\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 259, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2016/259\hfil 
 Controllabillity of impulsive integrodifferential systems]
{Controllabillity of second-order Sobolev-type neutral
impulsive integrodifferential systems in Banach spaces}

\author[B. Radhakrishan, P. Anukokila \hfil EJDE-2016/259\hfilneg]
{Bheeman Radhakrishan, Paraman Anukokila}

\address{Bheeman Radhakrishan \newline
Department of Mathematics,
PSG College of Technology,
Coimbatore - 641004, TN, India}
\email{radhakrishnanb1985@gmail.com}

\address{Paraman Anukokila \newline
Department of Mathematics,
PSG College of Arts and Science,
Coimbatore 641014, TN, India}
\email{anuparaman@gmail.com}

\thanks{Submitted February 1, 2016. Published September 22, 2016.}
\subjclass[2010]{93B05, 47H10, 34K40}
\keywords{Controllability; neutral integrodifferential system;
\hfill\break\indent impulsive differential equation; fixed point theorem}

\begin{abstract}
 In this article, we prove sufficient conditions for the controllability
 of second-order Sobolev-type nonlinear neutral impulsive integrodifferential
 systems in Banach spaces. The results are obtained by using strongly
 continuous cosine families of operators and the fixed point approach.
 An application is provided to illustrate the theory.
\end{abstract}

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

\section{Introduction}

The field of differential equations is very rich and contains a large 
variety of species. However, there is one basic feature common to all problems 
defined by a differential equation: the equation relates a function to 
its derivatives in such a way that the function itself can be determined. 
In many applications, one assumes the system under consideration is 
governed by a principle of causality; that is, the future state of the system 
is independent of the past states and is determined solely by the present. 
If it is also assumed that the system is governed by an equation involving 
the state and rate of change of the state, then, generally, one is 
considering either ordinary or partial differential equations. 
However, under closer scrutiny, it becomes apparent that the principle 
of causality is often only a first approximation to the true situation and 
that a more realistic model would include some of the past states of the system.

 A dynamical system may evolve through an observable quantity rather than 
the state of the system, a general class of evolutionary equations is defined. 
This class includes standard ordinary and partial differential equations as 
well as functional differential equations of retarded and neutral type. 
In this way, the theory serves as a unification of these classical problems. 
Dynamical systems theory holds the supreme position among all mathematical 
disciplines as it provides the foundation for unlocking many of the mysteries 
in nature and the universe which involve the evolution of time.
The dynamics of many evolving processes are subject to abrupt changes, 
such as shocks, harvesting, and natural disasters. These phenomena involve 
short-term perturbations from continuous and smooth dynamics, whose duration 
is negligible in comparison with the duration of an entire evolution. 
In models involving such perturbations, it is natural to assume these 
perturbations act instantaneously or in the form of ``impulses''.
� As a consequence, impulsive differential equations have been developed 
in modeling impulsive problems in physics, population dynamics, ecology, 
biotechnology, industrial robotics, pharmacokinetics, optimal control, 
and so forth. Again, associated with this development, a theory of 
impulsive differential equations has been given extensive attention.

A neutral functional differential equation is one in which the derivatives
of the past history or derivatives of functionals of the past history 
are involved as well as the present state of the system \cite{H2, H4}. 
The theory of impulsive differential equations \cite{H2, R2, S1} has seen 
considerable development by the monographs of Bainov and Simeonov \cite{B1}. 
Sobolev type equation appears in variety of physical problems such as flow of 
fluid through rocks, thermodynamics, propagation of long waves of small 
amplitude and shear in second order fluid and so on \cite{A1, Ba1}. 
Balachandran and Dauer \cite{KB3} provide some sufficient conditions for 
controllability of integer functional evolution equations of Sobolev type by 
the theory of semigroup theory via the techniques of fixed point 
theorem \cite{KB1, Ra1, S3}.

 The concept of impulsive control and its mathematical foundation called 
impulsive differential  equations, or differential equations with impulse effects, 
or differential equations with discontinuous right hand sides have a long history. 
In fact, in mechanical systems impulsive phenomena had been studied for 
a long time under different names such as: mechanical systems with impacts. 
The study of impulsive control systems (control systems with impulse effects) 
has also a long history that can be traced back to the beginning of modern 
control theory. Many impulsive control methods were successfully developed 
under the framework of optimal control and were occasionally called impulse control.

Controllability is an important property of a control system, and the 
controllability property plays a crucial role in many control problems, 
such as stabilization of unstable systems by feedback, or optimal control. 
A state $x$ is controllable at time $t$ if for some finite time $t$ there exists 
an input $u(t)$ that transfers the state $x(t)$ from $x$ to the origin at time $t$. 
That is a system is called controllable at time $t$ if every state $x$ in the 
state-space is controllable. It means a system with internal state vector $x$ 
is called controllable if and only if the system states can be changed by 
changing the system input. The concept of controllability plays a major role 
in finite dimensional control theory, so that it is natural to try to generalize 
it to infinite dimensional system. The nonlinear system of controllability 
represented by differential equations in a finite dimensional space is discussed 
many authors by means of fixed point approach \cite{Hu1,S2}. Second order 
nonlinear differential and integrodifferential equations arise in problems 
connected with many other physical phenomena. So it is quite significant to 
study controllability problem for such systems in Banach spaces \cite{KB4,P1,r2}. 
An abstract linear second order differential equations are related to strongly 
continuous cosine families of bounded linear operators \cite{T1,T2,T3}.

From the above literature, it should be noted that there are several contributions 
on the existence and controllability of differential equations, existence and 
controllability of integrodifferential equations with and without randomness 
using one or more parameter families. Till now, the exact controllability of 
second order Sobolev-type neutral impulsive integrodifferential systems untreated 
in the literature.

Motivated by this fact, in this article, we make an attempt to fill this gap by 
studying controllability of second order Sobolev-type neutral impulsive 
integrodifferential systems in Banach spaces.

\section{Preliminaries}

Consider the nonlinear impulsive neutral integrodifferential systems with Sobolev 
type of the form
\begin{gather} \label{e1.1}
\begin{aligned}
&\frac {d}{dt}\big[(Bx(t))'+f(t,x(t),x'(t))\big]\\
&=    Ax(t) +  \int_0^t    \mathscr{D}(t-s)x(s)ds+Gu(t)+g(t,x(t),x'(t)) \\
 &\quad +\int^t_0  k(t,s,x(s),x'(s))ds, \quad  t\in \mathcal{I},\; t\neq t_k,
\end{aligned}\\
\label{e1.2}
  x(0)= x_0 \quad x'(0)=y_0\\
\label{e1.3}
  \Delta x(t_k)=I_k\big(x(t_k), x'(t_k)\big),\quad
 \Delta x'(t_k)=  J_k\big(x(t_k), x'(t_k)\big), \quad k=1,2,\dots, m,
\end{gather}
where the state $x(\cdot)$ takes the values in the Banach space $X$, 
$x_0, y_0\in X$, $A$ is the infinitesimal generator of a strongly continuous 
cosine family $\{C(t), t \in \mathcal{I}\}$ of bounded linear operators
in the Banach space $X$, the interval $\mathcal{I}=[0, b]$, $G$ is a bounded
linear operator from $U$ to $X$ and the control function $u(\cdot)$ is given in 
${\mathscr{L}}^2(\mathcal{I},U)$, a Banach space of admissible control functions
with $U$ as a Banach space. $B$ is a linear operator with domain and range 
contained in a Banach space $X$.  $\mathscr{D}(t-s)$ is closed operator on 
$X$ with dense domain $X$ which is independent of $t$  and the nonlinear operators 
$f,g: \mathcal{I}\times X \times X \to X$, $k:\mathcal{I}^2\times X \times X \to X$
and $\mathcal{I}_k, J_k:X\times X\to X$, $k=1,2,\dotsm$, are given appropriate
functions and the symbol $\Delta x(t)$ represent the jump of the function $x$ 
at $t$, which is defined by $\Delta x(t)=x(t^+)-x(t^-)$.

Through out this paper,  $X$ is a Banach spaces endowed with the norm $\|\cdot\|$.
In what follows, we put $t=0$, $t_{n+1}=b$ and we denote by $\mathcal{PC}$ the 
space formed by the functions $u:\mathcal{I} \to X$ such that $u(\cdot)$ is 
continuous at $t\neq t_i$, $x(t_i^-)=x(t_i)$ and $x(t_i^+)$ exist for all 
$i=1,2,\dots,m$. It is clear that $\mathcal{PC}$, endowed with the norm
 $\|x\|_{\mathcal{PC}}:=\sup_{t \in \mathcal{I}}\|x(t)\|$, is a Banach space. 
Similarly, $\mathcal{PC'}$ will be the space of the functions 
$x(\cdot)\in \mathcal{PC}$ such that $x(\dot)$ is continuously
differentiable on $I$, $t_i$, $i=1,2,\dots,n$ and the derivatives 
\[
u_R'(t)=\lim_{s \to 0}\frac{u(t+s)-u(t^+)}{s}, \quad
u_L'(t)=\lim_{s \to 0}\frac{u(t+s)-u(t^-)}{s}
\]
are continuous on $[0,b[$ and $]0,b]$, respectively. Next, for 
$x \in \mathcal{PC'}$, we represent, by $u'(t)$, the left derivative at 
$t \in ]0,b]$ and, by $u'(0)$, the right derivative at zero. 
It easy to see that $\mathcal{PC'}$, provided with the norm 
$$
\|u\|_{\mathcal{PC'}}:=\|u\|_{\mathcal{PC}}+\|u'\|_{\mathcal{PC}}
$$ 
is a Banach space.

The operator-valued function
$ \mathcal{H}(t)=\begin{bmatrix} C(t) & S(t)\\ 	AS(t) & C(t) \end{bmatrix}$
is strongly continuous group of linear operators on the space $E \times X$ 
generated by the operator $\mathcal A=\begin{bmatrix}	0 & I \\
 	A & 0 	\end{bmatrix}$ defined on $D(A)\times E$. From this, 
it follows that $AS(t):E \to X$ is bounded linear operator and that 
$AS(t)x \to 0$ as $t \to 0$ for each $x \in E$. Furthermore, if
 $x:[0,\infty[ \to X$ is locally integrable, then 
$z(t):=\int_0^tS(t-s)x(s)ds$ defines an E-valued continuous function, 
which is a consequence of the fact that
\[
\int_0^t \mathcal{H}(t-s)\begin{bmatrix}	0 \\
x(s) 	\end{bmatrix} ds
=\begin{bmatrix}	\int_0^tS(t-s)x(s)ds \\
\int_0^tC(t-s)x(s)ds
	\end{bmatrix}
\]
defines an $E \times X$-valued continuous function.

 To prove our main theorem we assume certain conditions on the operators $A$ and
$B$. Let $X$ and $Y$ be Banach spaces with norm $|\cdot|$ and 
$\|\cdot\|$ respectively. The operators $A:\mathcal D(A)\subset X \to Y$ and 
$B:\mathcal D(A)\subset X \to Y$ satisfy the following hypothesis:
\begin{itemize}
\item[(1)] $A$ and $B$ are closed linear operators,
\item[(2)] $\mathcal D(B)\subset \mathcal D(A)$ and $B$ is bijective,
\item[(3)] $B^{-1} : Y \to  \mathcal D(B)$ is continuous.
\end{itemize}
 These hypothesis and the closed graph theorem imply the boundedness 
of the linear operator $AB^{-1} : Y \to Y$. 
Let  $\mathbb B_r=\{x \in X:\|x\| \leq r \}$ for some $r\geq1$.

\begin{definition} \label{def2.1a} \rm 
 A one parameter family $\{C(t), t \in \mathcal{I}\}$ of bounded linear operators 
in the Banach space $X$ is called a strongly continuous cosine family if 
\begin{itemize}
\item[(i)] $C(s+t)+C(s-t)=2C(s)C(t), $ for all $s,t \in \mathcal{I}$;
\item[(ii)] $C(0)=I$;
\item[(iii)] $C(t)x$ is continuous in $t$ on $\mathcal{I}$, for each $x \in X$.
\end{itemize} 
\end{definition}

Define the associated sine family $S(t), t \in \mathcal{I}$ by
$$
S(t)x:=\int^t_0C(s)xds, \quad x \in X,\; t \in \mathcal{I}
$$
The infinitesimal generator of a strongly continuous cosine family 
$\{C(t), t \in \mathcal{I}\}$ is the operator $A: X \to X$, defined by
$$
Ax=\frac{d^2}{dt^2}C(t)x|_{t=0}, \quad x \in D(A),
$$
where $D(A):=\{x \in X:C(t)x  \text{ is twice continuously differentiable in } t \}$.

 Define $E:=\{x \in X:C(t)x   \text{ is twice continuously differentiable in } t \}$. 
We assume
\begin{itemize}
\item[(A1)] 
A is the infinitesimal generator of a strongly continuous cosine family 
$\{C(t), t \in \mathcal{I}\}$ of bounded linear operators in the Banach space $X$.
\end{itemize}

 To establish our main theorem, we need the following lemmas.

\begin{lemma} \label{lem2.1} 
Let {\rm (A1)} hold. Then
\begin{itemize}
\item[(i)] there exist constants $M\geq 1$ and $\omega \geq 0$ such that 
$\|C(t)\|\leq Me^{\omega |t|}$ and
$$
\|S(t)-S(t^*)\|\leq M|\int_t^{t^*}e^{\omega |s|}ds|,\quad \text{for } 
 t, t^* \in \mathcal{I};
$$

\item[(ii)] $S(t)X\subset E$ and $S(t)E \subset D(A)$, for $t \in \mathcal{I}$;

\item[(iii)] $\frac{d}{dt}C(t)x=AS(t)x$, for $x \in E$ and $t \in \mathcal{I}$;

\item[(iv)] $\frac{d^2}{dt^2}C(t)x=AC(t)x$, for $x \in D(A)$ and 
$t \in \mathcal{I}$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{T3}] \label{lem2.2} 
Let {\rm (A1)} hold and $v:\mathcal R \to X$ be such that $v$ is continuous 
and let $q(t)=\int_0^t S(t-s)v(s)ds$. Then $q$ is twice continuously 
differentiable and, for $t \in \mathcal{I}$: $q(t)\in D(A)$, 
$q'(t)= \int_0^t C(t-s)v(s)ds$ and 
\[
q''(t)=\int_0^t C(t-s)v'(s)ds+C(t)v(0)=Aq(t)+v(t).
\]
\end{lemma}

 First we study the following Sobolev type neutral impulsive integrodifferential 
system
\begin{gather} \label{e2.1}
\begin{aligned}
\frac {d}{dt}\big[(Bx(t))'+f(t,x(t))\big]
&=Ax(t)+\int_0^t \mathscr{D}(t-s)x(s)ds+Gu(t)+g(t,x(t)) \\
&\quad +\int^t_0  k(t,s,x(s))ds, \quad  t\in(0, b],\; t\neq t_k,
\end{aligned}\\
\label{e2.2}
 x(0)=x_0 , \quad x'(0)=y_0\\
\label{e2.3}
 \Delta x(t_k)=I_k(x_{t_k}),\quad  \Delta x'(t_k)=J_k(x_{t_k}), \quad k=1,2,\dots, m.
\end{gather}

\begin{definition} \label{def2.1} \rm  
 A continuous solution $x(\cdot)$ of the integral equation
\begin{equation} \label{e2.4}
\begin{aligned}
x(t) 
&= B^{-1}S(t)[By_0+f(0,x(0))]+B^{-1}C(t)Bx_0 \\
&\quad -\int_0^t B^{-1}C(t-s)f(s,x(s))ds  \\
&\quad  +\int^t_0  B^{-1}S(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds \\
&\quad +\int^t_0  B^{-1}S(t-s)Gu(s)ds
 +\int^t_0  B^{-1}S(t-s)\Big[g(s,x(s)) \\
&\quad +\int_0^sk(s,\tau,x(\tau))d\tau\Big]ds 
  +\sum_{0<t_k<t}B^{-1}C(t-t_k)I_kx(t_k) \\
&\quad +\sum_{0<t_k<t}B^{-1}S(t-t_k)J_kx(t_k)
\end{aligned}
\end{equation}
is said to be a mild solution of problem \eqref{e2.1}-\eqref{e2.3} on $\mathcal{I}$.

 If $x(\cdot)$ is a mild solution of  \eqref{e2.1}-\eqref{e2.3}, then by the
 properties of a second order differential equation and Lemma \ref{lem2.2}, we have
\begin{align*} %\label{e2.5}
x'(t) &= B^{-1}C(t)[By_0+f(0,x(0))]+B^{-1}AS(t)Bx_0-B^{-1}f(t,x(t))\\
&\quad -\int_0^t B^{-1}AS(t-s)f(s,x(s))ds 
+ \int^t_0  B^{-1}C(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad +\int^t_0  B^{-1}C(t-s)Gu(s)ds+\int^t_0  B^{-1}C(t-s)\Big[g(s,x(s))\\
&\quad +\int_0^sk(s,\tau,x(\tau))d\tau\Big]ds
  +\sum_{0<t_k<t}B^{-1}AS(t-t_k)I_kx(t_k) \\
&\quad +\sum_{0<t_k<t}B^{-1}C(t-t_k)J_kx(t_k), \quad t\in \mathcal{I}.
\end{align*}
\end{definition}

To study the controllability problem, we assume the following hypotheses:
\begin{itemize}
\item[(H1)] $A$ is the infinitesimal generator of a strongly continuous 
cosine family $\{C(t), \; t \in \mathcal{I} \}$ of bounded linear operators 
in the Banach space $X$. There exist constants $\mathscr{M}_1\geq1$ and 
$\mathscr{M}_2,  \mathscr{L}_D \geq 0$ such that $\|C(t)\| \leq \mathscr{M}_1$,
 $\|S(t)\|\leq \mathscr{M}_2$, and $\|\mathscr{D}(t-s)\| \leq\mathscr{L}_D$, 
for every $t \in \mathcal{I}$. Furthermore we take 
$ \mathscr{M}_3=\sup_{t \in \mathcal{I}}\|AS(t)\|$,
$\mathscr{N}_1=\|B^{-1}\|$, and $\mathscr{N}_2=\|B\|$.

\item[(H2)] The linear operator $\mathscr{W}_1:\mathscr{L}^2(\mathcal{I}, U)\to X$ 
defined by 
\[
\mathscr{W}_1u=\int^{b}_{0}B^{-1}S(b-s)Gu(s)ds
\]
has an inverse operator $\mathscr{W}_1^{-1}$ which takes values in 
$\mathscr{L}^2(\mathcal{I}, U)/\ker \mathscr{W}_1$ and there exists a 
positive constant $\mathscr{K}_1$ such that
$\|G\mathscr{W}_1^{-1}\|\leq \mathscr{K}_1$.

\item[(H3)] The linear operator $\mathscr{W}_2:\mathscr{L}^2(\mathcal{I}, U)\to X$ 
defined by 
\[
\mathscr{W}_2u=\int^{b}_{0}B^{-1}C(b-s)Gu(s)ds
\]
has an inverse operator $\mathscr{W}_2^{-1}$ which takes values in 
$\mathscr{L}^2(\mathcal{I}, U)/\ker \mathscr{W}_2$ and there exists 
a positive constant $\mathscr{K}_2$ such that 
$\|G\mathscr{W}_2^{-1}\|\leq \mathscr{K}_2$.

\item[(H4)] $\mathscr{W}_1\mathscr{W}_2^{-1}x=\mathscr{W}_2\mathscr{W}_1^{-1}x=0$, 
for every $x\in X$.

\item[(H5)] The function $f:\mathcal{I}\times X\to X$ is continuous for a.e. 
$t\in \mathcal{I}$. and the function $f(.,x):\mathcal{I}\times X\to X$ is 
strongly measurable, for each $x\in X$. Then there exist positive constants 
$\mathscr{L}_f >0,\ {\mathscr{F}_0}>0$ such that
$$
\|f(t,x_1(t))-f(s,x_2(t))\|\leq \mathscr{L}_f[|t-s|+\|x_1-x_2\|],
$$
 for $t, s\in \mathcal{I}$ and $x_i \in X$, $i=1,2$,
and 
$$ 
\max_{t\in \mathcal{I}}\|f(t,0)\|={\mathscr{F}_0}.
$$

\item[(H6)] The function $g:\mathcal{I}\times X\to X$ satisfies the following 
conditions:
\begin{itemize}
\item[(i)] For each $t \in \mathcal{I}$, the function 
 $g(t,\cdot):\mathcal{I} \times X \to X$ is continuous and for each $x \in X$, 
 the function $g(\cdot, x):\mathcal{I} \times X \to X$ is strongly measurable.
\item[(ii)] There exist a constants $\mathscr{L}_g>0, {\mathscr G_0}$ such that
$$
\|g(t,x_1)-g(s,x_2)\|\leq \mathscr{L}_g[|t-s|+\|x_1-x_2\|],\quad
 \text{for } t\in I \text{ and } x_i \in X,\ i=1,2
$$
and
$$ 
\max_{t\in \mathcal{I}}\|g(t,0)\|\leq {\mathscr G_0 },\quad \text{for } t\in I.
$$
\end{itemize}

\item[(H7)] The function $k:\mathcal{I}^2\times X \to X$ satisfies the following 
condition:
\begin{itemize}
\item[(i)] For each $t,s \in \mathcal{I}$, the function 
 $k(t,s,\cdot):\mathcal{I}^2\times X \to X$ is continuous and for each 
 $x \in X$, the function $k(\cdot,\cdot,x): \mathcal{I}^2\times X \to X$ is 
 strongly measurable.
\item[(ii)] There exists a constant $\mathscr{L}_k>0,\ {\mathscr{K}_0} $ such that
$$
\|k(t,s,x_1)-k(t,s,x_2)\|\leq\mathscr{L}_k[\|x_1-x_2\|],\quad
 \text{for } t,s \in \mathcal{I}\text{ and } x_i \in X,\; i=1,2
$$ 
and
$$ 
\max_{t\in \mathcal{I}}\|k(t,s,0)\|\leq {\mathscr{K}_0}, \quad \text{for }
t,s \in \mathcal{I}.
$$
\end{itemize}

\item[(H8)] $I_k, J_k:X \to X$, $k=1,2,\dots,m$, are continuous and there 
exist constants $\mathscr{L}_I>0$, $ \mathscr{L}_J>0$, $ {\mathcal{I}_0}>0$ 
and $\mathcal{J}_0>0$ such that
\begin{gather*}
\|I_k(x_1)-I_k(x_2)\|\leq \mathscr{L}_I\|x_1-x_2\|, \\
\|J_k(x_1)-J_k(x_2)\|\leq \mathscr{L}_J\|x_1-x_2\|, \\
\mathcal{I}_0=\|I_k(0)\|, \quad \mathscr J_0=\|J_k(0)\|,\quad  k=1,2,\dotsm.
\end{gather*}
for all $x_1,x_2 \in X$ and $k=1,2,\dots, m$.

\item[(H9)] There exist constants $\rho>0$,  $\widehat{\rho}>0$ such that
\begin{align*}
&\mathscr{N}_1\mathscr{M}_2[\mathscr{N}_2\|y_0\|
 +\mathscr{F}_0]+\mathscr{N}_1\mathscr{N}_2 \mathscr{M}_1\|x_0\|
 +b\mathscr{N}_1\mathscr{M}_1[r\mathscr{L}_f+\mathscr{F}_0] \\
& +b^2r\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D 
 +b\mathscr{N}_1\mathscr{M}_2\mathcal S_0
  +b\mathscr{R}_1\mathscr{M}_2[r\mathscr{L}_g+\mathscr G_0
 +b\{r\mathscr{L}_k+\mathscr{K}_0\}] \\
&+\mathscr{R}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_I+ \mathcal{I}_0]
 +\mathscr{R}_1\mathscr{M}_2\sum_{k=0}^m[r\mathscr{L}_J+ \mathscr J_0]
\leq \rho
 \end{align*}
 and
 \begin{align*}
 &\mathscr{N}_1\mathscr{M}_1[\mathscr{N}_2\|y_0\|+\mathscr{F}_0]
 +\mathscr{N}_1\mathscr{M}_3\mathscr{N}_2\|x_0\|
 +\mathscr{N}_1[r\mathscr{L}_f+\mathscr{F}_0]
 +b\mathscr{N}_1\mathscr{M}_3[r\mathscr{L}_f+\mathscr{F}_0]\\
&+b^2r\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D  
 +b\mathscr{N}_1\mathscr{M}_1\mathcal S_0
 + b\mathscr{N}_1\mathscr{M}_1[r\mathscr{L}_g+\mathscr G_0+b\{r\mathscr{L}_k
 +\mathscr{K}_0\}] \\
&+\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m[r\mathscr{L}_I
 + \mathcal{I}_0]+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_J
 + \mathscr J_0]
 \leq  \widehat {\rho},
 \end{align*}
where
\begin{align*}
&\mathcal S_0  \\
&=\mathscr{K}_1\Big[\|x_b\|+\mathscr{N}_1\mathscr{M}_2[\mathscr{N}_2\|y_0\|
 +\mathscr{F}_0]+\mathscr{N}_1\mathscr{M}_1\mathscr{N}_2\|x_0\|
 + b\mathscr{M}_1\mathscr{N}_1[r\mathscr{L}_f +\mathscr{F}_0]\\
&\quad +b^2r\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D
 +b\mathscr{N}_1\mathscr{M}_2[r\mathscr{L}_g+\mathscr G_0
 +b\{r\mathscr{L}_k+\mathscr{K}_0\}]
 +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_I+ \mathcal{I}_0]\\
&\quad+\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m[r\mathscr{L}_J+\mathscr J_0]\Big]
 +\mathscr{K}_2\Big[\|y_b\|+\mathscr{N}_1\mathscr{M}_1[\mathscr{N}_2\|y_0\|
 +\mathscr{F}_0]+\mathscr{N}_1\mathscr{M}_3\mathscr{N}_2\|x_0\|\\
&\quad +\mathscr{N}_1[r\mathscr{L}_f+\mathscr{F}_0]
 + b\mathscr{M}_3\mathscr{N}_1[r\mathscr{L}_f
 +\mathscr{F}_0] +b^2r\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D
 +\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m[r\mathscr{L}_I+ \mathcal{I}_0]\\
&\quad +b\mathscr{N}_1\mathscr{M}_1[r\mathscr{L}_g+\mathscr G_0+b\{r\mathscr{L}_k
 + \mathscr{K}_0\}] +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_J
 +\mathscr J_0] \Big].
 \end{align*}
\end{itemize}

\begin{definition}[\cite{H1}] \label{def} \rm  
System \eqref{e2.1}-\eqref{e2.3} is said to be controllable on the interval 
$\mathcal{I}$, if for every initial functions $x_0,\ x_b \in X $ and 
$y_0,\ y_b \in X$, there exists a control $u \in \mathscr{L}^2(\mathcal{I}, U)$ 
such that the solution $x(\cdot)$ of \eqref{e2.1}-\eqref{e2.3} satisfies 
$x(0)=x_0$, $x(b)=x_b$ and $x'(0)=y_0,\ x'(b)=y_b$.
\end{definition}

\section{Controllability result}

\begin{theorem} \label{thm3.1} 
If  assumptions {\rm (H1)--(H9)} hold and if $0\leq \Lambda_1$,
$\Lambda_2 <1$, then system \eqref{e2.1}--\eqref{e2.3} is controllable on 
$\mathcal{I}$, provided that there exist constants
\begin{align*}
\Lambda_1
&=(1 +b\mathscr{N}_1\mathscr{M}_2 \mathscr{K}_1)
\Big[b\mathscr{M}_1\mathscr{N}_1\mathscr{L}_f
 +b^2\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D
 +b\mathscr{N}_1\mathscr{M}_2[\mathscr{L}_g+b\mathscr{L}_k]\\
&\quad +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_I
 +\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m\mathscr{L}_J\Big]
 +b\mathscr{N}_1\mathscr{M}_2\mathscr{K}_2   
 \Big[\mathscr{N}_1\mathscr{L}_f+ b\mathscr{M}_3\mathscr{N}_1\mathscr{L}_f\\
&\quad +b^2\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D
 +b\mathscr{N}_1\mathscr{M}_1\mathscr{L}_g 
+b\mathscr{L}_k+\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m\mathscr{L}_I
+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_J \Big]
\end{align*}
and
\begin{align*}
\Lambda_2
&=b\mathscr{N}_1\mathscr{M}_1 \mathscr{W}_1
 \Big[b\mathscr{M}_1\mathscr{N}_1\mathscr{L}_f 
 +b^2\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D 
 +b\mathscr{N}_1\mathscr{M}_2[\mathscr{L}_g+b\mathscr{L}_k]
 +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_I\\
 &\quad +\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m\mathscr{L}_J\Big]
 +(1 +b\mathscr{N}_1\mathscr{M}_1)\mathscr{W}_2 \Big[\mathscr{N}_1\mathscr{L}_f
 + b\mathscr{M}_3\mathscr{N}_1\mathscr{L}_f
 +b^2\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D\\
&\quad+b\mathscr{N}_1\mathscr{M}_1\mathscr{L}_g +b\mathscr{L}_k
 +\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m\mathscr{L}_I
 +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_J \Big].
 \end{align*}
\end{theorem}

\begin{proof}
Using {\rm (H2), (H3)} for an arbitrary function $x(\cdot)$, define the control
\begin{align*}
  u(t)
&= \mathscr{W}_1^{-1}\Big[x_b-B^{-1}S(b)[By_0+f(0,x(0))]-B^{-1}C(b)Bx_0 \\
&\quad +\int_0^b B^{-1}C(b-s)f(s,x(s))ds -\int^b_0  B^{-1}S(b-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad -\int^b_0  B^{-1}S(b-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]ds\\
 &\quad -\sum_{0<t_k<b}B^{-1}C(b-t_k)I_kx(t_k)+\sum_{0<t_k<b}B^{-1}S(b-t_k)J_kx(t_k)\Big](t)\\
 &\quad +\mathscr{W}_2^{-1}\Big[y_b-B^{-1}C(b)[By_0+f(0,x(0))]-B^{-1}AS(b)Bx_0+f(b,x(b))\\
 &\quad +\int_0^b B^{-1}AS(b-s)f(s,x(s))ds -\int^b_0  B^{-1}C(b-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad -\int^b_0  B^{-1}C(b-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]ds\\
 &\quad -\sum_{0<t_k<b}B^{-1}AS(b-t_k)I_kx(t_k)+\sum_{0<t_k<b}B^{-1}C(b-t_k)J_kx(t_k)\Big](t).
\end{align*}

Now we have to show that, when using this control $u(t)$, the nonlinear operator
$$
\mathscr P:\mathcal {PC} \to \mathcal {PC}$$ defined by
 \begin{align*}
&(\mathscr Px)(t) \\
&= B^{-1}S(t)[By_0+f(0,x(0))]+B^{-1}C(t)Bx_0 \\
&\quad -\int_0^t B^{-1}C(t-s)f(s,x(s))ds +\int^t_0  B^{-1}S(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad +\int^t_0  B^{-1}S(t-s)\Big\{G\mathscr{W}_1^{-1}\Big[x_b-B^{-1}S(b)[By_0+f(0,x(0))]-B^{-1}C(b)Bx_0 \\
&\quad +\int_0^b B^{-1}C(b-s)f(s,x(s))ds -\int^b_0 B^{-1}S(b-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad -\int^b_0  B^{-1}S(b-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]ds\\
 &\quad -\sum_{0<t_k<b}B^{-1}C(b-t_k)I_kx(t_k)+\sum_{0<t_k<b}B^{-1}S(b-t_k)J_kx(t_k)\Big](s)\\
 &\quad + G\mathscr{W}_2^{-1}\Big[y_b-B^{-1}C(b)[By_0+f(0,x(0))]-B^{-1}AS(b)Bx_0+B^{-1}f(b,x(b))\\
 &\quad +\int_0^b B^{-1}AS(b-s)f(s,x(s))ds -\int^b_0 B^{-1}C(b-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad -\int^b_0  B^{-1}C(b-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]ds\\
 &\quad -\sum_{0<t_k<b}B^{-1}AS(b-t_k)I_kx(t_k)+\sum_{0<t_k<b}B^{-1}C(b-t_k)J_kx(t_k)\Big](s) \Big\}ds\\
 &\quad +\int^t_0  B^{-1}S(t-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]ds\\
 &\quad +\sum_{0<t_k<t}B^{-1}C(t-t_k)I_kx(t_k)+\sum_{0<t_k<t}B^{-1}S(t-t_k)J_kx(t_k)
\end{align*}
 has a fixed point $x(\cdot)$, which is the solution of the system 
\eqref{e2.1}--\eqref{e2.3}. Clearly $x(b)=x_b,\ x^\prime(b)=y_b$, which imply 
that the system is controllable. Since all the functions involved in the 
operator are continuous, $\mathscr P$ is continuous.
For convenience, let 
\begin{align*}
&\mathscr S(s, x) \\
&=G \mathscr{W}_1^{-1}\Big[x_b-B^{-1}S(b)[By_0+f(0,x(0))]-B^{-1}C(b)Bx_0 \\
&\quad +\int_0^b B^{-1}C(b-s)f(s,x(s))ds -\int^b_0  B^{-1}S(b-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad -\int^b_0  B^{-1}S(b-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]ds\\
 &\quad -\sum_{0<t_k<b}B^{-1}C(b-t_k)I_kx(t_k)+\sum_{0<t_k<b}B^{-1}S(b-t_k)J_kx(t_k)\Big](s)\\
 &\quad + G\mathscr{W}_2^{-1}\Big[y_b-B^{-1}C(b)[By_0+f(0,x(0))]-B^{-1}AS(b)Bx_0+B^{-1}f(b,x(b))\\
 &\quad +\int_0^b B^{-1}AS(b-s)f(s,x(s))ds -\int^b_0 B^{-1}C(b-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
 &\quad -\int^b_0  B^{-1}C(b-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]ds\\
 &\quad -\sum_{0<t_k<b}B^{-1}AS(b-t_k)I_kx(t_k)+\sum_{0<t_k<b}B^{-1}C(b-t_k)J_kx(t_k)\Big](s).
\end{align*}
From assumptions {\rm (H1)--(H9)}, we have
\begin{align*}
&\|\mathscr S(s, x)\| \\
&\leq \mathscr{K}_1\Big[\|x_b\|+\mathscr{N}_1\mathscr{M}_2[\mathscr{N}_2\|y_0\|+\mathscr{F}_0]+\mathscr{N}_1\mathscr{M}_1\mathscr{N}_2\|x_0\|+ b\mathscr{M}_1\mathscr{N}_1[r\mathscr{L}_f
+\mathscr{F}_0]\\
&\quad +b^2r\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D+b\mathscr{N}_1\mathscr{M}_2[r\mathscr{L}_g+\mathscr G_0+b\{r\mathscr{L}_k+ \mathscr{K}_0\}]+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_I+ \mathscr I_0]\\
&\quad +\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m[r\mathscr{L}_J+\mathscr J_0]\Big]+\mathscr{K}_2\Big[\|y_b\|+\mathscr{N}_1\mathscr{M}_1[\mathscr{N}_2\|y_0\|+\mathscr{F}_0]+\mathscr{N}_1\mathscr{M}_3\mathscr{N}_2\|x_0\|\\
&\quad +\mathscr{N}_1[r\mathscr{L}_f+\mathscr{F}_0]+ b\mathscr{M}_3\mathscr{N}_1[r\mathscr{L}_f +\mathscr{F}_0]
+b^2r\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D+b\mathscr{N}_1\mathscr{M}_1[r\mathscr{L}_g+\mathscr G_0\\
&\quad +b\{r\mathscr{L}_k+ \mathscr{K}_0\}]+\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m[r\mathscr{L}_I+ \mathcal{I}_0]+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_J+\mathscr J_0] \Big]\\
&=\mathcal S_0
\end{align*}
and
\begin{align*}
&\|\mathscr S(s, x_1)-\mathscr S(s, x_2)\|\\
 &\leq \Big\{\mathscr{K}_1\Big[b\mathscr{M}_1\mathscr{N}_1\mathscr{L}_f
 +b^2\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D 
 +b\mathscr{N}_1\mathscr{M}_2\mathscr{L}_g+b\mathscr{L}_k
 +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_I\\
 &\quad +\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m\mathscr{L}_J\Big]
 +\mathscr{K}_2\Big[\mathscr{N}_1
\mathscr{L}_f+ b\mathscr{M}_3\mathscr{N}_1\mathscr{L}_f
 +b^2\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D
 +b\mathscr{N}_1\mathscr{M}_1\mathscr{L}_g\\
 &\quad +b\mathscr{L}_k+\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m\mathscr{L}_I
 +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_J \Big]\Big\}\|x_1-x_2\|.
\end{align*}
First we show that $\mathscr P$ maps $\mathcal {PC}$ into itself. Now
\begin{align*}
\|(\mathscr Px)(t)\|
&\leq \|B^{-1}S(t)[By_0+f(0,x(0))]\|+\|B^{-1}C(t)Bx_0\| \\
&\quad +\int_0^t \|B^{-1}C(t-s)f(s,x(s))\|ds +\int^t_0 \| B^{-1}S(t-s)\mathscr S(s, x)\|ds\\
 &\quad +\int^t_0 \| B^{-1}S(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau\| ds\\
 &\quad+\int^t_0  \|B^{-1}S(t-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]\|ds\\
 &\quad +\sum_{0<t_k<t}\|B^{-1}C(t-t_k)I_kx(t_k)\|+\sum_{0<t_k<t}\|B^{-1}S(t-t_k)J_kx(t_k)\|\\
&\leq \mathscr{N}_1\mathscr{M}_2[\mathscr{N}_2\|y_0\|+\mathscr{L}_f]
 +\mathscr{N}_1\mathscr{M}_1\mathscr{N}_2\|x_0\|
 +b\mathscr{N}_1\mathscr{M}_1[r\mathscr{L}_f+\mathscr{F}_0] \\
&\quad +b^2r\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D
 +b\mathscr{N}_1\mathscr{M}_2\mathcal S_0 
 +b\mathscr{N}_1\mathscr{M}_2[r\mathscr{L}_g+\mathscr G_0 \\
&\quad +b\{r\mathscr{L}_k+\mathscr{K}_0\}]
 +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_I+ \mathcal{I}_0]
 +\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m[r\mathscr{L}_J+ \mathscr J_0]\\
&< \rho
\end{align*}
and
\begin{align*}
&\|(\mathscr Px)'(t)\| \\
&\leq \|B^{-1}C(t)[By_0+f(0,x(0))]\|+\|B^{-1}AS(t)Bx_0\|+\|B^{-1}f(t,x(t))\| \\
&\quad +\int_0^t \|B^{-1}AS(t-s)f(s,x(s))\|ds+\int^t_0 \|B^{-1}C(t-s)\mathscr S(s, x)\|ds \\
 &\quad +\int^t_0 \| B^{-1}C(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau\| ds\\
 &\quad+\int^t_0  \|B^{-1}C(t-s)\big[g(s,x(s))+\int_0^sk(s,\tau,x(\tau))d\tau\big]\|ds\\
 &\quad +\sum_{0<t_k<t}\|B^{-1}AS(t-t_k)I_kx(t_k)\|+\sum_{0<t_k<t}\|B^{-1}C(t-t_k)J_kx(t_k)\|\\
 &\leq \mathscr{N}_1\mathscr{M}_1[\mathscr{N}_2\|y_0\|+\mathscr{F}_0]
 +\mathscr{N}_1\mathscr{M}_3\mathscr{N}_2\|x_0\|
 +\mathscr{N}_1[r\mathscr{L}_f+\mathscr{F}_0] \\
&\quad +b\mathscr{N}_1\mathscr{M}_3[r\mathscr{L}_f+\mathscr{F}_0]
 +b^2r\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D 
 +b\mathscr{N}_1\mathscr{M}_1\mathcal S_0
 +b\mathscr{N}_1\mathscr{M}_1[r\mathscr{L}_g+\mathscr G_0 \\
&\quad +br\{\mathscr{L}_k+\mathscr{K}_0\}]
 +\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m[r\mathscr{L}_I
+ \mathcal{I}_0] +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m[r\mathscr{L}_J
 + \mathscr J_0]
<\widehat{\rho}.
\end{align*}
Therefore $\mathscr P$ maps from $\mathcal {PC}$ into itself. 
Moreover, if $x_1, x_2 \in \mathcal {PC}$, then
\begin{align*}
&\|(\mathscr Px_1)(t)-(\mathscr Px_2)(t)\|\\
&\leq \Big\|\int_0^t B^{-1}C(t-s)[f(s,x_1(s))-f(s,x_2(s))]ds\Big\| \\
 &\quad +\Big\|\int^t_0 B^{-1}S(t-s)\int_0^s \mathscr{D}(s-\tau)[x_1(\tau)-x_2(\tau)]d\tau ds\Big\|\\
 &\quad +\Big\|\int^t_0  B^{-1}S(t-s)[\mathscr S(s, x_1)-\mathscr S(s, x_2)]ds\Big\|\\
 &\quad+\Big\|\int^t_0  B^{-1}S(t-s)[g(s,x_1(s))-g(s,x_2(s))]ds\Big\|\\
 &\quad +\Big\|\int^t_0  B^{-1}S(t-s)\int_0^s [(k(s,\tau,x_1(\tau))-k(s,\tau,x_2(\tau)))d\tau] ds\Big\|\\
 &\quad +\Big\|\sum_{0<t_k<t}B^{-1}C(t-t_k)[I_kx_1(t_k)-I_kx_2(t_k)]\Big\|\\
 &\quad+\Big\|\sum_{0<t_k<t}B^{-1}S(t-t_k)[J_kx_1(t_k)-J_kx_2(t_k)]\Big\|\\
&\leq \Big\{b\mathscr{N}_1\mathscr{M}_1\mathscr{L}_F+b^2\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D
+b\mathscr{N}_1\mathscr{M}_2[\mathscr{L}_g+b\mathscr{L}_k]+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m
\mathscr{L}_I\\
&\quad +\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m\mathscr{L}_J+ b\mathscr{N}_1\mathscr{M}_2 \mathscr{K}_1\Big[b\mathscr{M}_1\mathscr{N}_1\mathscr{L}_f+b^2\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D +b\mathscr{N}_1\mathscr{M}_2\mathscr{L}_g\\
  &\quad +b\mathscr{L}_k+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_I+\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m\mathscr{L}_J\Big]+b\mathscr{N}_1\mathscr{M}_2\mathscr{K}_2 \Big[\mathscr{N}_1\mathscr{L}_f\\
&\quad+ b\mathscr{M}_3\mathscr{N}_1\mathscr{L}_f+b^2\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D+b\mathscr{N}_1\mathscr{M}_1\mathscr{L}_g +b\mathscr{L}_k+\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m\mathscr{L}_I\\
&\quad+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_J
 \Big]\Big\}\|x_1-x_2\|= \Lambda_1 \|x_1-x_2\|.
\end{align*}
Also
\begin{align*}
&\|(\mathscr Px_1)^\prime(t)-(\mathscr Px_2)^\prime(t)\| \\
&\leq \Big \|B^{-1}[f(s,x_1(s))-f(s,x_2(s))]\Big\|\\
&\quad +\Big\|\int_0^t B^{-1}AS(t-s)[f(s,x_1(s))-f(s,x_2(s))]ds\Big\| \\
 &\quad +\Big\|\int^t_0 B^{-1}C(t-s)\int_0^s \mathscr{D}(s-\tau)[x_1(\tau)-x_2(\tau)]d\tau ds\Big\|\\
 &\quad +\Big\|\int^t_0  B^{-1}C(t-s)[\mathscr S(s, x_1)-\mathscr S(s, x_2)]ds\Big\|\\
 &\quad+\Big\|\int^t_0  B^{-1}C(t-s)[g(s,x_1(s))-g(s,x_2(s))]ds\Big\|\\
 &\quad +\Big\|\int^t_0  B^{-1}C(t-s)\int_0^s [(k(s,\tau,x_1(\tau))-k(s,\tau,x_2(\tau))d\tau)] ds\Big\|\\
 &\quad +\Big\|\sum_{0<t_k<t}B^{-1}AS(t-t_k)[I_kx_1(t_k)-I_kx_2(t_k)]\Big\|\\
 &\quad+\Big\|\sum_{0<t_k<t}B^{-1}C(t-t_k)[J_kx_1(t_k)-J_kx_2(t_k)]\Big\|\\
&\leq \Big\{\mathscr{N}_1\mathscr{L}_f+b\mathscr{N}_1\mathscr{M}_3\mathscr{L}_f+b^2\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D+b\mathscr{N}_1\mathscr{M}_1[\mathscr{L}_g +b\mathscr{L}_k]\\
 &\quad+\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m\mathscr{L}_I +\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_J + b\mathscr{N}_1\mathscr{M}_1 \mathscr{K}_1\Big[b\mathscr{M}_1\mathscr{N}_1\mathscr{L}_f\\
 &\quad+b^2\mathscr{N}_1\mathscr{M}_2\mathscr{L}_D +b\mathscr{N}_1\mathscr{M}_2[\mathscr{L}_g+b\mathscr{L}_k]+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_I\\
 &\quad+\mathscr{N}_1\mathscr{M}_2\sum_{k=0}^m\mathscr{L}_J\Big]+b\mathscr{N}_1\mathscr{M}_1\mathscr{K}_2    \Big[\mathscr{N}_1\mathscr{L}_f+ b\mathscr{M}_3\mathscr{N}_1\mathscr{L}_f+b^2\mathscr{N}_1\mathscr{M}_1\mathscr{L}_D\\
 &\quad+b\mathscr{N}_1\mathscr{M}_1[\mathscr{L}_g +b\mathscr{L}_k]+\mathscr{N}_1\mathscr{M}_3\sum_{k=0}^m\mathscr{L}_I+\mathscr{N}_1\mathscr{M}_1\sum_{k=0}^m\mathscr{L}_J \Big]\Big\}\|x_1-x_2\|\\
&= \Lambda_2 \|x_1-x_2\|.
\end{align*}
Since $\Lambda_1<1$ and $\Lambda_2<1$, the operator $\mathscr P$ is a contraction.
 Consequently by the  Banach contraction fixed point theorem, there exists a 
unique fixed point $x \in \mathcal {PC}$ such that $(\mathscr Px)(t)=x(t)$. 
This fixed point is then the solution of the problem \eqref{e2.1}-\eqref{e2.3}. 
Then clearly, $(\mathscr Px)(b)=x(b)=x_b,\ (\mathscr Px)'(b)=x'(b)=y_b$ which 
implies that the system \eqref{e2.1}-\eqref{e2.3} is controllable on $\mathcal{I}$. 
Thus the proof is complete.
\end{proof}

 Now to study  the controllability of \eqref{e1.1}-\eqref{e1.3},
 we impose the following additional hypotheses:
\begin{itemize}
\item[(H10)] The function $f:\mathcal{I}\times X \times X \to X$ is continuous 
for a.e. $t\in \mathcal{I}$. and the function 
$f(.,x,y):\mathcal{I}\times X\times X\to X$ is strongly measurable, for each 
$x\in X$. Then there exist positive constants $\mathscr{L}_F >0,\ F_0>0$ such that
$$
\|f(t,x_1(t),y_1(t))-f(s,x_2(t),y_2(t))\|
\leq \mathscr{L}_F[|t-s|+\|x_1-x_2\|+\|y_1-y_2\|],
$$
for $t, s\in \mathcal{I},\ x_i,y_i \in X,\ i=1,2$ and 
$$ 
\max_{t\in \mathcal{I}}\|f(t,0,0)\|= F_0.
$$

\item[(H11)] 
The function $g:\mathcal{I}\times X\times X\to X$ satisfies the following conditions:
\begin{itemize}
\item[(i)] For each $t \in \mathcal{I}$, the function $g(t,\cdot. \cdot):\mathcal{I}
 \times X \times X \to X$ is continuous and for each $x \in X$, the function
 $g(\cdot, x, \ y):\mathcal{I} \times X \times X \to X$ is strongly measurable.

\item[(ii)] There exist a constants $\mathscr{L}_G>0,\ G_0>0$ such that
$$\|g(t,x_1,y_1)-g(s,x_2,y_2)\|\leq \mathscr{L}_G[|t-s|+\|x_1-x_2\|+\|y_1-y_2\|],$$
for $t, s\in \mathcal{I}$, and $x_i,y_i \in X,\ i=1,2$,
and
$$ 
\max_{t\in \mathcal{I}}\|g(t,0,0)\|\leq  G_0,\quad \text{for}\ t\in I.
$$
\end{itemize}

\item[(H12)] The function $k:\mathcal{I}^2\times X \times X \to X$ satisfies 
the following conditions:
\begin{itemize}
\item[(i)] For each $t,s \in \mathcal{I}$, the function 
 $k(t,s,\cdot.\cdot):\mathcal{I}^2\times X \times X \to X$ is continuous and 
 for each $x \in X$, the function 
 $k(\cdot,\cdot,x,y): \mathcal{I}^2\times X \times X \to X$ is strongly measurable.

\item[(ii)] There exists a constant $\mathscr{L}_K>0,\ K_0>0 $ such that
$$
\|k(t,s,x_1,y_1)-k(t,s,x_2,y_2)\|\leq\mathscr{L}_K[\|x_1-x_2\|+\|y_1-y_2\|],
$$
for $t, s\in \mathcal{I}$, and $x_i,y_i \in X$, $i=1,2$,
and
$$ 
\max_{t\in \mathcal{I}}\|k(t,s,0,0)\|\leq K_0, \quad \text{for } t,s \in \mathcal{I}
$$
\end{itemize}

\item[(H13)] $I_k, J_k:X \times X \to X, k=1,2,\dots,m$, are continuous and 
there exist constants $\mathcal{L}_I>0$,
 $\mathcal{L}_J>0$, $I_0>0$ and $J_0>0$ such that
\begin{gather*}
\|I_k(x_1,y_1)-I_k(x_2,y_2)\|\leq \mathcal{L}_I[\|x_1-x_2\|+\|y_1-y_2\|], \\
\|J_k(x_1,y_1)-J_k(x_2,y_2)\|\leq\mathcal{L}_J[\|x_1-x_2\|+\|y_1-y_2\|]
\end{gather*}
for all $x_1,x_2,y_1,y_2 \in X$ and $k=1,2,\dots, m$,
and
$$
I_0=\|I_k(0)\|, \quad J_0=\|J_k(0)\|,\quad  k=1,2,\dots, m.
$$
\end{itemize}

\begin{definition} \label{def3.1} \rm  
 A continuous solution $x(\cdot)$ of the integral equation
\begin{equation} \label{e3.4}
\begin{aligned}
x(t) &= B^{-1}S(t)[By_0+f(0,x(0),x'(0))]+B^{-1}C(t)Bx_0 \\
&\quad -\int_0^t B^{-1}C(t-s)f(s,x(s),x'(s))ds \\
&\quad+\int^t_0  B^{-1}S(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds +\int^t_0  B^{-1}S(t-s)Gu(s)ds\\
 &\quad+\int^t_0  B^{-1}S(t-s)\big[g(s,x(s),x'(s))+\int_0^sk(s,\tau,x(\tau),x'(\tau))d\tau\big]ds\\
 &\quad +\sum_{0<t_k<t}B^{-1}C(t-t_k)I_k(x(t_k),x'(t_k))\\
&\quad +\sum_{0<t_k<t}B^{-1}S(t-t_k)J_k(x(t_k),x'(t_k))
\end{aligned}
\end{equation}
is said to be a mild solution of  \eqref{e1.1}-\eqref{e1.3} on $\mathcal{I}$.
\end{definition}

 If $x(\cdot)$ is a mild solution of  \eqref{e1.1}-\eqref{e1.3}, then by the 
properties of a second order differential equation and Lemma \ref{lem2.2}, we have
\begin{equation} \label{e3.5}
\begin{aligned}
x'(t) 
&= B^{-1}C(t)[By_0+f(0,x(0),x'(0))]+B^{-1}AS(t)Bx_0 \\
&\quad -B^{-1}f(t,x(t),x'(t))
 -\int_0^t B^{-1}AS(t-s)f(s,x(s),x'(s))ds  \\
&\quad + \int^t_0 B^{-1}C(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds\\
&\quad +\int^t_0  B^{-1}C(t-s)\big[g(s,x(s),x'(s))
 +\int_0^sk(s,\tau,x(\tau),x'(\tau))d\tau\big]ds\\
&\quad +\int^t_0  B^{-1}C(t-s)Gu(s)ds
 +\sum_{0<t_k<t}B^{-1}AS(t-t_k)I_k(x(t_k),x'(t_k))\\
&\quad +\sum_{0<t_k<t}B^{-1}C(t-t_k)J_k(x(t_k),x'(t_k)), \quad t\in \mathcal{I}.
\end{aligned}
\end{equation}

\begin{theorem} \label{thm3.3}
If assumptions {\rm (H1)--(H4), (H10)--(H13)} hold, 
then system \eqref{e1.1}-\eqref{e1.3} is controllable on $\mathcal{I}$.
\end{theorem}

The proof of the above is similar to Theorem \ref{thm3.1} and hence, is omitted.

\section{Nonlocal Initial Conditions}

The study of abstract nonlocal initial value problems was initiated by 
Byszewski \cite{by2}. Because it is demonstrated
that the nonlocal problems have better effects in applications than the classical 
Cauchy problems. Several authors have discussed the nonlocal problem in
 abstract spaces \cite{KB1, KB2}. The importance of nonlocal is studied 
in \cite{KB3,  by2}. In this section we consider a second order Sobolev 
type neutral integrodifferential equations with nonlocal initial condition
\begin{align} \label{e4.1}
  &x(0)+\sum_{i=1}^{n}p(x_i)=x_0 \quad x'(0)+\sum_{i=1}^nw(x_i)=y_0
\end{align}
In addition the assumptions in Section 2 and 3, we also assume the 
following hypotheses.
\begin{itemize}
\item[(H14)] 
 The function $p, w:\mathcal {PC}(\mathcal{I}, X) \to X$ is continuous function,
and then there exist positive constants 
 $\mathscr P_{\alpha} >0, \mathscr Q_{\alpha}>0$ such that
\begin{gather*}
\|\sum_{i=1}^{n}p(x_i)\|\leq \mathscr P_{\alpha}, \quad 
 \|\sum_{i=1}^nw(x_i)\|\leq \mathscr Q_{\alpha} \\
\|\sum_{i=1}^{n}p(x_i)-\sum_{i=1}^{n}p(y_i)\|\leq \mathscr P_{\alpha}[\|x-y\|],\\
\|\sum_{i=1}^{n}w(x_i)-\sum_{i=1}^{n}w(y_i)\|\leq \mathscr Q_{\alpha}[\|x-y\|],
\end{gather*}
for $x_i,y_i \in X$, $i=1,2,\dots,n$.
\end{itemize}

\begin{definition}   \label{def4.1} \rm
 A continuous solution $x(\cdot)$ of the integral equation
\begin{equation} \label{e4.4}
\begin{aligned}
x(t) &= B^{-1}S(t)\Big[B\{y_0-\sum_{i=1}^nw(x_i)\}+f(0,x(0),x'(0))\Big] \\
&\quad +B^{-1}C(t)B[x_0-\sum_{i=1}^np(x_i)] 
 -\int_0^t B^{-1}C(t-s)f(s,x(s),x'(s))ds \\
&\quad +\int^t_0  B^{-1}S(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds \\
&\quad +\int^t_0  B^{-1}S(t-s)\big[g(s,x(s),x'(s))+\int_0^sk(s,\tau,x(\tau),x'(\tau))d\tau\big]ds\\
 &\quad +\int^t_0  B^{-1}S(t-s)Gu(s)ds+\sum_{0<t_k<t}B^{-1}C(t-t_k)I_k(x(t_k),x'(t_k))\\
 &\quad+\sum_{0<t_k<t}B^{-1}S(t-t_k)J_k(x(t_k),x'(t_k))
\end{aligned}
\end{equation}
is said to be a mild solution of \eqref{e1.1}-\eqref{e1.3} and \eqref{e4.1} 
on $\mathcal{I}$.
\end{definition}

 If $x(\cdot)$ is a mild solution of  \eqref{e1.1}-\eqref{e1.3} and \eqref{e4.1},
 then by the properties of a second order differential equation and 
Lemma \ref{lem2.2},
 we have
\begin{equation} \label{e4.5}
\begin{aligned}
x'(t) 
&= B^{-1}C(t)\Big[B\big\{y_0-\sum_{i=1}^nw(x_i)\big\}+f(0,x(0),x'(0))\Big]\\
&\quad +B^{-1}AS(t)B[x_0-\sum_{i=1}^np(x_i)] -B^{-1}f(t,x(t),x'(t)) \\
&\quad -\int_0^t B^{-1}AS(t-s)f(s,x(s),x'(s))ds \\
&\quad + \int^t_0  B^{-1}C(t-s)\int_0^s \mathscr{D}(s-\tau)x(\tau)d\tau ds
+\int^t_0 B^{-1}C(t-s)Gu(s)ds \\
&\quad +\int^t_0  B^{-1}C(t-s)\big[g(s,x(s),x'(s)) 
 +\int_0^sk(s,\tau,x(\tau),x'(\tau))d\tau\big]ds\\
 &\quad +\sum_{0<t_k<t}B^{-1}AS(t-t_k)I_k(x(t_k),x'(t_k))\\
 &\quad+\sum_{0<t_k<t}B^{-1}C(t-t_k)J_k(x(t_k),x'(t_k)), \quad t\in \mathcal{I}.
\end{aligned}
\end{equation}

\begin{theorem} \label{thm4.2}
If assumptions {\rm (H1)--(H4),(H10)--(H14)} hold, then system 
\eqref{e1.1}-\eqref{e1.3} and \eqref{e4.1} is controllable on $\mathcal{I}$.
\end{theorem}

The of the above theorem is similar to Theorem \ref{thm3.1} and hence, is omitted.

\section {Example}

Consider the partial integrodifferential equation 
\begin{gather} \label{e5.1}
\begin{aligned}
&\frac{\partial}{\partial t}\Big[z_t(t,y) -\frac{1}{2}\cos z(t,y)\Big] \\
&= \frac{\partial^2}{\partial y^2}z(t,y)+\mu(t,y)  
 + b_1(s,y)\Big(t,\frac{1}{2}e^{-t}\sin z_t(t,y)), \\
&\quad \int_0^t \sin z_s(s,y)e^{-s}{z_{s} (\sin s,y)}ds\Big) 
\int^a_0 l(t,\tau)z_{\tau}(t,y)d\tau \\
&\quad +\int^t_{-\infty}b_2(s,y)\sin z_t(s,y)ds,\quad  y \in [0,\pi],\; t\in \mathcal{I},
\end{aligned} \\
 \label{e5.2}
z(t,0)=  z(t,\pi)=0,\quad t\in \mathcal{I},\\
z(0,y)+\sum^m_{i=1}\gamma_i\Phi_{t_i}(s,y)=  z_0(y)\quad 0<y<1,\quad
 t\in \mathcal{I}; \label{e5.3}\\
\Delta z|_{t=t_k}= I_k(z(y))  =   \int^{\pi}_0 \gamma_k(y,s)\cos^2z(s,y)ds,\quad
 z\in X,  \;  1\leq k\leq p,\label{e5.4}
\end{gather}
where $\mu(t, y):\mathcal{I} \times [0,\pi]\to [0,\pi]$ is continuous on 
$0\leq y\leq \pi, t\in \mathcal{I}$ and the
constant $\gamma_i $ are small.
 Let  $X={\mathcal{L}}^2[0,\pi]$ be  endowed with the usual
norm $\| \cdot \|_{\mathcal{L}_2}$, and let $x(t)=z(t,y)$ be continuous,
\begin{gather*}
f(t,x(t),x'(t))=\frac{1}{2}\cos z(t,y), \\
g(t,x(t),x'(t)) =b_1(s,y)\Big(t,\frac{1}{2}e^{-t}\sin z_t(t,y)),
\int_0^t \sin z_s(s,y)e^{-s}{z_{s} (\sin s,y)}ds\Big), \\
\int^t_0k(t,s,x_s)ds =\int^a_0 l(t,\tau)z_{\tau}(t,y)d\tau
 +\int^t_{-\infty}b_2(s,y)\sin z_t(s,y)ds, \\
\sum^n_{i=1}p(x_i) = \sum^m_{i=1}\gamma_i\Phi_{t_i}(s,y),\\
I_i(z(x)) =\int^{\pi}_0 \gamma_k(y,s)\cos^2z(s,y)ds.
\end{gather*}
Define the operator $A:\mathcal{D}(A)\subset X \to X$ and
$E:\mathcal{D}(E)\subset X \to X$ by
\[
  Az=-z_{xx}, \quad  Ez=z-z_{xx},
\]
where each domain $\mathcal{D}(A)$ and $\mathcal{D}(E)$ is given by
\[
  \{z\in X: z,z_x \text{ are absolutely continuous, }
z_{xx} \in  X, \; z(0)=z(\pi)=0\}.
\]
Then $A$ and $E$ can be written, respectively, as
\[
  Az = \sum^{\infty}_{n=1}n^2\langle z,z_n\rangle z_n,  \;
 z\in \mathcal{D}(A),\quad
 Ez = \sum^{\infty}_{n=1}(1+n^2)\langle z,z_n\rangle z_n, \; z\in \mathcal{D}(E),
\]
where $z_n(x)=\sqrt{2/\pi}\sin (nx)$, $n=1,2,\dots $, is the orthogonal
set of vectors of $A$. Furthermore for $z\in X$, we have
\begin{gather*}
  E^{-1}z = \sum^{\infty}_{n=1}\frac {1}{1+n^2}\langle z,z_n\rangle z_n,\quad
  -AE^{-1}z = \sum^{\infty}_{n=1}\frac {-n^2}{1+n^2}\langle z,z_n\rangle z_n, \\
S(t)z =  \sum^{\infty}_{n=1}\exp\Big(\frac {-n^2 t}{1+n^2}\Big)\langle z,z_n
\rangle z_n.
\end{gather*}
Further, the linear operators 
$\mathscr{W}_1,\ \mathscr{W}_2:\mathscr{L}^2(\mathcal{I}, U)\to X$ defined by
\[
\mathscr{W}_1u=\int^{b}_{0}B^{-1}S(b-s)Gu(s)ds, \quad
\mathscr{W}_2u=\int^{b}_{0}B^{-1}C(b-s)Gu(s)ds
\]
has a bounded inverse operators and satisfies the condition (H2) and (H3).

We see that \eqref{e5.1}--\eqref{e5.4}  can be formulated abstractly as
 \eqref{e1.1}--\eqref{e1.3}.
Hence all the conditions stated in the Theorem \ref{thm3.1} are satisfied and 
it is possible choose $b_1, b_2, \gamma_i$.
Hence by the Theorem \ref{thm3.1}, equation \eqref{e5.1}--\eqref{e5.4} is controllable 
on $\mathcal{I}$.

\subsection*{Acknowledgements}
The authors would like to special thank the editor and the anonymous
referees for their valuable suggestions that led to the improvement of the article. 
B. Radhakrishan was supported by Council of Scientific and Industrial
 Research (CSIR) of India (Grant No. 25(0232)/14/EMR-II).

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\end{document}