\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 138, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/138\hfil Controllability of nonlinear systems]
{Controllability of nonlinear differential evolution systems
in a separable Banach space}
\author[B. Radhakrishnan, K. Balachandran \hfil EJDE-2012/138\hfilneg]
{Bheeman Radhakrishnan, Krishnan Balachandran} % in alphabetical order
\address{Bheeman Radhakrishnan \newline
Department of Applied Mathematics \& Computational Sciences\\
PSG College of Technology\\
Coimbatore - 641 004, Tamilnadu, India}
\email{radhakrishnanb1985@gmail.com}
\address{Krishnan Balachandran \newline
Department of Mathematics\\
Bharathiar University\\
Coimbatore - 641 004, Tamilnadu, India}
\email{kb.maths.bu@gmail.com}
\thanks{Submitted April 19, 2012. Published August 19, 2012.}
\subjclass[2000]{93B05, 47D06}
\keywords{Controllability; semilinear differential system;
evolution system;\hfill\break\indent measure of noncompactness}
\begin{abstract}
In this article, we study the controllability of semilinear evolution
differential systems with nonlocal initial conditions in a separable
Banach space. The results are obtained by using Hausdorff measure of
noncompactness and a new calculation method.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
In various fields of science and engineering, many problems that are related to
linear viscoelasticity, nonlinear elasticity and Newtonian or non-Newtonian
fluid mechanics have mathematical models. Popular models essentially fall into
two categories: the differential models and the integrodifferential models. A
large class of scientific and engineering problems is modelled by partial
differential equations, integral equations or coupled ordinary and partial
differential equations which can be described as differential equations in
infinite dimensional spaces using semigroups. In general functional differential
equations or evolution equations serve as an abstract formulation of many
partial integrodifferential equations which arise in problems connected with
heat-flow in materials with memory and many other physical phenomena.
It is well known that the systems described by partial differential equations
can be expressed as abstract differential equations \cite{p1}. These equations
occur in various fields of study and each system can be represented by different
forms of differential or integrodifferential equations in Banach spaces. Using
the method of semigroups, various solutions of nonlinear and semilinear
evolution equations have been discussed by Pazy \cite{p1} and the nonlocal
problem for the same equations has been first studied by Byszewskii
\cite{by2,by3}. There have been appeared a lot of papers concerned with the
existence of semilinear evolution equations with nonlocal conditions
\cite{l1,x1}.
Motivated by the fact that 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 unifier of these classical
problems. Included in this general formulation is a general theory for the
evolution of temperature in a solid material. In the general case, temperature
is transmitted as waves with a finite speed of propagation. Special cases
include a theory of delayed diffusion. When physical problems are simulated, the
model often takes the form of semilinear evolution equations. Such problems in
the control fluid flow can be modelled by a semilinear evolution system in a
Banach space. For actual flow, control problems leading to this kind of model
and the resulting model equation are discussed in \cite{f1}. Control theory, on
the other hand, is that branch of application-oriented mathematics that deals
with the basic principles underlying the analysis and design of control systems.
To control an object implies the influence of its behaviour so as to accomplish
a desired goal. In order to implement this influence, practitioners build
devices and their interaction with the object being controlled is the subject of
control theory. In control theory, one of the most important qualitative aspects
of a dynamical system is controllability. 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. Roughly the concept of controllability denotes the
ability to move a system around in its entire configuration space using only
certain admissible manipulations.
Controllability of linear and nonlinear systems represented by ordinary
differential equations in finite-dimensional spaces has been extensively
investigated. The problem of controllability of linear systems represented by
differential equations in Banach spaces has been extensively studied by several
authors \cite{c2}. Several papers have appeared on finite dimensional
controllability of linear systems \cite{K1} and infinite dimensional systems in
abstract spaces \cite{c1}. Of late the controllability of nonlinear systems in
finite-dimensional spaces is studied by means of fixed point principles
\cite{B1}. Several authors have extended the concept of controllability to
infinite-dimensional spaces by applying semigroup theory \cite{ch1,p1,y1,z1}.
Controllability of nonlinear systems with different types of nonlinearity has
been studied by many authors with the help of fixed point principles \cite{B2}.
Naito \cite{Na3} discussed the controllability of nonlinear Volterra
integrodifferential systems and in \cite{Na1,Na2} he studied the controllability
of semilinear systems whereas Yamamoto and Park \cite{Y1} investigated the same
problem for a parabolic equation with a uniformly bounded nonlinear term.
A standard approach is to transform the controllability problem into a fixed
point problem for an appropriate operator in a function space. Most of the above
mentioned works require the assumption of compactness of the semigroups.
Balachandran and Kim \cite{B3} pointed out that controllability results are only
true for ordinary differential systems in finite dimensional spaces if the
corresponding semigroup is compact. However, controllability results maybe true
for abstract differential systems in infinite dimensional spaces if the
compactness of the corresponding operator semigroup is dropped.
Consider the semilinear evolution differential system with nonlocal
conditions
\begin{gather}
x'(t) = A(t)x(t)+Bu(t)+ f(t, x(t)),\quad t\in J, \label{e1}\\
x(0) = g(x), \label{e2}
\end{gather}
where the state variable $x(\cdot)$ takes values in a separable Banach
space $X$ with norm $\|\cdot\|$, $A(t):D_t\subset X \to X$ generates
an evolution system $\{U(t,s)\}_{0\leq s\leq t\leq b}$ on the separable
Banach space $X$. The control function $u(\cdot)$ is given in $L^2(J, U)$,
a Banach space of admissible control functions with $U$ as a Banach
space and the interval $J=[0, b]$. The functions $ g:\mathcal{C}(J,X)\to X$
and $f:J\times X \to X$ are continuous and $B$ is a bounded linear operator
from $U$ into $X$.
In this paper, we give conditions guaranteeing the controllability for nonlocal
evolution system \eqref{e1}--\eqref{e2} without assumptions on the compactness of $f, g$ and
the evolution system $\{U(t,s)\}$ is strongly continuous. The results obtained
are based on the new calculation method which employs the technique of a measure
of noncompactness.
\section{Preliminaries}
In this section, we collect some definitions, notation, lemmas and results
which are used later. Let $(X, \|\cdot\|)$ be a real Banach space with zero
element $\theta$. Denote by $\mathbb{B}(y,r)$ the closed ball in $X$ centered at
$y$ and with radius $r$. The collections of all linear and bounded operators
from $X$ into itself will be denoted by $\mathcal{B}(X)$. If $Y$ is a subset of
$X$ we write $\overline Y, \operatorname{Conv}Y$ to denote the closure and
convex closure of $Y$ respectively.
Moreover we denote by $\mathcal{F}_X$ the family of all nonempty and bounded
subsets of $X$ and by $\mathcal{G}_X$ its subfamily consisting of relatively
compact sets.
\begin{definition}[\cite{ban1}] \rm
A function $\chi:\mathcal{F}_X\to \mathbb R_+ $ is said to be a measure
of noncompactness if it satisfies the following conditions:
\begin{enumerate}
\item[(i)] The family $\ker \chi = \{Y\in \mathcal{F}_X: \chi(Y)=0 \}$
is nonempty and $\ker \chi\subset \mathcal{G}_X$.
\item[(ii)] $Y\subset Z \Rightarrow \chi(Y)\leq \chi(Z)$.
\item[(iii)] $\chi(\operatorname{Conv}Y)=\chi(Y)$.
\item[(iv)] $\chi(\lambda Y+(1-\lambda)Z)\leq \lambda\chi(Y)+(1-\lambda)\chi(Z)$,
for $\lambda\in [0,1]$.
\item[(v)] If $\{Y_n\}_{n=1}^{\infty}$ is a sequence of nonempty, bounded
and closed subsets of $X$ such that $Y_{n+1}\subset Y_n$ $(n=1,2,\dots )$
and if $\lim_{n\to\infty} \chi(Y_n)=0$, then the intersection
$Y_{\infty}=\cap_{n=1}^{\infty}Y_n$ is nonempty and compact in $X$.
\end{enumerate}
\end{definition}
The family $\ker \chi$ defined in $(i)$ is called the \emph{kernel} of
the measure of noncompactness $\chi$.
\begin{remark}\rm \label{rmk2.1}
Let us notice that the intersection set $Y_{\infty}$ described in axiom
(v) is a member of the kernel of the measure of noncompactness $\chi$.
In fact the inequality $\chi(Y_{\infty})\leq\chi(Y_n)$, for $n=1,2,\dots$
implies that $\chi(Y_{\infty})=0$. Hence $Y_{\infty}\in \ker \chi$.
This property of the set $Y_{\infty}$ will be important in our investigations.
\end{remark}
Throughout this paper, $\{A(t): t \in\mathbb R\}$ is a family of closed linear
operators defined on a common domain $\mathcal D$ which is dense in $X$
and we assume that the linear non-autonomous system
\begin{equation} \label{e3}
\begin{gathered}
u'(t)=A(t)u(t), \quad s\leq t\leq b, \\
u(s)=x\in X,
\end{gathered}
\end{equation}
has associated evolution family of operators $\{U(t,s):0 \leq s \leq t \leq b\}$.
In the next definition, $\mathcal L(X)$ is a space of bounded linear
operators from $X$ into $X$ endowed with the uniform convergence topology.
\begin{definition} \label{def2.2}\rm
A family of operators $\{U(t,s):0 \leq s \leq t \leq b\} \subset \mathcal L(X)$
is called a evolution family of operators for $(3)$ if the following properties hold:
\begin{itemize}
\item[(a)] $U(t,s)U(s,\tau)=U(t,\tau)$ and $U(t,t)x=x$, for every $ s \leq \tau\leq t $
and all $x \in X$;
\item[(b)] For each $x \in X$, the function for $(t,s)\to U(t,s)x$ is continuous
and $U(t,s) \in \mathcal L(X)$, for every $t \geq s$, and
\item[(c)] For $0 \leq s \leq t \leq b$, the function $t\to U(t,s)$,
for $(s,t]\in \mathcal L(X)$, is differentiable with
$\frac{\partial}{\partial t}U(t,s)=A(t)U(t,s)$.
\end{itemize}
\end{definition}
The most frequently applied measure of noncompactness is defined in the
following way
$$
\beta(Y)=\inf\{r>0: Y\ \text{can be covered by a finite number
of balls with radii }r\}.
$$
The measure $\beta$ is called the Hausdorff measure
of noncompactness.
In the sequel, we work in the space $\mathcal{C}(J,X)$ consisting of all
functions defined and continuous on $J$ with values in the Banach space $X$.
The space $\mathcal{C}(J,X)$ is furnished with the standard norm
$$
\|x\|_{\mathcal{C}}=\sup\{\|x(t)\|: t \in J=[0,b]\}.
$$
To define the measure, let us fix a nonempty bounded subset $Y$
of the space $\mathcal{C}(J,X)$ and a positive number $t\in J$.
For $y\in Y$ and $\epsilon\geq 0$ denote by $\omega^t(y,\epsilon)$
the \emph{modulus of continuity} of the function $y$ on the interval $[0, t]$;
that is,
$$
\omega^t(y,\epsilon)= \sup\{\|y(t_2)-y(t_1)\|: t_1, t_2\in [0,t],\
|t_2-t_1|\leq {\epsilon}\}.
$$
Further let us put
$$
\omega^t(Y,\epsilon)= \sup\{\omega^t(y,\epsilon):y\in Y\},\quad
\omega_0^t(Y)=\lim_{\epsilon\to 0+}\omega^t(Y,\epsilon).
$$
Apart from this, put
$$
\overline{\beta}(Y)=\sup\{\beta(Y(t)): t\in J\},
$$
where $\beta$ denotes the Hausdroff measure of noncompactnesss in $X$.
Finally we define the function $\chi$ on the family
$\mathcal{F}_{\mathcal{C}(J,X)}$ by putting
$$
\chi(Y)=\omega_0^t(Y)+\overline{\beta}(Y).
$$
It may be shown that the function $\chi$ is the measure of noncompactness
in the space $\mathcal{C}(J,X)$ (see \cite{ban2,ban1}).
The kernel $\ker \chi$ is the family of all nonempty and bounded
subsets $Y$ such that functions belonging to $Y$ are equicontinuous on $J$
and the set $Y(t)$ is relatively compact in $X$, for $t\in J$.
Next, for a given set $Y\in \mathcal{F}_{\mathcal{C}(J,X)}$, let us denote
\begin{gather*}
\int _0^t Y(s)ds=\Big\{\int_0^t y(s)ds: y\in Y\Big\},\quad t\in J,\\
Y([0,t])=\{y(s): y\in Y,\ s\in[0,t]\}.
\end{gather*}
\begin{lemma}[\cite{h1}] \label{lem2.1}
If the Banach space $X$ is \emph{separable} and a set $Y\subset \mathcal{C}(J,X)$
is bounded, then the function $t\to\beta(Y(t))$ is measurable and
$$
\beta\Big(\int _0^t Y(s)ds\Big)\leq \int _0^t\beta(Y(s))ds, \quad
\text{for each } t\in J.
$$
\end{lemma}
\begin{remark}\rm \label{rmk2.2}
Observe that in the above lemma we do not require the equicontinuity of
functions from the set $Y$.
\end{remark}
\begin{lemma} \label{lem2.2}
Assume that a set $Y\subset \mathcal{C}(J,X)$ is bounded. Then
\begin{equation} \label{e4}
\beta(Y([0,t]))\leq\omega_0^t(Y)+\sup_{s\leq t}\beta(Y(s)),\quad
\text{for } t\in J.
\end{equation}
\end{lemma}
\begin{proof}
Let $\delta>0$ be arbitrary. Then there exists $\epsilon>0$ such that
\begin{equation} \label{e5}
\omega^t(Y,\epsilon)\leq \omega_0^t(Y)+\delta/2.
\end{equation}
Let us take a partition $0=t_0 0,\ N_0 >0$
such that
$$
\|U(t,s)\|\leq N_1, \ \ \ \text{ for}\ \ 0\leq s\leq t\leq b,
$$
and $N_0=\sup\{\|U(s,0)\|:0\leq s\leq t \}$.
\item [(H2)] The linear operator $W:L^2(J, U)\rightarrow X$ defined by
$$
Wu=\int^{b}_{0}U(b,s)Bu(s)ds
$$
has an inverse operator $W^{-1}$ which takes values in $L^2(J, U)/\ker W$
and there exists a positive constant $K_1$ such that $\|BW^{-1}\|\leq K_1$.
\item [{(H3)}]
\begin{itemize}
\item[(i)] The mapping $f:J\times X\to X$ satisfies the Carathe\'odory condition,
that is, $f(\cdot, x)$ is measurable for $x\in X$ and $f(t, \cdot)$ is continuous
for a.e. $t\in J$.
\item[(ii)] The mapping $f$ is bounded on bounded subsets of $\mathcal{C}(J,X)$.
\item[(iii)] There exists a constant $m_f>0$ such that, for any bounded set
$Y\subset \mathcal{C}(J,X)$, the inequality
$$
\beta(f([0,t]\times Y))\leq m_f\beta(Y([0,t]))
$$
holds for $t\in J$, where $f([0,t]\times Y)=\{f(s,x(s)):0\leq s\leq t,\ x\in Y \}$.
\end{itemize}
\item [(H4)] The function $g:\mathcal{C}(J,X)\to X$ is continuous and there
exists a constant $m_g\geq 0$ such that
$$
\beta(g(Y))\leq m_g\beta(Y(J)),
$$
for each bounded set $Y\subset \mathcal{C}(J,X)$.
\item [(H5)] There exists a constant $r>0$ such that
$$
(1+bN_1K_1)\Big[N_0\sup_{x\in \mathbb{B}(\theta, r)}\|g(x)\| + N_1\sup_{x\in \mathbb{B}(\theta, r)}\int_0^b \|f(\tau,x(\tau))\|d{\tau}\Big] + bN_1K_1\|x_1\|\leq r,$$ for $t\in J$,
where $\mathbb{B}(\theta, r)$ is a closed ball in $\mathcal{C}(J,X)$ centered at $\theta$ and with radius $r$.
\item [{(H6)}]
\begin{align*}
&\min\{3m_g N_0(b)+3m_f b N_0(b)+ 3bm_g N_0(b)N_1(b)K_1\\
&+2m_f b^2 N_0(b)N_1^2(b)K_1\}<1.
\end{align*}
\end{itemize}
\begin{definition}[\cite{r1,r2}] \label{def2.4} \rm
System \eqref{e1}--\eqref{e2} is said to be \emph{controllable} on the interval
$J$, if for every initial functions $x_0 \in X$ and $x_1 \in X$, there
exists a control $u \in L^2(J, U)$ such that the solution $x(\cdot)$
of \eqref{e1}--\eqref{e2} satisfies $x(0)=x_0$ and $x(b)=x_1$.
\end{definition}
\section{Controllability Result}
Mathematical control theory is the area of application oriented mathematics
that deals with the basic principles underlying the analysis and design
of control systems. To control an object means to influence its behavior
so as to achieve a desired goal. In this section, we study the controllability
results for the semilinear differential system \eqref{e1}--\eqref{e2}.
Using (H2) for an arbitrary function $x(\cdot)\in \mathcal{C}(J,X)$,
we define the control
\begin{equation} \label{e9}
u(t) = W^{-1} \Bigl[x_1-U(b,0)g(x)-\int_0^b U(b,s)f\big(s,x(s)\big)ds \Bigr](t).
\end{equation}
Consider the Banach space $\mathcal Z=\mathcal{C}(J,X)$ with norm
$\|x\|=\sup\{|x(t)|: t \in J\}$.
We shall show that when using the control $u(t)$, the operator
$\Psi:\mathcal Z\to\mathcal Z $ defined by
\begin{align*}
(\Psi x)(t) &= U(t,0)g(x)+\int_0^t U(t,s) f\big(s,x(s)\big)ds \\
&\quad+\int_0^t U(t,s)BW^{-1} \Bigl[x_1-U(b,0)g(x)
-\int_0^b U(b,s) f\big(s,x(\tau)\big)d{\tau}\Bigr](s)ds
\end{align*}
has a fixed point $x(\cdot)$. This fixed point is a mild solution of
the system \eqref{e1}--\eqref{e2} and this implies that the system
is controllable on $J$.
Next consider the operators $v_1,\ v_2,\ v_3:\mathcal{C}(J,X)\to \mathcal{C}(J,X)$ defined by
\begin{gather*}
(v_1x)(t)= U(t,0)g(x),\\
(v_2x)(t)= \int_0^t U(t,s) f\big(s,x(s)\big)ds,\\
(v_3x)(t)= \int_0^t U(t,s)BW^{-1} \Bigl[x_1-U(b,0)g(x)
-\int_0^b U(b,\tau) f\big(\tau,x(\tau)\big)d{\tau}\Bigr](s)ds.
\end{gather*}
\begin{lemma} \label{lem3.1}
Assume that {\rm (H1), (H3)} are satisfied and a set $Y\subset \mathcal{C}(J,X)$
is bounded. Then
$$
\omega_0^t(v_2Y)\leq 2bN_1\beta(f([0,b]\times Y)),\quad \text{for } t\in J.
$$
\end{lemma}
\begin{proof}
Fix $t\in J$ and denote $Q=f([0,t]\times Y)$,
$$
q^t(\epsilon)=\sup\Big\{\|(U(t_2,s)-U(t_1,s))q\|:0\leq s \leq t_1\leq t_2\leq t,\;
t_2-t_1\leq\epsilon,\ q\in Q\Big\}.
$$
At the beginning, we show that
\begin{equation} \label{e10}
\lim_{\epsilon\to 0+}q^t(\epsilon)\leq 2N_1\beta(Q).
\end{equation}
Suppose the contrary. Then there exists a number $d$ such that
\begin{equation} \label{e11}
\lim_{\epsilon\to 0+}q^t(\epsilon)> d > 2N_1\beta(Q).
\end{equation}
Fix $\delta > 0$ such that
\begin{equation} \label{e12}
\lim_{\epsilon\to 0+}q^t(\epsilon)>d+\delta >d > 2N_1(\beta(Q)+\delta).
\end{equation}
Condition \eqref{e11} yields that there exist sequences
$\{t_{2,n}\}, \{t_{1,n}\}, \{s_{n}\}\subset J$ and $\{q_{n}\}\subset Q$
such that $t_{2,n}\to t'$, $t_{1,n}\to t',\ s_{n}\to s $ and
\begin{equation} \label{e13}
\|(U(t_{2,n},s_n)-U(t_{1,n},s_n))q_n\|>d.
\end{equation}
Let the points $z_1,z_2,\dots ,z_k\in X$ be such that
$Q\subset \cup_{i=1}^k B(z_i, \beta(Q)+\delta)$. Then there exists a point
$z_i$ and a subsequence $\{q_{n}\}$ such that $\{q_{n}\}\in B(z_i, \beta(Q)+\delta)$;
that is,
$$
\|z_j-q_n\|\leq \beta(Q)+\delta,\ \text{for}\ n=1,2,\dots
$$
Further we obtain
\begin{align*}
&\|U(t_{2,n},s_n)q_n-U(t_{1,n},s_n)q_n\|\\
&\leq \|U(t_{2,n},s_n)q_n - U(t_{1,n},s_n)z_j\|
+\|U(t_{2,n},s_n)z_j - U(t_{1,n},s_n)z_j\|\\
&\quad\times\|U(t_{2,n},s_n)z_j-U(t_{1,n},s_n)q_n\|\\
&\leq N_1\|q_n-z_j\|+N_1\|z_j-q_n\|+\|U(t_{2,n},s_n)z_j - U(t_{1,n},s_n)z_j\|\\
&\leq 2N_1(\beta(Q)+\delta)+\|U(t_{2,n},s_n)z_j - U(t_{1,n},s_n)z_j\|.
\end{align*}
Letting $n\to \infty$ and using the properties of the evolution system
$\{U(t,s)\}$ we obtain
$$
\limsup_{n\to\infty}\|U(t_{2,n},s_n)q_n-U(t_{1,n},s_n)q_n\|
\leq 2N_1(\beta(Q)+\delta).
$$
This contradicts \eqref{e11} and \eqref{e12}.
Now fix $\epsilon>0$ and $t_1,\ t_2\in[0,t]$ such that $0\leq t_2-t_1\leq \epsilon$.
Applying (H3), we obtain
\begin{align*}
&\|(v_2x)(t_2)-(v_2x)t_1\| \\
&\leq \int_0^{t_1} \|(U(t_2,s)-U(t_1,s))f(s,x(s))\|ds
+\int_{t_1}^{t_2} \|U(t_2,s)f(s,x(s))\|ds\\
&\quad +\int_0^t\|(U(t_2,s)-U(t_1,s))f(s,x(s))\|ds
+\epsilon N_1\sup\{\|f(s,x(s))\|: x\in Y\}.
\end{align*}
Hence we derive the inequality
\begin{align*}
\omega^t(v_1Y, \epsilon)
&\leq \sup\Big\{ \int_0^t \|(U(t_2,s)-U(t_1,s))f(s,x(s))\|ds:
t_1,t_2\in [0,t],\\
&\quad 0\leq t_2-t_1\leq \epsilon,\ x\in Y \Big\}
+\epsilon N_1\sup\{\|f(s,x(s))\|: x\in Y\}.
\end{align*}
Letting $\epsilon \to 0+$, we obtain the result.
\end{proof}
\begin{lemma} \label{lem3.2}
Assume that the assumptions {\rm (H1), (H4)} are satisfied and a set
$Y\subset\mathcal{C}(J,X)$ is bounded. Then
$$
\omega_0^t(v_1 Y)\leq 2N_0(t)\beta(g(Y)),\quad \text{for } t\in J.
$$
\end{lemma}
The proof of the above lemma simple and is omitted.
\begin{lemma} \label{lem3.3}
Assume that the assumptions {\rm (H1)--(H4)} are satisfied and a set
$Y\subset\mathcal{C}(J,X)$ is bounded. Then
$$
\omega_0^t(v_3 Y)\leq 2bN_1K_1\Big(\|x_1\|+N_0\beta(g(Y))
+bN_1\beta(f(Q))\Big),\quad \text{for } t\in J.
$$
\end{lemma}
\begin{proof} As in the Lemmas \ref{lem3.1} and \ref{lem3.2}, also
fix $\epsilon>0$ and $t_1, t_2\in[0,t]$, $0\leq t_2-t_1\leq \epsilon$.
Applying (H3) and (H4), we obtain
\begin{align*}
&\|(v_3x)(t_2)-(v_3x)(t_1)\|\\
&\leq \int_0^{t_1} \Big\|(U(t_2,s)-U(t_1,s))BW^{-1}
\Big[x_1-U(b,0)g(x)-\int_0^bU(b,\tau)f(\tau,x(\tau))d\tau\Big]\Big\|ds\\
&\quad +\int_{t_1}^{t_2} \Big\|U(t_2,s))BW^{-1}\Big[x_1-U(b,0)g(x)
-\int_0^b U(b,\tau)f(\tau,x(\tau))d\tau\Big]\Big\|ds\\
&\leq K_1\int_0^{t_1} \|U(t_2,s)-U(t_1,s)\|\Big[\|x_1\| + \|U(b,0)g(x)\| \\
&\quad + \int_0^b \|U(b,\tau)f(\tau,x(\tau))d\tau\|\Big]ds\\
&\quad +\epsilon K_1N_1\Big[\|x_1\|+N_0\sup\{\|g(x)\|:x\in Y\}
+N_1\sup\{\|f(s,x(s))\|: x\in Y\}\Big].
\end{align*}
Hence we derive the inequality
\begin{align*}
&\omega_0^t(v_3 Y)\\
&\leq \sup\Big\{K_1\int_0^{t_1} \|U(t_2,s)-U(t_1,s)\|\Big[\|x_1\|
+ \|U(b,0)g(x)\|\\
&\quad + \int_0^b \|U(b,\tau)f(\tau,x(\tau))d\tau\|\Big]ds: t_1,t_2\in[0,b],
\ 0\leq t_2-t_1\leq \epsilon, x\in Y\Big\}\\
&\quad +\epsilon K_1N_1\Big[\|x_1\|+N_0\sup\{\|g(x)\|:x\in Y\}+N_1\sup\{\|f(s,x(s))\|: x\in Y\}\Big].
\end{align*}
Letting $\epsilon\to 0+$, we obtain
$$
\omega_0^t(v_3 Y)\leq 2bN_1K_1\Big(\|x_1\|+N_0\beta(g(Y))+b N_1\beta(f(Q))\Big).
$$
The proof is complete.
\end{proof}
Our main result is as follows.
\begin{theorem} \label{thm3.1}
If the Banach space $X$ is separable and assumptions {\rm (H1)-(H4)}
are satisfied then system \eqref{e1}--\eqref{e2} is controllable on $J$.
\end{theorem}
\begin{proof} Consider the operator $\mathcal{P}$ defined by
\begin{align*}
(\mathcal{P} x)(t) &= U(t,0)g(x)+\int_0^t U(t,s) f\big(s,x(s)\big)ds \\
&\quad +\int_0^t U(t,s)BW^{-1} \Bigl[x_1-U(b,0)g(x)
-\int_0^b U(b,s) f\big(s,x(\tau)\big)d{\tau}\Bigr](s)ds.
\end{align*}
For an arbitrary $x \in \mathcal{C}(J,X)$ and $t\in J$, we obtain
\begin{align*}
\|(\mathcal{P} x)(t)\|
&\leq N_0\|g(x)\|+N_1\int_0^t \|f(s,x(s))\|ds\\
&\quad +N_1K_1\int_0^t\Big[\|x_1\|+N_0\|g(x)\|
+N_1\int_0^b \|f(\tau,x(\tau))\|d{\tau}\Big]ds \\
&\leq (1+bN_1K_1)\Big[N_0\|g(x)\|+N_1\int_0^b \|f(\tau,x(\tau))\|d{\tau}\Big]
+bN_1K_1\|x_1\|.
\end{align*}
From the above estimate and assumption (H5) we infer that there exists a
constant $r>0$ such that the operator $\mathcal{P}$ transforms
closed ball $\mathbb{B}$ into itself.
Now we prove that the operator $\mathcal{P}$ is continuous on
$\mathbb{B}(\theta, r)$.
Let us fix $x\in \mathbb{B}(\theta, r)$ and take an arbitrary sequence
$\{x_n\}\in \mathbb{B}(\theta, r)$ such that $x_n\to x$ in $\mathcal{C}(J,X)$.
Next we have
\begin{align*}
&\|\mathcal{P} x_n-\mathcal{P} x\|\\
&\leq N_0\|g(x_n)-g(x)\|+N_1\int_0^t\|f(s,x_n(s))-f(s,x(s))\|ds\\
&\quad +K_1\int_0^t \|U(t,s)\|\Big[N_0\|g(x_n)-g(x)\|
+ N_1 \int_0^b \|f(\tau,x_n(\tau)) - f(\tau,x(\tau))\|d{\tau}\Big]ds\\
&\leq (1+bN_1K_1)\Big[N_0\|g(x_n)-g(x)\| + N_1 \int_0^b \|f(\tau,x_n(\tau))
- f(\tau,x(\tau))\|d{\tau}\Big].
\end{align*}
Applying Lebesgue dominated convergence theorem, we derive that $\mathcal{P}$
is continuous on $\mathbb{B}(\theta, r)$.
Now we consider the sequence of sets $\{\Omega_n\}$ defined by induction as follows:
$$
\Omega_0 =\mathbb{B}(\theta, r),\ \Omega_{n+1}
= \operatorname{Conv}(\mathcal{P} \Omega_n ),\quad \text{for }n=1,2,\dots .
$$
This sequence is decreasing; that is,
$\Omega_n\supset \Omega_{n+1}$, for $n=1,2,\dots$.
Further let us put
$$
v_n(t)=\beta(\Omega_n([0,t])),\quad
w_n(t)=\omega_0^t(\Omega_n).
$$
Observe that each of the functions $v_n(t)$ and $w_n(t)$ is nondecreasing,
while sequences $\{v_n(t)\}$ and $\{w_n(t)\}$ are non-increasing at
any fixed $t\in J$. Put
$$
v_{\infty}(t)=\lim_{n\to\infty}v_n(t),\quad
w_{\infty}(t)=\lim_{n\to\infty}w_n(t),\quad \text{for } t\in J.
$$
Using Lemmas \ref{lem2.2}, \ref{lem3.2} and (H4), we obtain
\begin{align*}
\beta(v_1\Omega_n([0,t]))
&\leq \omega_0^t(v_1\Omega_n)+\sup_{s\leq t}\beta(v_1\Omega_n(s))\\
&\leq 2N_0(t)\beta(g(\Omega_n))+\sup_{s\leq t}N_0(s)\beta(g(\Omega_n))\\
&\leq 3N_0(t)\beta(g(\Omega_n))\\
&\leq 3m_gN_0(t)\beta(\Omega_n([0,b]))\\
&=3m_gN_0(t)v_n(b);
\end{align*}
that is,
\begin{equation} \label{e14}
\beta(v_1\Omega_n([0,t]))\leq3m_g N_0(t)v_n(b).
\end{equation}
Moreover,
\begin{align*}
\beta(v_2\Omega_n([0,t]))
&\leq \omega_0^t(v_2\Omega_n)+\sup_{s\leq t}\beta(v_2\Omega_n(s))\\
&\leq 2bN_1(t)\beta(f([0,t]\times \Omega_n ))+\sup_{s\leq t}\beta
\Big(\int_0^s U(s,\tau)f(\tau,\Omega_n(\tau))d\tau\Big)\\
&\leq 2m_f b N_1(t)\beta(\Omega_n([0,t]))+\sup_{s\leq t}N_1(t)
\int_0^s\beta(f(\tau,\Omega_n(\tau)))d\tau\\
&\leq 2m_fb N_1(t)v_n(t)+m_f N_1(t)\int_0^{t}v_n(\tau)d\tau
\end{align*}
and
\begin{align*}
&\beta(v_3\Omega_n([0,t]))\\
&\leq \omega_0^t(v_3\Omega_n)+\sup_{s\leq t}\beta(v_3\Omega_n(s))\\
&\leq 2bN_1(t)K_1\Big(\|x_1\|+N_0\beta(g(\Omega_n))+bN_1\beta(f(Q))\Big)\\
&\quad +\sup_{s\leq t}\beta\Big\{ \int_0^t U(t,s)BW^{-1}
\Bigl[x_1 - U(b,0)g(\Omega_n) \\
&\quad - \int_0^b U(b,\tau) f\big(\tau,\Omega_n(\tau)\big)d{\tau}\Bigr](s)ds \Big\}\\
&\leq 2bN_1(t)K_1\Big(\|x_1\|+N_0(t)\beta(g(\Omega_n))+bN_1(t)\beta(f([0,t]\times \Omega_n )))\Big\}\\
&\quad +\sup_{s\leq t}bN_1(s)K_1\Big\{ \|x_1\|+N_0\beta(g(\Omega_n))+ N_1(t)\int_0^s\beta(f(\tau,\Omega_n(\tau)))d\tau\Big\}\\
&\leq 3bN_1(t)K_1\Big(\|x_1\|+m_gN_0(t)v_n(b)\Big)
+ bm_f N_1(t)K_1\Big(2 b N_1(t) v_n(t) \\
&\quad + N_1\int_0^{t} v_n(\tau)d\tau\Big).
\end{align*}
Linking this estimate with \eqref{e13}, we obtain
\begin{align*}
v_{n+1}(t)
&= \beta(\Omega_{n+1}([0,t]))\\
&= \beta(\mathcal{P}\Omega_n([0,t]))\\
&\leq \beta(v_1\Omega_n([0,t]))+\beta(v_2\Omega_n([0,t]))+\beta(v_3\Omega_n([0,t]))\\
&\leq 3m_gN_0(t)v_n(b)+2m_f b N_1(t)v_n(t)+m_f N_1(t)\int_0^{t}v_n(\tau)d\tau\\
&\quad +3bN_1(t)K_1\Big(\|x_1\|+m_gN_0(t)v_n(b)\Big)\\
&\quad + bm_f N_1(t)K_1\Big(2 b N_1(t) v_n(t) + N_1\int_0^{t} v_n(\tau)d\tau\Big).
\end{align*}
Letting $n\to \infty$, we obtain
\begin{align*}
v_{\infty}(t)
&\leq 3m_gN_0(t)v_{\infty}(b)+2m_fb N_1(t)v_{\infty}(t)
+m_f N_1(t)\int_0^{t}v_{\infty}(\tau)d\tau\\
&\quad +bm_f N_1(t)K_1\Big(2 b N_1(t) v_{\infty}(t)
+ N_1\int_0^{t} v_{\infty}(\tau)d\tau\Big)\\
&\quad + 3bm_gN_0(t)N_1(t)K_1v_{\infty}(b).
\end{align*}
Hence putting $t=b$, in view of (H6), we obtain
\begin{equation} \label{e15}
v_{\infty}(b)=0.
\end{equation}
Furthermore, applying Lemmas \ref{lem3.1}, \ref{lem3.2}, \ref{lem3.3}, we have
\begin{align*}
&w_{n+1}(t)\\
&= \omega_0^t(\Omega_{n+1})\\
&= \omega_0^t(\mathcal{P} \Omega_n)\\
&\leq \omega_0^t(v_1\Omega_n)+\omega_0^t(v_2\Omega_n)+\omega_0^t(v_3\Omega_n)\\
&\leq 2m_gN_0 v_n(b)+2m_f b N_1 v_n(t)+2bN_1K_1\Big(\|x_1\|
+N_0\beta(g(Y))+b N_1\beta(f(Q))\Big)\\
&\leq 2m_gN_0 v_n(b)+2m_f b N_1 v_n(t)+2bN_1K_1\Big(\|x_1\|+m_g N_0 v_n(b)
+b m_f N_1 v_n(t)\Big)\\
&\leq (2+bN_1K_1)[m_gN_0 v_n(b)+m_f b N_1v_n(t)].
\end{align*}
Letting $n\to \infty$, we obtain
$$
w_{\infty}(t)\leq(2+bN_1K_1)[m_g N_0 v_{\infty}(b)+m_f b N_1v_{\infty}(t)].
$$
Putting $t=b$ and applying \eqref{e15}, we conclude that $w_{\infty}(b)=0$.
This fact together with \eqref{e15} implies that
$\lim_{n\to \infty}\chi(\Omega_n)=0$. Hence, in view of the
Remark \ref{rmk2.1},
we deduce that the set $\Omega_{\infty}=\cap_{n=0}^{\infty}\Omega_n$ is nonempty,
compact and convex. Finally. linking the above obtained facts concerning the
set $\Omega_{\infty}$ and the operator
$\mathcal{P}:\Omega_{\infty}\to\Omega_{\infty}$ and using the classical
Schauder fixed point theorem, we infer that the operator $\mathcal{P}$
has at least one fixed point $x$ in the set $\Omega_{\infty}$.
Obviously the function $x=x(t)$ is a mild solution of \eqref{e1}--\eqref{e2}
satisfying $x(b)=x_1$. Hence the given system is controllable on $J$.
\end{proof}
\begin{remark}\rm \label{rmk3.1}
Let us consider the case when the mapping $g$ is
$$
g(x)= \sum_{i=1}^n d_i x(t_i),
$$
where $0\le t_1