\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 146, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/146\hfil Output-feedback stabilization]
{Output-feedback stabilization and control optimization for
 parabolic equations with Neumann boundary control}

\author[A. Elharfi \hfil EJDE-2011/146\hfilneg]
{Abdelhadi Elharfi}

\address{Abdelhadi Elharfi \newline
Department of Mathematics,
Cadi Ayyad University, Faculty of Sciences Semlalia\\
B.P. 2390, 40000 Marrakesh, Morocco}
\email{a.elharfi@ucam.ac.ma}

\thanks{Submitted September 7, 2010. Published November 2, 2011.}
\subjclass[2000]{34K35}
\keywords{$C_0$-semigroup; feedback theory for regular linear systems}

\begin{abstract}
 Both of feedback stabilization and optimal control problems are
 analyzed for a parabolic partial differential equation with
 Neumann boundary control. This PDE serves as a model of heat
 exchangers in a conducting rod. First, we  explicitly construct
 an output-feedback operator which exponentially stabilizes  the
 abstract control system representing the model. Second, we derive
 a controller which, simultaneously, stabilizes the associated
 output an minimizes a suitable cost functional.
\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 this article, we study the parabolic equation
 \begin{equation}\label{CPE}
\begin{gathered}
z_t(t,x)=[\varepsilon(x) z_x(t,x)]_x+b(x)
z_x(t,x)+a(x) z(t,x), \quad\text{in }(0,\infty)\times(0,1),\\
z_x(t,0)=\rho z(t,0),\quad z_x(t,1)=u(t),  \quad\text{in }(0,\infty), \\
z(0,x)= z^0(x), \quad\text{in }(0,1),
\end{gathered}
\end{equation}
with a control $u(t)$ placed at the extremity  $x=1$,
via Neumann boundary condition,  where the parameters
$\varepsilon, a, b, \rho$, satisfy the assumptions
\begin{equation}\label{H}
 -\infty < \rho\leq +\infty,\quad
 a\in C^1[0,1],\quad b,\varepsilon \in C^2[0,1],
   \quad \inf_{x\in[0,1]}\varepsilon(x)>0.
\end{equation}
Equation \eqref{CPE} can be interpreted, in thermodynamic point
of view, as a model of heat conducting rod in which not only the
heat is being diffused and bifurcated  ($(\varepsilon z_x)_x+bz_x$) but
also a  destabilizing heat is generating ($az$).
System \eqref{CPE} also represents  very
well a linearized model of chemical tubular reactor  \cite{BK1}
 and it can further approximates  a linearized model of unstable
burning in solid propellant rockets \cite{BK3}.

The stabilization problem of parabolic systems is treated by
several authors with different approaches. Stability by boundary
control in the optimal control setting is discussed by Bensoussan
et al. \cite{{BDDM1}}.  In \cite{LT2,T}, the open-loop system is
separated into an infinite-dimensional stable part and a
finite-dimensional unstable part. A boundary control stabilizing
the unstable part and leaving the stable part stable is derived.
In  \cite{Weis5,Weis6}, the stabilizability problem  for parabolic
systems is approached using the feedback theory for (autonomous)
regular linear systems.  In time depend setting, the
stabilizability and the controllability for  non-autonomous
parabolic systems  are discussed in \cite{Sch3} by developing the
so called non-autonomous regular linear systems. The
finite-dimensional backstepping is applied  in \cite{BK2} to the
discritized version of \eqref{CPE}, and shown to be convergent in
$L^\infty$. The backstepping method with continuous kernel  is
investigated in \cite{ELharfi,KS,Liu} to construct boundary
feedback laws making the closed-loop systems exponentially stable.
The backstepping idea is to convert the parabolic system into a
well known one using an integral  transformation with a kernel
satisfying an adequate PDE.

In this paper, we combine the feedback theory for regular linear
system \cite{Weis5} and the backstepping method to design an
output-feedback which exponentially stabilizes the abstract
control system representing system \eqref{CPE}.  To be more
precise,   system \eqref{CPE} is written in a suitable state space
as an abstract  control system; ${z}_t(t)=Az(t)+Bu(t),
\,\,t>0,\,\,z(0)=0$, where $A$ represents the evolution of the
open-loop  system and $B$ is an appropriate control operator. For
any  $\lambda>0$, we explicitly construct  an admissible observation
operator $C^\lambda$ which exponentially  stabilizes $(A,B)$ at the
desired rate of $\lambda$.    The stabilizing observation operator is
given  in term of the solution of an adequate kernel PDE which
depends on $\lambda$.  On the other hand,  we erect a controller which
solves, simultaneously,  both the stabilization and the control
optimization problems
 associated with \eqref{CPE}. In particular, we design a controller which not only stabilizes the output of the concerned control system but also  minimizes
an adapted cost functional.

The paper is organized as follows: In Section \ref{s2}, we present
the stabilizability concept associated with regular linear
systems. The abstract control system representing  \eqref{CPE} is
derived in Section \ref{abst-control-system}. In Section
\ref{s-obse-oper}, an explicit construction of the observation
operator stabilizing \eqref{CPE} is given.  Section
\ref{clo-loop-sta} is devoted to study the $\lambda$-exponential
stability of the closed-loop system. Finally, the optimal control
problem of system\eqref{CPE} is treated in Section \ref{opt contr
pro}.

\section{Preliminaries}\label{s2}

Throughout this paper, $U,X,Y$, are  Hilbert spaces.
$A:D(A)\subset X\to X$ is the generator  of a $C_0$-semigroup $T$.
We denote by $X_1$ the Hilbert space $D(A)$ endowed with the graph
norm;  $\|x\|_1=\|x\|+\|Ax\|$.

We further set $R(\lambda,A)=(\lambda-A)^{-1}$ for $\lambda$ in the resolvent
set $\varrho(A)$. The Hilbert space $X_{-1}$ is the completion of
$X$ with respect to the norm $\|x\|_{-1}:=\|R(\lambda,A)x\|$ for
some $\lambda\in\varrho(A)$. Then,  $T$ is extended to a
$C_0$-semigroup $T_{-1}$ on $X_{-1}$. The generator of $T_{-1}$ is
denoted by $A_{-1}$ which is an extension of $A$ to $X$. For more
detail on extrapolation theory we refer to \cite{EN}.

Let $B\in\mathcal{L}(U,X_{-1})$, $C\in\mathcal{L}(X_1,Y)$ and
 on $X_{-1}$ consider the abstract linear  system
\begin{gather}
    z_t(t)=A_{-1}z(t)+Bu(t),\quad z(0)=z^0,\label{cont-equ}\\
    y(t)=Cz(t),\quad t>0,\label{obser-equ}
\end{gather}
where $u\in L^2_{loc}([0,\infty),U)$. 


The well-posedness of system \eqref{cont-equ}--\eqref{obser-equ} 
requires a certain regularity of the triplet  $(A,B,C)$, due to 
\cite{Weis5,Weis6}. 
Moreover, if one relates the output $y$ to the input $u$ by an adequate
(feedback) operator $K$;  $u =Ky$, $K\in \mathcal{L}(Y,U)$,
we obtain a new system called the closed-loop system.
From \cite{Weis5}, the well-posedness of the closed-loop system
requires  that the feedback operator should be admissible for the
transfer function $H(\cdot):=CR(\cdot,A_{-1})B$; i.e.,  the operator
$I_Y-H(\cdot)K$ is uniformly  invertible in some half plan
$\mathbb{C}_s:=\{\lambda\in \mathbb{C}:~\Re  \lambda>s\}$. If it is the case,
then due to Weiss \cite{Weis5}, the  operator representing the
closed-loop system.
\begin{align}\label{A^I}
 A^I:=A_{-1}+BC_{L}\quad \text{with}\quad
D(A^I):=\{x\in X: (A_{-1}+BC_{L})x\in X\},
\end{align}
generates a $C_0$ semigroup $T^I$.

In practice, many control systems are unstable. However, if one
feeds back the output of an unstable system to the  input by an
appropriate  feedback law $u=Ky$, it is possible to obtain a stable
closed-loop system. This is called  the   \emph{feedback
stabilizability} of the open-loop system. An extensive survey on
the stabilizability concept of linear systems can be found in
\cite{Russell}. Here,  we are concerned  with the concept of
exponential stabilizability  as presented in \cite{Weis6}.

\begin{definition}[\cite{Weis6}] \label{defi-stab} \rm
Consider an abstract  control system with open-loop  generator
$A$ and control operator $B\in\mathcal{L}(U,X_{-1})$. We say that
$C\in \mathcal{L}(X,U)$
stabilizes $(A,B)$  if\
\begin{itemize}
\item[(a)] $(A,B,C)$  is a regular triple,
\item[(b)] $I_U$ is an admissible feedback operator for
 $H(\cdot)=CR(\cdot,A_{-1})B$,
\item[(c)] the operator $A^I$, defined in  \eqref{A^I},
generates an exponentially stable semigroup.
\end{itemize}
\end{definition}


\section{The abstract control system associated with \eqref{CPE}}
 \label{abst-control-system}

Without loss of generality we set in what follows $b\equiv 0$
since it can be  eliminated from  equation \eqref{CPE}  using
the transformation
\begin{equation}\label{initial-transf}
\tilde{ z}(t,x):=\exp\Big(\int_0^{x}\frac{b(s)}{2\varepsilon(s)}ds\Big) z(t,x)
\end{equation}
with the compatible changes of parameters
\begin{equation} \label{comp-ch-var}
\begin{gathered}
  \tilde{\varepsilon}(x):=\varepsilon(x),\quad
  \tilde{a}(x):=a(x)-\frac{b'(x)}{2}-\frac{b^2(x)}{4\varepsilon(x)},\\
  \widetilde{\rho}:=\rho+\frac{b(0)}{2\varepsilon(0)},\quad
  \tilde{u}(t)  :=\exp\Big(\int_0^1\frac{b(s)}{2\varepsilon(s)}ds\Big)u(t),
\end{gathered}
\end{equation}
In fact, one can easily see that
\[
   {\widetilde{z}}_t-(\widetilde{\varepsilon}\widetilde{
  z}_x)_x-\widetilde{a}\widetilde{z}= \{ {z}_t-(\varepsilon
  z_x)_x-b z_x-a z\}\exp\Big(\int_0^{x}\frac{b(s)}{2\varepsilon(s)}ds \Big).
\]
Then, $z$ satisfies \eqref{CPE}  if and only if $\widetilde{z}$
satisfies \eqref{CPE}  with the parameters
$\widetilde{\varepsilon}, 0, \widetilde{a}, \widetilde{\rho}, \widetilde{u}$,
instead of $\varepsilon, b, a, \rho, u$. Moreover, provided that
$b\in C^2$, the parameters
$\widetilde{\varepsilon}, 0, \widetilde{a}, \widetilde{\rho}$, satisfy
 \eqref{H}.

To present system \eqref{CPE} as an abstract control system,
we define  on the state space $X=L^2(0,1)$ the operators
\begin{equation}
\begin{gathered}\label{A-B}
 Af:=(\varepsilon f_x)_x+af,\quad D(A):=\{f\in H^2(0,1): f_x(0)=\rho f(0),\,
f_x(1)=0\},\\
Bu:=-uA_{-1}\psi,\quad B\in \mathcal{L}(\mathbb{C},X_{-1}).
\end{gathered}
\end{equation}
where $\psi$  is the unique $H^2$-solution of the ordinary
differential equation
\begin{equation} \label{f1-pro}
\begin{gathered}
(\varepsilon \psi_x)_x+a\psi=0,\quad  0\le x\leq 1, \\
\psi_x(0)=\rho \psi(0), \quad \psi_x(1)= 1.
\end{gathered}
\end{equation}
 The smoothness of the solution of \eqref{f1-pro} is shown as
in \cite[VIII.4]{Brezis}.  We first confirm the well-posedness
of the evolution equation corresponding to $A$ and the admissibility
of the control operator $B$ (for $A$).

\begin{lemma}\label{lemma-R(s,A)B}
\begin{itemize}
\item[(i)] $A$ generates an analytic semigroup $T$ on $X$;
\item[(ii)] $B$ is an admissible control operator for $T$.
\end{itemize}
Further,  there exist constants $\theta, \alpha_0 >0$ such that
\begin{equation}\label{est-racinlamd}
 \|R(s,A_{-1})B\|_{\mathcal{L}(\mathbb{C},X)}
\leq \frac{\theta}{\sqrt{\Re s}}
\end{equation} for $\Re s >\alpha_0$.
\end{lemma}

\begin{proof}
(i)  Observe that $A$ is self-adjoint. Then  $A$ generates
an analytic semigroup $T$ on $X$; see e.g. \cite{EN}.
(ii) Since $T$ is analytic on the Hilbert space $X$, then due to
De Simon \cite{DS},
$$
\int_0^{t_0}u(t_0-\sigma)T(\sigma)f d\sigma\in D(A),
$$
for a.e. $t_0>0$,  all $f\in X$, and $u\in L^2([0,t_0],\mathbb{C})$.
Hence,
\[
    \Phi(t_0)u:=\int_0^{t_0}T_{-1}(t_0-\sigma)Bu(\sigma)d\sigma
=-A\int_0^{t_0}u(t_0-\sigma)T(\sigma)\psi d\sigma \in  X
\]
for some $t_0>0$. Therefore, $B$ is an admissible control
operator for $T$. Finally,   the estimate \eqref{est-racinlamd}
is a consequence of the admissibility of $B$ for an  analytic
semigroup, see \cite{HW}.
\end{proof}

\section{The observation operator}\label{s-obse-oper}

The idea of constructing the observation operator  is to convert
 \eqref{CPE} into a well known equation by using the
following  transformation.

 \begin{lemma}[\cite{Liu}]\label{isomor}   Let
$k\in H^2(\Delta)$, $\Delta:=\{(x,y):0\leq y \leq x \leq 1\}$,
and define the linear bounded operator
$\mathcal{T}_k:H^{i}(0,1)\to H^{i}(0,1)$, by
\begin{align*}\label{transf}
  (\mathcal{T}_kv)(x):=v(x)+\int_0^{x}k(x,y)v(y)dy.
\end{align*}
 Then, $\mathcal{T}_k$ has a linear bounded inverse
$\mathcal{T}_k^{-1}:H^{i}(0,1)\to H^{i}(0,1)$, $i=0,1,2$.
\end{lemma}

 Next, assume that $z(t)$ satisfies \eqref{CPE} and
for $t\geq 0$, $x\in[0,1]$, set
\[ %\label{Etat-Transf}
 w(t,x):=(\mathcal{T}_k z(t))(x)= z(t,x)+\int_0^{x}k(x,y) z(t,y)dy.
\]
Then,
\begin{align*}
{w}_t(t,x)
&= {z}_t(t,x)+\int_0^{x}k(x,y) {z}_t(t,y)dy\\
&= {z}_t(t,x)+\int_0^{x}k(x,y)\big[[\varepsilon(y)
z_{y}(t,y)]_{y}+a(y) z(t,y)\big]dy
\end{align*}
 By integrating by parts
from $0$ to $x$,  for $t>0$ and $\lambda>0$, we obtain
\begin{equation} \label{relation}
\begin{aligned}
&{w}_t-[\varepsilon w_x]_x+\lambda w\\
&=\big[(\lambda+a(x))-2\varepsilon(x)\frac{d}{dx}(k(x,x))-\varepsilon'(x)k(x,x)\big]
z(t,x)\\
&\quad +\int_0^{x}\big[(\lambda +a(y))
 k(x,y)+\big([\varepsilon(y)k_{y}(x,y)]_{y}-[\varepsilon(x)k_x(x,y)]_x\big)\big]
 z(t,y)dy\\
&\quad +[k_{y}(x,0)-\rho k(x,0)]\varepsilon(0) z(t,0).
\end{aligned}
\end{equation}
Then
${w}_t-[\varepsilon(x)w_x]_x+\lambda w=0$,  in
$(0,\infty)\times (0,1)$, if and only if  the kernel $k$ satisfies
the PDE
\begin{equation}\label{PDE}
 \begin{gathered}
[\varepsilon(x)k_x(x,y)]_x-[\varepsilon(y)k_{y}(x,y)]_{y}=a_{\lambda}(y)k(x,y),\quad
0\leq y \leq x\leq 1,\\
 k_{y}(x,0)=\rho k(x,0),\quad  0\leq x\leq 1,\\
k(x,x)=\frac{1}{2\sqrt{\varepsilon(x)}}\int_0^{x}
\frac{a_{\lambda}(s)}{\sqrt{\varepsilon(s)}}ds=:g(x),\quad 0\leq x\leq 1,
\end{gathered}
\end{equation}
where $a_{\lambda}(x):=a(x)+\lambda$. We note that the third (boundary)
equation of \eqref{PDE} is obtained by solving the first order
differential equation
\[ \label{ordinaire equa}
2\varepsilon(x)\frac{d}{dx}(k(x,x))+\varepsilon'(x)k(x,x)=a_{\lambda}(x)
\]
with the initial condition $k(0,0)=0$.
The following well-posedness result  of the kernel PDE \eqref{PDE}
is proved in \cite{ELharfi} which generalizes the one
obtained in \cite{KS} for $\varepsilon$ constant.

\begin{lemma}\label{WP}
Assume that \eqref{H} holds. Then  the kernel  equation \eqref{PDE}
has a unique solution $k\in H^2(\Delta)$.
\end{lemma}

 Now, let $k^\lambda$ be the solution of the PDE  \eqref{PDE}
associated with some $\lambda>0$. From \eqref{relation}, we obtain
\[
{w}_t=[\varepsilon(x)w_x]_x-\lambda w \quad {\rm in~}
(0,\infty)\times(0,1).
\]
Moreover, it follows from the boundary conditions of \eqref{CPE}
that
\begin{align*}
  w_x(t,0)=\rho w(t,0), \quad
w_x(t,1)=u(t)+k_0(1)z(t,1)+\langle k_{1}^\lambda, z(t)\rangle,
\end{align*}
 where $\langle \cdot,\cdot \rangle$ denotes the inner product
on $X$ and $k_0^\lambda(y)=k^\lambda(1,y)$, $k_1^\lambda(y)=k_x^\lambda(1,y)$.
Thus,  $w_x(t,1)=0$  if and only if  $u$ satisfies the  control  law
\begin{equation}\label{BFL}
 u(t)=-k_0^\lambda(1)z(t,1)-\langle k_1^\lambda,z(t)\rangle.
 \end{equation}
This means that $\mathcal{T}_k$ converts the closed-loop system
\eqref{CPE},\eqref{BFL}, into
\begin{equation}\label{NS}
\begin{gathered}
{w}_t(t,x)=[\varepsilon(x)w_x(t,x)]_x-\lambda w(t,x),
 \quad\text{in }(0,\infty)\times(0,1),\\
w_x(t,0)=\rho w_x(t,0), \quad w(t,1)=0,\quad\text{in }(0,\infty),\\
 w(0,x)=w^0(x), \quad\text{in }(0,1),
\end{gathered}
\end{equation}
where $w^0(x):= z^0(x)+\int_0^{x}k(x,y) z^0(y)dy$.

The following theorem states the well-posedness of the closed-loop
system \eqref{CPE}, \eqref{BFL} and also gives an estimation of
the solution.

\begin{theorem}\label{z-stab}
For any $z^0\in L^2(0,1)$, the closed-loop system
\eqref{CPE},\eqref{BFL} has a unique solution $z(t,x)\in
C^{1,2}:=C^1\big((0,\infty)\times C^2[0,1]\big)$  such that
\begin{equation}\label{ES-z}
\|z(t)\|\leq Me^{-\lambda t}\|z^0\|,
\end{equation}
where $M$ is a positive constant independent of $z^0$.
\end{theorem}

\begin{proof}
It remains  to show that the equivalent system \eqref{NS}
has a unique solution $w$ satisfying
\begin{equation}\label{ES-w}
\| w(t)\|\leq e^{-\lambda t}\| w^0\|.
\end{equation}
In fact, consider on the state space $X$ the operator
\begin{gather*}%\label{Operator-K}
   D(G):=\{f\in H^2(0,1):~ f_x(0)=\rho f(0), ~
f_x(1)=0\},\\
 Gf:=(\varepsilon f_x)_x-\lambda f,\quad {\rm for ~} f\in D(G).
\end{gather*}
Observe that $G$ is self adjoint. Moreover, by integrating by
parts over $[0,1]$, we get
\[
    \langle Gf,f\rangle ~\leq -\lambda \|f\|^2,
\]
for every $f\in D(G)$. Then, see e.g. \cite[p.\,55]{BDDM1}, $G$
generates a bounded analytic semigroup $S$ such that
\begin{equation}\label{ESa1}
\|S(t)\|\leq e^{-\lambda t}, \quad t\geq 0.
\end{equation}
This means  that for any $w^0\in X$  system \eqref{NS}
has a unique solution $w=S(\cdot)w^0\in C([0,\infty),X)$.
Since  $S$ is analytic,
$S(\cdot)w^0\in C^1((0,\infty),D(G^\infty))$ for all $t>0$,
where $D(G^{\infty}):=\cap_{n=0}^{\infty}D(G^n)$;
 see e.g. \cite[p. 93]{EN}.
 Now, the Sobolev embedding theorem leads us to conclude
that $w\in C^{1,2}$. Moreover,   \eqref{ES-w} is an
immediate consequence of \eqref{ESa1}.

System \eqref{CPE}, \eqref{BFL} is well posed, since  it can
 be transformed via the isomorphism $\mathcal{T}_k$ to the well
posed system \eqref{NS}.
Further, the fact that $\mathcal{T}_k^{-1}$ and $\mathcal{T}_k$
are bounded, then there exists a constant $\delta>0$ such that
\begin{equation} \label{2est}
\| z(t)\|\leq \delta\|w(t)\| \quad \text{and}\quad
\|w^0\|\leq \delta\|z^0\|,
\end{equation}
for $t\geq 0$. Finally,  \eqref{ES-z}, follows from
\eqref{ES-w} combined with \eqref{2est}.
\end{proof}

Theorem \ref{z-stab} shows that  the feedback law \eqref{BFL}
forces the   the open-loop system \eqref{CPE} to exhibit a behavior
akin to $e^{-\lambda t}$  with  $L^2$-norm  (as $t\to \infty$).
This  leads us to choose  as observation operator
\begin{align}\label{C^la}
 C^\lambda f:=-k_0^\lambda(1)f(1)-\langle k_{1}^\lambda,f\rangle,\quad
C^\lambda\in\mathcal{L}(X,\mathbb{C}),
\end{align}
where $k^\lambda$ is  the solution of the kernel PDE \eqref{PDE}
corresponding to  some $\lambda>0$. We will show in the following
section that $C^\lambda$ is an appropriate observation operator
to create a stabilizing  controller  with respect to the
open-loop system corresponding  the aforesaid operators
$(A,B)$.

\section{The closed-loop stability}\label{clo-loop-sta}

We confirm in this  section that $C^\lambda$ is a suitable
stabilizing output  operator  for the  abstract control system
represented by $(A,B)$. The following theorem constitutes the
first main result of this paper.

\begin{theorem}\label{coro-rersult}
Consider $(A,B)$ with representation \eqref{A-B} and  define
$C^\lambda$  by \eqref{C^la}. Then
\begin{itemize}
\item[(i)] $C^\lambda$ stabilizes $(A,B)$,
\item[(ii)] the operator $A^I:=A_{-1}+BC^\lambda$ with the domain
$D(A^I):=\{f\in X: A_{-1}f+BC^\lambda f\in X\}$, generates a
$C_0$-semigroup $T^I$ such that
\begin{equation}\label{estm-main-res}
   \| T^I(t)z^0\|\leq Me^{-\lambda t}\|z^0\|,
\end{equation}
for $t\geq 0$ and any $ ~z^0\in X$, where $M$ is a positive
constant independent of $z^0$.
\end{itemize}
\end{theorem}


\begin{proof}
 Since $C^\lambda$ is a bounded perturbation of the Dirichlet trace,
it follows  that it is an admissible observation operator for the
open-loop semigroup  $T$ and that its degree of unboundedness
is $1/4 $, see e.g. \cite{OS}. Taking into account the analyticity of the
open-loop semigroup $T$, the feedthrough operator
is equal to zero and the control operator $B$ also has 
the same degree of unboundedness $1/4$. 
\cite[Example 7.7.5]{OS}
then  shows  that $C^\lambda$ is an admissible state feedback operator.
Thus due to \cite{Weis5},  $(A,B,C^\lambda)$  is a regular triple and
the transfer function is given by
\[
H(s)=C^\lambda R(s,A_{-1})B,
\]
 for a sufficiently large $\Re s$.    On the other hand,
due to Lemma \ref{lemma-R(s,A)B}, there exist $\alpha,\,\theta>0$
 such that
\[
\|H(s)\|=\|C^\lambda R(s,A_{-1})B\|\leq
 \frac{\theta \|C^\lambda \|_{\mathcal{L}(X,\mathbb{C})}}{\sqrt{\Re s}},
\quad \text{for } s\in \mathbb{C}_\alpha.
\]
   Which  implies that there  exists $s_0>\alpha$ such that
$|H(s)|<1$ for $s\in\mathbb{C}_{s_0}$. Consequently,
$I_{\mathbb{C}}$ is an admissible feedback for $H$.
According to Section \ref{s2}, $A^I$ generates a $C_0$-semigroup
$T^I$. Which means  that $T^I(\cdot)z^0$ is the unique classical
solution of the evolution equation
\begin{gather*}
 {z}_t(t)=A_{-1}z(t)+BC^\lambda z(t), t>0,\\
 z(0)=z^0;
\end{gather*}
i.e.,  $T^I(\cdot)z^0$ is the unique solution of the closed-loop
system
\begin{equation} \label{C-Loop1}
\begin{gathered}
 {z}_t(t)=A_{-1}z(t)+Bv(t), z(0)=z^0,\\
 y(t)=C^\lambda z(t),\\
 v(t)=y(t),\quad t>0.
\end{gathered}
\end{equation}
On the other hand, in view of Theorem \ref{z-stab},
for a given $z^0\in X$  the system \eqref{CPE}, \eqref{BFL}
has a unique solution $z=z(t,x,z^0)\in C^{1,2}$.
Observe that, $z(t)-u(t)\psi\in D(A)$, for $t>0$, and
\[
    {z}_t(t)=A(z(t)-u(t)\psi)=A_{-1}z(t)+Bu(t).
\]
Moreover, the control law \eqref{BFL} means that
$$
u(t)=C^\lambda z(t)=y(t).
$$
This shows that $z$ is also a solution of \eqref{C-Loop1}.
Thus,  $z(\cdot,z^0)=T^{I}(\cdot) z^0$. Finally,
the estimate \eqref{estm-main-res} is an immediate consequence of
\eqref{ES-z}.
\end{proof}

Alternatively, instead of invoking  \cite [Example 7.7.5]{OS},
one can use  in the above proof, that the impulse response is
in $L^1(0;1)$ (which follows from analyticity of the semigroup
and the degrees of unboundedness) and then use  the
reasoning involving the concept of well-posedness radius
from  \cite{Weis5}  to show that $C^\lambda$ is an admissible
state feedback operator.

The scheme of Figure \ref{figurekz} makes understood the meaning
of the stability  result  stated in  Theorem \ref{coro-rersult},
 and  shows how the controller \eqref{BFL}  affects  in  a closed
form the open-loop system \eqref{CPE},

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1}
\caption {The closed-loop system
of \eqref{CPE}  associated with the control law \eqref{BFL}}
\label{figurekz}
\end{center}
\end{figure}

In view of the scheme  of Figure \ref{figurekz}, in order to
stabilize \eqref{CPE} in a closed form, for a given rate $\lambda$,
one computes, for example by a numerical calculator,  the
quantity  $q:=-k_0^\lambda(1)z(t,1)-\langle k_1^{\lambda},z(t)\rangle$,
and  one  injects, intermediary a dispositive described by the
control operator $B$ , the sum  $q$ at the extremity $x=1$. The
state of the resulting closed-loop system exhibits a behavior akin
to $e^{-\lambda t}$ as $t\to \infty$.

\begin{remark} \label{rmk7} \rm
Although of the  results in the above sections are given for $b=0$.
However, if  $b\neq0$, one may  consider, in view
of \eqref{initial-transf}--\eqref{comp-ch-var},
the observation operator
 $$
\widetilde{C}^{\lambda} f:=\big(-\widetilde{k}_0^\lambda(1)f(1)-\langle
{\widetilde{k}^\lambda}_1,f\rangle\big) e^{-\int_0^1\frac{b(s)}{2\varepsilon(s)}},
\quad f\in X,
$$
where $\widetilde{k}$ is the solution of the kernel PDE given for
$\widetilde{\varepsilon}$, $\widetilde{a}$, $\widetilde{\rho}$ instead
of  $\varepsilon$, $a$, $\rho$.
\end{remark}

\section{Optimal control problem for \eqref{CPE}}\label{opt contr pro}

In some applications, it is not benefic to stabilizes a system by
a large cost. So, by stabilizing a system, a  question should de asked.
What is the cost of stabilizing the system?
To this purpose, we devote this section to deal with the
 optimal control of  system  \eqref{CPE}
coupled with the adequate output function
\begin{equation}\label{output}
    y(t):=2\sqrt{1+\lambda}\langle k_0,z(t)\rangle,
\end{equation}
where $k$ is the solution of the kernel PDE \eqref{PDE} and
$z(t)=z(t,u,z^0)$ is the  solution of the  system \eqref{CPE}
corresponding to the initial condition $z^0$ and the control  $u$.
The optimal control problem that we address  here, is to
design a control $u$  which, simultaneously, stabilizes the
output function $y$ and minimizes the cost functional
\begin{equation}\label{cost}
    J(u):=\int_0^{\infty}y(t)^2dt+\int_0^{\infty}
\big\{\varepsilon_{2}z_x(t,1)-Q(u)\big\}^2dt
\end{equation}
with
\[
    Q(u):=\varepsilon_{1}z(t,1)-\langle p,z(t)\rangle,
\]
where $\varepsilon_{1}:=\varepsilon(1)k_0^{'}(1), \varepsilon_{2}:=\varepsilon(1)k_0(1)$ and
$p(y):= \big{[\varepsilon(x)k_x(x,y)\big]_x}_{|_{x=1}}$.
We note here that $J$  can be written as
$\int_0^\infty \big( y(t)^2+ \|Ku(t)\|^2\big )dt$,
where $K$ is a linear operator chosen appropriately.
Which shows that \eqref{cost} has the usual  form of a cost
functional. The second main result of
this paper is given by the following theorem.

\begin{theorem}\label{cost-theorem}
 The  controller
\begin{equation}\label{u^n}
    \varepsilon_{2}u^{\rm opt}(t)=\varepsilon_{1}z(t,1)+\langle 2k_0-p,z(t)\rangle,
\end{equation}
applied to  \eqref{CPE}, stabilizes the output function $y$
and minimizes the cost $J$. Moreover, the optimal value
for $J$ is given by
\[ %\label{opt-value}
    J^{\rm opt}=2\langle k_0,z^0\rangle ^2.
\]
\end{theorem}

\begin{proof} For $t\geq 0$, set
\[ %\label{functional}
    V(t):=\frac{1}{2}\langle k_0,z(t)\rangle ^2.
\]
 By integrating by parts and using  \eqref{PDE}, we obtain
\begin{align*}
\dot{V}(t)&=\langle k_0,z(t)\rangle
     \big\{\varepsilon_{2}z_x(t,1)-\varepsilon_{1}z(t,1)
     +\langle p-\lambda k_0,z(t)\rangle\big\}\\
  &=-\lambda \langle k_0,z(t)\rangle ^2+\langle k_0,z(t)\rangle
  \big[\varepsilon_{2}z_x(t,1)-Q(u)\big],
\end{align*}
which can be written as
\begin{equation} \label{estim1for V}
\begin{aligned}
   \dot{V}(t)=&\big\{\langle k_0,z(t)\rangle +\frac{1}{2}
   \big[\varepsilon_{2}z_x(t,1)-Q(u)\big]\big\}^2\\ &-(1+\lambda)\langle k_0,z(t)\rangle ^2 -\frac{1}{4}\big[\varepsilon_{2}z_x(t,1)-Q(u)\big]^2.
\end{aligned}
\end{equation}
So,
\begin{equation}\label{estim for J}
    \frac{1}{4}J(u)=V(0)-V(\infty)
 +\int_0^{\infty}\big\{\langle k_0,z(t)\rangle+\frac{1}{2}
   \big[\varepsilon_{2}z_x(t,1)-Q(u)\big]\big\}^2dt.
\end{equation}
Choosing now the control $u^{\rm opt}$
 as in \eqref{u^n}, then the control law $z_x(t,1)=u^{\rm opt}(t)$ is
 equivalent to
\begin{equation}\label{K+Q=0}
  \langle k_0,z(t)\rangle+\frac{1}{2}
   \big[\varepsilon_{2}z_x(t,1)-Q(u)\big]=0.
\end{equation}
Substituting \eqref{K+Q=0} in \eqref{estim1for V}, we  obtain
$ \dot{V}(t)\leq -2(1+\lambda)V(t)$,
which implies
\begin{equation} \label{v-infty}
V(t)\leq e^{-2(1+\lambda)t}V(0)  \quad \text{and}\quad
y(t)^2\leq e^{-2(1+\lambda)t}y(0)^2.
\end{equation}
This proves that the control law $u^{\rm opt}(t)=z_x(t,1)$
stabilizes  the output $y$.

On the other hand, from \eqref{v-infty}, one has
$V(\infty)=0$. Substituting \eqref{K+Q=0}
in \eqref{estim for J}, we obtain
\begin{equation*}
    J(u^{\rm opt})=4V(0)=J^{\rm opt}.
\end{equation*}
This completes the the proof.
\end{proof}

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\end{document}