqrt{L_3}+K\sqrt{TL_1}(\sqrt{T}+1)+KT\sqrt{TL_2}\\
&\quad +\frac{1}{\alpha}NK^2\Big[2\sqrt{TL_1}+l_0\sqrt{TL_3}
  +T\sqrt{L_2}\Big]\|x(s)-y(s)\|_*\,.
\end{align*}
Here we used the inequality \eqref{e2.11} and \eqref{e2.12}. So, if
 \begin{equation}
\label{e3.5}
\begin{aligned}
&Kl+(1+KTl_0)\sqrt{L_3}+K\sqrt{TL_1}(\sqrt{T}+1)\\
&+KT\sqrt{TL_2} +\frac{1}{\alpha}NK^2\Big[2\sqrt{TL_1}+l_0\sqrt{TL_3}
+T\sqrt{L_2}\Big] <1
\end{aligned}
\end{equation}
Thus $\mathbb{P}_1^\alpha$ is a contraction mapping.

\textbf{Step 4.} Now we prove $\mathbb{P}_2^\alpha$ is
continuous on $C(I,H)$. To apply the Nussbaum fixed-point theorem
it remains to show that $\mathbb{P}_2^\alpha$ is continuous on
$C(I,L_2)$. Let $\{x^n(\cdot)\}\subset C(I,L_2)$ with
$x^n(\cdot)\to x(\cdot) \in C(I,L_2)$. Then the
Lebesgue-dominated convergence theorem implies
\begin{align*}
&\|\mathbb{P}_2^\alpha x^n(t)-\mathbb{P}_2^\alpha x(t)\| \\
&\leq  \frac{1}{\alpha}NK\Big[\big\|S(T)(h(x^n)-h(x))\big\|
 +\big\|q(t,x^n(t))-q(t,x(t))\big\| \\
&\quad +\mathbf{E}\int_0^T\big\|
AS(T-s)\Big(q(s,x^n(s))-q(s,x(s)\Big)\big\|ds\\
&\quad +\mathbf{E}\int_0^T\big\| S(T-s)\Big(f(s,x^n(s))
 -f(s,x(s)\Big)\big\|ds\\
&\quad + \mathbf{E}\int_0^T \big\|S(T-s)\int_0^s
\Big(g(s,\tau,x^n(\tau))-g(s,\tau,x(\tau))\Big)d\tau\big\|ds \Big]\\
&\leq \frac{1}{\alpha}NK(Kl+\sqrt{L_3})+\frac{1}{\alpha}NK^2\sqrt{T}
 \Big[l_0^2\int_0^T\mathbf{E}\|q(s,x^n(s))-q(s,x(s))\|^2ds\\
&\quad +\int_0^T\mathbf{E}\|f(s,x^n(s))-f(s,x(s))\|^2ds\\
&\quad +\int_0^T\Big(\int_0^s\mathbf{E}\big\| g(s,\tau,x^n(\tau))
  -g(s,\tau,x(\tau))\big\|^2d\tau\Big)ds\Big]^{1/2}\\
&\leq \frac{1}{\alpha}NK(Kl+\sqrt{L_3})+\frac{1}{\alpha}NK^2\sqrt{T}
 \Big(l_0^2L_3+L_1+TL_2\Big)\\
&\quad \times \Big[\int_0^T\mathbf{E}\Big(\|x^n(s)-x(s)\|^2ds
 +\|x^n(s)-x(s)\|^2ds+\|x^n(s)-x(s)\|^2ds\Big)\Big]^{1/2}\\
&\leq \frac{1}{\alpha}NK(Kl+\sqrt{L_3})+\frac{3}{\alpha}NK^2T
 \Big(l_0^2L_3+L_1+TL_2\Big)\|x^n-x\| \to 0
\end{align*}
as $n\to\infty$. Thus $\mathbb{P}_2^\alpha$ is continuous
on $C(I,L_2)$.

\textbf{Step 5.} From the Nussbaum fixed point theorem
$\mathbb{P}_\alpha$ has a fixed point provided that the inequality
\eqref{e3.5} is satisfied. It is easily seen that this fixed point is a
solution of the system \eqref{e1.2}. The extra condition
\eqref{e3.5} can easily be removed by considering \eqref{e1.2} on
intervals $[0,\tilde T],[\tilde T,2\tilde T],\dots $, with $\tilde T$
satisfying \eqref{e3.5}. Let $x_\alpha^*(\cdot)$ be a fixed point of the
operator $\mathbb{ P}^\alpha$ in $Y_{r_0}$. Any fixed point of
$\mathbb{P}^\alpha$ is a mild solution of \eqref{e1.1} on $[0,T]$
under the control $u_\alpha(t)$ defined by \eqref{e3.1}, where $x$ is
replaced by $x_\alpha ^*$ and, by Lemma 2.2 satisfies
\begin{equation}\label{e3.6}
\begin{aligned}
(\mathbb{P}^\alpha x_\alpha^*)(T)
&= x_\alpha^*(T)\\
&=b+\alpha R(\alpha,\Pi_0^T)
 \Big(S(T)[x_0-h(x)-q(0,x(0))]+q(T,x(T))\\
&\quad +\int_0^TA S(T-s)q(s,x_\alpha^*(s))ds
 +\int_0^T S(T-s)f(s,x_\alpha^*(s))ds\\
&\quad +\int_0^TS(T-s)\Big[\int_0^s g(s,\tau,x_\alpha^*(\tau))d\tau\Big]ds\\
&\quad +\int_0^T S(T-s)\sigma(s,x_\alpha^*(s))dw(s)-b\Big).
\end{aligned}
 \end{equation}
 Set
\begin{align*}
z_\alpha
&= S(T)[x_0-h(x)-q(0,x(0))]+q(T,x(T))\\
&\quad +\int_0^TA S(T-s)q(s,x_\alpha^*(s))ds
  +\int_0^T S(T-s)f(s,x_\alpha^*(s))ds\\
&\quad +\int_0^TS(T-s)\Big[\int_0^s
 g(s,\tau,x_\alpha^*(\tau))d\tau\Big]ds+\int_0^T S(T-s)
 \sigma(s,x_\alpha^*(s))dw(s)-b\Big).
\end{align*}
By (H2), and then there is a subsequence, still denoted
by
$$
\{ f(s,x_\alpha^*(s)), q(s,x_\alpha^*(s)),
 \int_0^s g(s,\tau,x_\alpha^*(\tau))d\tau,
\sigma(s,x_\alpha^*)\},
$$
 weakly converging to, say,
$(f(s,\omega), (q(s,\omega), \sigma(s,\omega))$ in
 $H  \times L_0^2$ and $g(s,\tau,\omega) $ in
 $H  \times H\times L_0^2$. The compactness of $S(t)$, $t>0$ implies
\begin{gather*}
S(T-s)f(s,x_\alpha^*(s)) \to  S(T-s)f(s,\omega),\\
S(T-s)q(s,x_\alpha^*(s)) \to  q(T-s)q(s,\omega),\\
S(T-s)g(s,\tau,x_\alpha^*(\tau)) \to  S(T-s)g(s,\tau,\omega),\\
S(T-s)\sigma(s,x_\alpha^*(s)) \to  S(T-s)\sigma(s,\omega)\quad
\text{a.e. in } I \times \Omega.
\end{gather*}
On the other hand
\begin{gather*}
\|S(T-s)f(s,x_\alpha^*(s))\|^2+\|S(T-s)\sigma(s,x_\alpha^*(s))\|^2
\leq K^2L_4,\\
\|S(T-s) g(s,\tau, x_\alpha^*(\tau))\|^2 \leq  K^2L_5, \\
\|S(T-s)q(s,x_\alpha^*(s))\|^2 \leq  K^2L_6\quad
\text{a.e.  in } I \times\Omega.
\end{gather*}
Thus by the Lebesgue-dominated convergence theorem
$$
\mathbf{E}\|z_\alpha-z\|^2\to 0\quad\text{as } \alpha\to 0^+,
$$
where
\begin{align*}
z &= S(T)[x_0-h(x)-q(0)]+q(T)+\int_0^TA S(T-s)q(s)ds
  +\int_0^T S(T-s)f(s)ds\\&\quad +\int_0^TS(T-s)
  \Big[\int_0^s  g(s,\tau)d\tau\Big]ds+\int_0^T S(T-s)\sigma(s)dw(s)-b.
\end{align*}
Then having in mind that $\mathbf{E}\|\alpha R(\alpha,\Pi_0^T)\|^2\leq
1$ and $\alpha R(\alpha,\Pi_0^T)\to 0$ strongly by the
assumption (AC), from \eqref{e3.6} we obtain
\begin{align*}
\sqrt{\mathbf{E}\|x_\alpha^*(T)-h\|^2}
&\leq \sqrt{\mathbf{E}\|\alpha R(\alpha,\Pi_0^T)(z_\alpha-z)\|^2}
 +\sqrt{\mathbf{E}\|\alpha R(\alpha,\Pi_0^T)(z)\|^2}\\
&\leq \sqrt{\mathbf{E}\|z_\alpha-z\|^2}
 +\sqrt{\mathbf{E}\|\alpha R(\alpha,\Pi_0^T)(z)\|^2}\to 0
\end{align*}
as $\alpha\to 0^+$. This gives the approximate
controllability of \eqref{e1.2}. Hence the
proof is complete.
\end{proof}

\begin{corollary} \label{coro3.1}
Assume that  {\rm (H2)} holds.
If the semigroup $S(t)$ is analytic and the deterministic linear
system corresponding to \eqref{e1.1} is approximately controllable
on $[0,T]$ then the stochastic system \eqref{e1.1} is approximately
controllable on $[0,T]$.
\end{corollary}

\begin{proof} It is known that (see \cite[Theorem 4.3]{m1})
when the semigroup $S(t)$ is analytic the linear stochastic system
\eqref{e1.3} is approximately controllable on $[0,T]$ if and only if the
corresponding deterministic linear system is approximately
controllable on $[0,T]$. Then by Theorem 3.1, the system
\eqref{e1.1} is approximately controllable on $[0,T]$.
\end{proof}

\section{Applications}

Consider the following stochastic classical heat equation for
material with memory
\begin{equation} \label{e4.1}
\begin{gathered}
\begin{aligned}
&d[z(t,\theta)-m(t,z(t,\theta))]\\
&= \Big[z_{\theta\theta}(t,\theta)+Bu(t,\theta)+p(t,z(t,\theta))
  +\int_0^t q(t,s,z(s,\theta)ds\Big]dt
 +k(t,z(t,\theta)dw(t),\\
&\quad  \text{for } (t,\theta)  \in  I \times[0,\pi]=\Omega,
\end{aligned} \\
 z(t,\theta) = 0\quad \text{for } I \times \partial\Omega ,\\
z(0,\theta)+\sum_{i=1}^pc_iz(t_i,\theta) = z_0(\theta) \quad
\text{for } \theta \in \Omega,\; 0<t_i\leq T,
\end{gathered}
\end{equation}
where $\Omega$ is an open  bounded subset of $R^n$ with smooth boundary
$\partial\Omega$, and $B$ is a bounded linear operator from a
Hilbert space $U$ into $H$. We assume that $p:I \times H \to H$,
$m:I \times H \to H$,   $k:I \times H \to L_2^0$,
$q:I \times I\times H \to H$, and $c_i\in C(I,H)$ are all continuous
and uniformly bounded, $u(t)$ is a feedback control and $w$ is an
$Q$-Wiener process. Let $H=L_2[0, \pi]$, and let $A:H\to H$ be an operator
defined by
$$
Ax=x_{\theta \theta}
$$
with domain
$$
D(A)=\{x \in H \ |x, x_\theta\text{ are  absolutely  continuous, }
 x_{\theta\theta} \in H, x(0)=x(\pi)=0\}.
$$
Let $f :I \times H \to H$ be defined by
$$
f(t,x)(\theta)=p(t,x(\theta)),\quad (t,x) \in I\times H, \quad
 \theta  \in [0,\pi]
$$
Let $q :I \times H \to H$ be defined by
$$
q(t,x)(\theta)=m(t,x(\theta)),
$$
Let $g :I \times I\times H \to H$ be defined by
$$
g(t,s,x)(\theta)=q(t,s,x(\theta)),
$$
Let $h:C(I,H) \to H$ be defined by
$$
h(x)(\theta)=\sum_{i=1}^pc_ix(t_i)(\theta),
$$
Let $\sigma :I \times H \to L_2^0 $ be defined by
$$
\sigma(t,x)(\theta)=k(t,x(\theta)).
$$
With this choice of $A, B, f, q, g, h$ and $\sigma$, \eqref{e1.2} is the
abstract formulation of \eqref{e4.1}, be such that the condition in (H2)
is satisfied.
Then
$$
Ax=\sum_{n=1}^{\infty}\Big(-n^2\Big) (x,e_n)e_n(\theta),\quad x \in  D(A),
$$
where $e_n(\theta)=\sqrt{\frac{2}{\pi}}\sin n\theta$,
$0\leq \theta \leq \pi$, $n=1,2,\dots $
It is known that $A$ generates an analytic semigroup $S(t)$, $t>0$ in $H$
and is given by
$$
S(t)x=\sum_{n=1}^{\infty}e^{-n^2t}(x,e_n)e_n(\theta),\quad x\in H.
$$
Now define an infinite-dimensional space
$$
U=\Big\{u=\sum_{n=2}^{\infty}u_n e_n(\theta) : \sum_{n=2}^{\infty}u_n^2
<\infty\Big\}
$$
with a norm defined by $\|u\|=(\sum_{n=2}^{\infty}u_n^2)^{\frac{1}{2}}$
and a linear continuous mapping B from U to H as follows:
$$
Bu=2u_2e_1(\theta)+\sum_{n=2}^{\infty}u_n e_n(\theta).
$$
It is obvious that for
$u(t,\theta,\omega)=\sum_{n=2}^{\infty}u_n(t,\omega)e_n(\theta)
\in L_2^\mathcal{F}(I,U)$
$$
Bu(t)=2u_2(t)e_1(\theta)+\sum_{n=2}^{\infty}u_n(t)e_n(\theta)
 \in L_2^\mathcal{F}(I,H).
$$
Moreover,
\begin{gather*}
B^* v=(2 v_1 + v_2)e_2(\theta)+\sum_{n=3}^{\infty} v_n e_n(\theta),\\
B^*S^*(t)z=\Big(2z_1e^{-t}+z_2e^{-4t}\Big)e_2(\theta)
  +\sum_{n=3}^{\infty}z_n e^{-n^2t}e_n(\theta)
\end{gather*}
for $v=\sum_{n=1}^{\infty}v_n e_n(\theta)$ and
$z=\sum_{n=1}^{\infty}z_ne_n(\theta).$ Let
$$
\|B^*S^*(t)z\|=0,\quad t\in[0,T],
$$
it follows that
\begin{gather*}
\big\|2z_1e^{-t}+z_2e^{-4t} \Big \|^2+\sum_{n=3}^{\infty}
\big\|z_n e^{-n^2t}\big\|^2=0,\quad t\in[0,T]\\
\Longrightarrow z_n=0,\quad n=1,2,\dots \Longrightarrow z=0.
\end{gather*}
Thus, by \cite[Theorem 4.1.7]{c1}, the deterministic linear system
corresponding to \eqref{e4.1} is approximately controllable
on $[0,T]$ and by Corollary 3.1, the system \eqref{e4.1}
is approximately controllable on $[0,T]$ provided that
$f$, $q$, $g$, $h$ and $\sigma$ satisfies the assumptions (H2).

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