\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 38, 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-2011/38\hfil Feedback stabilization]
{Feedback stabilization of  parabolic systems with bilinear controls}

\author[M. Ouzahra\hfil EJDE-2011/38\hfilneg]
{Mohamed Ouzahra}

\address{Mohamed Ouzahra \newline
 Department of Mathematics and informatics\\
 P.O. Box 5206, ENS, Fes, Morocco}
\email{m.ouzahra@yahoo.fr}

\thanks{Submitted January 14, 2011. Published March 7, 2011.}
\subjclass[2000]{93D15, 93D09}
\keywords{Bilinear systems; constrained controls;
feedback stabilization;\hfill\break\indent  robustness}

\begin{abstract}
 In this article, we study infinite-bilinear systems, and consider
 a decomposition of the state space via the spectral properties
 of the systems. Then we apply this approach to strong and
 exponential stabilization problem using quadratic and constrained
 feedbacks. We present also some applications.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

In this work, we  study the  stabilization
of bilinear systems governed by the abstract equation
\begin{equation}\label{SS}
\frac{d z(t)}{d t}= Az(t) + v(t)Bz(t),\quad z(0) = z_0,
\end{equation}
on a separable Hilbert space $H$ with the inner product
$\langle\cdot,\cdot\rangle$
and corresponding norm $ \|\cdot\|$, where
$A:D(A)\subset H \to H$ is a linear operator with domain
$ D(A) $, the operator $A$ generates a $C_0$-semigroup $ S(t) $ on
$H$, the operator $B : H \to H $ is  linear, and the scalar
valued function $ t \mapsto v(t) $ represents the control.

In \cite{Bal1}, the  quadratic control
\begin{equation}\label{feedbal}
v(t) = - \langle z(t),Bz(t)\rangle,
\end{equation}
was proposed  to study  the feedback stabilization of
\eqref{SS}, and a weak stabilization result was established
under the condition
\begin{equation} \label{adbal}
\langle BS(t)y,S(t)y\rangle=0, \; \forall t \ge 0 \Rightarrow
y=0\,.
\end{equation}
In \cite{berr0,berr2}, it has been  proved that under \eqref{adbal},
the same quadratic control \eqref{feedbal} ensures the strong
stabilization for a class of semilinear systems.

If the assumption \eqref{adbal} is replaced by
\begin{equation}\label{coerejc}
\int_0^T|\langle BS(t)y,S(t)y\rangle| dt
\ge \delta \|y\|^2,\;\forall \;y\in H, (\text{for some } T, \delta >0),
\end{equation}
then we have strong stability of \eqref{SS} with the decay
estimate $\|z(t)\| = O(1/\sqrt{t})$, (see \cite{berr0,ouz2}).

Recently,  the exponential stabilization problem of distributed
bilinear systems has been resolved (see \cite{ouz4}).
Then it has been proved that under the assumption
\eqref{coerejc}, the feedback defined by
\begin{equation} \label{et}
v(t)=\begin{cases}
-\frac{\langle Bz(t),z(t)\rangle }{\|z(t)\|^2}, & \text{if }
z(t)\ne 0 \\
 0, &\text{if } z(t)= 0,
\end{cases}
\end{equation}
guarantees the exponential stabilization. The finite-dimensional
case has been treated in \cite{chen}.

It is interesting to investigate the relation between the
stability of a distributed parameter system and that of a
finite-dimensional system. In \cite{kat}, it has been showed
that if the spectrum $\sigma(A)$ of $A$ can be decomposed
into  $ \sigma_u(A)=\{\lambda: \operatorname{Re}(\lambda )\geq -\eta \}$
and $ \sigma_s(A) = \{\lambda:\operatorname{Re}(\lambda ) < -\eta \}$
for some $\eta >0$, such that
$\sigma_u(A) $ can be separated from  $ \sigma_s(A)$
by a simple and closed curve $C$, then the state space $ H $ can
be decomposed according to
\begin{equation}\label{H}
H=H_u\oplus H_s,
\end{equation}
where  $ H_u = P_uH$, $H_s=P_sH$, and $ P_u $ is the projection
operator
$$
P_u=\frac{1}{2\pi i}\int_C{(\lambda I-A)^{-1}}d\lambda
$$
and $ P_s=I-P_u$. Then the operator $A$ can be decomposed as
$ A = A_u \oplus A_s $ with $A_u = P_uA $ and $A_s=P_sA$.
For linear systems and based on the above decomposition \eqref{H},
it has been showed that the whole system can be divided into two
uncoupled subsystems, one of which is exponentially stable without
applying controls, while another one is unstable.
Then under the spectrum growth assumption:
\begin{equation}\label{sga}
 \lim_{t\to +\infty} \frac {\ln \|S_s(t)\|}{t}
= \sup \operatorname{Re}(\sigma  (A_s)),
\end{equation}
where  $ S_s(t) $ denotes the  semigroup generated by $A_s$, it
has been proved that stabilizing a linear system turns  out to
stabilizing its unstable part (see \cite{tri}). This technique has
been used to study weak and strong stabilization of \eqref{SS}
using the quadratic control
\begin{equation}\label{feed}
 v_u(t) = - \langle z(t),P_uBP_u z(t)\rangle,
\end{equation}
where it  has been assumed that  the  operator $B$  can be
decomposed as
\begin{equation}\label{bu}
B=B_u\oplus  B_s,
\end{equation}
with $B_u=P_uBP_u$ and $B_s=P_sBP_s$ (see \cite{ouz3}).
The aim of this work consists on exploring the decomposition
\eqref{H}  of the state space, to study the strong and exponential
stabilization of the system \eqref{SS} using quadratic and
constrained controls. In the second section, we show that
one can achieve the strong stabilization of \eqref{SS} under the
condition \eqref{adbal}.  If in
addition \eqref{bu} holds, then under a weaker version of
\eqref{coerejc}, one obtains exponential stabilization.
In the third section, we study the question of robustness. The last
section concerns some situations that illustrate the established
results.

\section{Stabilization results}

\subsection{Strong stabilization}

The next result concerns the strong stabilization of \eqref{SS}.


\begin{theorem} \label{thm1}
 Let (i) $A$ generate a linear $C_0$-semigroup $S(t)$ of
contractions on $H$,
(ii)  $A$ allow the decomposition \eqref{H} of $H$ with
$\dim H_u < +\infty$  such that \eqref{sga} holds, and
(iii) $B $ be  compact such that
\begin{equation} \label{ad} \langle BS(t)y,S(t)y\rangle=0, \;
\forall t \ge 0 \Rightarrow y=0\,.
\end{equation}
Then the system \eqref{SS} is strongly stabilizable by the
feedbacks
\begin{equation}\label{feedrho}
v(t) = -\rho \langle z(t),Bz(t)\rangle,
\end{equation}
and
\begin{equation} \label{etrho}
v(t)=-\rho\frac{\langle z(t),Bz(t)\rangle}{\|z(t)\|^2}\mathbf{1}_E
\end{equation}
where $\rho >0$ and $\mathbf{1}_E $ is the characteristic function
 of the set $E=\{t\ge0: z(t)\ne 0\}$.
\end{theorem}

\begin{proof}
 System \eqref{SS}, controlled by \eqref{feedrho} or \eqref{etrho},
possesses a unique mild
solution $z(t)$ defined on a maximal interval $[0,t_{max}[$ and given by  the variation of constants formula
\begin{equation}\label{zt}
z(t) = S(t)z_{0} -
 \int_0^t S(t-s)F(z(s))ds,
 \end{equation}
where $F(z)=\rho \langle BZ,z\rangle Bz$ corresponds to
 \eqref{feedrho} and
$F(z)=\rho\frac{\langle BZ,z\rangle }{\|z\|^2}Bz$,
 for all $z\ne 0$, $F(0)=0$ corresponds to \eqref{etrho}.
Since $S(t) $ is a semigroup of contractions,
\begin{equation}\label{cv}
 \frac{d \|z(t)\|^2}{dt} \le -
2\langle F(z(t)),z(t)\rangle , \quad \forall z_0\in D(A)\,.
\end{equation}
It follows that
\begin{equation}\label{zborn}
\|z(t)\| \le \|z_{0}\|\,.
\end{equation}
Based on \eqref{zt} and using the fact  that $S(t)$ is a
semigroup of contractions and Gronwall inequality, we deduce  that
the map $z_0\to z(t)$ is continuous from $H $ to $H$. We
deduce that \eqref{zborn} holds for all $z_0\in H$ and hence
$t_{\rm max}=+\infty$ (see \cite{Pazy}).

Now let us show that $z(t)\rightharpoonup 0$, as
$t\to +\infty$. Let  $ t_n\to +\infty$ such that $ z(t_n)$
weakly converge in $H$, and let $y\in H$ such that
$ z(t_n)\rightharpoonup y, $ as $n\to \infty$.
(The existence of such $(t_n)$ and  $y$ is ensured by \eqref{zborn}
and by the fact that $H$ is reflexive).
\end{proof}

Let us recall the following existing result.

\begin{lemma}[\cite{ouz2, ouz4}]
Let $A$ generate a semigroup of contractions $S(t)$ on $H$ and
 let $ B$ be linear and bounded. Then \eqref{SS}, controlled by
\eqref{feedrho} or \eqref{etrho}, possesses a unique mild
 solution $z(t) $ on $\mathbb{R}^+$ for each $z_0\in H $
which satisfies
\begin{equation}\label{estim-bst}
\int_0^T|\langle BS(t)z_0,S(t)z_0\rangle| dt
\le \mathcal{C} \|z_0\|
\Big\{\int_0^{T}\langle F(z(t)),z(t)\rangle
dt\Big\}^{1/2},
\end{equation}
for all $T>0$ and for some $\mathcal{C} = \mathcal{C}(T,\|z_0\|) >0$.
\end{lemma}

Taking $z(t_n)$ as initial state in  \eqref{estim-bst}
and using superposition property of the solution we obtain,
via  the dominated convergence theorem,
$\langle BS(t)y,S(t)y\rangle=0$, for all $t\ge 0$.
 It follows from  \eqref{ad}  that $y=0$.
Hence $z(t)\rightharpoonup 0, $ as $t\to +\infty$, and since
$\dim H_u < +\infty$, we have
$$
z_u(t) \to 0, \quad \text{as }   t\to +\infty.
$$
For the component $z_s(t)$ of $z(t)$, we  have
\begin{equation}\label{zs2}
z_s(t)=S_s(t)z_{0s}-\int_{0}^t{S_s(t-\tau )F(z(\tau
))}d\tau\,.
\end{equation}
The semigroup  $ S_s(t) $ satisifes, by  \eqref{sga}, the
inequality
\begin{equation}\label{01}
\|S_s(t)\|\leq \alpha e^{-\eta t}, \quad \forall t \ge 0,
\quad(\text{for some } \alpha, \eta > 0).
\end{equation}
Then for all $0\le t_1 \le t$, we have
$z_s(t)=S_s(t-t_1)z_s(t_1)-\int_{t_1}^t{S_s(t-\tau )F(z(\tau))}d\tau$.
It follows that
\begin{equation}\label{z-s}
\|z_s(t)\|\le \alpha e^{{-\eta (t-t_1)}} \|z_s(t_1)\|+ \alpha
\int_{t_1}^t{e^{-\eta(t-\tau )}}
\|F(z(\tau))\|d\tau\,.
\end{equation}
Since $F$  is sequentially continuous,
$$
F(z(t))\to 0, \quad \text{as }  t\to +\infty\,.
$$
Let  $\varepsilon > 0$ and  let $t_1>0$ such that
$\|F(z(t))\| < \varepsilon$, for all $t\ge t_1$. It follows that
$$
\|z_s(t)\|\le \alpha e^{{-\eta (t-t_1)}} \|z_s(t_1)\|+ \frac{\alpha
\varepsilon}{\eta} ,\quad  \forall \; t\ge t_1\,.
$$
Hence $z(t)=z_u(t)+z_s(t)\to 0, \quad \text{as }  t\to+\infty$.


\begin{remark} \label{rks1} \rm

(1) Note that the  feedback \eqref{etrho} is a bounded function
in time and is uniformly bounded with
respect to  initial states
$$
|v(t)| \le \rho \|B\|, \quad  \forall t\ge 0, \;\forall z_0\in H\,.
$$

(2) For finite-dimensional systems, the conditions  \eqref{adbal}
and \eqref{coerejc}  are equivalent (see \cite{chen, quin}).
However, in infinite-dimensional
case and  if $B$ is compact, then the condition \eqref{coerejc}
is impossible. Indeed, if $(\varphi_j)$
is an orthonormal basis of $H$, then applying \eqref{coerejc}
for $y=\varphi_j$ and using the fact that
$\varphi_j \rightharpoonup 0, $ as $j\to +\infty$,
we obtain the contradiction $\delta=0$.


(3) In \cite{Bal1} (resp. \cite{ouz4}), the case $H_u=H$ has
been considered and it has been shown that \eqref{feedbal}
(resp. \eqref{et}) guarantees the weak stability of \eqref{SS}.

(4) Since $\|z(t)\| $ decreases, then we have  $\exists t_0 \ge 0$;
$z(t_0)=0  \Leftrightarrow z(t)=0,\; \forall t\ge t_0$.
In this case we have $ v(t)=0, \; \forall t\ge t_0$.

(5)  Note that for finite-dimensional
bilinear systems, unlike the linear case, the strong stability
is not equivalent to the exponential one (see \cite{ouz3}).

(6) We can extend the estimate \eqref{estim-bst} to the case
where $B$ is nonlinear and locally Lipschitz, so we can obtain
a semi-linear version of the above theorem.
\end{remark}


\subsection{Exponential stabilization}

In this section, we will associate between the exponential
stabilizability of the whole system \eqref{SS} and the one of its
unstable part. In the sequel we suppose that \eqref{bu} holds,
 so that the system \eqref{SS} can be decomposed
in the  following two subsystems
\begin{gather}\label{s1}
\frac{d z_u(t)}{d t}= A_uz_u(t) + v(t)B_uz_u(t), \quad
z_u(0)=z_{0u}\in H_u, \\
\label{s2}
\frac{d z_s(t)}{d t}= A_sz_s(t) + v(t)B_sz_s(t),
\quad z_s(0)= z_{0s} \in H_s,
\end{gather}
in the state spaces $H_u$ and $H_s$ respectively.

In the following result, we study the exponential stabilizability
of system \eqref{SS}.


\begin{theorem} \label{thm2}
 Let (i) $A$ generate a linear $C_0-$semigroup $S(t)$ such
that $S_u(t)$ is of isometries and \eqref{sga} holds,
(ii)  $A$ allow the decomposition \eqref{H} of $H$ with
 $\dim H_u < +\infty$,
(iii)   $B \in \mathcal{L}(H)$ such that for all $y_u\in H_u$,
we have
\begin{equation}\label{balu}
\langle B_u e^{tA_u}y_u,e^{tA_u}y_u\rangle=0, \quad
 \forall t \ge 0 \Rightarrow y_u=0\,.
\end{equation}
Then there exists $\rho >0$ such that  the feedback
\begin{equation} \label{eut}
v_u(t)=-\rho \frac{\langle z_u(t),B_uz_u(t)\rangle}
{\|z_u(t)\|^2}\mathbf{1}_{E_u},
\end{equation}
where $E_u=\{t\ge 0: z_u(t)\ne 0\}$, exponentially
stabilizes \eqref{SS}.
\end{theorem}

\begin{proof}
 Let us consider  the  system
\begin{equation}\label{SNL}
\frac{d z_u(t)}{d t}= A_uz_u(t) +
f_\rho(z_u(t))B_uz_u(t),\;z_u(0)=z_{0u},
\end{equation}
where
$$
f_\rho(z_u)= \begin{cases}
-\rho \frac{\langle z_u,B_uz_u\rangle }{\|z_u\|^2}, &z_u\ne 0 \\
 0, &z_u=0\\
\end{cases}
$$
The system \eqref{SNL} possesses a unique  mild
solution $z_u(t)$ defined for all $t\ge 0$, and the map
$T_u(t)z_{0u}=z_u(t)$ defines a nonlinear
semigroup on $H_u$.

Integrating the inequality over the interval $[k,k+1]$,
$k\in \mathbb{N}$,
\begin{equation}\label{cv2}
\frac{d \|z_u(t)\|^2}{dt} \le -
2|f_\rho(z_u(t))|^2\|z_u(t)\|^2,
\end{equation}
we obtain
$$
\|z_u(k+1)\|^2-\|z_u(k)\|^2\le
-2\int_k^{k+1}|f_\rho(z_u(\tau))|^2\|z_u(\tau)\|^2d\tau
$$
and since $\|z_u(t)\|$ decreases, we deduce that
\begin{equation}\label{zk}
\|z_u(k+1)\|^2-\|z_u(k)\|^2\le
-2\|z_u(k+1)\|^2\int_k^{k+1}|f_\rho(z_u(\tau))|^2d\tau\,.
\end{equation}
Let $\delta :=2 \inf_{\|z_{0u}\|=1}\int_0^{1}|f_\rho(T_u(\tau)z_{0u})
|^2d\tau \ge 0$. Since $B_u$ is linear, then
$f_\rho(\lambda z_u)= f_\rho(z_u), \forall \lambda\in \mathbb{C},
z_u\in H_u$.
Then by an argument of uniqueness of the mild solution,
we deduce that $T_u(t)(\lambda z_u)=\lambda T_u(t)z_u,\; \forall t\ge 0, \lambda\in \mathbb{C}, z_u\in H_u$. It follows that $2\int_0^{1}|f_\rho(T_u(\tau)z_{0u})|^2d\tau\ge \delta, \; \forall z_{0u}\in H_u-\{0\}$.
Using the superposition property of the semigroup $T_u(t)$ and the fact that $\|T_u(t)z_{0u}\|\le \|z_{0u}\| $ we
deduce that $ 2\int_k^{k+1}|f_\rho(T_u(\tau)z_{0u})|^2d\tau\ge
\delta, \; \forall k\in \mathbb{N}$. Then \eqref{zk} implies
$$
\|z_u(k+1)\|^2-\|z_u(k)\|^2\le -\delta \|z_u(k+1)\|^2\,.
$$
It follows that
$$
\|z_u(k)\|\le \frac{\|z_{0u}\|}{(1+\delta)^{\frac{k}{2}}}\,.
$$
Finally, using the fact that $\|z_{u}(t)\|$ decreases,
we deduce that
$$
\|z_{u}(t)\|\le \|z_{u}(k)\|\le Me^{-\sigma t}\|z_{0u}\|,
$$
where $M=e^{\frac{\ln (1+\delta)}{2}}$ and $\sigma = \frac{\ln
(1+\delta)}{2}$.
Let us show that $\delta >0$. Assume that $\delta =0$.
Since $\int_0^{1}|f_\rho(T_u(\tau)z_{0u})|^2d\tau$ depends
continuously on $z_{0u}$ and $\dim H_u < +\infty$,
 then there exists $y_u \in H_u$ such that
$ \|y_{u}\|=1$ and $ \int_0^{1}|f_\rho(T_u(\tau)y_{u})|^2d\tau=0$.
Then $f_\rho(T_u(t)y_{u})=0, \; \forall 0\le t\le 1$,
which imply that $T_u(t)y_{u}=S_u(t)y_{u}, \; \forall 0\le t\le 1$
and hence $<B_uS_u(t)y_{u},S_u(t)y_{u}\rangle=0, \;\forall 0\le t\le 1$.
 Now from \cite{chen, quin}, we have
\[
\int_0^1|<B_uS_u(t)z_{u},S_u(t)z_{u}>| dt
\ge \delta \|z_{u}\|^2,\; \forall z_{u}\in H_u,
\]
 which gives the contradiction $y_u=0$.

For the component $z_s(t)$, we shall show that $z_s(t)$ is defined
for all $t\ge 0$ and exponentially converges to $0$, as $t\to
+\infty. $ The system \eqref{SS}, excited by the control \eqref{eut},
admits a unique mild solution defined for all  $\; t $ in a
maximal interval $ [0,t_{\rm max}[$ by
$$
z(t) = S(t)z_0 +\int_0^t v_u(\tau )S(t-\tau)Bz(\tau)d\tau\,.
$$
Thus
\begin{equation}\label{zs}
z_s(t)=S_s(t)z_{0s}+\int_{0}^t{v_u(\tau )S_s(t-\tau
)B_sz_s(\tau )}d\tau, \quad \forall t\in [0,t_{\rm max}[\,.
\end{equation}
It follows from \eqref{01} and \eqref{zs} that
$$
\|z_s(t)\|  \le \alpha \:e^{-\eta t}\|z_{0s}\|+ \alpha L
 \int_{0}^t{e^{-\eta (t-\tau)}|v_u(\tau)|\|z_s(\tau
 )}\|d\tau
$$
 for all $t\in [0,t_{\rm max}[$ with $L=\|B\|$.
Then the scalar function $y(t)=\|z_s(t)\| \:e^{\eta t}$ satisfies
$$
y(t)   \le  \alpha \|z_{0s}\|+ \rho \alpha L^2
 \int_{0}^{t}y(\tau)d\tau\,.
$$
Gronwall inequality then yields
$ y(t)\le  \alpha \|z_{0s}\|e^{\rho \alpha L^2 t}$.
In other words,
\begin{equation}\label{estimzs}
\|z_s(t)\| \le  \alpha \|z_{0s}\|
 e^{(\rho \alpha L^2-\eta )t}.
\end{equation}

Taking  $\rho>0$ such that $\rho \alpha L^2-\eta < 0$, it follows
that $z_s(t)$ is bounded on  $ [0,t_{\rm max}[$ so that
$t_{\text{max}}=+\infty$, and  the estimate \eqref{estimzs} holds
for all $t\ge 0$.
We conclude that
\begin{equation}\label{z-exp}
\|z(t)\| \le N  e^{-\beta t}\|z_0\|, \quad \forall t\ge 0,
\end{equation}
where $N>0$ and $\beta=\min(\sigma,\eta-\rho \alpha
L^2)>0$.
\end{proof}


\begin{remark} \label{rm3.2} \rm

(1)  The feedback \eqref{eut} depends only on the
unstable part $z_u(t)$ and is uniformly bounded with respect to
initial states.

(2) The quadratic control \eqref{feed} does not guarantee
the exponential stability.

(3) We note that  \eqref{balu} is weaker than both  \eqref{adbal}
and \eqref{coerejc} .

(4) The rate of exponential convergence $\beta$ (given in \eqref{z-exp}) can be explicitly expressed, and from \cite{ouz4},
one can calculate the parameter $\rho$ corresponding to the
optimal value of $\beta$.

(5) If $H=H_u$ is of finite-dimension, then we retrieve  the result of
\cite{chen}.

(6) In the case $\dim H_u =+\infty$ or if $B$ is nonlinear and
locally Lipschitz, then following the techniques used in \cite{ouz4},
we can obtain  the result of  Theorem \ref{thm2}
if \eqref{balu} is changed to
$$
\int_0^T |\langle B_u e^{tA_u}y_u,e^{tA_u}y_u\rangle| dt
\ge \delta \|y_u\|^2, \quad \forall y_u
\in H_u, \;  (T, \delta> 0)\,.
$$


(7) It is easily verified that the condition \eqref{bu} holds
in the case of commutative systems (i.e. $B$ commutes with $A$).

(8) Note that the condition \eqref{bu} also holds in the
case  $H_u=H$. This special case has been treated in \cite{ouz4}.
\end{remark}


\section{Robustness}

In this section, we study the robustness of the
controls \eqref{feedrho}, \eqref{etrho} and \eqref{eut}
under a class of perturbations on the dynamic $A$ of \eqref{SS}.

\subsection{Strong robustness}
In this part, we consider the strong robustness of the
feedbacks \eqref{feedrho} and
\eqref{etrho}. We show that the stability property
of the system \eqref{SS} remains invariant under a certain
class of bounded perturbations. Consider the perturbed system
\begin{equation}\label{SSp1}
\frac{d z(t)}{d t}= (A+E)z(t) + v(t)Bz(t),\quad
z(0) = z_0\,.
\end{equation}
Where $E$ is a perturbation of $A$.  Let us define
\begin{align*}
\Lambda &=\big\{ E\in \mathcal{L}(H); E\text{ commutes with } P_u,
\; E=FB \text{ for some } F\in \mathcal{L}(H),\\
&\quad   A+E \text{ is dissipative and }
\|E_s\| < \frac{\eta}{\alpha}\big\}.
\end{align*}
where $\alpha, \eta$ are given by \eqref{01}.

\begin{proposition}\label{robust}
Let
\begin{itemize}
\item[(i)] $A$ generate a linear $C_0$-semigroup $S(t)$ of
contractions on $H$ such that \eqref{sga} holds,
\item[(ii)]  $B$ be self-adjoint, positive and compact such that \eqref{ad}
holds, and
\item[(iii)] $E\in \Lambda$.
\end{itemize}
Then both the feedbacks \eqref{feedrho} and
\eqref{etrho}  strongly stabilize the system
\eqref{SSp1}.
\end{proposition}


\begin{proof}
 Since   $ A+E $ is dissipative,
 the operator $A+E$ generates a semigroup of contractions
$T(t)$ (see \cite{Pazy}). Let us show that \eqref{ad}
$ \Rightarrow (\langle BT(t)y,T(t)y\rangle =0$ for all
$t\geq 0 \Longrightarrow y = 0)$.
Suppose that $\langle BT(t)y,T(t)y\rangle =0$ for all $t\geq 0$.
Since $B\ge  0$, we have $B^{1/2}T(t)y=0$ and so $BT(t)y=0$ for all
$t\ge 0$. Using  the variation of constant formula, we obtain
\begin{equation}\label{ts}
T(t)y = S(t)y+
 \int_0^t S(t-s)ET(s)yds,\quad \forall y\in H,
\end{equation}
 which implies that $T(t)y = S(t)y, $ and so
$\langle BS(t)y,S(t)y\rangle > =0$ for all $t\geq 0$.
It follows from \eqref{ad} that $y=0$.

Taking $y=z_{0s}$ in \eqref{ts} and using the fact that $E$
commutes with $P_u$, we obtain the formula
$$
T_s(t)z_{0s} = S_s(t)z_{0s} +  \int_0^t S_s(t-s)E_sT_s(s)z_{0s}ds,
$$
where $T_s(t)$ is the restriction of $T(t)$ to $H_s$. Then,
using the fact that $S_s(t)$ verifies \eqref{01}, we obtain
$$
\|T_s(t)z_{0s}\| \le  \alpha \|z_{0s}\| \exp({-\eta t}) +
 \alpha\|E_s\|\int_0^t \exp({-\eta (t-s)})\|T_s(s)z_{0s}\|ds\,.
$$
Applying the Gronwall's inquality, we deduce that
$$
\|T_s(t)z_{0s}\| \le \alpha\exp({(\alpha\|E_s\|-\eta)
t})\|z_{0s}\|\,.
$$
Taking $\|E_s\|  <  \frac{\eta}{\alpha}$,
we conclude that  $T_s(t)$ satisifes \eqref{01}.
The result of Theorem \ref{thm1} completes the proof.
\end{proof}

\subsection{Exponential robustness}

Consider the perturbed system
\begin{equation}\label{SSp}
\frac{d z(t)}{d t}= (A+E)z(t) + v(t)Bz(t),\quad
z(0) = z_0,
\end{equation}
where $E$ is a perturbation of $A$. In this  part, we consider the
robustness of the feedback control law \eqref{eut}.

Let us define the set of admissible operator of perturbations
\begin{align*}
\Lambda &= \big\{E\in \mathcal{L}(H); E \text{ commutes with } P_u,\;
E_u=B_uF_u, \text{ for some } F_u\in \mathcal{L}(H_u),\\
&\quad A_u+E_u \text{ is dissipative and } \|E_s\|  \le
\epsilon\frac{\eta}{\alpha},\; (0< \epsilon <1)\big\}.
\end{align*}

\begin{proposition}\label{robust2}
Let \begin{itemize}
\item[(i)] $A_u$ generates a linear $C_0-$semigroup of isometries
$S_u(t)$  on $H$ such that \eqref{sga} holds,
\item[(ii)]  $B\in \mathcal{L}(H)$ and $B_u$ is self-adjoint,
positive such that \eqref{balu} holds, and
\item[(iii)] $E\in \Lambda$.
\end{itemize}
Then there exists  $\rho >0$ such
that the feedback \eqref{eut} is exponentially robust.
\end{proposition}


\begin{proof}
 Since $E$ commutes with $P_u$,  the perturbed system
\eqref{SSp} can be decomposed as
$$
\frac{d z_u(t)}{d t}= (A_u+E_u)z_u(t) +
v(t)B_uz_u(t), \quad  z_u(0)=z_{0u}\in H_u,
$$
and
\begin{equation}\label{zps}
\frac{d z_s(t)}{d t}=
(A_s+E_s)z_s(t) + v(t)B_sz_s(t), \quad z_s(0)= z_{0s} \in H_s.
\end{equation}
Using the same techniques as in the proof of  Proposition \ref{robust},
we can show that
$$
\langle B_uT_u(t)y_u,T_u(t)y_u\rangle =0,\;
 \forall \;t\geq 0 \Longrightarrow y_{u} = 0
$$
and
$$
 \|T_s(t)z_{0s}\| \le \alpha\exp({(\alpha\|E_s\|-\eta)t})\|z_{0s}\|\,.
$$
We conclude that  $T_s(t)$ verifies \eqref{01} and hence the
solution of \eqref{zps} verifies the estimate \eqref{estimzs}.
Taking  $\rho>0$ such that $\rho < (1-\epsilon)\frac{\eta}{\alpha
\|B\|^2}$, we deduce that the feedback
law \eqref{eut} exponentially stabilizes
the perturbed system.
\end{proof}

\section{Applications}

Let us consider the system defined by
\begin{equation} \label{lap}
 \begin{gathered}
\frac{\partial z(x,t)}{\partial t} =
 \frac{\partial^2z(x,t) }{\partial x^2} + v(t)
 Bz(t), \quad \forall x\in ]0,1[, \;\forall t>0,\\
 z'(0,t) = z'(1,t)= 0, \quad \forall t>0
 \end{gathered}
\end{equation}
where  the state space is $H=L^2(0,1)$, the operator is
$Az = \frac{\partial^2 z}{\partial x^2}$,
for
$z\in \mathcal{D}(A) = \{ z\in H^2(0,1 ):z'(0) = z'(1) = 0\}$.
The  spectrum of $A$ is given by the simple eigenvalues
$\lambda_j = -\pi^2(j-1)^2$, $j\in \mathbb{N}^*$ and eigenfunctions
$\varphi_1 (x) = 1$ and $\varphi_j (x) = \sqrt{2}\cos ((j-1)\pi
x)$ for all $j \ge 2$.


For the operator of control $B$, we  consider two  situations:

\textbf{Case 1:} $B=I$. This case has been considered in \cite{ouz3}
to obtain  strong stabilizability  of \eqref{lap} using the
quadratic feedback \eqref{feed}. However, in this way one does not
obtain a better convergence than of order of $1/\sqrt{t}$. In the
following, we show that the feedback \eqref{eut} ensures the
exponential stability. In the case
$\int_0^1z(x,t)dx=0$, we have $z_u(t)=0$ for all $t\ge 0$,
then  $v_u(t)=0$ and $z(t)=z_s(t)=S_s(t)z_{0s}$.

Let us suppose that $\int_0^1z(x,t)dx\ne 0$. Here we
can take $\eta=\alpha=1$. Then applying the result of
Theorem \ref{thm2}, we deduce that for all $0< \rho < 1$, the control
$v_u(t)= - \rho$ achieves the exponential stabilization of
\eqref{lap} with the rate of convergence
$\beta=\min(\frac{1}{2}\ln(\frac{11}{9}),1-\rho)$.

Note that $z_u(t)$ can be directly expressed
 $z_u(t)= e^{-t}z_{0u},\; \forall t\ge 0$.
This  shows that the rate of exponential convergence $\beta$
can be improved since we have
 $1 > \min(\frac{1}{2}\ln(\frac{11}{9}),1-\rho)$.

Now let us examine the robustness of the control \eqref{eut}.
 Let us reconsider the above system with the
perturbation $Ez=\epsilon (z-\int_0^1z(x)dx)$,
$0<\epsilon<1$.  The perturbed open-loop
system remains unstable.  However, for all $0< \rho < 1-\epsilon$,
the control $v(t)= -\rho $ exponentially stabilizes the perturbed
system.


\textbf{Case 2:}
$Bz = \sum_{j=1}^{+\infty}\alpha_j<z$, $\varphi_{j}>\varphi_{j}$,
where $\alpha_j \ge 0$ for all $j\ge 1$ and
$\sum_{j=1}^{+\infty}\alpha^2_j <\infty$.
This case was considered in \cite{ouz4}, where it has been showed
that if, $\alpha_j > 0$ for all $j\ge 1$ then  \eqref{et}
weakly stabilizes \eqref{lap}. Here we show that the stability is
in the strong sense. Clearly $B$ is a linear and compact operator,
and from the relation
$$
\langle BS(t)y,S(t)y\rangle=\sum_{j=1}^{+\infty}\alpha_j|
\langle z,\varphi_{j}\rangle |^2,
$$
 we can see that  \eqref{ad} holds if $\alpha_j > 0$ for all
$j\ge 1$.
In this case, Theorem \ref{thm1} applies and the system
\eqref{lap} is strongly stabilizable using  the
controls \eqref{feedrho} and \eqref{etrho}; i.e.,
$$
v_1(t)= -\rho\sum_{j=1}^{+\infty}\alpha_j|
\langle z(\cdot,t),\varphi_{j}\rangle|^2
$$
and
$$
v_2(t)= \frac{-\rho}{\|z(\cdot,t)\|^2}
\sum_{j=1}^{+\infty}\alpha_j|\langle z(\cdot,t),\varphi_{j}\rangle|^2,
\quad \forall z_0\ne 0\,.
$$
Note that \eqref{coerejc} does not hold, so the existing result of
\cite{ouz4} is not applicable to establish exponential
stabilization by the control \eqref{et}.

It is clear that we can  apply the result of Theorem
\ref{thm2} to deduce that if $\alpha_1>0$ and
$0< \rho < 1/\alpha_1$, then the control defined by
$v_u(t)= -\alpha_1 \rho$, for all $z_0$ such that
$z_{0u}\ne 0$ ensures the exponential stabilizability
of \eqref{lap} with the rate of convergence
$\beta=\min(\ln(11/9)/2,1-\rho)$.
With  the perturbation
$$
Ez=\epsilon(z-\int_0^1z(x)dx), \quad 0<\epsilon\le 1
$$
on the open-loop system, both  controls \eqref{feedrho}
 and \eqref{etrho} are strongly robust,
and for $0<\epsilon<1$, the control \eqref{eut} is
exponentially robust.

\subsection*{Conclusion}
In this work we have considered the problem  of strong
stabilization of a constrained parabolic bilinear system under
the conventional ad-condition \eqref{adbal}. Under a weaker
condition than  \eqref{coerejc}, an exponential stabilization
result has been established. Also the question of robustness
of the stabilizing controls is discussed.

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\end{document}