\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \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 $$\label{CPE} \begin{gathered} z_t(t,x)=[\varepsilon(x) z_x(t,x)]_x+b(x) z_x(t,x)+a(x) z(t,x), \quad\text{in }(0,\infty)\times(0,1),\\ z_x(t,0)=\rho z(t,0),\quad z_x(t,1)=u(t), \quad\text{in }(0,\infty), \\ z(0,x)= z^0(x), \quad\text{in }(0,1), \end{gathered}$$ 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 $$\label{H} -\infty < \rho\leq +\infty,\quad a\in C^1[0,1],\quad b,\varepsilon \in C^2[0,1], \quad \inf_{x\in[0,1]}\varepsilon(x)>0.$$ 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 $$\label{initial-transf} \tilde{ z}(t,x):=\exp\Big(\int_0^{x}\frac{b(s)}{2\varepsilon(s)}ds\Big) z(t,x)$$ with the compatible changes of parameters $$\label{comp-ch-var} \begin{gathered} \tilde{\varepsilon}(x):=\varepsilon(x),\quad \tilde{a}(x):=a(x)-\frac{b'(x)}{2}-\frac{b^2(x)}{4\varepsilon(x)},\\ \widetilde{\rho}:=\rho+\frac{b(0)}{2\varepsilon(0)},\quad \tilde{u}(t) :=\exp\Big(\int_0^1\frac{b(s)}{2\varepsilon(s)}ds\Big)u(t), \end{gathered}$$ 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 $$\begin{gathered}\label{A-B} Af:=(\varepsilon f_x)_x+af,\quad D(A):=\{f\in H^2(0,1): f_x(0)=\rho f(0),\, f_x(1)=0\},\\ Bu:=-uA_{-1}\psi,\quad B\in \mathcal{L}(\mathbb{C},X_{-1}). \end{gathered}$$ where $\psi$ is the unique $H^2$-solution of the ordinary differential equation $$\label{f1-pro} \begin{gathered} (\varepsilon \psi_x)_x+a\psi=0,\quad 0\le x\leq 1, \\ \psi_x(0)=\rho \psi(0), \quad \psi_x(1)= 1. \end{gathered}$$ 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 $$\label{est-racinlamd} \|R(s,A_{-1})B\|_{\mathcal{L}(\mathbb{C},X)} \leq \frac{\theta}{\sqrt{\Re s}}$$ 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 \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} 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 $$\label{PDE} \begin{gathered} [\varepsilon(x)k_x(x,y)]_x-[\varepsilon(y)k_{y}(x,y)]_{y}=a_{\lambda}(y)k(x,y),\quad 0\leq y \leq x\leq 1,\\ k_{y}(x,0)=\rho k(x,0),\quad 0\leq x\leq 1,\\ k(x,x)=\frac{1}{2\sqrt{\varepsilon(x)}}\int_0^{x} \frac{a_{\lambda}(s)}{\sqrt{\varepsilon(s)}}ds=:g(x),\quad 0\leq x\leq 1, \end{gathered}$$ 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 $$\label{BFL} u(t)=-k_0^\lambda(1)z(t,1)-\langle k_1^\lambda,z(t)\rangle.$$ This means that $\mathcal{T}_k$ converts the closed-loop system \eqref{CPE},\eqref{BFL}, into $$\label{NS} \begin{gathered} {w}_t(t,x)=[\varepsilon(x)w_x(t,x)]_x-\lambda w(t,x), \quad\text{in }(0,\infty)\times(0,1),\\ w_x(t,0)=\rho w_x(t,0), \quad w(t,1)=0,\quad\text{in }(0,\infty),\\ w(0,x)=w^0(x), \quad\text{in }(0,1), \end{gathered}$$ 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 $$\label{ES-z} \|z(t)\|\leq Me^{-\lambda t}\|z^0\|,$$ 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 $$\label{ES-w} \| w(t)\|\leq e^{-\lambda t}\| w^0\|.$$ 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 $$\label{ESa1} \|S(t)\|\leq e^{-\lambda t}, \quad t\geq 0.$$ 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 $$\label{2est} \| z(t)\|\leq \delta\|w(t)\| \quad \text{and}\quad \|w^0\|\leq \delta\|z^0\|,$$ 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 $$\label{estm-main-res} \| T^I(t)z^0\|\leq Me^{-\lambda t}\|z^0\|,$$ 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 $$\label{C-Loop1} \begin{gathered} {z}_t(t)=A_{-1}z(t)+Bv(t), z(0)=z^0,\\ y(t)=C^\lambda z(t),\\ v(t)=y(t),\quad t>0. \end{gathered}$$ 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 $$\label{output} y(t):=2\sqrt{1+\lambda}\langle k_0,z(t)\rangle,$$ 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 $$\label{cost} J(u):=\int_0^{\infty}y(t)^2dt+\int_0^{\infty} \big\{\varepsilon_{2}z_x(t,1)-Q(u)\big\}^2dt$$ 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 $$\label{u^n} \varepsilon_{2}u^{\rm opt}(t)=\varepsilon_{1}z(t,1)+\langle 2k_0-p,z(t)\rangle,$$ 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 \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} So, $$\label{estim for J} \frac{1}{4}J(u)=V(0)-V(\infty) +\int_0^{\infty}\big\{\langle k_0,z(t)\rangle+\frac{1}{2} \big[\varepsilon_{2}z_x(t,1)-Q(u)\big]\big\}^2dt.$$ 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 $$\label{K+Q=0} \langle k_0,z(t)\rangle+\frac{1}{2} \big[\varepsilon_{2}z_x(t,1)-Q(u)\big]=0.$$ Substituting \eqref{K+Q=0} in \eqref{estim1for V}, we obtain $\dot{V}(t)\leq -2(1+\lambda)V(t)$, which implies $$\label{v-infty} V(t)\leq e^{-2(1+\lambda)t}V(0) \quad \text{and}\quad y(t)^2\leq e^{-2(1+\lambda)t}y(0)^2.$$ 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} \begin{thebibliography}{00} \bibitem{BK2} {A. Balogh, M. Krstic}; \emph{Infinite dimensional backstipping-styl feedback for a heat equation with arbitrarily level of instability}, European J. Control, 8 (2002), pp. 165-176. \bibitem{BDDM1} A. Bensoussan, G. Da Prato, M. C. Delfour, S. K. Mitter; \emph{Representation and Control of Infinite Dimensional Systems: Volume I}, Birkh\"auser 1992. \bibitem{BK1} D. M. Boskovic, M. Kristic; \emph{Backstepping control of chemical tubular reactors}, Comput. Chem. Eng. \textbf{26} (2002), pp. 1077-1085. \bibitem{BK3} D. M. Boskovic, M. Kristic; \emph{Stabilization of a solid propellant rocket instability by state feedback}, int. J. Robust Nonlinear control, \textbf{13} (2003), pp. 483-495. \bibitem{Brezis} H. 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