\documentclass[reqno]{amsart}
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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}