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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 256, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/256\hfil Controllability of integro-differential systems]
{Controllability of neutral stochastic
integro-differential systems with \\ impulsive effects}

\author[A. Benchaabane \hfil EJDE-2015/256\hfilneg]
{Abbes Benchaabane}

\address{Abbes Benchaabane \newline
Laboratoire LAMED, 8 May 1945 University,
BP401, Guelma 24000, Algeria}
\email{abbesbenchaabane@gmail.com}

\thanks{Submitted June 15, 2014. Published October 2, 2015.}
\subjclass[2010]{93B05, 93E03, 37C25}
\keywords{Complete controllability; fixed point theorem; 
\hfill\break\indent stochastic neutral impulsive systems}

\begin{abstract}
 This article concerns the complete controllability for nonlinear
 neutral impulsive stochastic integro-differential system in finite
 dimensional spaces. Sufficient conditions ensuring the complete
 controllability are formulated and proved under the natural assumption that
 the associated linear control system is completely controllable. The results
 are obtained by using the Banach fixed point theorem. A numerical example is
 provided to illustrate our technique. 
\end{abstract}

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

\section{Introduction}

The problem of controllability is one of the fundamental concept in
mathematical control theory and engineering. The problem of controllability
is to show the existence of a control function, which steers dynamical
control systems from its initial state to the final state, where the initial
and final states may vary over the entire space. The controllability of
nonlinear deterministic systems in a finite dimensional space has been
extensively studied, 
\cite{balachandran1987controllability,klamka2000schauder}.

Stochastic differential equations have been considered extensively through
discussion in the finite dimensional spaces. As a matter of fact, there
exist broad literature on the related to the topic and it has played an
important role in many ways such as option pricing, forecast of the growth
of population, etc., and as an applications which cover the generalizations
of stochastic differential equations arising in the fields such as
electromagnetic theory, population dynamics, and heat conduction in material
with memory and stochastic differential equations are obtained by including
random fluctuations in ordinary differential equations which have been
deduced from phenomenological or physical laws. Random differential and
integral equations play an important role in characterizing numerous social,
physical, biological and engineering problems. For more details reader may
refer 
\cite{bharucha1972random,mao2007stochastic,oksendal2003stochastic} 
and reference therein. For a dynamic system the
simplest continuous stochastic perturbation is naturally considered to be a
Brownian motion (BM). In general, a continuous stochastic perturbation will
be modeled as some stochastic integral with respect to the (BM). However,
the (BM) has the strange property that even though its trajectory is
continuous in $t$, it is not differentiable for all $t$ . So for a
stochastic integral with respect to (BM) one has to use a different
approach, Ito approach is used to define it 
(see \cite{situ2005brownian,peszat2007stochastic} for details).

Controllability of non-linear stochastic systems in finite-dimensional
spaces has been investigated by many authors. Klamka and Socha 
\cite{klamka1977some} derived sufficient conditions for the stochastic
controllability of linear and nonlinear systems using a Lyapunov technique.
Mahmudov and Zorlu \cite{mahmudov2005controllability} derived sufficient
conditions for complete and approximate controllability of semilinear
stochastic systems with non-Lipschitz coefficients via Picard-type
iterations. Balachandran et al. 
\cite{balachandran2007controllability,balachandran2008controllability} 
studied the controllability of semilinear
stochastic integrodifferential systems using the Banach fixed point theorem.

The theory of impulsive differential equations has provided a natural
framework for mathematical modeling of many real world phenomena, namely in
control, biological and medical domains 
\cite{yang2001impulsive,bainov1993impulsive,balachandran2009existence}.
 In these models, the processes are characterized by the fact that they 
 undergo abrupt changes
of state at certain moments of time between intervals of continuous
evolution. The presence of impulses implies that the trajectories of the
system do not necessarily preserve the basic properties of the non-impulsive
dynamical systems. To this end the theory of impulsive differential systems
has emerged as an important area of investigation in applied sciences 
\cite{lakshmikantham1989theory}. Yang, Xu and Xiang 
\cite{yang2006exponential}
established the exponential stability of non-linear impulsive stochastic
differential equations with delays. More recently, Liu and Liao 
\cite{liu2007existence} studied the existence, uniqueness and stability of
stochastic impulsive systems using Lyapunov-like functions.

Many of the physical systems may also contain some information about the
derivative of the state component and such systems are called neutral
systems. Therefore, the investigation of stochastic impulsive neutral
differential equations attracts great attention, especially as regards to
controllability 
\cite{sivasundaram2008controllability,xu2007exponential}.

In this article, we consider the  impulsive neutral semilinear
stochastic integrodifferential system
\begin{equation}
\begin{gathered}
\begin{aligned}
&d\{ x(t)-G(t,x(t),g(\eta x(t))\}\\
& =A(t)x(t)dt+B(t)u(t)dt 
  +F_1\left( t,x(t),f_{1,1}(\eta x(t)),f_{1,2}
  (\delta x(t)),f_{1,3}(\xi x(t))\right) dt \\
&\quad +F_2\left( t,x(t),f_{2,1}(\eta x(t)),f_{2,2}(\delta
x(t)),f_{2,3}(\xi x(t))\right) dw(t), 
\end{aligned}\\
t\in [ 0,T] ,\quad t\neq t_k, \\
\Delta x(t_k)=I_k(x(t_k^{-})),\quad t=t_k,\;  k=1,2,\dots,r, \\
x(0)=x_0\in \mathbb{R}^n,
\end{gathered}  \label{eq1}
\end{equation}
where, for $i=1,2$:
\begin{gather*}
f_{i,1}(\eta x(t))=\int_0^{t}f_{i,1}(t,s,x(s))ds, \quad
f_{i,2}(\delta x(t))=\int_0^{T}f_{i,2}(t,s,x(s))ds, \\
g(\eta x(t))=\int_0^{t}g(t,s,x(s))ds, \quad
f_{i,3}(\xi x(t))=\int_0^{t}f_{i,3}(t,s,x(s))dw(s).
\end{gather*}
Here $A(t)$ and $B(t)$ are continuous matrices of dimensions $n\times n$,
and $n\times m$ respectively
\begin{gather*}
F_1:[ 0,T] \times \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^n\times
\mathbb{R}^n\to \mathbb{R}^n, \\
F_2:[ 0,T] \times \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^n\times
\mathbb{R}^n\to R^{n\times n}, \\
G:[ 0,T] \times \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n, \quad
f_{i,1},f_{i,2}:[ 0,T] \times [ 0,T] \times
\mathbb{R}^n\to \mathbb{R}^n, \\
f_{i,3}:[ 0,T] \times [ 0,T] \times \mathbb{R}^n\to
R^{n\times n}, \quad
g:[ 0,T] \times [ 0,T] \times \mathbb{R}^n\to \mathbb{R}^n.
\end{gather*}
$I_k\in C(\mathbb{R}^n,\mathbb{R}^n)$, $u(t)$ is a feedback control and $w$ is a
$n$-dimensional standard (BM). Furthermore, 
$0=t_0<t_1<\dots<t_{r}<t_{r+1}=T$, $x(t_k^{+})$ and
$x(t_k^{-})$ represent the right and left limits of
$x(t)$ at $t=t_k$, respectively. Also
$\Delta x(t_k)=x(t_k^{+})-x(t_k^{-})$, represents the jump in the state
$x$ at time $t_k$ with $I_k$ determining the size of the jump, the initial
value $x_0$ is $\mathcal{F}_0$-measurable with $\mathbf{E}\|
x_0\| ^2<\infty $.

The system \eqref{eq1} is in a very general form and it covers many possible
models with various definitions of $f_{1,1}$, $f_{1,2}$,
$f_{1,3}$, $f_{2,1}$,
$f_{2,2}$, $f_{2,3}$ and $g$. We would like to mention that Balachandran and
Karthikeyan \cite{balachandran2008controllability} studied the case 
$I_k=g=f_{1,3}=f_{2,3}=0$. The controllability problem with
$g=f_{1,3}=f_{2,3}=0$ was studied by Sakthivel 
\cite{sakthivel2009controllability}. The system \eqref{eq1} with 
$f_{1,3}=f_{2,3}=0$ was investigated by Karthikeyan and Balachandran 
\cite{karthikeyan2009controllability}.

Motivated by the above references, we extend the results to obtain the complete
controllability for wide class of impulsive neutral integro-differential
equations under basic assumptions on the system operators. In particular, we
assume the complete controllability of the associated linear system. To
prove the main results, Theorem \ref{T1}, we use stochastic analysis and
fixed point theorem. Our work is organized as follows. The next section
contains definitions, preliminary results and a mathematical model of
impulsive stochastic systems with control. Section 3 is devoted to analyzing
complete controllability results of the problem \eqref{eq1} via a fixed
point technique. Section 4 contains an illustrative example.

\section{Preliminaries}

The problem of controllability of a linear stochastic system of the form
\begin{equation}
\begin{gathered}
dx(t)=[ A(t)x(t)+B(t)u(t)] dt+\sigma (t)dw(t),\quad t\in [0,T] , \\
x(0)=x_0,
\end{gathered}  \label{eq0}
\end{equation}
has been studied by various authors 
\cite{zabczyk1981controllability,ehrhardt1982controllability,
 mahmudov2001controllability} where 
 $\sigma :[ 0,T] \to R^{n\times n}$. In this article, the
following notation is adopted.
\begin{itemize}
\item $(\Omega ,\mathcal{F},P)$ is the probability space with
probability measure $P$ on $\Omega $.

\item $\{\mathcal{F}_{t}\ |\ t\in [ 0,T]\}$ is the filtration generated
by $\{w(s):0\leq s\leq t\}$ and $\mathcal{F=F}_T$.

\item $L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n)$ is the Hilbert space of
all $\mathcal{F}_T$-measurable square integrable variables with values in
$\mathbb{R}^n$.

\item $U_{ad}=L_2^{\mathcal{F}}([0,T],R^{m})$ is the Hilbert space
of all square integrable and $\mathcal{F}_{t}$-measurable processes with
values in $R^{m}$.

\item $PC([0;T];\mathbb{R}^n)$ is the space of function from $[0;T]$ into $
\mathbb{R}^n$ such that $x(t)$ is continuous at $t\neq t_k$\ and left continuous
at $t=t_k$ and the right limit $X(t_k^{+})$ exists for $k=1,2,\dots r$.

\item $\mathbf{H}_2:=PC_{\mathcal{F}_{t}}^{b}([0,T],L_2(\Omega ,\mathcal{
F}_{t},\mathbb{R}^n))$ is the Banach space of all bounded $\mathcal{F}_{t}$
-measurable, $PC([0;T];\mathbb{R}^n)$ valued random variables $\varphi $ satisfying
\begin{equation*}
\| \varphi \| ^2=\sup_{t\in [ 0,T]}\mathbf{E}
\| \varphi (t)\| ^2.
\end{equation*}

\item $\mathcal{L(}X,Y)$ is the space of all linear bounded operators from a
Banach space $X$ to a Banach space $Y$,

\item $\phi (t)=\exp (At)$.
\end{itemize}

Now we introduce the following operators and sets.
\begin{itemize}
\item The operator $L_0^{T}\in \mathcal{L}\left( L_2^{\mathcal{F}
}([0,T],R^{m}\mathbf{),}L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n)\right)
$ is defined by
\begin{equation*}
L_0^{T}=\int_0^{T}\phi (T-s)B(s)u(s)ds.
\end{equation*}
Clearly, the adjoint $\left( L_0^{T}\right) ^{\ast }:L_2(\Omega ,
\mathcal{F}_T,\mathbb{R}^n\mathbf{)\to }L_2^{\mathcal{F}}([0,T],R^{m}
)$ is defined by
\begin{equation*}
(L_0^{T})^{\ast }z=B^{\ast }\phi ^{\ast }(T-t)\mathbf{E(}z|
\mathcal{F}_{t}).
\end{equation*}

\item The controllability operator $\Pi _0^{T}$ associated with \eqref{eq0}
 is
\begin{equation*}
\Pi _0^{T}(.)=\int_{s}^{T}\phi (T-t)BB^{\ast }\phi ^{\ast }(T-t)\mathbf{
E(.\ |\ }\mathcal{F}_{t})dt,
\end{equation*}
which belongs to $\mathcal{L}\left( L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n
\mathbf{),}L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n)\right) $ and the
controllability matrix $\Gamma _{s}^{T}\in \mathcal{L(}\mathbb{R}^n,\mathbb{R}^n)$
\begin{equation*}
\Gamma _{s}^{T}=\int_{s}^{T}\phi (T-t)BB^{\ast }\phi ^{\ast }(T-t)dt,\ \ \
0\leq s\leq t.
\end{equation*}

\item The set of all states attainable from $x_0$ in time $T>0$ is
\begin{equation*}
\mathcal{R}_{t}(x_0)=\{ x(t,x_0,u):u\in U_{ad}\} ,
\end{equation*}
where $x(t,x_0,u)$ is the solution of \eqref{eq1} corresponding to $
x_0\in \mathbb{R}^n$ and $u(.)\in U_{ad}$.
\end{itemize}

\begin{definition} \label{def1} \rm
System \eqref{eq1} is completely controllable on $[0,T]$ if
\begin{equation*}
\mathcal{R}_T(x_0)=L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n\mathbf{),}
\end{equation*}
that is, if all the points in $L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n)$
can be reached from the point $x_0$ at time $T$. See Example \eqref{x1}
for Motivated application.
\end{definition}

\section{Controllability}

In this section we derive controllability conditions for the non-linear
stochastic system \eqref{eq1} using the contraction mapping principle.

We impose the following conditions on data of the problem
\begin{itemize}
\item[(H1)] The functions $F_i$, $f_{i,j}$, $G$, $g$, $i=1,2$, $j=1,3$
satisfies the Lipschitz condition:
there exist constants $L_1$, $N_1$, $K_1$, $C_1$, $q_k>0$ for $x_h$,
 $y_h$, $v_h$, $z_h\in \mathbb{R}^n$, $h=1,2$ and $0\leq s\leq t\leq T$
such that
\begin{gather*}
\begin{aligned}
&\|F_i(t,x_1,y_1,v_1,z_1)-F_i(t,x_2,y_2,v_2,z_2)\|^2 \\
&\leq L_1\left( \| x_1-x_2\| ^2+\|
y_1-y_2\| ^2+\| v_1-v_2\|
^2+\| z_1-z_2\| ^2\right),
\end{aligned} 
\\
\| G(t,x_1,y_1)-G(t,x_2,y_2)\| ^2\leq N_1\left(
\| x_1-x_2\| ^2+\| y_1-y_2\| ^2\right) ,
\\
\| f_{i,j}(t,s,x_1(s))-f_{i,j}(t,s,x_2(s))\| ^2\leq
K_1\| x_1-x_2\| ^2,
\\
\| g(t,s,x_1(s))-g(t,s,x_2(s))\| ^2\leq C_1\| x_1-x_2\| ^2,
\\
\| I_k(x)-I_k(y)\| ^2\leq q_k\| x-y\| ^2,\quad  k\in \{ 1,\dots,r\} .
\end{gather*}

\item[(H2)] The functions $F_i$, $f_{i,j}$, $G$, $g$, $i=1,2$, $j=1,3$ are
continuous and there are constants $L_2$, $N_2$, $K_2$, $C_2$, $d_k>0$
for $x, y, v, z\in \mathbb{R}^n$ and $0\leq t\leq T$ such that
\begin{gather*}
\| F_i(t,x,y,v,z)\| ^2\leq L_2\left( 1+\|
x\| ^2+\| y\| ^2+\| v\|^2+\| z\| ^2\right) ,
\\
\| G(t,x,y)\| ^2\leq N_2\left( 1+\|x\| ^2+\| y\| ^2\right) ,
\\
\| f_{i,j}(t,s,x(s))\| ^2\leq K_2\left( 1+\|x\| ^2\right) ,
\\
\| g(t,s,x)\| ^2\leq C_2\left( 1+\|x\| ^2\right) ,
\\
\| I_k(x)\| ^2\leq d_k\left( 1+\|x\| ^2\right) ,\quad \ k\in \{ 1,\dots,r\} .
\end{gather*}

\item[(H3)] The linear system \eqref{eq0} is completely controllable.
\end{itemize}

Now for our convenience, let us introduce the following notation:
\begin{gather*}
l_1=\max \{ \| \phi (t)\| ^2,\; t\in [0,T]\} ,\quad
l_2=\max \{ \| A(t)\| ^2,\; t\in [ 0,T]\} , \\
M=\max \{ \| \Gamma _{s}^{T}\| ^2,\; s\in [0,T]\} .
\end{gather*}
The following lemma will play an important role in the proofs of our main
results (see \cite{oksendal2003stochastic}).

\begin{lemma}[Ito isometry] \label{lemmeizo}
Let $\Psi :J\times \Omega \to \mathbb{R}^n$ be measurable
and $\mathcal{F}_{t}$-adapted mapping and such that $\mathbf{E}
\int_0^{T}\| \Psi (s,\omega )\| ^2ds<\infty $. Then
\begin{equation*}
\mathbf{E}\| \int_0^{t}\Psi (s)dw(s)\| ^2
=\mathbf{E}
\Big( \int_0^{t}\| \Psi (s)\| ^2ds\Big) ,\quad \text{for }t\in [ 0,T]
\end{equation*}
\end{lemma}

\begin{lemma}[\cite{mahmudov2003controllability}]\label{lemme1}
For every $z\in L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n)$
\begin{itemize}
\item $\mathbf{E}\| \Pi _0^{t}z\| ^2\leq M\mathbf{E}\| z\| ^2$.

\item Assume (H3) holds, then there exist $l_{3}>0$ such that
\begin{equation*}
\mathbf{E}\| \mathbf{(}\Pi _0^{T})^{-1}\|
^2\leq l_{3}.
\end{equation*}
\end{itemize}
\end{lemma}

We define the operator $V$ from $\mathbf{H}_2$ to $\mathbf{H}_2$ as
follows:
\begin{align*}
(Vx)(t) 
&=(\widehat{G}x)(t)+\int_0^{t}A\phi (t-s)(\widehat{G}
x)(s)ds+\int_0^{t}\phi (t-s)(\widehat{F}_1x)(s)ds \\
&\quad +\int_0^{t}\phi (t-s)(\widehat{F}_2x)(s)dw(s)+\sum_{0<t_k<t}
\phi (t-t_k)I_k(x(t_k^{-})),  
\end{align*}
where, for $i=1,2$,
\begin{gather*}
(\widehat{F_i}x)(t)=F_i\left( t,x(t),f_{i,1}(\eta x(t)),f_{i,2}(\delta
x(t)),f_{i,3}(\xi x(t))\right) , \\
(\widehat{G}x)(t)=G\left( t,x(t),g(\eta x(t))\right) .
\end{gather*}
The following results will be used throughout this paper.

\begin{lemma}\label{lemme3}
Under conditions {\rm  (H1)} and {\rm (H2)}, there exist real constants
$M_1,M_2>0$ such that for $x,y\in \mathbf{H}_2$, we have
\begin{gather}
\mathbf{E}\| (Vx)(t)-(Vy)(t)\| ^2\leq M_1\Big(
\sup_{s\in [ 0,T] }\mathbf{E}\| x(s)-y(s)\|^2\Big) ,  \label{ini2}
\\
\mathbf{E}\| (Vx)(t)\| ^2\leq M_2\Big( 1+T\sup_{s\in
[ 0,T] }\mathbf{E}\| x(s)\| ^2\Big) .
\label{ini3}
\end{gather}
\end{lemma}

\begin{proof}
First, we prove inequality \eqref{ini2}, since \eqref{ini3} can be established 
in a similar way. For $i=1,2$, let $x, y\in \mathbf{H}_2$. 
It follows from condition (H1), Holder inequality and Ito isometry  that
\begin{align*}
&\| (\widehat{F}_ix)(t)-(\widehat{F}_iy)(t)\| ^2\\
&\leq L_1\Big( \| x(t))-y(t))\| ^2+\| f_{i,1}(\eta
x(t))-f_{i,1}(\eta y(t))\| ^2  \\
&\quad  +\| f_{i,2}(\delta x(t))-f_{i,2}(\delta y(t))\|
^2+\| f_{i,3}(\xi x(t))-f_{i,3}(\xi y(t))\|
_{Q}^2\Big) , \\
&\leq L_1(1+2T^2K_1+TK_1)\sup_{s\in [ 0,T] }\| x(s)-y(s)\| ^2,
\end{align*}
from which it follows that
\[
\mathbf{E}\Big( \int_0^{t}\| (\widehat{F}_ix)(s)-(\widehat{F}
_iy)(s)\| ^2ds\Big) 
\leq L_1T(1+2T^2K_1+TK_1)\sup_{s\in [ 0,T] }\mathbf{E}
\| x(s)-y(s)\| ^2.
\]
We have
\begin{align*}
\| (\widehat{G}x)(t)-(\widehat{G}y)(t)\| ^2
&\leq N_1\Big( \| x(t)-y(t)\| ^2+\| g(\eta
x(t))-g(\eta y(t))\| ^2\Big) , \\
&\leq N_1(1+T^2C_1)\Big( \sup_{s\in [ 0,T] }\|x(s)-y(s)\| ^2\Big) ,
\end{align*}
Then we obtain 
\begin{equation}
\mathbf{E}\Big( \int_0^{t}\| (\widehat{G}x)(s)-(\widehat{G}
y)(s)\| ^2ds\Big) 
\leq N_1T(1+T^2C_1)\Big( \sup_{s\in
[ 0,T] }\mathbf{E}\| x(s)-y(s)\| ^2\Big) .
\end{equation}
It follows from the above inequality, Holder inequality and Ito isometry
that
\begin{align*}
&\mathbf{E}\| (Vx)(t)-(Vy)(t)\| ^2\\
&\leq 5\mathbf{E} \| \int_0^{t}A\phi (t-s)[ (\widehat{G}x)(s)-(\widehat{G}y)(s)
] ds\| ^2 \\
&\quad +5\mathbf{E}\| \int_0^{t}\phi (t-s)[ (\widehat{F}_1x)(s)-(
\widehat{F}_1y)(s)] ds\| ^2 \\
&\quad +5\mathbf{E}\| \int_0^{t}\phi (t-s)[ (\widehat{F}_2x)(s)-(
\widehat{F}_2y)(s)] dw(s)\| ^2 \\
&\quad +5\mathbf{E}\| \sum_{0<t_k<t}\phi (t-t_k)[
I_k(x(t_k^{-}))-I_k(y(t_k^{-}))] \| ^2 
+5\mathbf{E}\| (\widehat{G}x)(t)-(\widehat{G}y)(t)\| ^2,
\end{align*}
then, we have
\begin{align*}
&\mathbf{E}\| (Vx)(t)-(Vy)(t)\| ^2\\
&\leq 5Tl_1l_2\mathbf{E}\int_0^{t}\| (\widehat{G}x)(s)-(\widehat{G}y)(s)\|^2ds 
 +5Tl_1\mathbf{E}\int_0^{t}\| (\hat{F}_1x)(s)-(\hat{F}_1y)(s)\| ^2ds \\
&\quad +5l_1\mathbf{E}\int_0^{t}\| (\hat{F}_2x)(s)-(\hat{F}
_2y)(s)\| ^2ds 
+5l_1r\sum_{k=1}^{r}\mathbf{E}\| I_k(x(t_k^{-}))-I_k(y(t_k^{-}))\| ^2 \\
&\quad +5\mathbf{E}\| (\widehat{G}x)(t)-(\widehat{G}y)(t)\| ^2.
\end{align*}
Thus we have
\begin{align*}
&\mathbf{E}\| (Vx)(t)-(Vy)(t)\| ^2\\
&\leq \Big( 10T^2l_1l_2N_1(1+T^2C_1)+15l_1(T+1)L_1T(1+2T^2K_1+TK_1)
\\
&\quad +5l_1r\left( \sum_{k=1}^{r}q_k\right)
+10N_1(1+T^2C_1)\Big) \sup_{s\in [ 0,T] }\mathbf{E}
\| x(s)-y(s)\| ^2 \\
&=M_1\sup_{s\in [ 0,T] }\mathbf{E}\|
x(s)-y(s)\| ^2,
\end{align*}
where
\begin{align*}
M_1&=5T^2l_1l_2N_1(1+T^2C_1)+5l_1r\left(
\sum_{k=1}^{r}q_k\right) +5N_1(1+T^2C_1) \\
&\quad +5l_1(T+1)L_1T(1+2T^2K_1+TK_1).
\end{align*}
\end{proof}

For a given initial condition and any $u\in U_{ad}$ for 
$t\in [ 0,T] $, one can prove the existence and uniqueness of solution 
$x(t,x_0,u)$ of the of the nonlinear impulsive stochastic
integrodifferential state equations \eqref{eq1} based on the fixed point
technique \cite{murge1986explosion}. The solution of the which can be
represented in the following integral form:
\begin{equation}
\begin{aligned}
x(t)&=\phi (t)[ x_0-G(0,x_0,0)] +(\widehat{G}
x)(t)+\int_0^{t}A\phi (t-s)(\widehat{G}x)(s)ds   \\
&\quad +\int_0^{t}\phi (t-s)\left( Bu(s)+(\widehat{F}_1x)(s)\right)
ds+\int_0^{t}\phi (t-s)(\widehat{F}_2x)(s)dw(s)   \\
&\quad +\sum_{0<t_k<t}\phi (t-t_k)I_k(x(t_k^{-})),   \\
&=\phi (t)[ x_0-G(0,x_0,0)] +(Vx)(t)+\int_0^{t}\phi
(t-s)Bu(s)ds.
\end{aligned} \label{eq2}
\end{equation}
The following lemma gives a formula for a control steering the state $x_0$
to an arbitrary final point $x_T$.

\begin{lemma}\label{lemme2}
Assume $\Pi _0^{T}$ is invertible, then for
arbitrary $x_T\in L_2(\Omega ,\mathcal{F}_T,\mathbb{R}^n)$ the
control
\begin{equation}
u(t)=B^{\ast }\phi ^{\ast }(T-t)\mathbf{E}\{ (\Pi _0^{T})^{-1}\left(
x_T-\phi (T)[x_0-G(0,x_0,0)]-(Vx)(T)\right) |\mathcal{F}
_{t}\}  \label{eqc}
\end{equation}
transfers the system \eqref{eq2} from $x_0\in \mathbb{R}^n$ to $x_T\in \mathbb{R}^n$
at time $T$.
\end{lemma}

\begin{proof}
By substituting \eqref{eqc} in \eqref{eq2}, we obtain
\begin{equation}
\begin{aligned}
x(t)&=\phi (t)[ x_0-G(0,x_0,0)] +(Vx)(t) \\
&\quad +\int_0^{t}\phi (t-s)BB^{\ast }\phi ^{\ast }(t-s)\phi ^{\ast }(T-t)
\\
&\quad \times \mathbf{E}\{ (\Pi _0^{T})^{-1}\left( x_T-\phi
(T)[x_0-G(0,x_0,0)]-(Vx)(T)\right) |\mathcal{F}_{t}\}
  \\
&=\phi (t)[ x_0-G(0,x_0,0)] +(Vx)(t)+\Pi _0^{t}(\phi ^{\ast
}(T-t)(\Pi _0^{T})^{-1}   \\
&\quad \times \left( x_T-\phi (T)[x_0-G(0,x_0,0)]-(Vx)(T)\right) ).
\end{aligned}\label{eq5}
\end{equation}
Writing $t=T$ in \eqref{eq5}, we see that the control $u(.)$ transfers the
system \eqref{eq2} from $x_0$ to $x_T$.
\end{proof}

To apply the contraction mapping principle, we define the nonlinear operator
$\Upsilon :\mathbf{H}_2\to \mathbf{H}_2$ by
\begin{equation*}
(\Upsilon x)(t)=\phi (t)[ x_0-G(0,x_0,0)] +(Vx)(t)+
\int_0^{t}\phi (t-s)Bu(s)ds,
\end{equation*}
where
\begin{equation}
u(t)=B^{\ast }\phi ^{\ast }(T-t)\mathbf{E}\{ (\Pi _0^{T})^{-1}\left(
x_T-\phi (T)[x_0-G(0,x_0,0)]-(Vx)(T)\right) |\mathcal{F}
_{t}\} .  \label{eq4}
\end{equation}
From Lemma \eqref{lemme2}, the control \eqref{eq4} transfers the system 
\eqref{eq2} from the initial state $x_0$ to the final state $x_T$ provided
that the operator $\Upsilon $ has a fixed point. So, if the operator $
\Upsilon $ has a fixed point then the system \eqref{eq1} is completely
controllable.\ As mentioned above, to prove the complete controllability it
is enough to show that $\Upsilon $ has a fixed point in $\mathbf{H}_2$.
To do this, we use the contraction mapping principle. To apply the
contraction principle, first we show that $\Upsilon $ maps $\mathbf{H}_2$
into itself.

\begin{theorem}\label{T1}
Assume that conditions {\rm (H1)--(H3)} hold. If the
inequality
\begin{equation}
2M_1(1+Ml_1l_{3})<1  \label{ini1}
\end{equation}
holds, then the stochastic control system \eqref{eq1} is completely
controllable on $[ 0,T] $.
\end{theorem}

\begin{proof}
To prove the complete controllability of the stochastic system 
\eqref{lemme2} it is enough to show that $\Upsilon $ has a fixed point in 
$\mathbf{H}_2$. To apply the contraction principle, first we show that 
$\Upsilon $ maps $\mathbf{H}_2$ into itself.
Let $x\in \mathbf{H}_2$. Now by Lemma \eqref{lemme2},  for $t\in[ 0,T] $
we have
\begin{align*}
&\mathbf{E}\| (\Upsilon x)(t)\| ^2\\
&=\mathbf{E}\| \phi (t)[ x_0-G(0,x_0,0)] +(Vx)(t) \\
&\quad  +\Pi _0^{t}\phi ^{\ast }(T-t)(\Pi _0^{T})^{-1}\left( x_T-\phi
(T)[x_0-G(0,x_0,0)]-(Vx)(T)\right) \| ^2, \\
&\leq 3\mathbf{E}\| \phi (t)[ x_0-G(0,x_0,0)]
\| ^2+3\mathbf{E}\| (Vx)(t)\| ^2 \\
&\quad +3\mathbf{E}\| \Pi _0^{t}\phi ^{\ast }(T-t)(\Pi
_0^{T})^{-1}\left( x_T-\phi (T)[x_0-G(0,x_0,0)]-(Vx)(T)\right)\| ^2.
\end{align*}
From Lemma \eqref{lemme1} it follows that
\begin{align*}
&\mathbf{E}\| (\Upsilon x)(t)\| ^2\\
&\leq 6l_1\left(\| x_0\| ^2+\| G(0,x_0,0)\|^2\right) 
+3\mathbf{E}\| (Vx)(t)\| ^2 \\
&\quad +9Ml_1l_{3}\left( \mathbf{E}\| x_T\| ^2+2l_1[\| x_0\| ^2+\| G(0,x_0,0)\| ^2
] +\mathbf{E}\| (Vx)(T)\| ^2\right) , \\
&\leq 6l_1\left( \| x_0\| ^2+\|G(0,x_0,0)\| ^2\right) \\
&\quad +9Ml_1l_{3}\left( \mathbf{E}\| x_T\| ^2+2l_1[
\| x_0\| ^2+\| G(0,x_0,0)\| ^2] \right) \\
&\quad +3(1+3Ml_1l_{3})M_2\Big( 1+T\sup_{s\in [ 0,T] }\mathbf{E}
\| x(s))\| ^2\Big) ,
\end{align*}
therefore, we obtain that 
$\| (\Upsilon x)(t)\| _{\mathbf{H}_2}^2<\infty $. Since $\Upsilon $ 
maps $\mathbf{H}_2$\ into itself.

Secondly, we show that $\Upsilon $ is a contraction mapping. To see this let
$x, y\in \mathbf{H}_2$, so for $t\in [ 0,T] $ we have
\begin{align*}
&\mathbf{E}\| (\Upsilon x)(t)-(\Upsilon y)(t)\| ^2\\
&=\mathbf{E}\| (Vx)(t)-(Vy)(t)+\Pi _0^{t}\phi ^{\ast }(T-t)\mathbf{(}\Pi
_0^{T})^{-1}\left( (Vx)(T)-(Vy)(T)\right) \| ^2, \\
&\leq 2\mathbf{E}\| (Vx)(t)-(Vy)(t)\| ^2+2Ml_1l_{3}
\mathbf{E}\| (Vx)(T)-(Vy)(T)\| ^2, \\
&\leq 2(1+Ml_1l_{3})\sup_{s\in [ 0,T] }\mathbf{E}\|
V(x(s))-V(y(s))\| ^2, \\
&\leq 2(1+Ml_1l_{3})M_1\Big( \sup_{s\in [ 0,T] }\mathbf{E}
\| x(s)-y(s)\| ^2\Big) .
\end{align*}
It result from Lemma \eqref{lemme1} and inequality \eqref{ini2} that
\begin{equation*}
\sup_{s\in [ 0,T] }\mathbf{E}\| (\Upsilon x)(s)-(\Upsilon
y)(s)\| ^2\leq 2M_1(1+Ml_1l_{3})\Big( \sup_{s\in [ 0,T
] }\mathbf{E}\| x(s)-y(s)\| ^2\Big) .
\end{equation*}
Therefore $\Upsilon $ is contraction mapping if the inequality \eqref{ini1}
holds. Then the mapping $\Upsilon $\ has a unique fixed point $x(.)$ in 
$\mathbf{H}_2$ which is the solution of the equation \eqref{eq1}. Thus the
system \eqref{eq1} is completely controllable.
\end{proof}

\section{Applications}

\begin{example}\label{x1} \rm
A rocket in vertical motion may be modeled by
\begin{equation} \label{eqmo}
\begin{gathered}
\dot{h}=v   \\
m\dot{v}=-mg+f\,,
\end{gathered}
\end{equation}
where $h$ is altitude, $v$ is velocity; $m$ is mass, $f$ is thrust force.
Let $x_1=h$, $x_2=v$, $u=\frac{f}{m}-g$, then \eqref{eqmo} becomes
\begin{gather*}
\dot{x_1}=x_2 \\
\dot{x_2}=u
\end{gather*}
The complete controllability is to find a (continuous) control $u(t)$ over
the period $[t_0,t_{f}]$ to move the state of the system from a given
initial state $x(t_0)=x_0$ to a desired final state $x(t_{f})=x_{f}$.
\end{example}

\begin{example} \label{x2} \rm
Consider the nonlinear impulsive neutral stochastic systems in the form of 
\eqref{eq1},
\begin{equation}
\begin{gathered}
\begin{aligned}
d\{ x(t)-(\widehat{G}x)(t)\} 
&=-5x(t)dt+\{ e^{-2t}u(t)+(
\widehat{F}_1x)(t)\} dt\\
&\quad +(\widehat{F}_2x)(t)dw(t),\quad t\in [0,T] ,\; t\neq t_k,
\end{aligned} \\
\Delta x(t_k)=0,24e^{0,03}(x(t_k^{-1})),\quad t=t_k, \text{ where }
t_k=t_{k-1}+0,5\text{ for }k=1,2,\dots r.
 \\
x(0)=x_0\in \mathbb{R}^n.
\end{gathered}  \label{qe}
\end{equation}
where $w(.)$ is one-dimensional Brownian motion. (BM) is any of various
physical phenomena in which some quantity is constantly undergoing small,
random fluctuations. If a number of particles subject to (BM) are present in
a given medium and there is no preferred direction for the random
oscillations, then over a period of time the particles will tend to be
spread evenly throughout the medium. (BM) is a Gaussian process with
independent increments which are normally distributed. Here
\begin{equation*}
A(t)=-5,\ B(t)=e^{-2t}.
\end{equation*}
Moreover,
\begin{gather*}
\begin{aligned}
(\widehat{F}_1x)(t)
& =x(t)+2t^2e^{-t}+\int_0^{t}se^{-s}x(s)ds \\
&\quad +\int_0^{T}\arctan (x(s))ds+\int_0^{t}\cos (x(s))dw(s),
\end{aligned} \\
\begin{aligned}
(\widehat{F}_2x)(t) 
&=e^{-t}\sin (x(t))+\int_0^{t}(2s^2+3)x(s)ds \\
&\quad +\int_0^{T}\frac{1}{\sqrt{1+| x(s)| }}
ds+\int_0^{t}\log (1+| x(s)| )dw(s),
\end{aligned} \\
(\widehat{G}x)(t)=\log [ e^{2t}\big|
\int_0^{t}e^{-s}(x(s)+1)ds\big| +1] ,
\end{gather*}
Note that the above functions satisfy the hypotheses (H1)--(H2). For this
system, we have
\begin{align*}
\Gamma _0^{T} 
&=\int_0^{T}\phi (T,s)B(s)B^{\ast }(s)\phi ^{\ast}(T,s)ds, \\
&= \int_0^{T}e^{-4T}ds=Te^{-4T}>0,\quad\text{for some }T>0.
\end{align*}
Hence, the stochastic system \eqref{eq1} is completely controllable on $
[ 0,T] $.
\end{example}

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