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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 86, 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}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/86\hfil Controllability of neutral systems]
{Controllability of neutral impulsive stochastic quasilinear
 integrodifferential systems with nonlocal conditions}

\author[K. Balachandran, R. Sathya \hfil EJDE-2011/86\hfilneg]
{Krishnan Balachandran, Ravikumar Sathya}  % in alphabetical order

\address{Department of Mathematics,
Bharathiar University, Coimbatore - 641046, India}
\email[K. Balachandran]{kb.maths.bu@gmail.com}
\email[R. Sathya]{sathyain.math@gmail.com}

\thanks{Submitted June 3, 2011. Published June 29, 2011.}
\subjclass[2000]{93B05, 34A37, 34K50}
\keywords{Controllability; neutral equation; fixed point;
\hfill\break\indent  impulsive stochastic integrodifferential system}

\begin{abstract}
 We establish sufficient conditions for controllability of
 neutral impulsive stochastic quasilinear integrodifferential
 systems with nonlocal conditions in Hilbert spaces.
 The results are obtained by using semigroup theory, evolution
 operator and a fixed point technique.
 An example is provided to illustrate the obtained results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Abstract differential systems in infinite-dimensional spaces
appear in many bran\-ches of science and engineering, such as heat
flow in materials with memory, viscoelasticity and other physical
phenomena. In these fields many stochastic differential equations
are obtained by including random fluctuations in ordinary
differential equations which have been deduced from phenomological
or physical laws. Quasilinear evolution equations forms a very
important class of evolution equations as many time dependent
phenomena in physics, chemistry and biology can be represented by
such evolution equations. Some examples of quasi-stochastic
systems are the system of price fluctuations in financial markets,
earth climate or the seismic activity of the earth crust and a
dice game. Of particular interest the following
integrodifferential equation arises in the theory of
one-dimensional viscoelasticity \cite{Kim, Xie} and also a
special model for one-dimensional heat flow in materials with
memory. \begin{equation} \label{eqvis1}
\begin{gathered}
 u_t(t,x)=\int_0^tk(t-s)(\sigma(u_x))_x(s,x)ds +f(t,x), \quad
  t\geq 0,\; x\in (0,1),\\
u(0,x)=u_0(x),\quad  x\in[0,1],\quad u(t,0)=u(t,1)=0,\; t>0.
\end{gathered}
\end{equation}
In many of the papers, the mathematical model for certain
problems in nonlinear viscoelasticity is discussed in the form
\begin{equation} \label{eqvis}
\begin{gathered}
u_{tt}(t,x)=\phi(u_x(t,x))_x +\int_0^t a(t-s)\psi(u_x(s,x))_{x}ds
+g(t,x),\quad  t\geq 0,\\
u(0,x)=u_0(x),\quad x \in \mathbb{R}.
\end{gathered}
\end{equation}
which is the same as \eqref{eqvis1} if $\phi=\psi=\sigma$, $k(0)=1$
and $a=k'$ (see \cite{Gust}). In \cite{Heard}, the following
equation occurred during the study of the nonlinear behavior
of elastic strings \cite{Nara}.
\begin{equation} \label{eqmem}
\begin{gathered}
 u_{tt}(t,x) + c(t)u_t(t,x) -   M\Big(\int_{-\infty} ^\infty
    |u_x(t,s)|^2ds\Big)u_{xx}(t,x)+u(t,x)=h(t,x,u(t,x)),\\
0\leq t< \infty,\\
u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),\quad  x\in \mathbb{R}.
\end{gathered}
\end{equation}
The above equations take the abstract form as
\begin{equation}\label{abs}
 \frac{du(t)}{dt}=A(u)u(t)+f(t,u(t)),\quad u(0)=u_0.
\end{equation}
where $A$ is a linear operator in a Hilbert space $H$ and $f$
is a real function. Hence the natural generalization of
 \eqref{abs} is the following quasilinear integrodifferential
equation
\begin{equation}\label{eqqu}
\begin{gathered}
 u'(t)=A(t,u)u(t)+f(t,u(t))+\int_0^t g(t,s,u(s))ds,\\
u(0)=u_0.
\end{gathered}
\end{equation} Systems with short-term perturbations are often naturally
described  by impulsive differential equations. The theory of
impulsive differential equations is much richer than the
corresponding theory of differential equations without impulse
effects \cite{Lakshmi,Samo}. For instance, impulsive interruptions
are observed in mechanics, radio engineering, communication
security, control theory, optimal control, biology, mechanics,
medicine, bio-technologies, electronics, neural networks and
economics. The introduction of non-local conditions can improve
the qualitative and quantitative characteristics of the problem
which lead to good results concerning existence, uniqueness
\cite{Bys} and regularity of the solution. Problems related to non
local conditions have applications such as in the theory of heat
conduction, thermoelasticity, plasma physics, control theory etc.
Many real systems are quite sensitive to sudden changes. This fact
may suggest that proper mathematical models of systems should
consist of some neutral equations. Indeed, we may find that
neutral term effects can be quite significant in real mathematical
models. The neutral equations find numerous applications in
applied mathematics, natural sciences, biological and physical
systems. For this reason these type of equations have received
much attention in recent years.

Several authors have studied the existence of solutions of
abstract quasilinear evolution equations in Banach spaces
\cite{Amann, Bahuguna, Uchiyama, Chandra, Dong,  Kato, Oka,
Tanaka}. Park et al. \cite{Park}, Balachandran and Paul Samuel
\cite{Paul} studied the regularity of solutions and the existence
of solutions of quasilinear delay integrodifferential equations
respectively. Controllability of quasilinear systems has gained
renewed interests and few papers appeared \cite{Balasubramaniam,
Anandhi, KB}. The controllability of nonlinear stochastic systems
in  finite and infinite-dimensional spaces have been extensively
studied by many authors \cite{Dauer, Klamka, Mah}. Park et al.
\cite{Bala} discussed the controllability of neutral stochastic
functional integrodifferential infinite delay systems in abstract
spaces. Karthikeyan and Balachandran \cite{Karthi} studied the
controllability of nonlinear stochastic neutral impulsive systems.
Subalakshmi and Balachandran \cite{Suba, Suba1} investigated the
approximate controllability of neutral and impulsive stochastic
integrodifferential systems in Hilbert spaces. Moreover, the
controllability of neutral impulsive stochastic quasilinear
integrodifferential systems is an untreated topic in the
literature so far. Motivated by this fact, in this paper we study
the controllability of neutral impulsive stochastic quasilinear
integrodifferential systems with nonlocal conditions. For that, we
impose neutral, impulse and nonlocal condition with random
perturbations in \eqref{eqqu} which gives the form
\begin{equation}\label{eq1}
\begin{gathered}
\begin{aligned}
&d\big[x(t)-q(t,x(t))\big]\\
&=\Big[ A(t,x)x(t)+Bu(t)+f(t,x(t))+\int_0^t g(t,s,x(s))ds\Big]dt\\
&\quad +  \sigma(t,x(t))dw(t),\quad   t \in J:=[0,a], \quad t\neq t_k,
\end{aligned}\\
\Delta x(t_k)=x(t_k^+)-x(t_k^-)= I_k(x(t_k^-)), \quad
 k=1,2,\dots,m, \\
x(0)+h(x)=x_0.
\end{gathered}
\end{equation}
Here, the state variable $x(\cdot)$ takes values in a real separable
 Hilbert space $H$ with
inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$ and the control
function $u(\cdot)$ takes values in $L^2(J,U)$, a Banach space of
admissible control functions for a separable Hilbert space $U$.
Also, $A(t,x)$ is the infinitesimal generator of a $C_0$-semigroup
in $H$ and $B$ is a bounded linear operator from $U$ into $H$. Let
$K$ be another separable Hilbert space with inner product
$(\cdot,\cdot)_K$ and the norm $\|\cdot\|_K$.  We employ the same
notation $\|\cdot\|$ for the norm $\mathcal{L}(K,H)$, where
$\mathcal{L}(K,H)$ denotes the space of all bounded linear
operators from  $K$ into $H$. Further, $q:J\times H \to H$,
$f:J\times H \to H$, $g:\Lambda\times H \to H$, $\sigma:J\times
H\to  \mathcal{L}_Q(K,H)$ are measurable mappings in $H$-norm and
$\mathcal{L}_Q(K,H)$ norm respectively, where $\mathcal{L}_Q(K,H)$
denotes the  space of all $Q$-Hilbert-Schmidt operators from $K$
into $H$ which will be defined in Section 2 and $\Lambda
=\{(t,s)\in J\times J:s\leq t\}$. Here, the nonlocal function
$h:\mathcal{PC}[J:H] \to H$ and impulsive function
$I_k \in C(H,H)$ $(k=1,2,\dots,m)$ are bounded functions.
Furthermore, the fixed
times $t_k$ satisfies $0=t_0<t_1<t_2<\dots<t_m<a$, $x(t_k^+)$ and
$x(t_k^-)$ denote the right and left limits of $x(t)$ at $t=t_k$.
And $\Delta x(t_k) =x(t_k^+)-x(t_k^-)$ represents the jump in the
state $x$ at time $t_k$, where $I_k$ determines the size of the
jump.

\section{Preliminaries}

Let $(\Omega, \mathcal{F},P;\mathbf{F})
\{\mathbf{F}=\{ \mathcal{F}_t\}_{t \geq 0}\}$ be a complete
filtered probability space satisfying that $\mathcal{F}_0$
contains all $P$-null sets of $\mathcal{F}$.
 An $H$-valued random variable is an $\mathcal{F}$-measurable
function $x(t):\Omega  \to  H$ and the collection of random variables
$S = \{x(t,\omega) : \Omega  \to H\setminus t \in J \}$
is called a stochastic process. Generally, we just write $x(t)$
instead of $x(t,\omega)$
and $x(t):J \to H$ in the space of $S$.
Let $\{e_i\}_{i=1 }^\infty$ be a complete orthonormal basis of $K$.
Suppose that $\{w(t):t\geq 0\}$ is a
cylindrical $K$-valued wiener process with a finite trace nuclear
covariance operator $Q \geq 0$, denote
$\operatorname{Tr}(Q) = \sum_{i=1}^\infty \lambda_i = \lambda  <
\infty$, which satisfies that $Q e_i = \lambda_i e_i$. So, actually, $\omega(t)=\sum_{i=1}^\infty \sqrt{\lambda_i}\omega_i(t)e_i$, where
$\{\omega_i(t)\}_{i=1}^\infty$ are mutually independent one-dimensional standard Wiener processes. We assume that $\mathcal{F}_t = \sigma\{\omega(s):0 \leq s\leq t\}$
is the $\sigma$-algebra generated by $\omega$ and $\mathcal{F}_a=\mathcal{F}$. Let $\Psi \in \mathcal{L}(K,H)$ and define
$$
\|\Psi\|^2_Q =\operatorname{Tr}(\Psi Q \Psi^*)
= \sum_{n=1}^\infty \|\sqrt{\lambda_n} \Psi e_n\|^2.
$$
If $\|\Psi\|_Q <\infty$, then $\Psi$ is called a
$Q$-Hilbert-Schmidt operator. Let $\mathcal{L}_Q(K,H)$ denote the
space of all $Q$-Hilbert-Schmidt operators $\Psi:K \to H$. The
completion $\mathcal{L}_Q(K,H)$ of $\mathcal{L}(K,H)$ with respect
to the topology induced by the norm $\|\cdot\|_Q$ where
$\|\Psi\|^2_Q =\langle \Psi,\Psi\rangle$ is a Hilbert
space with the above norm topology. For more details in this
section refer \cite{Prato}. $L_2^{\mathcal{F}}(J,H)$ is the space
of all $\mathcal{F}_t$ - adapted, $H$-valued measurable square
integrable processes on $J\times\Omega$. Denote $J_0=[0,t_1], J_k
=(t_k, t_{k+1}],k=1,2,\dots,m,$ and define the following class of
functions:
\begin{align*}
&\mathcal{PC}(J,L_2(\Omega,\mathcal{F},P;H))\\
&=\Big\{x:J\to L_2: x(t)
 \text{ is continuous everywhere except for some
$t_k$  at  which} \\
&\quad\text{$ x(t_k^-)$  and $x(t_k^+)$ exists  and }
 x(t_k^-)=x(t_k),\; k=1,2,3,\dots,m\Big\}
\end{align*}
is the Banach space of piecewise continuous maps from $J$ into
$L_2(\Omega,\mathcal{F},P;H)$ satisfying the condition
$\sup_{t \in J} E\|x(t)\|^2 <\infty$.
Let $\mathcal{Z}\equiv\mathcal{PC}(J,L_2)$ be the closed subspace
of $\mathcal{PC}(J,L_2(\Omega,\mathcal{F},P;H) )$ consisting
of measurable,
$\mathcal{F}_t$ - adapted and $H$-valued processes $x(t)$. Then
$\mathcal{PC}(J,L_2)$ is a Banach space endowed with the norm
$$
\|x\|_{\mathcal{PC}}^2 = \sup_{t \in J}\quad \big\{E \|x(t)\|^2
:x \in \mathcal{PC}(J,L_2)\big\}.
$$

Let $H$ and $Y$ be two Hilbert spaces such that $Y$ is densely
and continuously embedded in $H$. For any Hilbert space
$\mathcal{Z}$ the norm of $\mathcal{Z}$ is denoted by
$\|\cdot\|_{\mathcal{PC}}$ or $\|\cdot\|$. The space of all
bounded linear operators from $H$ to $Y$ is denoted by $B(H,Y)$
and $B(H,H)$ is written as $B(H)$. We recall some definitions
and known facts from Pazy \cite{Pazy}.

\begin{definition} \label{def2.1} \rm
Let $S$ be a linear operator in $H$ and let $Y$ be a subspace of $H$.
The operator $\tilde S $ defined by
$D(\tilde S)=\{x \in D(S)\cap Y:Sx \in Y\}$ and $\tilde Sx=Sx$
for $x \in D(\tilde S)$ is called the part of $S$  in $Y$.
\end{definition}

\begin{definition} \label{def2.2} \rm
Let $Q$ be a subset of $H$ and for every $0\leq t\leq a$ and
$b\in Q$, let $A(t,b)$ be the infinitesimal generator of a $C_0$
semigroup $S_{t,b}(s), s \geq0$ on $H$. The family of operators
$\{A(t,b)\}$, $(t,b) \in J\times Q$, is stable if there are constants
$M\geq 1$ and $\omega$ such that
\begin{gather*}
\rho(A(t,b))\supset (\omega,\infty) \quad \text{for }
 (t,b) \in J \times Q,\\
\|\prod_{j=1}^k R(\lambda:A(t_j,b_j))\|
\leq M (\lambda-\omega)^{-k}\quad \text{for } \lambda >\omega
\end{gather*}
and every finite sequence $0\leq t_1 \leq t_2 \leq \dots \leq t_k
\leq a$, $b_j \in Q, 1 \leq j\leq k$. The stability of
$\{A(t,b)\}$, $(t,b) \in J\times Q$, implies \cite{Pazy} that
$$
\|\prod_{j=1}^k S_{t_j,b_j}(s_j)\|\leq M \exp
\{\omega \sum _{j=1}^k s_j\} \quad \text{for } s_j\geq 0
$$
and any finite sequences $0\leq t_1 \leq t_2 \leq \dots \leq t_k
\leq a$,
$b_j \in Q, 1 \leq j\leq k$. $k=1,2, \dots$.
\end{definition}

\begin{definition} \label{def2.3} \rm
Let $S_{t,b}(s)$, $s\geq 0$ be the $C_0$ semigroup generated by
$A(t,b),(t,b) \in J\times Q$. A subspace $Y$ of $H$ is called
$A(t,b)$-admissible if $Y$ is invariant subspace of $S_{t,b}(s)$
and the restriction of $S_{t,b}(s)$ to $Y$ is a $C_0$-semigroup in $Y$.
\end{definition}

Let $Q \subset H$ be a subset of $H$ such that for every
$(t,b) \in J\times Q$, $A(t,b)$ is the infinitesimal generator
of a $C_0$-semigroup $S_{t,b}(s), s\geq 0$ on $H$.
We make the following assumptions:
\begin{itemize}
\item[(E1)] The family $\{A(t,b)\}, (t,b) \in J \times Q$ is stable.
\item[(E2)]$Y$ is  $A(t,b)$- admissible for $(t,b) \in J \times Q$
 and the family $\{\tilde A(t,b)\},(t,b)\in J\times Q$ of parts
 $\tilde A(t,b)$ of $A(t,b)$ in $Y$, is stable in $Y$.
\item[(E3)]For $(t,b)\in J\times Q$, $D(A(t,b))\supset Y$, $A(t,b)$
 is a bounded linear operator from $Y$ to $H$ and $t \to A(t,b)$
 is continuous in the $B(Y,H)$ norm $\|\cdot\|$ for every $b \in Q$.
\item[(E4)]There is a constant $L>0$ such that
$$
\|A(t,b_1)-A(t,b_2)\|_{Y\to H} \leq L\|b_1-b_2\|_H
$$
holds for every $b_1,b_2 \in Q$ and $0\leq t\leq a$.
\end{itemize}
Let $Q$ be a subset of $H$ and let $\{A(t,b)\}$,
$(t,b) \in J\times Q$ be a family of operators satisfying the
conditions $(E1)-(E4)$. If $x\in \mathcal{PC}(J,L_2)$ has values
in $Q$ then there is a unique evolution system
$U(t,s ;x), 0\leq s\leq t\leq a$ in $H$ satisfying
(see \cite{Pazy})
\begin{itemize}
\item[(i)] $\|U(t,s; x)\|\leq Me^{\omega (t-s)}$ for
 $0\leq s\leq t\leq a$, where $M$ and $\omega$ are stability constants.
\item[(ii)] $\frac{\partial^+}{\partial t}U(t,s; x)y=A(s,x(s))U(t,s; x)y$
  for $y\in Y$, $0\leq s\leq t\leq a$.
\item[(iii)] $\frac{\partial}{\partial s}U(t,s; x)y
 = -U(t,s; x)A(s,x(s))y$  for $y \in Y$,  $0\leq s\leq t\leq a$.
\end{itemize}
Further we assume that
\begin{itemize}
\item[(E5)] For every $x \in \mathcal{PC}(J,L_2)$ satisfying
 $x(t) \in Q$ for $0\leq t\leq a$, we have
$$
U(t,s; x)Y\subset Y,\quad 0\leq s\leq t\leq a
$$
and $U(t,s; x)$ is strongly continuous in $Y$ for
$0\leq s\leq t\leq a$.
\item[(E6)] Closed bounded convex subsets of $Y$ are closed in $H$.
\item[(E7)]For every $(t,b)\in J\times Q$, $q(t,b)\in Y$
and $f(t,b)\in Y$, $((t,s),b)\in \Lambda\times Q, g(t,s,b)\in Y$
and  $(t,b)\in J\times Q$, $\sigma(t,b)\in Y$.
\end{itemize}

\begin{definition}[\cite{Dauer}] \label{def2.4} \rm
A stochastic process $x$ is said to be a mild solution of \eqref{eq1}
 if the following conditions are satisfied:
\begin{itemize}
 \item[{(a)}] $x(t,\omega)$ is a measurable function from
 $J \times \Omega$ to $H$ and $x(t)$ is $\mathcal{F}_t$-adapted,
 \item[{(b)}] $E\|x(t)\|^2 < \infty$ for each $t \in J$,
  \item[{(c)}] $\Delta x(t_k)= x(t_k^+)-x(t_k^-)=I_k(x(t_k^-))$,
 $k=1,2,\dots,m,$
 \item [{(d)}] For each $u \in L_2^\mathcal{F}(J,U)$, the process
 $x$ satisfies the following integral equation
 \begin{equation}\label{eq2}
\begin{gathered}
\begin{aligned}
x(t)&=   U(t,0 ;x)\big[x_0 -h(x) -q(0,x(0))\big]+q(t,x(t))\\
&\quad+\int_0^t U(t,s;x)A(s,x(s))q(s,x(s))ds\\
&\quad +\int_0^t   U(t,s; x)\big[Bu(s)+f(s,x(s))\big]ds\\
&\quad+\int_0^t U(t,s; x)\Big[\int_0^s g\big(s,\tau,x(\tau)d\tau\big)\Big]ds+\int_0^tU(t,s; x)\sigma(s,x(s))dw(s)\\
&\quad +\sum_{0<t_k<t} U(t,t_k; x)I_k(x(t_k^-)),\quad
\text{for  a.e. } t \in J,
\end{aligned}\\
x(0)+h(x)=x_0 \in H.
\end{gathered}
\end{equation}
\end{itemize}
\end{definition}

\begin{definition} \label{def2.5} \rm
System \eqref{eq1} is said to be controllable on the interval $J$,
if for every initial condition
$x_0$ and $x_1 \in H$, there exists a control $u \in L^2(J,U)$
such that the solution $x(\cdot)$ of \eqref{eq1}
 satisfies $x(a) = x_1$.
\end{definition}

Further there exists a constant $\mathcal N>0$ such that for
every $x,y \in \mathcal{PC}(J,L_2)$ and every $\tilde y\in Y$
we have
$$
\|U(t,s; x)\tilde y -U(t,s; y)\tilde y\|^2
\leq \mathcal N a^2\|\tilde y\|_Y^2 \|x-y\|_{\mathcal{PC}}^2.
$$

To establish our controllability result we assume the following
hypotheses:
\begin{itemize}
\item[(H1)] $A(t,x)$ generates a family of evolution operators
 $U(t,s; x)$ in $H$ and there exists a constant $\mathcal{C}_U>0$
such that
$$
\|U(t,s; x)\|^2\leq \mathcal{C}_U \quad \text{for } 0\leq s\leq t
\leq a,\; x \in \mathcal{Z}.
$$
\item[(H2)] The linear operator $W: L^2(J,U) \to H$ defined by
$$
Wu= \int_0^a U(a,s; x) Bu(s) ds
$$
is invertible with inverse operator $W^{-1}$ taking
values in $L^2(J,U)\setminus \ker W$ and there exists a positive
constant $\mathcal{C}_{W}$ such that
$$
\|BW^{-1}\|^2 \leq \mathcal{C}_{W}.
$$

\item[(H3)] \begin{itemize}
 \item[(i)] The function $q:J \times \mathcal{Z} \to \mathcal{Z}$
  is continuous and there exist constants $\mathcal{C}_q>0$,
  $\tilde{\mathcal{C}_q}>0$ for $s,t \in J$ and  $x,y \in \mathcal{Z}$
  such that the function $A(t,x)q$ satisfies the Lipschitz condition:
  $$
  E\|A(t,x(t))q(t,x)-A(t,y(t))q(t,y)\|^2 \leq C_q \|x-y\|^2,
  $$
  and $\tilde {\mathcal{C}_q} = \sup_{t \in J} \|A(t,0) q(t,0)\|^2$.
 \item[(ii)] There exist constants $\mathcal{C}_k>0, \mathcal{C}_1>0$
  and $\mathcal{C}_2>0$ such that
  \begin{gather*}
   E\|q(t,x)-q(t,y)\|^2 \leq \mathcal{C}_k[|t-s|^2+\|x-y\|^2] ,\\
   E\|q(t,x)\|^2 \leq  \mathcal{C}_1\|x\|^2+\mathcal{C}_2,
  \end{gather*}
  where $\mathcal{C}_2=\sup_{t \in J} \|q(t,0)\|^2$.
  \end{itemize}

\item[(H4)] The nonlinear function
$f:J\times \mathcal{Z} \to \mathcal{Z}$ is continuous and there
exist constants $\mathcal{C}_f >0$, $\tilde{\mathcal{C}_f}>0$
for $t \in J$ and  $x,y \in \mathcal{Z}$ such that
$$
E\|f(t,x)-f(t,y)\|^2 \leq \mathcal{C}_f \|x-y\|^2
$$
and $\tilde {\mathcal{C}_f} = \sup_{t \in J} \|f(t,0)\|^2$.

\item[(H5)] The  nonlinear function $g:\Lambda \times \mathcal{Z}
 \to \mathcal{Z}$ is continuous and there exist positive constants
$\mathcal{C}_g$, $\tilde{\mathcal{C}_g}$,  for $x,y \in \mathcal{Z}$
and $(t,s) \in \Lambda$ such that
$$
E\big\|g(t,s,x)-g(t,s,y)\big\|^2 \leq \mathcal{C}_g\|x-y\|^2
$$
and $\tilde{\mathcal{C}_g} = \sup _{(t,s) \in \Lambda} \|g(t,s,0)\|^2$.

\item[(H6)] The function $\sigma:J \times \mathcal{Z} \to
 \mathcal{L}_Q(K,H)$ is continuous and there exist constants
 $\mathcal{C}_\sigma>0$,  $\tilde{\mathcal{C}_\sigma}>0$ for
 $t\in J$ and $x,y \in \mathcal{Z}$ such that
$$
E\|\sigma(t,x)-\sigma(t,y)\|^2 _Q\leq \mathcal{C}_\sigma\|x-y\|^2
$$
 and $\tilde{\mathcal{C}_\sigma}=\sup_{t \in J} \|\sigma(t,0)\|^2$.

\item[(H7)] The nonlocal function $h:\mathcal{PC}(J:\mathcal{Z})
 \to \mathcal{Z}$ is continuous and there exist constants
 $\mathcal{C}_h>0$, $\tilde{\mathcal{C}_h}>0$  for
 $x,y \in \mathcal{Z}$ such that
$$
E\|h(x)-h(y)\|^2 \leq \mathcal{C}_h\|x-y\|^2 , \quad
E\|h(x)\|^2\leq \tilde{\mathcal{C}_h}.
$$

\item[(H8)] $I_k:\mathcal{Z}\to \mathcal{Z}$ is continuous and
 there exist constants $\beta_k>0$, $\tilde{\beta_k}>0$ for
 $x,y \in \mathcal{Z}$ such that
$$
E\|I_k(x)-I_k(y)\|^2\leq \beta_k\|x-y\|^2, \quad k=1,2,\dots,m
$$
and $\tilde{\beta_k}=\|I_k(0)\|^2, \ k=1,2,\dots,m$.

\item[(H9)] There exists a constant $r>0$ such that
\begin{align*}
&10\Big\{\mathcal{C}_U(\|x_0\|^2 +
 \tilde{\mathcal{C}_h})+a^2 \mathcal{C}_U \mathcal{G}+2\mathcal{C}_U\big[\mathcal{C}_1(\|x_0\|^2+\tilde{\mathcal{C}_h})+\mathcal{C}_2\big]+\mathcal{C}_1 r+\mathcal{C}_2\\
&+ 2a^2\mathcal{C}_U(\mathcal{C}_q r+\tilde{\mathcal{C}_q})+2a^2\mathcal{C}_U(\mathcal{C}_f r+\tilde{\mathcal{C}_f})
 +2a^3\mathcal{C}_U\big[ \mathcal{C}_g r+\tilde{\mathcal{C}_g}\big]\\
&+ 2a\ \mathcal{C}_U\  \operatorname{Tr}(Q)\big(\mathcal{C}_\sigma r+\tilde{\mathcal{C}_\sigma}\big)
 +2m\mathcal{C}_U\Big[\sum_{k=1}^m \beta_k r+\sum_{k=1}^m
 \tilde{ \beta_k}\Big]\Big\}\\
&\leq r
\end{align*}
and
\[
\nu =     10\Big\{ (1+18a^2\mathcal{C}_U\mathcal{C}_W)
(N_1+N_2+N_3+N_4+N_5+N_6+N_7) + 2a^3\mathcal N  \mathcal{G} \Big\}
\]
where
\begin{gather*}
N_1 =\mathcal Na^2\|x_0\|^2+2(\mathcal N
 a^2\tilde{\mathcal{C}_h}+\mathcal{C}_U \mathcal{C}_h),\\
N_2 =2\Big[2\mathcal N a^2\big(\mathcal{C}_1(\|x_0\|^2
 +\tilde{\mathcal{C}_h})+\mathcal{C}_2\big)
 +\mathcal{C}_U\mathcal{C}_k\mathcal{C}_h\Big]+\mathcal{C}_q,\\
N_3=2a^2\Big[2\mathcal N a\big(\mathcal{C}_q r
 +\tilde{\mathcal{C}_q}\big)+\mathcal{C}_U \mathcal{C}_q\Big],\\
N_4=2a^2\Big[2\mathcal N a\big(\mathcal{C}_f r
 +\tilde{\mathcal{C}_f}\big)+\mathcal{C}_U \mathcal{C}_f\Big],\\
N_5= 2a^3\Big[2\mathcal N a\big(\mathcal{C}_gr
 +\tilde{\mathcal{C}_g}\big)+\mathcal{C}_U\mathcal{C}_g\Big],\\
N_6=2a\Big[2\mathcal Na\operatorname{Tr}(Q)\big(\mathcal{C}_\sigma r
 +\tilde{\mathcal{C}_\sigma}\big)+\mathcal{C}_U
 \operatorname{Tr}(Q)\mathcal{C}_\sigma\Big],\\
N_7=2m\Big[2\mathcal N a^2\Big(\sum_{k=1}^m\beta_k r
+\sum_{k=1}^m \tilde{\beta_k}\Big)
+\mathcal{C}_U\sum_{k=1}^m\beta_k\Big].
\end{gather*}
\end{itemize}

\section{Controllability Result}

\begin{theorem}\label{main}
If the conditions {\rm (H1)-(H9)} are satisfied and if
 $0\leq\nu < 1$, then system \eqref{eq1} is controllable on $J$.
\end{theorem}

\begin{proof}
  Using  (H2) for an arbitrary function $x(\cdot)$, define the control
\begin{equation}
\begin{split}
u(t)&= W^{-1}\Big[x_1- U(a,0 ;x)\big[x_0 -h(x)-q(0,x(0))\big]
 -q(a,x(a))\\
&\quad -\int_0^a U(a,s;x)A(s,x(s))q(s,x(s))ds-\int_0^aU(a,s; x)
 \sigma(s,x(s))dw(s)\\
&\quad -\int_0^a U(a,s; x)\Big[f(s,x(s))+\int_0^s g(s,\tau,
 x(\tau))d\tau\Big]ds\\
&\quad -\sum_{0<t_k<a} U(a,t_k; x)I_k(x(t_k^-))\Big](t).
\end{split}
\end{equation}
Let $\mathcal{Y}_r$ be a nonempty closed subset of
$\mathcal{PC}(J,L_2)$ defined by
$$
\mathcal{Y}_r=\{x:x \in \mathcal{PC}(J,L_2) | E\|x(t)\|^2\leq r\}.
$$
Consider a mapping $\Phi: \mathcal{Y}_r \to \mathcal{Y}_r$ defined by
\begin{align*}
&(\Phi x)(t)\\
&=U(t,0 ;x)\big[x_0 -h(x)-q(0,x(0))\big]+q(t,x(t))\\
&\quad +\int_0^t  U(t,s;x)A(s,x(s))q(s,x(s))ds\\
&\quad +\int_0^t    U(t,s; x)BW^{-1}\Big[x_1- U(a,0 ;x)
 \big[x_0 -h(x)-q(0,x(0))\big]-q(a,x(a))\\
&\quad -\int_0^a U(a,s;x)A(s,x(s))q(s,x(s))ds-\int_0^aU(a,s; x)
 \sigma(s,x(s))dw(s)\\
&\quad -\int_0^a U(a,s; x)\Big[f(s,x(s))+\int_0^s g(s,\tau,x(\tau))
 d\tau\Big]ds\\
&\quad -\sum_{0<t_k<a} U(a,t_k; x)I_k(x(t_k^-))\Big](s)ds
 + \int_0^t  U(t,s; x) f(s,x(s))ds\\
&\quad +\int_0^t   U(t,s; x)\Big[\int_0^sg(s,\tau,x(\tau))d\tau\Big]ds
 +\int_0^t  U(t,s; x)\sigma(s,x(s))dw(s)\\
&\quad +\sum_{0<t_k<t} U(t,t_k; x)I_k(x(t_k^-)).
\end{align*}
We have to show that by using the above control the operator
$\Phi$ has a fixed point. Since all the functions involved in
the operator are continuous therefore $\Phi$ is continuous.
For convenience let us take
\begin{align*}
V(\mu,x)
&=   BW^{-1}\Big[x_1- U(a,0 ;x)\big[x_0 -h(x)-q(0,x(0))\big]
 -q(a,x(a))\\
&\quad -\int_0^a U(a,s;x)A(s,x(s))q(s,x(s))ds-\int_0^aU(a,s; x)
 \sigma(s,x(s))dw(s)\\
&\quad -\int_0^a U(a,s; x)\Big[f(s,x(s))+\int_0^s g(s,\tau,x
 (\tau))d\tau\Big]ds\\
&\quad -\sum_{0<t_k<a} U(a,t_k; x)I_k(x(t_k^-))\Big](\mu).
\end{align*}
From our assumptions we have
\begin{align*}
E\|V(\mu,x)\|^2
&\leq  10\mathcal{C}_W \Big\{\|x_1\|^2+\mathcal{C}_U(\|x_0\|^2
 +\tilde{\mathcal{C}_h})
 +2\mathcal{C}_U\big[\mathcal{C}_1(\|x_0\|^2+\tilde{\mathcal{C}_h})
 +\mathcal{C}_2\big]
 +\mathcal{C}_1 r\\
&\quad+\mathcal{C}_2
 +2a^2\mathcal{C}_U(\mathcal{C}_q r+\tilde{\mathcal{C}_q})
 +2a^2\mathcal{C}_U(\mathcal{C}_f r+\tilde{\mathcal{C}_f})
 +2a^3\mathcal{C}_U\big[ \mathcal{C}_g r+\tilde{\mathcal{C}_g}\big]\\
&\quad +2a\ \mathcal{C}_U\  \operatorname{Tr}(Q)\big(\mathcal{C}_\sigma
 r+\tilde{\mathcal{C}_\sigma}\big)+2m\mathcal{C}_U\Big[\sum_{k=1}^m
 \beta_k r+\sum_{k=1}^m \tilde{ \beta_k}\Big]\Big\} := \mathcal{G}.
\end{align*}
and
\begin{align*}
&E\|V(\mu,x)-V(\mu,y)\|^2\\
&\leq  9\mathcal{C}_W\Big\{\mathcal Na^2\|x_0\|^2
 +2(\mathcal N a^2\tilde{\mathcal{C}_h}+\mathcal{C}_U \mathcal{C}_h)
 +2\Big[2\mathcal N a^2\big(\mathcal{C}_1(\|x_0\|^2+\tilde{\mathcal{C}_h})\\&\quad +\mathcal{C}_2\big)+\mathcal{C}_U\mathcal{C}_k\mathcal{C}_h\Big]+\mathcal{C}_q
 +2a^2\Big[2\mathcal N a\big(\mathcal{C}_q r+\tilde{\mathcal{C}_q}\big)
 +\mathcal{C}_U \mathcal{C}_q\Big]\\
&\quad  +2a^2\Big[2\mathcal N a\big(\mathcal{C}_f r
 +\tilde{\mathcal{C}_f}\big)+\mathcal{C}_U \mathcal{C}_f\Big]
 +2a^3\Big[2\mathcal N a \big(\mathcal{C}_gr
 +\tilde{\mathcal{C}_g}\big)+\mathcal{C}_U\mathcal{C}_g\Big]\\
&\quad +2a\Big[2\mathcal Na \ \operatorname{Tr}(Q)
 \big(\mathcal{C}_\sigma r+\tilde{\mathcal{C}_\sigma}\big)
 +\mathcal{C}_U \ \operatorname{Tr}(Q)\mathcal{C}_\sigma\Big]\\
&\quad +2m\Big[2\mathcal N a^2\Big(\sum_{k=1}^m\beta_k r
 +\sum_{k=1}^m \tilde{\beta_k}\Big)+\mathcal{C}_U
 \sum_{k=1}^m\beta_k\Big]\Big\}.
\end{align*}
First we show that the operator $\Phi$ maps $\mathcal{Y}_r$ into
itself. Now
\begin{align*}
& E\|(\Phi x)(t)\|^2\\
&\leq  10\Big\{E\big\|U(t,0 ;x)\big[x_0 -h(x)-q(0,x(0))\big]
 \big\|^2+E\|q(t,x(t))\|^2\\
&\quad +E\|\int_0^t     U(t,s;x)A(s,x(s))q(s,x(s))ds\|^2
 +E\big\|\int_0^t U(t,\mu; x)V(\mu,x)d\mu\big\|^2\\
&\quad +E\big\|\int_0^t  U(t,s; x) \Big[f(s,x(s))+\int_0^s
 g\big(s,\tau,x(\tau)\big)d\tau\Big]ds\big\|^2\\
&\quad +E\big\|\int_0^t  U(t,s; x)\sigma(s,x(s))dw(s)\big\|^2
 +E\big\|\sum_{0<t_k<t} U(t,t_k; x)I_k(x(t_k^-))\big\|^2\Big\}\\
&\leq  10\Big\{\mathcal{C}_U(\|x_0\|^2+\tilde{\mathcal{C}_h})
 +2\mathcal{C}_U\big[\mathcal{C}_1(\|x_0\|^2+\tilde{\mathcal{C}_h})
 +\mathcal{C}_2\big]+\mathcal{C}_1 r+\mathcal{C}_2\\
&\quad +2a^2\mathcal{C}_U(\mathcal{C}_q r+\tilde{\mathcal{C}_q})
 +a^2 \mathcal{C}_U \mathcal{G}+2a^2\mathcal{C}_U(\mathcal{C}_f r
 +\tilde{\mathcal{C}_f})
 +2a^3\mathcal{C}_U\big[ \mathcal{C}_g r+\tilde{\mathcal{C}_g}\big]\\
&\quad +2a\ \mathcal{C}_U\  \operatorname{Tr}(Q)
 \big(\mathcal{C}_\sigma r+\tilde{\mathcal{C}_\sigma}\big)
+2m\mathcal{C}_U\Big[\sum_{k=1}^m \beta_k r+\sum_{k=1}^m
 \tilde{ \beta_k}\Big]\Big\}\\
&\leq r.
\end{align*}
From (H9) we obtain $E\|(\Phi x)(t)\|^2\leq r$. Hence $\Phi$
maps $\mathcal{Y}_r$ into $\mathcal{Y}_r$.
Let $x,y \in \mathcal{Y}_r$, then
\begin{align*}
&E\|(\Phi x)(t)-(\Phi y)(t)\|^2 \\
&\leq  10\Big\{E\big\|U(t,0 ;x)\big[x_0 -h(x)-q(0,x(0))\big]
 -U(t,0 ;y)\big[x_0 -h(y)-q(0,y(0))\big]\big\|^2\\
&\quad   +E\big\|q(t,x(t))-q(t,y(t))\big\|^2 + E\big\|  \int_0^t  \Big[ U(t,s;x)A(s,x(s))q(s,x(s))\\
&\quad -U(t,s;y)A(s,y(s))q(s,y(s))\Big]ds\big\|^2\\
&\quad +E\big\|\int_0^t\Big[ U(t,\mu; x)V(\mu,x)-U(t,\mu; y)
 V(\mu,y)\Big]d\mu\big\|^2\\
&\quad +E\big\|\int_0^t \Big[ U(t,s; x) f(s,x(s))-U(t,s; y)
 f(s,y(s))\Big]ds\big\|^2 \\
&\quad +E\big\|\int_0^t \Big[U(t,s; x)
 \Big[\int_0^sg(s,\tau,x(\tau))d\tau\Big]
 -U(t,s; y)\Big[\int_0^sg(s,\tau,y(\tau))d\tau\Big]\Big]ds\big\|^2\\
&\quad +E\big\|\int_0^t\Big[U(t,s; x)\sigma(s,x(s))
 -U(t,s; y)\sigma(s,y(s))\Big]dw(s)\big\|^2\\
&\quad +E\big\|\sum_{0<t_k<t} \Big[U(t,t_k; x)I_k(x(t_k^-))
 - U(t,t_k; y)I_k(y(t_k^-))\Big]\big\|^2\Big\}\\
&\leq 10\Big\{(1+18a^2\mathcal{C}_U\mathcal{C}_W)
 (N_1+N_2+N_3+N_4+N_5+N_6+N_7)
 +2a^3\mathcal N  \mathcal{G}\Big\}\|x-y\|^2\\
&\leq \nu\|x-y\|^2.
\end{align*}
Since $\nu <1$, the mapping $\Phi$ is a contraction and hence
by Banach fixed point theorem there exists a unique fixed point
$x \in \mathcal{Y}_r$ such that $(\Phi x)(t) =x(t)$.
This fixed point is then the solution of the system \eqref{eq1}
and clearly, $x(a) =(\Phi x)(a)=x_1$ which implies that the
system \eqref{eq1} is controllable on $J$.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
 Consider the neutral impulsive stochastic quasilinear system
\begin{equation}\label{eqfu}
\begin{gathered}
\begin{aligned}
d\Big[x(t)-q(t,x(t))\Big]
&= \Big[ A(t,x)\big[x(t)-q(t,x(t))\big]+Bu(t)+f(t,x(t))\\
&\quad +\int_0^t g(t,s,x(s))ds\Big]dt+ \ \sigma(t,x(t))dw(t),\\
&\quad t \in J:=[0,a], \quad t\neq t_k,
\end{aligned} \\
\Delta x(t_k)= x(t_k^+)-x(t_k^-)= I_k(x(t_k^-)), \quad k=1,2,\dots,m, \\
x(0)+h(x)=x_0.
\end{gathered}
\end{equation}
\end{remark}
where $A, B, q, f, g, \sigma$ are as before.
The solution to the above equation is
\begin{align*}
x(t)&= U(t,0 ;x)\big[x_0 -h(x) -q(0,x(0))\big]+q(t,x(t))
 +\int_0^t   U(t,s; x)Bu(s)ds\\
&\quad +\int_0^t   U(t,s; x)\Big[f(s,x(s))
 +\int_0^s g\big(s,\tau,x(\tau)d\tau\big)\Big]ds\\
&\quad +\int_0^tU(t,s; x)\sigma(s,x(s))dw(s)
 +\sum_{0<t_k<t} U(t,t_k; x)I_k(x(t_k^-)),
\end{align*}
 for a.e. $t \in J$.
If the functions involved in \eqref{eqfu} satisfy the lipschitz
condition then the suitable control function will steer the
system \eqref{eqfu} from $x_0$ to $x_1$ provided the above
equation is satisfied.

\section{Neutral Stochastic Quasilinear Integrodifferential Systems}

Consider the neutral stochastic quasilinear integrodifferential
system
\begin{equation} \label{eqin}
\begin{gathered}
\begin{aligned}
&d\Big[x(t) - Q\Big(t,x(t),\int_0^t   q(t,s,x(s))ds\Big)\Big]\\
&=\Big[A(t,x)x(t) + Bu(t) + F\Big(t,x(t),\int_0^t
 f\big(t,s,x(s))ds\Big)\Big]dt\\
&\quad +G\Big(t,x(t),\int_0^t    \sigma\big(t,s,x(s))ds\Big)dw(t),\quad
t \in J,\;  t\neq t_k,
\end{aligned}
\\
\Delta x(t_k)= x(t_k^+)-x(t_k^-)= I_k(x(t_k^-)),\quad k=1,2,\dots,m, \\
x(0)+h(x)=x_0.
\end{gathered}
\end{equation}
where $A, B, I_k, h$ are defined as before. Further,
\begin{gather*}
Q:J \times H \times H \to H ,\quad
F:J\times H \times H \to H ,\quad
G:J\times H \times H \to \mathcal{L}_Q(K,H),\\
q:\Lambda\times H \to H ,\quad
f:\Lambda \times H \to H, \quad
\sigma:\Lambda \times H \to H.
\end{gather*}
are measurable mappings in $H$-norm and $\mathcal{L}_Q(K,H)$-norm ,
respectively. The solution of the above equation is
\begin{equation}\label{eqsol2}
\begin{split}
&x(t)\\
&=U(t,0 ;x)\Big[x_0 -h(x) -Q(0,x(0),0)\Big]+Q\Big(t,x(t),
 \int_0^t   q(t,s,x(s))ds\Big)\\
&\quad +\int_0^t U(t,s;x)A(s,x(s))Q\Big(s,x(s),\int_0^s
 q(s,\tau,x(\tau))d\tau\Big)ds\\
&\quad +\int_0^t U(t,s; x)Bu(s)ds
   +\int_0^tU(t,s; x)F\Big(s,x(s),
  \int_0^sf\big(s,\tau, x(\tau))d\tau\Big)ds \\
&\quad +\int_0^t U(t,s; x) G\Big(s,x(s),\int_0^s\sigma\big(s,\tau,
 x(\tau))d\tau\Big)dw(s)\\
&\quad +\sum_{0<t_k<t} U(t,t_k; x)I_k(x(t_k^-)),
\quad \text{for  a.e. } t \in J.
\end{split}
\end{equation}
Concerning the operators $Q,q,F,f,G,\sigma$ we assume the
following hypotheses:
\begin{itemize}
\item[(H10)]\begin{itemize}
  \item[(i)] The function $Q:J \times \mathcal{Z} \times
  \mathcal{Z}\to \mathcal{Z}$ is continuous and there exist
   constants $\mathcal{C}_Q>0$, $\tilde{\mathcal{C}_Q}>0$ for
  $s,t \in J$ and  $x, y, x_1, y_1 \in \mathcal{Z}$  such that the
  function $A(t,x)Q$ satisfies the Lipschitz condition
  $$
  E\|A(t,x(t))Q(t,x,x_1)-A(t,y(t))Q(t,y,y_1)\|^2
  \leq C_Q\big(\|x-y\|^2+\|x_1-y_1\|^2\big),
  $$
  and $\tilde {\mathcal{C}_Q} = \sup_{t \in J} \|A(t,0) Q(t,0,0)\|^2$.
  \item[(ii)] There exist constants $ Q_k>0, Q_1>0$ and $Q_2>0$
  such that
  \begin{gather*}
   E\|Q(t,x,x_1)-Q(t,y,y_1)\|^2 \leq
    Q_k\big(|t-s|^2+\|x-y\|^2+\|x_1-y_1\|^2\big),  \\
   E\|Q(t,x,y)\|^2 \leq  Q_1\big(\|x\|^2+\|y\|^2\big)+ Q_2,
  \end{gather*}
  where $Q_2=\sup_{t \in J} \|Q(t,0,0)\|^2$.
  \end{itemize}

\item[(H11)] The  nonlinear function
$q:\Lambda \times \mathcal{Z} \to \mathcal{Z}$ is continuous
and there exist positive constants $\mathcal{C}_q$,
$\tilde{\mathcal{C}_q}$,  for $x,y \in \mathcal{Z}$ and
$(t,s) \in \Lambda$ such that
$$
E\|\int_0^t \big(q(t,s,x)-q(t,s,y)\big)ds\|^2
\leq \mathcal{C}_q \|x-y\|^2
$$ and $\tilde{\mathcal{C}_q}
= \sup _{(t,s) \in \Lambda} \|\int_0^t q(t,s,0)ds\|^2$.

\item [(H12)] The nonlinear function $F:J\times \mathcal{Z}
 \times \mathcal{Z} \to \mathcal{Z}$ is continuous and there
 exist constants $\mathcal{C}_F >0$, $\tilde{\mathcal{C}_F}>0$
 for $t \in J$ and  $x_1, x_2, y_1, y_2 \in \mathcal{Z}$ such that
$$
E\|F(t,x_1,y_1)-F(t,x_2,y_2)\|^2 \leq \mathcal{C}_F\big(\|x_1-x_2\|^2
 +\|y_1-y_2\|^2\big)
$$
and $\tilde {\mathcal{C}_F} = \sup_{t \in J} \|F(t,0,0)\|^2$.

\item[(H13)] The  nonlinear function $f:\Lambda \times \mathcal{Z}
\to \mathcal{Z}$ is continuous and there exist positive constants
$\mathcal{C}_f$, $\tilde{\mathcal{C}_f}$,  for $x,y \in \mathcal{Z}$
and $(t,s) \in \Lambda$ such that
$$
E\big\|\int_0^t \big(f(t,s,x)-f(t,s,y)\big)ds\big\|^2
 \leq \mathcal{C}_f \|x-y\|^2
$$
and $\tilde{\mathcal{C}_f} = \sup _{(t,s) \in \Lambda}
  \|\int_0^t f(t,s,0)ds\|^2$.

\item[(H14)] The nonlinear function $G:J\times \mathcal{Z}
 \times \mathcal{Z} \to \mathcal{L}_Q(K,H)$ is continuous and
 there exist constants $\mathcal{C}_G >0$, $\tilde{\mathcal{C}_G}>0$
for $t \in J$ and  $x_1, x_2, y_1, y_2 \in \mathcal{Z}$ such that
$$
E\|G(t,x_1,y_1)-G(t,x_2,y_2)\|^2 \leq \mathcal{C}_G
\big(\|x_1-x_2\|^2+\|y_1-y_2\|^2\big)$$ and
$\tilde {\mathcal{C}_G} = \sup_{t \in J} \|G(t,0,0)\|^2$.

\item[(H15)] The  nonlinear function $\sigma:\Lambda \times
 \mathcal{Z} \to \mathcal{Z}$ is continuous and there exist
positive constants $\mathcal{C}_\sigma$, $\tilde{\mathcal{C}_\sigma}$,
for $x,y \in \mathcal{Z}$ and $(t,s) \in \Lambda$ such that
$$
E\big\|\int_0^t \big(\sigma(t,s,x)-\sigma(t,s,y)\big)ds\big\|^2
\leq \mathcal{C}_\sigma \|x-y\|^2 $$ and $\tilde{\mathcal{C}_\sigma}
 = \sup _{(t,s) \in \Lambda} \|\int_0^t \sigma(t,s,0)ds\|^2$.

\item[(H16)] There exists a constant $r^*>0$ such that
\begin{align*}
&9\Big\{\mathcal{C}_U(\|x_0\|^2 +  \tilde{\mathcal{C}_h})
+ a^2 \mathcal{C}_U \mathcal{G} + 2\mathcal{C}_U\big[Q_1(\|x_0\|^2
+ \tilde{\mathcal{C}_h}) + Q_2\big]\\
&+Q_1\big[(1 + 2\mathcal{C}_q)r + 2\tilde{\mathcal{C}_q}\big]
+Q_2 + 2a^2\mathcal{C}_U\big[\mathcal{C}_Q \big((1 + 2\mathcal{C}_q)r
 + 2\tilde{\mathcal{C}_q}\big) + \tilde{\mathcal{C}_Q}\big]\\
&+ 2a^2\mathcal{C}_U\big[\mathcal{C}_F \big((1 + 2\mathcal{C}_f)r
 + 2\tilde{\mathcal{C}_f}\big) + \tilde{\mathcal{C}_F}\big]\\
&+ 2a\mathcal{C}_U\operatorname{Tr}(Q)\big[\mathcal{C}_G \big((1
 + 2\mathcal{C}_\sigma)r + 2\tilde{\mathcal{C}_\sigma}\big)
 + \tilde{\mathcal{C}_G}\big]
 + 2m\mathcal{C}_U\Big[\sum_{k=1}^m
 \beta_k r+\sum_{k=1}^m  \tilde{ \beta_k}\Big]\Big\}\\
&\leq  r^*
\end{align*}
 and
\[
\nu^* =  9\Big\{ (1+16a^2\mathcal{C}_U\mathcal{C}_W)
 (N_1+N_2+N_3+N_4+N_5+N_6) + 2a^3\mathcal N  \mathcal{G} \Big\}
\]
where
\begin{gather*}
N_1= \mathcal Na^2\|x_0\|^2+2(\mathcal N a^2\tilde{\mathcal{C}_h}
 +\mathcal{C}_U \mathcal{C}_h)\\
N_2= 2\Big[2\mathcal N a^2\big(Q_1(\|x_0\|^2+\tilde{\mathcal{C}_h})
 +Q_2\big)+\mathcal{C}_U Q_k\mathcal{C}_h\Big]+Q_k(1+\mathcal{C}_q)\\
N_3= 2a^2\Big[2\mathcal N a\Big[\mathcal{C}_Q \big((1+2\mathcal{C}_q)r
 +2\tilde{\mathcal{C}_q}\big)+\tilde{\mathcal{C}_Q}\Big]
 +\mathcal{C}_U\mathcal{C}_Q(1+\mathcal{C}_q)\Big]\\
N_4= 2a^2\Big[2\mathcal N a\Big[\mathcal{C}_F \big((1+2\mathcal{C}_f)r
 +2\tilde{\mathcal{C}_f}\big)+\tilde{\mathcal{C}_F}\Big]
 +\mathcal{C}_U \mathcal{C}_F(1+\mathcal{C}_f)\Big]\\
N_5= 2a\Big[2\mathcal Na  \operatorname{Tr}(Q)
 \Big[\mathcal{C}_G\big((1 + 2\mathcal{C}_\sigma)r
 + 2\tilde{\mathcal{C}_\sigma}\big) + \tilde{\mathcal{C}_G}\Big]
 + \mathcal{C}_U \ \operatorname{Tr}(Q)C_G(1
 + \mathcal{C}_\sigma)\Big]\\
N_6= 2m\Big[2\mathcal N a^2\Big(\sum_{k=1}^m\beta_k r
+\sum_{k=1}^m \tilde{\beta_k}\Big)+\mathcal{C}_U\sum_{k=1}^m
 \beta_k\Big].
\end{gather*}
\end{itemize}
To apply the contraction mapping, we define the nonlinear operator
$\Phi^*: \mathcal{Y}_r \to \mathcal{Y}_r$ as
\begin{align*}
&(\Phi^* x)(t)\\
&=U(t,0 ;x)\Big[x_0 -h(x) -Q(0,x(0),0)\Big]+Q\Big(t,x(t),
  \int_0^t   q(t,s,x(s))ds\Big)\\
&\quad +\int_0^t
U(t,s;x)A(s,x(s))Q\Big(s,x(s),\int_0^s  q(s,\tau,x(\tau))d\tau\Big)ds\\
&\quad +\int_0^t U(t,s; x)Bu(s)ds
 +\int_0^t U(t,s; x)F\Big(s,x(s),\int_0^s f\big(s,\tau,x(\tau))
 d\tau\Big)ds\\
&\quad +\int_0^t U(t,s;x)G\Big(s,x(s),\int_0^s
 \sigma\big(s,\tau,x(\tau))d\tau\Big)dw(s)+\sum_{0<t_k<t}
U(t,t_k; x)I_k(x(t_k^-)).
\end{align*}
where
\begin{align*}
u(t)
&= W^{-1}\Big[x_1-U(a,0 ;x)\big[x_0 -h(x) -Q(0,x(0),0)\big]\\
&\quad -Q\Big(a,x(a),\int_0^a   q(a,s,x(s))ds\Big)\\
&\quad -\int_0^a U(a,s;x)A(s,x(s))Q\Big(s,x(s),\int_0^s
 q(s,\tau,x(\tau))d\tau\Big)ds\\
&\quad -\int_0^a U(a,s; x)F\Big(s,x(s),\int_0^s f\big(s,\tau,
 x(\tau))d\tau\Big)ds\\
&\quad -\int_0^aU(a,s;x)G\Big(s,x(s),\int_0^s
  \sigma\big(s,\tau,x(\tau))d\tau\Big)dw(s)\\
&\quad -\sum_{0<t_k<a} U(a,t_k; x)I_k(x(t_k^-))\Big](t).
\end{align*}
Clearly the above control transfers the system \eqref{eqin}
from the initial state $x_0$ to the final state $x_1$ provided that
the operator $\Phi^* x$ has a fixed point. Hence, if the operator
$\Phi^* x$ has a fixed point then the system \eqref{eqin}
 is controllable.

\begin{theorem} \label{thm4.1}
 If {\rm (H10)--(H16)} hold, then system \eqref{eqin} is controllable
provided that
$$
9\Big\{ (1+16a^2\mathcal{C}_U\mathcal{C}_W)(N_1+N_2+N_3+N_4+N_5+N_6)
 + 2a^3\mathcal N  \mathcal{G} \Big\}<1.
$$
\end{theorem}

 The proof of the above theorem is similar to that of
Theorem \ref{main} and hence it is omitted.

\section{Example}

Consider the  partial integrodifferential equation
\begin{equation}\label{ex}
\begin{gathered}
\begin{aligned}
{\partial}\Big({z(t,y)-\frac{1}{2}\cos z(t,y)}\Big)
&= \Big(\frac{\partial ^3}{\partial y^3} z(t,y)+z(t,y)
 \frac{\partial}{\partial y}z(t,y)+\mu(t,y)\\
&\quad +\frac{1}{2}e^{-t}\sin z(t,y)+\frac{z(t,y)}{t(1+t^2)}
 \Big[\int_0^te^{-z(s,y)}ds\Big]\Big){\partial t}\\
&\quad +\frac{1}{2}\cos t\ z(t,y)dw(t),\ t\in J:=[0,1],\quad
  t\neq t_k,
\end{aligned}\\
z(0,y)+\int_0^1 m(s) \log(1+|z(s,y)|)ds =z_0(y),\\
\Delta z|_{t=t_k}= I_k(z(y))=\int_\Omega d_k(y,s) \cos^2(z(s,y))
ds, \quad k=1,2,\dots,m.
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in
$\mathbb{R}^n$ with smooth boundary,
$m(\cdot) \in L^1([0,1];\mathbb{R})$ and
$d_k \in C(\bar \Omega\times \bar\Omega, \mathbb{R})$ for
$k = 1,2, \dots, m$. For every
real $s$ we introduce a Hilbert space $H^s(R)$ as follows
\cite{Pazy}. Let $z\in L^2(R)$ and set
$$
\|z\|_s=\Big(\int_R (1+\xi^2)^s|\widehat{z} (\xi)|^2d\xi\Big)^{1/2} ,
$$
where $\widehat z$ is the Fourier transform of $z$. The linear space
of functions $z \in L^2(R)$ for which $\|z\|_s$ is finite is a
pre-Hilbert space with the inner product
$$
(z,y)_s =\Big(\int_R (1+\xi^2)^s\widehat{z}
(\xi)\overline{\widehat{y}(\xi)}d\xi\Big)^{1/2}.
$$
The completion of this space with respect to the norm $\|\cdot\|_s$
is a Hilbert space which we denote by $H^s(R)$. It is clear
that $H^0(R)=L^2(R)$.

Take $H=U=K=L^2(R) =H^0(R)$ and $Y=H^s(R), s \geq 3$.
Define an operator $A_0$ by $D(A_0)=H^3(R)$ and $A_0z=D^3z$
for $z \in D(A_0)$ where $D=d/dy$. Then $A_0$ is the inifinitesimal
generator of a $C_0$-group of isometries on $H$.
Next we define for every $v \in Y$ an operator $A_1(v)$ by
$D(A_1(v))= H^1(R)$ and $z \in D(A_1(v))$, $A_1(v)z=vDz$.
Then for every $v \in Y$ the operator $A(v)=A_0+A_1(v)$
is the infinitesimal generator of $C_0$ semigroup $U(t,0;v)$
on $H$ satisfying $\|U(t,0;v)\| \leq e^{\beta t}$ for every
$\beta \geq c_0\|v\|_s$, where $c_0$ is a constant independent
of $v \in Y$. Let $\mathcal{Y}_r$ be the ball of radius $r>0$
in $Y$ and it is proved that the family of operators
$A(v), v \in \mathcal{Y}_r,$ satisfies the conditions (E1)--(E4) and
(H1) (see \cite{Pazy}).
Put $x(t)=z(t,\cdot)$ and $u(t)=\mu(t,\cdot)$ where
$\mu:J\times \mathbb{R}\to\mathbb{R}$ is continuous,
\begin{gather*}
 f(t,x(t))=\frac{1}{2}e^{-t}\sin z(t,y) ,\quad
 \sigma(t,x(t))=\frac{1}{2}\cos t\ z(t,y),\\
q(t,x(t))=\frac{1}{2}\cos z(t,y) ,\quad
h(x)=\int_0^1 m(s) \log(1+|z(s,y)|)ds\\
\int_0^tg(t,s,x(s))ds= \frac{z(t,y)}{t(1+t^2)}
\Big[\int_0^te^{-z(s,y)}ds\Big].
\end{gather*}
With this choice of $A(v)$, $I_k, q, f, g, h,\sigma$, $B =I$,
the identity operator and $w(t)$ denotes a one dimensional
standard wiener process, we see that \eqref{ex} is an
abstract formulation of the system \eqref{eq1}. Further we have
$$
\big\|\frac{z(t,y)}{t(1+t^2)} \Big[\int_0^te^{-z(s,y)}ds\Big]
\big\| \leq\frac{1}{1+t^2}\|z\|.
$$
Assume that the operator $W:L^2(J,U)/Ker W \to H$ defined by
$$
W u =\int_0^1 U(1,s;x)\mu(s,\cdot)ds
$$
has an inverse operator and satisfies (H2) for every
$x \in \mathcal{Y}_r$.
Further the other assumptions (H3)--(H9) are obviously
satisfied and it is possible to choose a suitable control
function $u(t)$ in such a way that the constant $\nu <1$
which will steer the system from $x_0$ to $x_1$.  Hence,
by Theorem \ref{thm4.1},  system \eqref{ex} is controllable on $J$.


\subsection*{Acknowledgements}
The second author is thankful to UGC, New Delhi, for providing a
BSR Fellowship during 2010.

\begin{thebibliography}{00}

\bibitem{Amann} H. Amann;
 Quasilinear evolution equations and parabolic systems,
\emph{Trans. Amer. Math. Soc.}, 29 (1986), 191-227.

\bibitem{Bahuguna} D. Bahuguna;
 Quasilinear integrodifferential equations in Banach spaces,
\emph{Nonlinear Anal.}, 24 (1995), 175-183.

\bibitem{Paul} K. Balachandran and F. Paul Samuel;
 Existence of solutions for quasilinear delay integrodifferential
equations with nonlocal conditions,
\emph{Electron. J. Differential Equations}, vol. 2009 (2009), no. 6 ,
 1-7.

\bibitem{Uchiyama} K. Balachandran and K. Uchiyama;
 Existence of solutions of quasilinear integrodifferential
equations with nonlocal condition, \emph{Tokyo J. Math.},
23 (2000), 203-210.

\bibitem{Balasubramaniam} K. Balachandran, P. Balasubramaniam
and J. P. Dauer;
 Controllability of quasilinear delay systems in Banach spaces,
 \emph{Optim. Contr. Appl. Meth.}, 16 (1995), 283-290.

\bibitem{Anandhi} K. Balachandran, J.Y. Park and E.R. Anandhi;
 Local controllability of quasilinear integrodifferential evolution
systems in Banach spaces, \emph{J. Math. Anal. Appl.}, 258 (2001),
309-319.

\bibitem{KB} K. Balachandran, J. Y. Park and S. H. Park;
 Controllability of nonlocal impulsive quasilinear integrodifferential
systems in Banach spaces, \emph{Rep. Math. Phys.}, 65 (2010), 247-257.

\bibitem{Bys} L. Byszewski and V. Lakshmikantham;
 Theorem about the existence and uniqueness of a solution of a
nonlocal abstract Cauchy problem in a Banach space,
\emph{Appl. Anal.}, 40 (1991), 11-19.

\bibitem{Chandra} M. Chandrasekaran;
 Nonlocal Cauchy problem for quasilinear integrodifferential
equations in Banach spaces, \emph{Electron. J. Differential Equations},
2007 (2007), no. 33, 1-6.

\bibitem{Prato} G. Da Prato and J. Zabczyk;
 \emph{Stochastic Equations in Infinite Dimensions},
Cambridge University Press, Cambridge, 1992.

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

\bibitem{Dong} Q. Dong, G. Li and J. Zhang;
 Quasilinear nonlocal integrodifferential equations in Banach spaces,
 \emph{Electron. J. Differential Equations}, 2008 (2008), no 19, 1-8.

\bibitem{Gust} G. Gripenberg;
 Weak solutions of hyperbolic-parabolic volterra equations,
\emph{Trans. American Math. Soc.}, 343 (1994), 675-694.

\bibitem{Heard} M. L. Heard;
 A quasilinear hyperbolic integrodifferential equation related
to a nonlinear string, \emph{Trans. American Math. Soc.},
285 (1984), 805-823.

\bibitem{Karthi} S. Karthikeyan and K. Balachandran;
 Controllability of nonlinear stochastic neutral impulsive systems,
\emph{Nonlinear Anal.}, 3 (2009), 266-276.

\bibitem{Kato} S. Kato;
 Nonhomogeneous quasilinear evolution equations in Banach spaces,
\emph{Nonlinear Anal.}, 9 (1985), 1061-1071.

\bibitem{Klamka} J. Klamka;
 Stochastic controllability of linear systems with state delays,
 \emph{Int. J. Appl. Math. Comput. Sci.}, 17(2007), 5-13.

\bibitem{Kim} J. H. Kim, On a stochastic nonlinear equation
in one-dimensional viscoelasticity,
\emph{Trans. American Math. Soc.}, 354 (2002), 1117-1135.

\bibitem{Lakshmi} V. Lakshmikantham, D. D. Bainov and P. S. Simeonov,
\emph{Theory of Impulsive Differential Equations}, World Scientific,
Singapore, 1989.

\bibitem{Mah} N. I. Mahmudov;
 Existence and uniqueness results for neutral stochastic differential
equations in Hilbert spaces, \emph{Stoch. Anal. Appl.}, 24 (2006), 79-95.

\bibitem{Nara} R. Narasimha;
 Nonlinear vibration of an elastic string, \emph{J. Sound Vibration},
8 (1968), 134 -146.

\bibitem{Oka} H. Oka;
 Abstract quasilinear Volterra integrodifferential equations,
\emph{Nonlinear Anal.}, 28 (1997), 1019-1045.

\bibitem{Tanaka} H. Oka and N. Tanaka;
 Abstract quasilinear integrodifferential equations of hyperbolic type,
 \emph{Nonlinear Anal.}, 29 (1997), 903-925.

\bibitem{Bala} J.Y. Park, P. Balasubramaniam and N. Kumerasan;
 Controllability for neutral stochastic functional integrodifferential
infinite delay systems in abstract space,
\emph{Numer. Funct. Anal. Optim.}, 28 (2007), 1369-1386.

\bibitem{Park} D. G. Park, K. Balachandran and F. Paul Samuel;
 Regularity of solutions of abstract quasilinear delay
integrodifferential equations, \emph{J. Korean Math. Soc.},
 48  (2011), 585-597.

\bibitem{Pazy} A. Pazy;
 \emph{Semigroups of Linear Operators and Applications to Partial
Differential Equations}, Springer, Berlin, 1983.

\bibitem{Samo} A.M. Samoilenko and N.A. Perestyuk;
 \emph{Impulsive Differential Equations}, World Scientific,
Singapore, 1995.

\bibitem{Suba} R. Subalakshmi and K. Balachandran;
Approximate controllability of neutral stochastic integrodifferential
systems in Hilbert spaces, \emph{Electron. J. Differential Equations},
vol. 2008 (2008), no. 162, 1-15.

\bibitem{Suba1} R. Subalakshmi and K. Balachandran;
 Approximate controllability of nonlinear stochastic impulsive
integrodifferential systems in Hilbert spaces,
\emph{Chaos, Solitons and Fractals}, 42 (2009), 2035-2046.

\bibitem{Xie} S. Xie;
Numerical algorithms for solving a type of nonlinear
integrodifferential equations, \emph{World Academy of Science,
 Engineering and Technology}, 65 (2010), 1083-1086.

\end{thebibliography}

\end{document}