\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 36, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2014/36\hfil Controllability without uniqueness]
{Controllability for semilinear functional differential equations
without uniqueness}
\author[T. D. Ke \hfil EJDE-2014/36\hfilneg]
{Tran Dinh Ke} % in alphabetical order
\address{Tran Dinh Ke \newline
Department of Mathematics, Hanoi National University of Education,
136 Xuan Thuy, Cau Giay, Hanoi, Vietnam}
\email{ketd@hnue.edu.vn}
\thanks{Submitted June 27, 2013. Published January 29, 2014.}
\subjclass[2000]{93B05, 93C10, 93C25, 47H04, 47H08, 47H10}
\keywords{Functional differential equation; reachable set; condensing map;
\hfill\break\indent non-convex valued multimap; measure of noncompactness;
approximate controllability;
\hfill\break\indent AR-space; ANR-space; $R_\delta$-map}
\begin{abstract}
We study the controllability for a class of semilinear control problems
in Hilbert spaces, for which the uniqueness is unavailable.
Using the fixed point theory for multivalued maps with nonconvex values,
we show that the nonlinear problem is approximately controllable provided
that the corresponding linear problem is. We also obtain some results on
the continuity of solution map and the topological structure of the solution
set of the mentioned problem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
We are concerned with the control problem
\begin{gather}
x'(t) = Ax(t) + F(t,x(t), x_t) + Bu(t),\quad t\in J:=[0,T],\label{e1}\\
x(s) = \varphi(s),\quad s\in [-h,0]\label{e2}
\end{gather}
where the state function $x(t)$ takes values in a Hilbert space $X$, $u(t)$ belongs to
a Hilbert space $V$, $F: J\times X\times C([-h,0];X)\to X$ is a
nonlinear mapping, $A$ is the generator of a strongly continuous
semigroup $\{S(t)\}_{t\ge 0}$ in $X$ and $B: L^2(J;V)\to L^2(J;X)$
is a bounded linear operator. The history $x_t$ is defined by $x_t(s):=x(t+s)$
for $s\in [-h,0]$.
The controllability problems governed by differential equations have been
presented in a great work of literature. In dealing with control systems
involving abstract differential equations in Banach spaces, semigroup
theory has been an effective tool. Following this approach, one can find
the comprehensive researches in the monographs \cite{Barbu, BPDM,EnNa,HuPa}.
To solve nonlinear control problems like \eqref{e1}-\eqref{e2}, a typical
assumption used in many works (see, e.g. \cite{ChangLi, CLN,Liu,OZ})
is that the operator
\begin{equation}\label{eas}
\mathcal W u = \int_0^T S(T-s)Bu(s)ds\quad \text{has a pseudo-inverse}
\end{equation}
(see \cite{QuiCar} for the definition). This requires $\mathcal W$ to be
surjective. In \cite{OZ}, the authors pointed out an affirmative example,
in which the semigroup $S(\cdot)$ is not compact. However, for many application
models, in which $X$ is infinite dimensional and the semigroup $S(\cdot)$
is compact (or $B$ is compact), the operator $\mathcal W$ is not surjective
(see \cite{Trig1,Trig2} for details). Then \eqref{eas} has been seen as a
too strong assumption.
In \cite{Naito2}, Naito introduced an approach to study the approximate
controllability for control systems in Hilbert spaces.
The goal is to find suitable conditions ensuring the positive invariance
of reachable set of control systems under nonlinear perturbations; that is
\begin{equation}\label{invar}
\mathcal R_T(0)\subset \overline{\mathcal R_T(F)},
\end{equation}
where $\mathcal R_T(F)$ and $\mathcal R_T(0)$ are the reachable sets of
the nonlinear system and corresponding linear system, respectively.
Using a similar approach, there were some works,
see e.g. \cite{JKP,KS,Naito1,Wang}, that proved the controllability results
for numerous semilinear differential and integro-differential equations.
A crucial assumption in these works is that, control systems have a unique
solution and the solution operator $ W(u) = x(\cdot;u) $ is compact.
This is usually obtained by the compactness of the semigroup $S(\cdot)$
and the Lipschitz continuity of nonlinearity. Then some well-known fixed
point theorems, such as the Schauder fixed point theorem and Banach
contraction principle, were employed to demonstrate \eqref{invar}.
In this article, we address the case when the uniqueness of
\eqref{e1}-\eqref{e2} is unavailable and this turns out that $W$, in general,
becomes a multivalued mapping. This fact leads to a difficulty in
proving \eqref{invar}, namely, fixed point theory for singlevalued maps does
not work in this situation. To overcome the obstacle, we will show that $W$
is an $R_\delta$-map. This property enables us to deploy the fixed point
theory for nonconvex valued maps to obtain the inclusion
$\mathcal R_T(0)\subset \mathcal R_T(F)$, that derives the controllability
results.
The rest of our paper is as follows. In the next section, we prove the
existence result in general case. Specifically, for the nonlinearity,
the Lipschitz condition is replaced by a regular assumption expressed by
measures of noncompactness. Sect. 3 shows the result describing the structure
of the solution set. In fact, the solution set is compact and, moreover,
it is an $R_\delta$-set. This enable us to prove the controllability results
in Sect. 4. Finally, we give a simple example to show that our approach allows
relaxing the Lipschitz condition and uniform boundedness of the nonlinearity.
\section{Existence results}
In this section, we consider system \eqref{e1}-\eqref{e2} under some regular
conditions which guarantee the existence but not provide the uniqueness of
solutions. To show the existence results, we make use of the fixed point
theory for condensing maps (see e.g. \cite{KaObZe}).
Let $\mathcal{E}$ be a Banach space. Denote by $\mathcal B(\mathcal E)$ the
collection of nonempty bounded subsets of $\mathcal E$. We will use the
following definition of measure of noncompactness.
\begin{definition}\label{def:mnc} \rm
Let $(\mathcal{A},\leq)$ be a partially ordered set. A function
$\beta: \mathcal{B}(\mathcal{E})\to \mathcal{A}$ is called a measure of
noncompactness (MNC) in $\mathcal{E}$ if
$$
\beta (\overline{{\rm co}}\;\Omega) = \beta(\Omega) \quad
\text{for }\Omega\in \mathcal{P}(\mathcal{E}),
$$
where $\overline{{\rm co}}\;\Omega$ is the closure of the convex hull of
$\Omega$. An MNC $\beta$ is called
\begin{itemize}
\item[(i)] monotone, if $\Omega_0,\Omega_1\in\mathcal{P}(\mathcal{E})$,
$\Omega_0\subset\Omega_1$ implies $\beta(\Omega_0)\le \beta(\Omega_1)$;
\item[(ii)] nonsingular, if $\beta(\{a\}\cup \Omega) = \beta(\Omega)$ for any
$a\in\mathcal{E},\Omega\in \mathcal{P}(\mathcal{E})$;
\item[(iii)] invariant with respect to union with compact sets, if
$\beta(K\cup \Omega) = \beta(\Omega)$ for every relatively compact set
$K\subset\mathcal{E}$ and $\Omega\in\mathcal{P}(\mathcal{E})$;
\end{itemize}
If $\mathcal{A}$ is a cone in a normed space, we say that $\beta$ is
\begin{itemize}
\item[(iv)] algebraically semi-additive, if
$\beta(\Omega_0+\Omega_1)\le \beta(\Omega_0) + \beta(\Omega_1)$
for any $\Omega_0,\Omega_1\in \mathcal{P}(\mathcal{E})$;
\item[(v)] regular, if $\beta(\Omega)=0$ is equivalent to the relative
compactness of $\Omega$.
\end{itemize}
\end{definition}
An important example of MNC is {\it the Hausdorff} MNC $\chi(\cdot)$,
which is defined as follows
$$
\chi (\Omega) = \inf \{\varepsilon: \Omega \text{ has a finite $\varepsilon$-net}\}.
$$
We also make use of the following MNCs: for any bounded set
$D\in C(J;\mathcal E)$,
\begin{itemize}
\item the modulus of fiber noncompactness of $D$ is given by
\begin{equation}\label{gammaL}
\gamma_L(D) = \sup_{t\in J}e^{-Lt}\chi(D(t)),
\end{equation}
where $L$ is a positive constant.
\item the modulus of equicontinuity of $D$ is
\begin{equation}\label{modc}
\omega_C(D) = \lim\limits_{\delta\to 0}\sup_{y\in D}
\max_{|t_1-t_2|<\delta}\|y(t_1)-y(t_2)\|_{\mathcal E};
\end{equation}
\end{itemize}
As remarked in \cite{KaObZe}, these MNCs satisfy all properties stated
in Definition \ref{def:mnc}, but regularity. Consider now the function
\begin{equation}
\chi^*: \mathcal{B} (C(J;\mathcal{E}))\to {\mathbb R}_+ ^2, \quad
\chi^*(\Omega) = \max_{D\in \Delta(\Omega)} (\gamma_L (D),
\omega_C (D)),
\label{chistar}
\end{equation}
where the MNCs $\gamma_L$ and $\omega_C$ are defined in \eqref{gammaL}
and \eqref{modc} respectively, $\Delta(\Omega)$ is the collection
of all countable subsets of $\Omega$ and the maximum is taken in the
sense of the partial order in the cone ${\mathbb R}^2_+$.
By the arguments given in \cite{KaObZe}, $\chi^*$ is well defined.
That is, the maximum is achieved in $\Delta(\Omega)$ and $\chi^*$
is an MNC in the space $C(J;\mathcal{E})$, which satisfies all properties
in Definition \ref{def:mnc} (see \cite[Example 2.1.3]{KaObZe} for details).
\begin{definition}\label{def2} \rm
A continuous map $\mathcal{F}: Z\subseteq \mathcal{E}\to
\mathcal{E}$ is said to be condensing with respect to an MNC $\beta$
($\beta$-condensing) if for any bounded set $\Omega\subset Z$, the relation
$$
\beta(\Omega) \leq \beta(\mathcal{F}(\Omega))
$$
implies the relative compactness of $\Omega$.
\end{definition}
Let $\beta$ be a monotone nonsingular MNC in $\mathcal E$.
The application of the topological degree theory for condensing maps
(see, e.g., \cite{KaObZe}) yields the following fixed point principle.
\begin{theorem}\cite[Corollary 3.3.1]{KaObZe}\label{thma}
Let $\mathcal{M}$ be a bounded convex closed subset of $\mathcal{E}$
and let $\mathcal{F}: \mathcal{M}\to \mathcal{M}$ be a $\beta$-condensing
map. Then the fixed point set of $\mathcal{F}$,
$\operatorname{Fix}(\mathcal{F}):=\{x=\mathcal F(x)\}$, is a nonempty
compact set.
\end{theorem}
Now we consider the nonlinearity $F:J\times X\times C([-h,0];X)\to X$
in our problem \eqref{e1}-\eqref{e2}. Denote
$$
\|\psi\|_h = \|\psi\|_{C([-h,0];X)} := \sup_{\tau\in [-h,0]}\|\psi(\tau)\|,
$$
here $\|\cdot\|:=\|\cdot\|_X$.
We assume that, $F$ satisfies the following:
\begin{itemize}
\item[(F1)] $F(\cdot, \zeta, \psi)$ is measurable for each
$\zeta\in X, \psi\in C([-h,0];X)$ and $F(t, \cdot, \cdot)$
is continuous for each $t\in J$;
\item[(F2)] there exist functions $a, b, c\in L^1(J)$ such that
$$
\|F(t,\zeta,\psi)\| \le a(t)\|\zeta\| + b(t)\|\psi\|_h + c(t),
$$
for all $(\zeta, \psi)\in X\times C([-h,0];X)$;
\item[(F3)] there are nonnegative functions $h, k: J\times J\to \mathbb R$
such that $h(t, \cdot), k(t, \cdot)\in L^1(0,t)$ for each $t>0$ and
$$
\chi(S(t-s)F(s,\Omega, \mathcal{Q}))\le h(t,s) \chi(\Omega)
+ k(t,s)\sup_{-h\le \theta\le 0} \chi(\mathcal{Q}(\theta)),
$$
for all bounded subsets $\Omega\subset X$,
$\mathcal{Q}\subset C([-h,0];X)$ and for a.e. $t,s\in J$.
\end{itemize}
\begin{remark}\label{rm.f3} \rm
Let us give a note on assumption (F3).
In the case $X=\mathbb R^m$, (F3) can be deduced from
(F2). That is the locally bounded property implies that the set
$S(t-s)F(s,\Omega, \mathcal{Q})$ is bounded in $\mathbb R^m$ and then
it is precompact for each $t,s\in J$.
Especially, if $S(t)$ is compact for $t>0$ then (F3) is testified
obviously with $k=h=0$.
\end{remark}
\begin{definition}\label{soldef} \rm
A function $x\in C([-h,T];X)$ is called a mild solution of \eqref{e1}-\eqref{e2}
corresponding to control $u$ if
\begin{equation*}
x(t) = \begin{cases}
\varphi(t), &\text{for }t\in [-h,0],\\
S(t)\varphi(0) + \int_0^t S(t-s)[F(s, x(s), x_s)
+ Bu(s)]ds,& \text{for } t\in J.
\end{cases}
\end{equation*}
\end{definition}
Let
\[
\mathcal C_\varphi = \{y\in C(J;X):\; y(0)=\varphi(0)\},
\]
and for $y\in \mathcal C_\varphi$, we set
\begin{equation*}
y[\varphi](t) =\begin{cases}
\varphi(t),&\text{for } t\in [-h,0],\\
y(t), &\text{for } t\in J.
\end{cases}
\end{equation*}
For each $u\in L^2(J;V)$, denote by $\mathcal F_u$ the operator
acting on $\mathcal C_\varphi$ such that
\begin{equation}\label{eq:solop}
\mathcal F_u(x)(t) =
S(t)\varphi(0) + \int_0^t S(t-s)[F(s, x(s), x[\varphi]_s) + Bu(s)]ds.
\end{equation}
Define a mapping $\mathbf{S}: L^1(J;X)\to C(J;X)$ as
\begin{equation} \label{CauchyOp}
\mathbf{S}(f)(t) = \int_0^t S(t-s)f(s)ds.
\end{equation}
Additionally, putting
\begin{equation}\label{Nemytski}
N_F(x)(t) = F(t, x(t), x[\varphi]_t),
\end{equation}
we have
\begin{equation*}
\mathcal F_u (x) = S(\cdot)\varphi(0) + \mathbf{S}(N_F(x) + Bu).
\end{equation*}
It is evident that $x\in \mathcal C_\varphi$ is a fixed point of
$\mathcal F_u$ if and only if $x[\varphi]$ is a mild solution of
\eqref{e1}-\eqref{e2}.
We have the first property of the solution operator $\mathcal F_u$.
\begin{lemma}\label{lm:equi}
Let {\rm (F1)} and {\rm (F2)} hold. Then $\mathcal F_u(\{y_n\})$ is
relatively compact for all bounded sequence $\{y_n\}\subset \mathcal C_\varphi$
satisfying $\gamma_L(\{y_n\})=0$. In particular, we have
$\omega_{C}(\mathcal F_u(\{y_n\}))=0$.
\end{lemma}
\begin{proof}
We use an assertion that, if $\{f_n\}\subset L^1(J;X)$ is a semicompact sequence,
that is
\begin{itemize}
\item there exists $q\in L^1(J)$ such that $\|f_n(t)\|\le q(t)$ for all $n$
and for a.e. $t\in J$;
\item $\chi(\{f_n(t)\})=0$ for a.e. $t\in J$,
\end{itemize}
then $\mathbf{S}(f_n)$ is relatively compact in $C(J;X)$ (see \cite{KaObZe}).
Now let $\{y_n\}\subset \mathcal C_\varphi$ be a bounded sequence such that
$\gamma_L(\{y_n\})=0$. Then it follows from (F2) that,
$f_n(t) = F(t,y_n(t), y_n[\varphi]_t)$ satisfies the estimate
\begin{align*}
\|f_n(t)\|
&\le a(t)\|y_n(t)\| + b(t)(\sup_{s\in [0,t]}\|y_n(s)\|
+ \|\varphi\|_{h}) + c(t)\\
& \le q(t):=M[a(t)+b(t)] + b(t)\|\varphi\|_h + c(t),
\end{align*}
where $M$ is an upper bound for $\{y_n\}$ in $C(J;X)$.
As $\gamma_L(\{y_n\})=0$ one has $\chi(\{y_n(t)\})=0$ for all $t\in J$;
i.e. $\{y_n(t)\}$ is relatively compact for each $t\in J$.
Then it is readily seen that $\{y_n[\varphi]_t\}$ is a relatively
compact set in $C([-h,0];X)$. Since $F(t, \cdot, \cdot)$ is continuous,
we get that $\{F(t, y_n(t), y_n[\varphi]_t)$\} is relatively compact for
a.e. $t\in J$. Thus $\{f_n\}$ is a semicompact sequence and then
$\mathcal F_u(\{y_n\})=\mathbf{S}(\{f_n\}) + \mathbf{S}(Bu) + S(t)\varphi(0)$
is relatively compact in $C(J;X)$. In particular, $\mathcal F_u(\{y_n\})$
is an equicontinuous set, or equivalently, $\omega_C(\mathcal F_u(\{y_n\}))=0$.
The proof is complete.
\end{proof}
Now choosing $L$ in the definition of $\gamma_L$ in \eqref{gammaL} such that
\begin{equation}\label{ell}
\ell:= 2\sup_{t\in J}\int_0^t e^{-L(t-s)}[h(t,s) + k(t,s)] ds <1,
\end{equation}
we will show that $\mathcal F_u$ is $\chi^*$-condensing. To this aim,
we need the following assertion, whose proof can be found in \cite{KaObZe}.
\begin{proposition}\label{pp:ineqmnc}
If $\{w_n\}\subset L^1(J;X)$ such that
$\|w_n(t)\|\le \nu(t)$, for a.e. $t\in J$,
and for some $\nu\in L^1(J)$, then
$$
\chi(\{\int_0^t w_n(s)ds\}) \le 2\int_0^t \chi(\{w_n(s)\}) ds, \quad
\text{for }t\in J.
$$
\end{proposition}
\begin{lemma}\label{lm:conde}
Let {\rm (F1)--(F3)} hold. Then the solution operator $\mathcal F_u$
is $\chi^*$-condensing.
\end{lemma}
\begin{proof}
By (F1)--(F2), it is clear that $\mathcal F_u$ is a continuous mapping.
Let $\Omega\subset \mathcal C_\varphi$ be a bounded set such that
\begin{equation}\label{eq:conde0}
\chi^*(\Omega)\le \chi^*(\mathcal F_u(\Omega)).
\end{equation}
Then we show that $\Omega$ is relatively compact in $C(J;X)$.
By the definition of $\chi^*$, there exists a sequence $\{y_n\}\subset \Omega$
such that
\begin{equation}\label{eq:conde1}
\chi^*(\mathcal F_u(\Omega)) = \Big(\gamma_L(\mathcal F_u(\{y_n\})),
\omega_C(\mathcal F_u(\{y_n\}))\Big)\ge \Big(\gamma_L(\{y_n\}),
\omega_C(\{y_n\})\Big).
\end{equation}
We first give an estimate for $\gamma_L(\mathcal F_u(\{y_n\}))$.
By Proposition \ref{pp:ineqmnc} and (F3), we have
\begin{align*}
\chi(\mathcal F_u(\{y_n\})(t))
&\le \chi\Big(\{\int_0^t S(t-s)F(t, y_n(s), y_n[\varphi]_s)ds\}\Big)\\
& \le 2\int_0^t \chi(\Big\{S(t-s)F(t, y_n(s), y_n[\varphi]_s)\}\Big)ds\\
& \le 2\int_0^t [h(t,s)\chi(\{y_n(s)\}) + k(t,s)\sup_{\tau\in [-h,0]}
\chi(\{y_n[\varphi](s+\tau)\})]ds\\
& \le 2 \int_0^t [h(t,s)\chi(\{y_n(s)\}) + k(t,s)\sup_{\rho\in [0,s]}
\chi(\{y_n(\rho)\})]ds\\
& \le 2\int_0^t [h(t,s)+k(t,s)]\sup_{\rho\in[0,s]}\chi(\{y_n(\rho)\})ds.
\end{align*}
Then
\begin{align*}
e^{-Lt}\chi(\mathcal F_u(\{y_n\})(t))
& \le 2\int_0^t e^{-L(t-s)}[h(t,s)+k(t,s)]\sup_{\rho\in[0,s]}
e^{-L\rho}\chi(\{y_n(\rho)\})ds\\
&\le 2\gamma_L(\{y_n\})\int_0^t e^{-L(t-s)}[h(t,s)+k(t,s)] ds.
\end{align*}
The last inequality implies
\begin{equation*}
\gamma_L(\mathcal F_u(\{y_n\})) \le \ell \gamma_L(\{y_n\}).
\end{equation*}
Taking into account \eqref{eq:conde1}, we have
$\gamma_L(\{y_n\}) \le \ell \gamma_L(\{y_n\})$. Then
$\gamma_L(\{y_n\}) = 0$ due to the fact that $\ell <1$ as chosen in \eqref{ell}.
This turns out that $\gamma_L(\mathcal F_u(\{y_n\}))=0$ and by
Lemma \ref{lm:equi}, we have $\omega_C(\mathcal F_u(\{y_n\}))=0$.
Using \eqref{eq:conde1} again, we have $\chi^*(\mathcal F_u(\Omega))=0$.
Hence $\chi^*(\Omega)=0$ due to \eqref{eq:conde0}. The proof is complete.
\end{proof}
We are in a position to state the existence result.
\begin{theorem}\label{th:exists}
Assume hypotheses {\rm (F1)--(F3)}. Then the solution set of problem
\eqref{e1}-\eqref{e2} is nonempty and compact. In addition, any
solution of \eqref{e1}-\eqref{e2} obeys the following estimate
\begin{equation}\label{solest}
\sup_{\tau\in [0,t]}\|x(\tau)\|_{X}\le C_0(M_\varphi
+ \|Bu\|_{L^1(J;X)})\exp\{C_0\int_0^t [a(s)+b(s)]ds\}, t\in J,
\end{equation}
where $C_0 = \sup_{t\in J}\|S(t)\|_{X\to X}, M_\varphi
=\|\varphi(0)\| + \|\varphi\|_h\|b\|_{L^1(J)} + \|c\|_{L^1(J)}$.
\end{theorem}
\begin{proof}
By the hypotheses, the solution operator $\mathcal F_u$ is
$\chi^*$-condensing due to Lemma \ref{lm:conde}. Let $\xi\in C(J;X)$
be the solution of the integral equation
\begin{align*}
\xi(t) &= C_0 \|\varphi(0)\| + C_0 \|\varphi\|_h\|b\|_{L^1(J)}
+ C_0 \|c\|_{L^1(J)} \\
&\quad + C_0\|Bu\|_{L^1(J;X)} +C_0 \int_0^t [a(s)+b(s)]\xi(s)ds,
\end{align*}
and
\begin{equation*}
\mathcal M = \{y\in \mathcal C_\varphi: \sup_{s\in [0,t]}\|y(s)\|
\le \xi(t), t\in J\},
\end{equation*}
where $C_0 = \sup_{t\in J}\|S(t)\|_{L(X)}$ ($\|\cdot\|_{L(X)}$
stands for operator norm).
Then it is easy to check that $\mathcal M$ is a bounded, closed and convex set.
In addition, if $y\in\mathcal M$ then
\begin{align*}
&\|\mathcal F_u(y)(t)\| \\
&\le \|S(t)\varphi(0)\| + \int_0^t \|S(t-s)[F(s,y(s),y[\varphi]_s) + Bu(s)]\| ds\\
& \le C_0\|\varphi(0)\| + C_0\|Bu\|_{L^1(J;X)}+C_0 \int_0^t [a(s)\|y(s)\| + b(s)\|y[\varphi]_s\|_{h} + c(s)]ds\\
& \le C_0 \|\varphi(0)\| + C_0\|Bu\|_{L^1(J;X)}+ C_0 \|c\|_{L^1(J)} \\
& \quad + C_0\int_0^t [a(s)\|y(s)\| + b(s)(\sup_{\tau\in [0,s]}\|y(\tau)\| + \|\varphi\|_{h})]ds\\
& \le C_0 \|\varphi(0)\| + C_0\|Bu\|_{L^1(J;X)}+ C_0 \|c\|_{L^1(J)} + C_0\|\varphi\|_h\|b\|_{L^1(J)} \\
& \quad + C_0\int_0^t [a(s)+b(s)]\sup_{\tau\in[0,s]}\|y(\tau)\| ds\\
& \le C_0 \|\varphi(0)\| + C_0\|Bu\|_{L^1(J;X)}+ C_0 \|c\|_{L^1(J)} + C_0\|\varphi\|_h\|b\|_{L^1(J)} \\
& \quad + C_0\int_0^t [a(s)+b(s)]\xi(s) ds= \xi(t).
\end{align*}
Due to the fact that $\xi$ is increasing, we have
\begin{equation*}
\|\mathcal F_u(y)(\rho)\|\le \xi(\rho)\le \xi(t),
\end{equation*}
for all $0\le \rho\le t$. Thus $\mathcal F_u(y)\in \mathcal M$;
that is $\mathcal F_u(\mathcal M)\subset \mathcal M$.
The application of Theorem \ref{thma} yields the conclusion of existence result.
Now let $x$ be a solution of \eqref{e1}-\eqref{e2} then by the same estimate
as that for $\mathcal F_u$, we have
\begin{align*}
\|x(t)\|
& \le C_0 \|\varphi(0)\| + C_0\|Bu\|_{L^1(J;X)}+ C_0 \|c\|_{L^1(J)}
+ C_0\|\varphi\|_h\|b\|_{L^1(J)} \\
& \quad + C_0\int_0^t [a(s)+b(s)]\sup_{\tau\in[0,s]}\|x(\tau)\| ds\\
& \le C_0(M_\varphi + \|Bu\|_{L^1(J;X)}) + C_0\int_0^t
[a(s)+b(s)]\sup_{\tau\in[0,s]}\|x(\tau)\| ds.
\end{align*}
Since the right hand side is increasing with respect to $t$, one has
\begin{equation*}
\sup_{\tau\in [0,t]}\|x(\tau)\|\le C_0(M_\varphi + \|Bu\|_{L^1(J;X)})
+ C_0\int_0^t [a(s)+b(s)]\sup_{\tau\in[0,s]}\|x(\tau)\| ds.
\end{equation*}
Hence we obtain estimate \eqref{solest} by using the Gronwall inequality.
The proof is complete.
\end{proof}
\section{Topological structure of the solution set}
Let $Y$ and $Z$ be metric spaces. A multi-valued map (multimap)
$G:Y\to \mathcal{P}(Z)$ is said to be: (i) \emph{upper semi-continuous
(u.s.c.)} if the set
$$
G^{-1}_+(V)= \{y\in Y: G(y)\subset V\}
$$
is open for any open set $V\subset Z;$ (ii) \emph{closed} if its graph
$\Gamma_G \subset Y \times Z,$
$$
\Gamma_G = \{(y,z): z \in G(y)\}
$$
is a closed subset of $Y \times Z.$
The multimap $G$ is called \emph{quasi-compact} if its restriction to any
compact set is compact. The following statement gives a sufficient
condition for upper semi-continuity.
\begin{lemma}[\cite{KaObZe}] \label{lmusc}
Let $Y$ and $Z$ be metric spaces and $G: Y\to \mathcal P (Z)$ a closed
quasi-compact multimap with compact values. Then $G$ is u.s.c.
\end{lemma}
Consider \emph{the solution multimap}
\begin{equation}
W: L^2(J;V)\to \mathcal P (C(J;X)), \quad
W (u) = \{x: x = \mathcal F_u (x)\}.\label{e6}
\end{equation}
We need an additional assumption on the semigroup $S(\cdot)$:
\begin{itemize}
\item[(A1)] The semigroup $S(\cdot)$ generated by $A$ is compact,
i.e. $S(t)$ is compact for all $t>0$.
\end{itemize}
\begin{proposition}\label{pp:compactCauchyOp}
Under assumption {\rm (A1)}, the restriction of operator $\mathbf{S}$,
given by \eqref{CauchyOp}, on $L^2(J;X)$ is compact; i.e.,
if $\Omega\subset L^2(J;X)$ is a bounded set, then $\mathbf{S}(\Omega)$
is relatively compact in $C(J;X)$.
\end{proposition}
The proof of the above proposition is standard and we omit it.
To obtain further properties of the solution multimap $W$, we justify (F2)
as follows
\begin{itemize}
\item[(F2A)] The nonlinearity $F$ satisfies (F2) with $a, b, c\in L^2(J)$.
\end{itemize}
\begin{lemma}\label{lm7}
Under assumptions {\rm (F1), (F2A), (A1)}, the solution
multimap $W$ defined by \eqref{e6} is completely continuous; i.e.,
it is u.s.c. and sends each bounded set into a relatively compact set.
\end{lemma}
\begin{proof}
Let $\mathcal Q\subset L^2(J;V)$ be a bounded set. We prove that
$W(\mathcal Q)$ is relatively compact in $C(J;X)$. Suppose that
$\{x_n\}\subset W(\mathcal Q)$. Then there exists a sequence
$\{u_n\}\subset \mathcal Q$ such that
\begin{equation}
x_n(t) = S(t)\varphi(0) + \int_0^t S(t-s)[F(s,x_n(s), x_n[\varphi]_s)
+ Bu_n(s)]ds.
\end{equation}
Equivalently, we can write
\begin{equation}\label{e7}
x_n(t) = S(t)\varphi(0) + \mathbf{S} (f_n + Bu_n)(t),
\end{equation}
where $\mathbf{S}$ is the operator defined in \eqref{CauchyOp},
$f_n(t) = F(t,x_n(t), x_n[\varphi]_t)$. We observe that $\{Bu_n\}$
is a bounded set in $L^2(J;X)$ since $B$ is a bounded linear operator.
This implies, by Proposition \ref{pp:compactCauchyOp}, that
$\{\mathbf{S} (Bu_n)\}$ is relatively compact in $C(J;X)$.
On the other hand, by some standard estimates we can obtain that
$\{x_n\}$ is a bounded sequence in $C(J;X)$. This together with (F2A)
deduce that $\{f_n\}$ is also bounded in $L^2(J;X)$ and one ensures that
$\{\mathbf{S} (f_n + Bu_n)\}$ is compact. In view of \eqref{e7},
we conclude that $\{x_n\}$ is compact as well.
To prove that $W$ is u.s.c., it remains to show, according
to Lemma \ref{lmusc}, that $W$ has a closed graph. Let $u_n\to u$ in
$L^2(J;V)$ and $x_n \in W(u_n), x_n\to x$ in $C(J;X)$. We
claim that $x\in W(u)$. Indeed, one has
\begin{equation}\label{exn}
x_n(t) = S(t)\varphi(0) + \int_0^t S(t-s) [F(s, x_n(s), x_n[\varphi]_s)
+ Bu_n(s)]ds.
\end{equation}
Since $F(t,\cdot, \cdot)$ is a continuous function, we have
$f_n(s) = F(s,x_n(s), x_n[\varphi]_s)$ converging to $
f(s) = F(s,x(s), x[\varphi]_s)$ for a.e. $s\in J$. Due to the fact that
$\{f_n\}$ is integrably bounded, the Lebesgue dominated convergence theorem
implies
$$
f_n - f \to 0\quad \text{in }L^1(J;X).
$$
In addition, since $B$ is bounded, one can assert that
$$
Bu_n - Bu\to 0\quad \text{in } L^1(J;X).
$$
Therefore, taking \eqref{exn} into account, we arrive at
$$
x(t) = S(t)\varphi(0) + \int_0^t S(t-s) [F(s, x(s), x[\varphi]_s)
+ Bu(s)]ds, t\ge 0.
$$
The proof is complete.
\end{proof}
Let us recall some notions which will be used in the sequel.
\begin{definition} \rm
A subset $B$ of a metric space $Y$ is said to be contractible in $Y$
if the inclusion map $i_B: B\to Y$ is null-homotopic; i.e., there
exists $y_0\in Y$ and a continuous map $h: B\times [0,1]\to Y$ such
that $h(y,0) = y$ and $h(y,1) = y_0$ for every $y\in B$.
\end{definition}
\begin{definition} \rm
A subset $B$ of a metric space $Y$ is called an $R_\delta$-set if
$B$ can be represented as the intersection of decreasing sequence of
compact contractible sets.
\end{definition}
A multimap $G: X\to \mathcal P(Y)$ is said to be an
$R_\delta$-map if $G$ is u.s.c. and for each $x\in X$, $G(x)$ is an
$R_\delta$-set in $Y$. Every single-valued continuous map can be
seen as an $R_\delta$-map.
The following lemma gives us a condition for a set of being $R_\delta$.
\begin{lemma}[\cite{BG}] \label{rdelta}
Let $X$ be a metric space, $E$ a Banach space and $g: X\to E$ a proper map;
i.e., $g$ is continuous and $g^{-1}(K)$ is compact for each compact set
$K\subset E$. If there exists a sequence $\{g_n\}$ of mappings from
$X$ into $E$ such that
\begin{enumerate}
\item $g_n$ is proper and $\{g_n\}$ converges to $g$ uniformly on $X$;
\item for a given point $y_0\in E$ and for all $y$ in a neighborhood
$\mathcal N (y_0)$ of $y_0$ in $E$, there exists exactly one solution
$x_n$ of the equation $g_n(x) = y$.
\end{enumerate}
Then $g^{-1}(y_0)$ is an $R_\delta$-set.
\end{lemma}
To use this Lemma, we need the following result, which is
called \emph{the Lasota-Yorke Approximation Theorem} (see e.g.,
\cite{Gorniewicz}).
\begin{lemma}\label{lmapprox}
Let $E$ be a normed space and $f: X\to E$ a continuous map. Then for
each $\epsilon >0$, there is a locally Lipschitz map $f_{\epsilon}:
X\to E$ such that:
$$
\|f_{\epsilon}(x)-f(x)\|_E<\epsilon
$$
for each $x \in X$.
\end{lemma}
The following theorem is the main result in this section.
\begin{theorem}\label{th3}
Assume the hypotheses of Lemma \ref{lm7}.
Then for each $u\in L^2(J;V)$, $W(u)$ is an $R_\delta$-set.
\end{theorem}
\begin{proof}
Since the nonlinearity $F(t,\cdot,\cdot)$ in our problem is continuous,
according to Lemma \ref{lmapprox}, one can take a sequence $\{F_n\}$
such that, $F_n(t,\cdot,\cdot)$ are locally Lipschitz functions and
\begin{equation*}
\|F_n(t,\zeta,\psi)-F(t,\zeta,\psi)\| < \epsilon_n,
\end{equation*}
for a.e. $t\in J$ and for all $\zeta\in X, \psi\in C([-h,0];X)$, where
$\epsilon_n\to 0$ as $n\to\infty$. Without loss of generality, we can
assume that
\begin{equation*}
\|F_n(t,\zeta,\psi)\| \le a(t)\|\zeta\| + b(t)\|\psi\|_{h} + c(t) + 1,
\end{equation*}
for all $n$.
Consider the equation
\begin{equation}\label{appeq}
x(t) = y^*(t) + \int_0^t S(t-s)[F_n(s, x(s), x[\varphi]_s) + Bu(s)]ds.
\end{equation}
Using the same arguments as in the previous section, one obtains the
existence result for \eqref{appeq}. In addition, since
$F_n(t,\cdot, \cdot)$ is locally Lipschitz, the solution of \eqref{appeq}
is unique.
Let
\begin{gather*}
\mathcal G (x) = (I - \mathcal F_u) (x),\\
\mathcal F_{un} (x) = S(t)\varphi(0) + \int_0^t S(t-s)[F_n(s,x(s), x[\varphi]_s)
+ Bu(s)] ds,\\
\mathcal G_n (x)(t) = (I-\mathcal F_{un}) (x).
\end{gather*}
Then one claims that the maps $\mathcal G$ and $\mathcal G_n$ are proper.
Indeed, we will prove this assertion, e.g., for $\mathcal G$. Let us show
that $\mathcal G^{-1}(K)$ is a compact set for each compact set
$K\subset C(J;X)$. Assume that
$$
(I-\mathcal F^u) (D) = K
$$
and $\{x_n\}\subset D$ is any sequence. Then there exists a sequence
$\{y_n\}\subset K$ such that
$$
x_n - \mathcal F^u(x_n) = y_n.
$$
That is,
\begin{equation}\label{eprop}
x_n(t) = S(t)\varphi(0) + y_n(t) + \int_0^t S(t-s)[f_n(s) + Bu(s)]ds,
\end{equation}
where $f_n(s) = F(s,x_n(s), x_n[\varphi]_s), s\in J$.
Using (F2A) and the fact that $\{y_n\}$ is bounded in $C(J;X)$, we see
that $\{x_n\}$ is also bounded in $C(J;X)$. Then $\{f_n\}$, in turn,
is bounded in $L^2(J;X)$. Thus $\{\mathbf{S}(f_n))\}$ is compact according
to Proposition \ref{pp:compactCauchyOp}. Thus $\{x_n\}$ is relatively
compact and therefore $D$ is a compact set.
On the other hand, $\{\mathcal G_n\}$ converges to $\mathcal G$ uniformly
in $C(J;X)$ and equation
$\mathcal G_n (x) = y$
has a unique solution for each $y\in \mathcal C_\varphi$ since
it is in the form of \eqref{appeq}.
Therefore, applying Lemma \ref{rdelta} we conclude
that
$$
W(u) = \mathcal G^{-1}(0)
$$
is an $R_\delta$-set. The proof is complete.
\end{proof}
\section{Controllability results}
Following the arguments in Section 2, the problem
\begin{gather}
y'(t) = Ay(t) + F(t,y(t),y_t) + z(t),\;t\in J,\label{e22}\\
y(s) = \varphi(s),\; s\in [-h,0]\label{e23}
\end{gather}
has at least one mild solution $y=y(\cdot;z)$, for each $z\in L^2(J;X)$.
Furthermore, by the same arguments in Section 3, we see that the solution map
$$
W(z) = \{y(\cdot;z):\;\text{the solution of \eqref{e22}-\eqref{e23}}\}
$$
is an $R_\delta$-map. Moreover, in view of ({\bf F1})-(F2), the map $N_F$
defined by \eqref{Nemytski} is continuous. Therefore, $N_F$ is also
an $R_\delta$-map.
Define a linear operator $\mathcal S_T: L^2(J;X)\to X$ by
\begin{equation*}
\mathcal S_T (v) = \int_0^T S(T-s)v(s) ds.
\end{equation*}
Denoting $\mathcal N=\{v\in L^2(J;X): \mathcal S_T v=0\}$, one has
that $\mathcal N$ is a closed subspace of $L^2(J;X)$.
Let $\mathcal N^\perp$ be the orthogonal space of $\mathcal N$ in $L^2(J;X)$
and $Q$ be the projection from $L^2(J;X)$ into $\mathcal N^\perp$.
Let $R[B]$ be the range of $B$, we make use of the following assumption
\begin{itemize}
\item[(B1)] For any $p\in L^2(J;X)$, there exists $q\in R[B]$ such that
$$
\mathcal S_T(p) = \mathcal S_T(q).
$$
\end{itemize}
This assumption implies that $\{y+\mathcal N\}\cap R[B]\neq \emptyset$
for any $y\in \mathcal N^\perp$. Hence, by the proof of \cite[Lemma 1]{Naito2},
the mapping $P$ from $\mathcal N^\perp$ to $R[B]$ by
\begin{align*}
Py = \Big\{& y^*: y^*\in \{y+\mathcal N\}\cap R[B], \text{and}\\
&\|y^*\|_{L^2(J;X)} = \min\{\|x\|_{L^2(J;X)}: x\in \{y+\mathcal N\}\cap R[B]\}\Big\}
\end{align*}
is well-defined. Moreover, $P$ is linear and bounded.
\begin{remark} \rm
We take assumption (B1) as the same one in \cite{Seidman}; i.e.,
it requires $q\in R[B]$, which is slightly stronger than that in
\cite{Naito2} ($q\in\overline{R[B]}$). In fact, this requirement is necessary
for our arguments when the solution to control system is not unique.
Moreover, in application, (B1) is much easier to verify than that
in \cite{Naito2}. In related works (e.g., \cite{Naito1,Naito2,Seidman,Wang}),
the authors checked (B1) only.
\end{remark}
For given $u_0\in L^2(J;V)$, we construct the operator
$\mathcal K: \mathcal N^\perp\to \mathcal P(\mathcal N^\perp)$ given by
\begin{equation}\label{Jop}
\mathcal Kv = QBu_0 - QN_FWPv.
\end{equation}
We will show that $\mathcal K$ has a fixed point. For this purpose,
we need the following notions and facts in the sequel.
\begin{definition}\rm
Let $Y$ be a metric space.
\begin{itemize}
\item[(1)] $Y$ is called an absolute retract (AR-space) if for any metric
space $X$ and any closed $A\subset X$, every continuous function
$f: A\to Y$ extends to a continuous $\tilde f: X\to Y$.
\item[(2)] $Y$ is called an absolute neighborhood retract (ANR-space)
if for any metric space $X$, any closed $A\subset X$ and continuous
$f: A\to Y$, there exists a neighborhood $U\supset A$ and a continuous
extension $\tilde f: U\to Y$ of $f$.
\end{itemize}
\end{definition}
Obviously, if $Y$ is an AR-space then $Y$ is an ANR-space.
\begin{proposition}[\cite{Dugundji}]\label{pp4}
Let $C$ be a convex set in a locally convex linear space $Y$.
Then $C$ is an AR-space.
\end{proposition}
In particular, the last proposition states that every Banach space and
its convex subsets are AR-spaces.
The following theorem is the main tool for this section.
For related results on fixed point theory for ANR-spaces, we refer
the reader to \cite{Gorniewicz, GD}.
\begin{theorem}[{\cite[Corollary 4.3]{GL}}]\label{fixpt}
Let $Y$ be an AR-space. Assume that $\phi: Y\to \mathcal P(Y)$
can be factorized as
$$
\phi = \phi_n\circ \phi_{n-1}\circ ...\circ \phi_1
$$
where
$$
\phi_i: Y_{i-1}\to\mathcal P (Y_i), \;i=1,..,N
$$
are $R_\delta$-maps and $Y_i, i=1,..,N-1$ are ANR-spaces, $Y_0=Y_{N}=Y$ are AR-spaces. If there is a compact set $D$ such that $\phi(Y)\subset D\subset Y$ then $\phi$ has a fixed point.
\end{theorem}
Using this theorem, we get the following result.
\begin{theorem}\label{thm5}
Let {\rm (F1), (F2A), (A1)} hold. Assume that {\rm (B1)} takes place.
Then the operator $\mathcal K$ defined in \eqref{Jop} has a fixed point
in $\mathcal N^\perp$ provided that
\begin{equation}\label{thm5-cond}
C_0\sqrt T \|P\| \|a+b\|_{L^2(J;X)}e^{C_0\|a+b\|_{L^1(J;X)}}< 1.
\end{equation}
\end{theorem}
\begin{proof}
The operator $\mathcal K$ can be factorized as
$$
\mathcal K = \mathcal T\circ Q\circ N_F \circ W \circ P
$$
where $\mathcal T (v) = QBu_0 - v$, a single-valued and continuous mapping.
It is easy to see that all component in above presentation is $R_\delta$-map.
Therefore, in order to use Theorem \ref{fixpt}, it suffices to show that
there exists a convex subset $D\subset \mathcal N^\perp$ such that
$\mathcal K(D)\subset D$ and $\mathcal K(D)$ is a compact set.
We look for $R>0$ such that $\|\mathcal K(v)\|_{L^2(J;X)}\le R$ provided
$\|v\|_{L^2(J;X)}\le R$ and then take
\begin{equation}\label{ARspace}
D=\overline B_R\cap \mathcal N^\perp,
\end{equation}
thanks to the fact that $\mathcal N^\perp$ is a convex subset of $L^2(J;X)$.
For $y\in W(Pv)$, it follows from Theorem \ref{th:exists} that
\begin{equation}\label{thm5-eq1}
\sup_{\tau\in [0,t]}\|y(\tau)\| \le C_0(M_\varphi + \|Pv\|_{L^1(J;X)})\exp\{C_0\int_0^t[a(s)+b(s)]ds\}
\end{equation}
Now using (F2) and \eqref{thm5-eq1}, we have
\begin{align*}
\|N_F(y)(t)\|
& =\|F(t, y(t), y_t)\| \le a(t)\|y(t)\| + b(t)\|y_t\|_h + c(t)\\
& \le [a(t)+b(t)]\sup_{\tau\in [0,t]}\|y(\tau)\| + b(t)\|\varphi\|_{h} + c(t)\\
& \le C_0\Big(M_\varphi + \|Pv\|_{L^1(J;X)}\Big)[a(t)+b(t)]
\exp\{C_0\int_0^t[a(s)+b(s)]ds\} \\
&\quad + b(t)\|\varphi\|_{h} + c(t).
\end{align*}
Taking into account that $\|Q\|\le 1$, for any $z\in \mathcal K(v)$ we get
\begin{align*}
\|z(t)\|
& \le \|Bu_0(t)\| + C_0\Big(M_\varphi
+ \|Pv\|_{L^1(J;X)}\Big)[a(t)+b(t)]\\
&\quad\times \exp\{C_0\int_0^t[a(s)+b(s)]ds\}
+ b(t)\|\varphi\|_{h} + c(t).
\end{align*}
This implies that
\begin{equation} \label{e26}
\begin{aligned}
\|z\|_{L^2(J;X)}
&\le \|Bu_0\|_{L^2(J;X)} \\
&\quad + C_0\Big(M_\varphi + \sqrt T\|P\|\|v\|_{L^2(J;X)}\Big)
\|a+b\|_{L^2(J;X)}e^{C_0\|a+b\|_{L^1(J;X)}}\\
&\quad + \|b\|_{L^2(J;X)}\|\varphi\|_{h} + \|c\|_{L^2(J;X)}.
\end{aligned}
\end{equation}
Thanks to assumption \eqref{thm5-cond}, \eqref{e26} ensures the existence
of a number $R>0$ such that $\|z\|_{L^2(J;X)}\le R$ provided
$\|v\|_{L^2(J;X)}\le R$. That is, $\mathcal K(D)\subset D$
with the closed bounded subset $D$ denoted in \eqref{ARspace}.
By Lemma \ref{lm7} the set $W\circ P(D)$ is compact and then
$K=\mathcal K (D)$ is a compact set. Thus we get the desired conclusion.
\end{proof}
\begin{remark} \rm
If the nonlinearity $F$ is uniformly bounded with respect to the second
and third arguments; that is
$$
\|F(t, \xi, \eta)\|\le c(t),\quad
\text{for a.e. $t\in J$ and all $\xi\in X$, $\eta\in C([-h,0];X)$}
$$
then condition \eqref{thm5-cond} can be relaxed since in this case $a=b=0$.
\end{remark}
Let
$$
\mathcal R_T(F) = \{x(T;u): u\in L^2(J;V)\},
$$
the set of all terminal state of solutions to the system \eqref{e1}-\eqref{e2}.
The set $\mathcal R_T(F)$ is called the \emph{reachable set} of the control
system \eqref{e1}-\eqref{e2}. When $F=0$, the notation $\mathcal R_T(0)$
stands for the reachable set of the corresponding linear system.
\begin{definition} \rm
The control system \eqref{e1}-\eqref{e2} is said to be exactly controllable
(or controllable) if $\mathcal R_T(F)=X$. It is called approximately
controllable if $\overline {\mathcal R_T(F)} = X$.
\end{definition}
It is shown in \cite[Lemma 2]{Naito2} that, by hypothesis (B1) one has
$\overline{\mathcal R_T(0)}=X$. That is, (B1) is a sufficient condition
for the approximate controllability of the linear system corresponding
to \eqref{e1}-\eqref{e2}. One can find in \cite{BM,JKP} for some other
conditions. The following theorem is our main result in this section.
\begin{theorem}
Under the hypotheses of Theorem \ref{thm5}, the control system
\eqref{e1}-\eqref{e2} is approximately controllable if the corresponding
linear system is.
\end{theorem}
\begin{proof}
We show that $\mathcal R_T(0)\subset \mathcal R_T(F)$.
Let $y_0\in \mathcal R_T(0)$, then there exists $u_0\in L^2(J;V)$ such that
$$
y_0 = S(T)\varphi(0) + \mathcal S_T B u_0.
$$
Let $v^*$ be a fixed point of $\mathcal K$, then we have
\begin{equation}\label{e27}
QBu_0 = QN_FWP v^* + v^*.
\end{equation}
By the definition of $P$, it is evident that
$Pv^* \in \{v^* + \mathcal N\}\cap R[B]$,
and then
\begin{equation}\label{e28}
\mathcal S_T Pv^* = \mathcal S_T v^*.
\end{equation}
On the other hand, $Q$ is the projection from $L^2(J;X)$
into $\mathcal N^\perp$, then
\begin{equation}\label{e29}
\mathcal S_T Q p = \mathcal S_T p,\;\text{for all}\; p\in L^2(J;X).
\end{equation}
Combining \eqref{e27}-\eqref{e29} yields
$$
\mathcal S_T Bu_0 = \mathcal S_T (f+Pv^*)
$$
where $f\in N_FWPv^*$. Therefore,
\[
y_0 = S(T)\varphi(0) + \mathcal S_T Bu_0
= S(T)\varphi(0) + \mathcal S_T (f+Pv^*)
= y(T;Pv^*)
\]
where $y$ is a solution of \eqref{e22}-\eqref{e23}. Since $Pv^*\in R[B]$,
there exists a function $u\in L^2(J;V)$ such that $Pv^*=Bu$.
Then we have $y(\cdot;Pv^*)=y(\cdot;Bu) = x(\cdot;u)$, where $x$ is a
mild solution of the system \eqref{e1}-\eqref{e2}. This implies that
$\mathcal R_T(0)\subset \mathcal R_T(F)$ and the proof is complete.
\end{proof}
\subsection*{An example}
We end this note with an application to the control system involving
a semilinear partial differential equation
\begin{gather}
\frac{\partial y}{\partial t}(t,x)
= \frac{\partial^2 y}{\partial x^2}(t,x) + f(t, y(t,x), y(t-h,x))
+ Bu(t,x),\; x\in [0,\pi], \quad t\in J,\label{app1}\\
y(t,0) = y(t,\pi) = 0,\quad t\in J,\label{app2}\\
y(s,x) = \varphi(s,x), \quad s\in [-h,0],\; x\in [0,\pi],\label{app3}
\end{gather}
where $y, u\in C(J; L^2(0,\pi))$ is the state function and the control
function, respectively. The feature in this example is that, the nonlinearity
has neither Lipschitz property nor uniform boundedness, in comparison with
the existing results in literature.
The operator $A=\dfrac{d^2}{d x^2}$, with the domain
$D(A)=H^2(0,\pi)\cap H_0^1(0,\pi)$, is the infinitesimal generator of a
compact semigroup $S(\cdot)$ in $L^2(0,\pi)$. Here $H^2(0,\pi)$ and
$H_0^1(0,\pi)$ are usual Sobolev spaces. The expression for $S(\cdot)$
is as follows:
$$
(S(t)h)(x) = \sum_{n=1}^{\infty} e^{-n^2 t}
\Big(\frac 2{\pi}\int_0^{\pi}h(x)\sin nx dx\Big)\sin nx,
$$
for $h\in L^2(0,\pi)$.
We select $B$ as in \cite{Naito2}, that is the intercept operator
$B_{\alpha, T}$ is
$$
B_{\alpha, T}v(t)=\begin{cases}
0, & 0\le t < \alpha, \\
v(t), & \alpha\le t\le T,
\end{cases}
$$
where $v\in L^2(0,T; L^2(0,\pi))$.
It is known that $B=B_{\alpha, T}$ satisfies (B1). Then the linear system
\begin{gather*}
\frac{\partial y}{\partial t}(t,x)
= \frac{\partial^2 y}{\partial x^2}(t,x) + Bu(t,x),\quad x\in [0,\pi], t\in J,\\
y(t,0) = y(t,\pi) = 0,\quad t\in J,\\
y(0,x) = y_0(x), \quad x\in [0,\pi],
\end{gather*}
is approximately controllable. That is $\overline{\mathcal R_T(0)}=L^2(0,\pi)$.
As far as nonlinearity $f$ is concerned, we assume that:
\begin{itemize}
\item[(N1)] $f: J\times \mathbb R \times\mathbb R \to \mathbb R$ is continuous;
\item[(N2)] there exist functions $a, b, c \in L^2(J)$ such that
$$
|f(t, \xi, \eta)|\le a(t)|\xi| + b(t)|\eta| + c(t),
$$
for all $t\in J, \xi, \eta \in \mathbb R$.
\end{itemize}
Applying the abstract results in previous sections, we conclude that the
control system \eqref{app1}-\eqref{app3} is approximately controllable
in $L^2(0,\pi)$ provided inequality \eqref{thm5-cond} holds.
Furthermore, the solution set depends upper-semicontinuously
in control function $u$ and it is an $R_\delta$-set.
\subsection{Acknowledgments}
The author acknowledges the support from Vietnam Ministry of Education
and Training under grant B2012-17-12.
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