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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 131, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/131\hfil Control of switched systems]
{$H_{\infty}$ control of switched linear parabolic systems}

\author[L. Bao, S. Fei, L. Chai \hfil EJDE-2014/131\hfilneg]
{Leping Bao, Shumin Fei, Lin Chai}  % in alphabetical order

\address{Leping Bao \newline
School of Automation, Southeast University, Nanjing, Jiangsu 210096, China.\newline
Department of Automation, Taiyuan Institute of Technology, Taiyuan,
Shangxi 030008, China}
\email{jsll68@126.com}

\address{Shumin Fei \newline
School of Automation, Southeast University, Nanjing, Jiangsu 210096, China}
\email{smfei@seu.edu.cn}

\address{Lin Chai \newline
School of Automation, Southeast University, Nanjing, Jiangsu 210096, China}
\email{Chailin-1@163.com}

\thanks{Submitted August 16, 2013. Published June 10, 2014.}
\subjclass[2000]{34K20, 35R15, 34D10}
\keywords{Switched systems; parabolic systems; $H_{\infty}$ control;
\hfill\break\indent linear matrix inequalities; average dwell time}

\begin{abstract}
 The $H_{\infty}$ control problem of a class of switched linear parabolic
 systems is considered. By applying the multiple Lyapunov function
 method and the average dwell time scheme, sufficient conditions for exponential
 stability and the $H_{\infty}$ control synthesis are established
 in terms of LMIs and a family of switching signals. The advantage in this work
 lies in the fact that sufficient conditions completely depend on the system
 parameters and the system can be analyzed by using numerical softwares.
 At the end of the paper, an example is given to illustrate the obtained result.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

During the previous decade, the study of switched systems
has attracted considerable attention because of its significance in both
theoretical research and practical applications 
\cite{a1,c1,d1,l1,l2,n1,p2,q1,z1,z2}.
A switched system is a dynamical system described by a family of 
continuous-time subsystems and a
rule that governs the switching among them.
In many realistic cases, switched systems can be described by partial
differential equations (PDE) or a combination of ordinary differential
equations (ODE) and PDE such as in chemical industry
process and biomedical engineering. We refer to these switched systems as 
distributed parameter switched systems (DPSS) or infinite dimensional 
switched systems \cite{c3,l4}.
However, there are very few works concerning DPSS (see, eg. 
\cite{a1,d2,h1,m1,o1,p3,s1} and the references cited therein). 
For example, Sasane generalized the results presented in
\cite{n1} to infinite dimensional Hilbert spaces \cite{s1}, and showed when
 all subsystems are stable and commutative pairwise, the switched linear system
is stable under an arbitrary switching via the common  Lyapunov function. 
Michel and Sun provided the stability conditions for switched nonlinear
systems on Banach spaces under constrained switching \cite{m1}. 
Hante and Sigalotti  gave necessary and sufficient conditions
in term of the existence of common Lyapunov functions for
DPSS \cite{a1,d2}. It seems that the majority of works deal with the 
stability of DPSS.

The $H_{\infty}$ control is an interesting research topic in the field of
switched systems. Up to now, most of results in the literature are regarding 
the $H_{\infty}$ control of switched systems which
 are described by ODEs \cite{d1,l3,z2}. For example, the stability, the $L_2$-gain
analysis and the $H_{\infty}$ control for switched systems via the 
multiple Lyapunov-like function methods is considered in \cite{z2}. 
The dynamic output feedback in $H_{\infty}$ control design of switched linear 
systems are studied  in terms of linear matrix inequalities (LMIs) in \cite{d1}.  
To the best of our knowledge, the $H_{\infty}$ control has not been 
investigated for DPSS.

Motivated by the above consideration, we study the $H_{\infty}$ control
 synthesis for switched linear parabolic systems in this paper.  
The main contributions of the present paper can be summarized as follows: 
Firstly, the concept of $H_{\infty}$ control is extended
to DPSS. Secondly, sufficient conditions for the exponential stabilization 
and the $H_{\infty}$ control
synthesis of DPSS are developed in terms of LMIs and a class of switching signals. 
Compared with the work in \cite{d2}, our sufficient conditions completely depend on 
the system parameters.

In this article, $L_2(\Omega, R^{n})$ denotes the Hilbert space of square
integrable $n$ dimensional vector-valued functions
$\nu(x), x\in\Omega$ with the norm $\|\nu\|_{L_2}=\int_{\Omega}\nu^T\nu dx$.
$L_2[t_0,\infty; L_2(\Omega, R^{n}))$ is the Hilbert space of square
integrable functions $\nu(t,\cdot)\in L_2[t_0,\infty)$ with values
$\nu(\cdot,x)\in L_2(\Omega, R^{n})$. $H^2(\Omega, R^{n})$ and
$H_0^2(\Omega, R^{n})$ denote the classical Sobolev spaces defined by
$H^2(\Omega, R^{n})=\{\nu\in L_2(\Omega, R^{n})
:\frac{\partial^2\nu}{\partial x^2}\in L_2(\Omega, R^{n})\}$ and
$H_0^1(\Omega, R^{n})=\{\nu\in L_2(\Omega, R^{n}):
\frac{\partial \nu}{\partial x}\in L_2(\Omega, R^{n}),\nu(\partial\Omega, t)=0\}$
respectively.
$\gamma_{M}(P)(\gamma_{m}(P))$ denotes the largest (smallest) eigenvalue of $P$. 
The symmetric elements of the matrix will be denoted by $T$.

\section{Problem formulation and preliminaries}

Consider the  switched linear parabolic systems
\begin{equation}
\begin{gathered}
\frac{\partial \nu(x,t)}{\partial t}=D_{\sigma(t)}\Delta\nu(x,t)
+A_{\sigma(t)}\nu(x,t)+B_{\sigma(t)}u(x,t)
+C_{\sigma(t)}\omega(x,t)\\
y(x,t)=E_{\sigma(t)}\nu(x,t)+F_{\sigma(t)}\omega(x,t)\\
\nu(t_0)=\nu_0\\
\nu(x,t)=0,\quad  (x,t)\in \partial\Omega\times [t_0, +\infty)
\end{gathered}\label{e1}
\end{equation}
with the static state feedback control
\begin{equation}
u(x,t)=K_{\sigma(t)}\nu(x,t)\label{e2}
\end{equation}
where $\nu(x,t)\in L_2[t_0,\infty; L_2(\Omega, R^{n}))$ is a vector-valued
function representing the state of the process,
$u(x,t)\in L_2[t_0,\infty; L_2(\Omega, R^{s}))$ denotes the manipulated
input, $\omega(x,t)\in L_2[t_0,\infty; L_2(\Omega, R^{p}))$
is the disturbance, and $y(x,t)\in L_2[t_0,\infty; L_2(\Omega, R^{q}))$
denotes the measured output with $(x,t)\in \Omega\times [t_0,+\infty)$.
 $\Omega=[0,\sqrt{2}]\times[0,\sqrt{2}]\subset R^2$ is a bounded domain with
the smooth boundary. $\Delta$ denotes the Laplace operator; i.e.,
 $\Delta=\sum^2_{k=1}\frac{\partial^2}
{\partial x_k^2}$. $D_i=\operatorname{diag}(d_{i1},d_{i2}, \dots, d_{in})$ represent
positive diagonal matrices, and $A_i, B_i, C_i,  E_i, F_i$ $(i=1, 2, \dots, m)$
represent constant matrices of compatible dimensions.
$\sigma(t): [t_0,\infty)\to \Theta$ is the switching signal mapping
time to some finite index set $\Theta=\{1,2,\dots ,m\}$, and the switching
signal $\sigma(t)$ is a piecewise continuous (from the right) function depending
on time or state or both. The discontinuities of $\sigma(t)$ are called switching
times or switching instants. The integer $m$ is the number of models
(called subsystems) of the switched system.

The objective of this article is to establish sufficient conditions of 
the $H_{\infty}$ control for the system \eqref{e1}--\eqref{e2}. 
That is,  we look for  controller gain matrices $K_i\ (i\in \Theta)$ and 
a class of switching signals $\sigma(t)$, such that
\begin{itemize}
\item[1.] When $\omega=0$,  the system \eqref{e1}
 is exponentially stabilized by the state feedback control \eqref{e2}.

\item[2.] The system \eqref{e1} is exponentially stabilized by \eqref{e2} 
with the $H_{\infty}$ disturbance level $\gamma>0$, i.e., for a prescribed 
scalar  $\gamma>0$, the performance index is
$$
J(\omega)=\int_{t_0}^{\infty}\int_{\Omega}[ y^T(x,s)y(x,s)-\gamma^2
\omega^T(x,s)\omega(x,s)]\,dx\,ds\leq0
$$  
for all non-zero $\omega(x,t)\in L_2[t_0,\infty; L_2(\Omega, R^{p}))$
under the zero initial condition.
\end{itemize}
The following is the definition of average dwell time (ADT) \cite{l1}.

\begin{definition} \label{def2.1} \rm
Given some family of switching signals $\sigma(t)\in \Theta$, for each 
$\sigma(t)$  and each $t>s\geq t_0$, let
$N_\sigma(s,t)$ denote the number of switching of $\sigma(t)$
in the open interval $(s,t)$. If
$N_\sigma(s,t)\leq N_0+\frac{t-s}{\tau_a}$ holds for $\tau_a>0$ and $N_0> 0$, 
then the positive constant  $\tau_a$ is called the ADT and $N_0$ 
is the chatter bound.
\end{definition}

\begin{lemma}[Poincare's inequality \cite{c2}]  \label{lem2.1}
Let the scalar function $u\in H_0^1(\bar{\Omega}, R)$ with
$\Omega\subseteq \Omega_1$, then we have
\begin{equation}
\int_{\Omega} u^2dx \leq \gamma^2\int_{\Omega}\sum^{n}_{i=1}
(\frac{\partial u}{\partial x_i})^2dx
=\gamma^2\int_{\Omega} \mid \nabla u\mid^2dx \label{e3}
\end{equation}
where $\Omega_1: 0 \leq x_i \leq \delta (i=1, 2, \dots, n)$,
$\gamma=\frac{\delta}{\sqrt{n}}$, and
$\nabla=(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\dots,
\frac{\partial}{\partial x_{n}})$.
\end{lemma}

\section{Stabilization analysis of switched parabolic systems}

In this section, we consider the exponential stabilization problem of the 
switched linear parabolic system
\begin{equation}
\begin{gathered}
\frac{\partial \nu(x,t)}{\partial t}=D_{\sigma(t)}\Delta\nu(x,t)
+A_{\sigma(t)}\nu(x,t)+B_{\sigma(t)}u(x,t)\\
\nu(t_0)=\nu_0\\
\nu(x,t)=0,\quad  (x,t)\in \partial\Omega\times [t_0, +\infty)
\end{gathered}\label{e4}
\end{equation}
with the static state feedback control \eqref{e2}.

We assume that the state of system does not jump at switching instants; 
i.e., the state trajectory is continuous and the
switching signal $\sigma(t)$ has the finite switching number in any finite 
time interval \cite{l1}.

We start with the well-posedness problem of the closed-loop system 
\eqref{e2}-\eqref{e4}. Define the state function $z(t)$ as
$z(t)=\nu(\cdot,t)$ on the Hilbert space $H=L_2(\Omega, R^{n})$ with the norm
$\|\cdot\|_{L_2}$,
then the equation of closed-loop \eqref{e2} and \eqref{e4} can be rewritten as
\begin{equation}
\dot{z}(t)=\widetilde{A}_{\sigma(t)} z(t)+f_{\sigma(t)}(t), \quad
t\geq t_0\label{e5}
\end{equation}
in $H$, where the infinitesimal operators $\widetilde{A}_i$
($\widetilde{A}_ix=D_i\Delta x$) have the dense domain
$W=D(\widetilde{A}_i)=\{\nu\in H^2(\Omega, R^{n})\bigcap H_0^1(\Omega, R^{n}):
\nu(x)=0, x\in \partial \Omega\}$,
 $f_{\sigma(t)}(t)=[A_{\sigma(t)} +B_{\sigma(t)}K_{\sigma(t)}]z(t)$.

As we know, the infinitesimal operators $\widetilde{A}_i$ generate analytical 
semigroups $T_i(t)$ \cite{p1}. Because the state of system \eqref{e2}-\eqref{e4}
 does not jump at switching instants,
for every initial value $\nu_0\in W$, there exists a unique solution 
for system \eqref{e2}-\eqref{e4}. Thus, the initial problem 
\eqref{e2}-\eqref{e4} turns out to be well-posed on the time interval
 $[t_0,\infty)$.

\begin{lemma} \label{lem3.1}
 For the given scalar $\lambda>0$, if there exist positive constants
 $\alpha_i,\beta_i$, a constant $\mu\geq 1$, and continuous functions
 $V_i\in C(H\times [t_0, +\infty), R^{+})$  such that the functions 
$V_i(t)=V_i(\nu,t)$ are absolutely continuous along the solutions $\nu$ 
of system \eqref{e2}-\eqref{e4} and satisfy
\begin{gather}
\alpha_i\|\nu(t)\|_{L_2} \leq V_i(t) \leq \beta_i\|\nu(t)\|_{L_2}\label{e6}\\
\dot{V_i}(t)+\lambda V_i(t)\leq 0.\label{e7}
\end{gather}
Furthermore, the Lyapunov function  of the system satisfies
\begin{equation}
V_i(t)\leq \mu V_{j}(t),\quad  \forall i,j\in \Theta. \label{e8}
\end{equation}
Then the closed-loop system \eqref{e2}-\eqref{e4} is exponentially stable
for the arbitrary switching signal $\sigma(t)$ with the ADT
$\tau_{a}>\frac{ln\mu}{\lambda}$.
\end{lemma}

\begin{proof}
For $ t\in [t_k,t_{k+1})(k=0,1\dots)$,  from \eqref{e7} it follows that
$$
V_{\sigma(t)}(t)\leq e^{-\lambda(t-t_k)}V_{\sigma(t_k)}(t_k).
$$
This, together with \eqref{e8} gives
$$
V_{\sigma(t)}(t)\leq \mu V_{\sigma(t^{-}_k)}(t^{-}_k)e^{-\lambda(t-t_k)}.
$$
It is easy to show that
\begin{align*}
&V_{\sigma(t)}(t)\leq \mu V_{\sigma(t^{-}_k)}(t^{-}_k)e^{-\lambda(t-t_k)}
\leq \mu V_{\sigma(t_{k-1})}(t_{k-1})e^{-\lambda(t-t_k)}e^{-\lambda(t_k-t_{k-1})}\\
&\leq \mu^2 V_{\sigma(t^{-}_{k-1})}(t^{-}_{k-1})e^{-\lambda(t-t_{k-1})}\leq\dots\dots
\leq \mu^{k} V_{\sigma(t_0})(t_0)e^{-\lambda(t-t_0)}
\end{align*}
for all $t\geq t_0$ and a constant $\mu\geq 1$.

Note that when $k{\tau_{a}} \leq t-t_0$, we have
\begin{equation}
V_{\sigma(t)}(t)\leq e^{-\lambda(t-t_0)}e^{kln\mu} V_{\sigma(t_0})(t_0)
\leq e^{-(\lambda-\frac{ln\mu}{\tau_{a}})(t-t_0)}V_{\sigma(t_0})(t_0).
\label{e9}
\end{equation}
Combing \eqref{e6} and \eqref{e9}, we obtain
$$
\|\nu(t)\|_{L_2}\leq \frac{V_{\sigma(t)}(t)}{\alpha}
\leq \frac{1}{\alpha}e^{-(\lambda-\frac{ln\mu}
{\tau_{a}})(t-t_0)}V_{\sigma(t_0})(t_0).
$$
Thus we have
$$
\|\nu(t)\|_{L_2}\leq \frac{1}{\alpha}e^{-(\lambda-\frac{ln\mu}{\tau_{a}})(t-t_0)}
\cdot\gamma\|\nu_0\|_{L_2}\leq \frac{\gamma}{\alpha}e^{-(\lambda
-\frac{ln\mu}{\tau_{a}})(t-t_0)}
\|\nu_0\|_{L_2}
$$
where $\gamma=\max_{i \in \Theta}\{\beta_i\}$ and
$\alpha=\min_{i \in \Theta}\{\alpha_i\}$. Let $h=\lambda-\frac{ln\mu}{\tau_{a}}>0$,
and we have $\tau_{a}>\frac{ln\mu}{\lambda}$.
It is obvious that the system \eqref{e2}-\eqref{e4} is exponentially stable
for the arbitrary switching signal $\sigma(t)$ with the ADT
$\tau_{a}>\frac{ln\mu}{\lambda}$.
\end{proof}

\begin{theorem} \label{thm3.1}
For the given scalar  $\lambda>0$, if there exist diagonal matrices $X_i>0$, 
and matrices $Y_i>0$ such that
\begin{equation}
\Pi_i=-2D_iX_i+A_iX_i+X_i^TA_i^T
+B_iY_i+Y_i^TB_i^T+\lambda X_i<0.\label{e10}
\end{equation}
Then  system \eqref{e4} can be exponentially stabilized by the state feedback
control \eqref{e2} with $K_i=Y_iX_i^{-1}$ for the
arbitrary switching signal $\sigma(t)$ with the ADT
$\tau_{a}>\ln(\mu)/\lambda$, where $\mu$ is determined by
\begin{equation}
\mu=\max_{\forall k,l\in \Theta} \big\{\frac{\gamma_{M}(X_k)}{\gamma_{m} (X_{l})}
\big\}.\label{e11}
\end{equation}
\end{theorem}

\begin{proof}
Choose the multiple Lyapunov function for the system \eqref{e2}-\eqref{e4}
\begin{equation}
V(t)=V_{\sigma(t)}(t)=\int_{\Omega}\nu^T(x,t)P_{\sigma(t)}\nu(x,t)\,dx \label{e12}
\end{equation}
with constant diagonal matrices $P_i>0$. It is not difficult to see that there
exist positive numbers $\alpha_i, \beta_i$ and a constant $\mu\geq1$ such that
inequalities \eqref{e6} and \eqref{e8} hold. For inequalities \eqref{e8},
we can choose
$\mu=\max_{\forall k,l\in \Theta}\{\frac{\gamma_{M}(P_k)}{\gamma_{m}(P_{l})}\}$
\cite{z1}.

Differentiating $V(t)$ with respect to $t$ along the trajectory of the 
closed-loop system \eqref{e2}-\eqref{e4}, we have
\begin{equation}
\begin{aligned}
&\dot{V_i}(t)+\lambda V_i(t)\\
&=\int_{\Omega}[\Delta\nu(x,t)]^TD_iP_i\nu(x,t)dx
+\int_{\Omega}\nu^T(x,t)P_iD_i\Delta\nu(x,t)dx \\
&\quad +\int_{\Omega}\nu^T(x,t)[P_iA_i+P_iB_iK_i]\nu(x,t)dx\\
&\quad 
+\int_{\Omega}\nu^T(x,t)[A_i^TP_i+K_i^TB_i^TP_i]\nu(x,t)dx
+\int_{\Omega}\nu^T(x,t)\lambda P_i\nu(x,t)dx.
\end{aligned} \label{e13}
\end{equation}
Because $D_i$ and $P_i$ are positive diagonal matrices, we find that
$P_iD_i=D_iP_i$. Thus we have
\begin{align*}
&\int_{\Omega}[\Delta\nu(x,t)]^TD_iP_i\nu(x,t)dx
+\int_{\Omega}\nu^T(x,t)P_iD_i\Delta\nu(x,t)dx \\
&\leq 2\lambda_{\rm max}(P_iD_i)\int_{\Omega} \Delta\nu^T(x,t)I \nu(x,t)dx
\\
&\leq2\lambda_{\rm max}(P_iD_i)\int_{\Omega}
[\Delta\nu_1(x,t), \dots,\Delta \nu_{n}(x,t)]
\begin{bmatrix}
\nu_1(x,t)\\
\dots\\
\nu_{n}(x,t)
\end{bmatrix}\, dx\\
&\leq-2\lambda_{\rm max}(P_iD_i)\int_{\Omega}
[\nu_1(x,t)\Delta\nu_1(x,t)
+\dots+\nu_{n}(x,t)\Delta\nu_{n}(x,t)]dx.
\end{align*}
According to Gauss's divergence theorem, Poincare's inequality \eqref{e3}
and taking into account the boundary condition in \eqref{e4}, we obtain
\begin{equation}
\begin{aligned}
&\int_{\Omega}[\Delta\nu(x,t)]^TD_iP_i\nu(x,t)dx
+\int_{\Omega}\nu^T(x,t)P_iD_i\Delta\nu(x,t)dx
\\
&\leq-2\lambda_{\rm max}(P_iD_i)\int_{\Omega}[\nu_1^2(x,t)+\dots
+\nu_{n}(^2x,t)]dx\\
&\leq -2\lambda_{\rm max}(P_iD_i)\int_{\Omega} \nu^T(x,t)I\nu(x,t)dx\\
&\leq\int_{\Omega} \nu^T(x,t)(-2P_iD_i)\nu(x,t)dx<0.
\end{aligned}\label{e14}
\end{equation}
Substituting \eqref{e14} into \eqref{e13} yields
\[
\dot{V_i}(t)+\lambda V_i(t)
\leq \int_{\Omega}\nu^T(x,t)\Gamma_i\nu(x,t)\,dx
\]
where
$$
\Gamma_i=-2P_iD_i+P_iA_i+A_i^TP_i
+P_iB_iK_i+K_i^TB_i^TP_i+\lambda P_i.
$$
When
\begin{equation}
\Gamma_i=-2P_iD_i+P_iA_i+A_i^TP_i
+P_iB_iK_i+K_i^TB_i^TP_i+\lambda P_i<0,
\label{e15}
\end{equation}
we have $\dot{V_i}(t)+\lambda V_i(t)<0$  (for all $\nu(x,t)\neq0$).
If the switching signal $\sigma(t)$ satisfies the ADT
$\tau_{a}>\ln(\mu)/\lambda$, all conditions in Lemma 3.1 hold.
Hence, the closed-loop system \eqref{e2}-\eqref{e4} is
exponentially stable.

Left- and right-  multiplying \eqref{e15} by  $P_i^{-1}$ and letting 
$X_i=P_i^{-1}$ and $Y_i=K_iP_i^{-1}$,
it is not difficult to see that equality \eqref{e15} is equivalent to \eqref{e10}
 and $\mu=\max_{\forall k,l\in \Theta}\{\frac{\gamma_{M}(P_k)}{\gamma_{m}
(P_{l})}\}$ leads to \eqref{e11} immediately. Consequently, 
the proof is completed.
\end{proof}

\section{$H_{\infty}$ control synthesis}

In the section, we consider the $H_{\infty}$ control problem for 
system \eqref{e1}--\eqref{e2}.

\begin{lemma} \label{lem4.1}
For given scalars $\lambda>0$ and $\gamma>0$, if there exist diagonal matrices 
$P_i>0$ such that
\begin{equation}
\begin{bmatrix}
\Gamma_i+E_i^TE_i   & P_iC_i+E_i^TF_i\\
C_i^TP_i+F_i^TE_i &-\gamma^2I+F_i^TF_i
\end{bmatrix}
<0,\label{e16}
\end{equation}
then for any $t\in[t_k,t_{k+1})$, along the trajectory of system
\eqref{e1}-\eqref{e2}, we have
\begin{equation}
V_i(t)\leq e^{-\lambda(t-t_k)} V_i(t_k)
-\int_{t_k}^{t}\int_{\Omega}e^{-\lambda(t-s)} \Upsilon(x,s)\,dx\, ds\label{e17}
\end{equation}
where
\begin{gather*}
\Gamma_i=-2P_iD_i+P_iA_i+A_i^TP_i
+P_iB_iK_i+K_i^TB_i^TP_i+\lambda P_i,\\
\Upsilon(x,s)=y^T(x,s)y(x,s)-\gamma^2
\omega^T(x,s)\omega(x,s).
\end{gather*}
\end{lemma}

\begin{proof}
Differentiating $V_i(t)$ with respect to $t$ along the trajectory of the 
closed-loop system \eqref{e1}--\eqref{e2} and using the similar argument 
described in the previous section, we have
\begin{equation}
\dot{V_i}(t)+\lambda V_i(t)
\leq\int_{\Omega}\eta^T(x,t)
  \begin{bmatrix}
\Gamma_i   & P_iC_i\\
C_i^TP_i & 0
\end{bmatrix}
\eta(x,t)dx,\label{e18}
\end{equation}
where
$\eta^T(x,t)=[\nu(x,t), \omega(x,t)]^T$. It follows from \eqref{e16}
and \eqref{e18} that
\[
\dot{V_i}(t)+\lambda V_i(t)
<-\int_{\Omega}\eta^T(x,t)
\begin{bmatrix}
E_i^TE_i & E_i^TF_i\\
F_i^TE_i & -\gamma^2I+F_i^TF_i
\end{bmatrix}
\eta(x,t)dx
=-\int_{\Omega}\Upsilon(x,s)dx.
\]
By calculation, we have
\begin{equation}
\frac{d}{dt}(e^{\lambda t}V_i(t))<-e^{\lambda t} \int_{\Omega}\Upsilon(x,s)dx.
\label{e19}
\end{equation}
Integrating  leads to \eqref{e17}.
\end{proof}

\begin{theorem} \label{thm4.1}
For given scalars $\lambda>0$ and $\gamma>0$, if there exist diagonal 
matrices $P_i>0$ such that
\begin{equation}
  \begin{bmatrix}
\Gamma_i+E_i^TE_i & P_iC_i+E_i^TF_i\\
C_i^TP_i+F_i^TE_i & -\gamma^2I+F_i^TF_i
\end{bmatrix} <0.\label{e20}
\end{equation}
Then, the system \eqref{e1} can be exponentially stabilized by the state
feedback control \eqref{e2} with
the $H_{\infty}$ disturbance level $\gamma>0$ for the arbitrary switching
signal $\sigma(t)$ with the ADT $\tau_{a}>\frac{ln\mu}{\lambda}$,
where $\mu$ is determined by
$\mu=\max_{\forall k,l\in \Theta}\{\frac{\gamma_{M}(P_k)}{\gamma_{m}
(P_{l})}\}$.
\end{theorem}

\begin{proof} It follows from \eqref{e20} that
$$
  \begin{bmatrix}
\Gamma_i+E_i^TE_i & P_iC_i+E_i^TF_i\\
C_i^TP_i+F_i^TE_i & -\gamma^2I+F_i^TF_i
\end{bmatrix}
=  \begin{bmatrix}
\Gamma_i & P_iC_i\\
\ast     & -\gamma^2I
\end{bmatrix}
+\begin{bmatrix}
E_i^TE_i & E_i^TF_i\\
F_i^TE_i & F_i^TF_i
\end{bmatrix} <0.
$$
Since
$$
\begin{bmatrix}
E_i^TE_i & E_i^TF_i\\
F_i^TE_i & F_i^TF_i
\end{bmatrix}
=\begin{bmatrix}
E_i^T\\
F_i^T
\end{bmatrix}
\begin{bmatrix}
E_i & F_i
\end{bmatrix}
\geq 0, 
$$
we have
$$
  \begin{bmatrix}
\Gamma_i & P_iC_i\\
\ast &  -\gamma^2I
\end{bmatrix}
<0.
$$
According to the Schur complement \cite{b1}, we have $\Gamma_i<0$.
By virtue of the proof of Theorem 3.1, the closed-loop system 
\eqref{e1}--\eqref{e2} is exponentially stable when $\omega=0$.

On the other hand, combining \eqref{e8} and \eqref{e17}, we obtain
$$
V_i(t)\leq\mu e^{-\lambda(t-t_k)} V_i(t_k^{-})
-\int_{t_k}^{t}\int_{\Omega}e^{-\lambda(t-s)}
\Upsilon(x,s)\,dx\,ds.
$$
Following \cite{z1} and repeating the above procedure, we obtain
\begin{equation}
\begin{aligned}
&V_i(t)\\
&\leq\mu^{k}e^{-\lambda t}V(\nu_0)
-\mu^{k}\int_{t_0}^{t_1}\int_{\Omega}e^{-\lambda(t-s)}
\Upsilon(x,s)\,dx\,ds\\
&\quad -\mu^{k-1}\int_{t_1}^{t_2}\int_{\Omega}e^{-\lambda(t-s)}
\Upsilon(x,s)\,dx\,ds
-\dots
-\int_{t_k}^{t}\int_{\Omega}e^{-\lambda(t-s)}
\Upsilon(x,s)\,dx\,ds\\
&=\mu^{k}e^{-\lambda t}V(\nu_0)-\int_{t_0}^{t}\int_{\Omega}
e^{-\lambda(t-s)+N_{\sigma}(s,t)ln\mu}
\Upsilon(x,s)\,dx\,ds.
\end{aligned}\label{e21}
\end{equation}
Because $V_i(t)>0$, the zero initial condition implies $V(\nu_0)=0$.
Using \eqref{e21} yields
$$
\int_{t_0}^{t}\int_{\Omega}e^{-\lambda(t-s)+N_{\sigma}(s,t)ln\mu}
\Upsilon(x,s)\,dx\,ds\leq0.
$$
It follows from $e^{N_{\sigma}(s,t)ln\mu}\geq1 (\mu\geq1,N_{\sigma}(s,t)>0)$ that
\begin{equation}
\int_{t_0}^{t}\int_{\Omega}e^{-\lambda(t-s)}
[y^T(x,s)y(x,s)-\gamma^2
\omega^T(x,s)\omega(x,s)]\,dx\,ds\leq 0. \label{e22}
\end{equation}
Notice that since $N_{\sigma}(t_0,s)\leq N_0+\frac{s-t_0}{\tau_a}$, $N_0>0$,
and $\tau_{a}>\frac{ln\mu}{\lambda}$, we derive that
$$
N_\sigma(t_0,s)ln\mu\leq N_0ln\mu+\lambda (s-t_0).
$$
Multiplying both sides of \eqref{e22} by $e^{-[N_0ln\mu+\lambda (s-t_0)]}$ gives
\begin{equation}
\begin{aligned}
&\int_{t_0}^{t}\int_{\Omega}e^{-\lambda(t-s)-[N_0ln\mu+\lambda (s-t_0)]}
y^T(x,s)y(x,s)\,dx\,ds\\
&\leq\int_{t_0}^{t}\int_{\Omega}e^{-\lambda(t-s)
-[N_0ln\mu+\lambda (s-t_0)]}\gamma^2
\omega^T(x,s)\omega(x,s)\,dx\,ds.
\end{aligned} \label{e23}
\end{equation}
Thus we obtain
\begin{equation}
\int_{t_0}^{t}\int_{\Omega}e^{-\lambda t}y^T(x,s)y(x,s)\,dx\,ds
\leq\int_{t_0}^{t}\int_{\Omega}e^{-\lambda t}\gamma^2
\omega^T(x,s)\omega(x,s)\,dx\,ds.
\label{e24}
\end{equation}
Integrating both sides  from $t=t_0$ to $\infty$ gives
\begin{equation}
\int_{t_0}^{\infty}\int_{\Omega}y^T(x,s)y(x,s)\,dx\,ds
\leq\int_{t_0}^{\infty}\int_{\Omega}\gamma^2
\omega^T(x,s)\omega(x,s)\,dx\,ds\; \label{e25}
\end{equation}
i.e., $J(\omega)\leq0$. This completes the proof.
\end{proof}

The matrix inequalities \eqref{e20} are not LMIs. Left- and right- 
 multiplying \eqref{e20} by  $\operatorname{diag}\{P_i^{-1},I\}$.
Let $X_i=P_i^{-1}$, $Y_i=K_iP_i^{-1}$. It follows from the Schur complement 
that \eqref{e20} is equivalent to the  LMIs
\begin{equation}
\begin{bmatrix}
\Pi_i & C_i+X_iE_i^TF_i    & X_i & 0 \\
\ast  & -\gamma^2I+F_i^TF_i & 0  & 0 \\
\ast  & \ast  & (-E_i^TE_i)^{-1} & 0 \\
\ast  & \ast  & \ast & -I
\end{bmatrix}
<0,\label{e26}
\end{equation}
where $\Pi_i=-2D_iX_i+A_iX_i+X_i^TA_i^T +B_iY_i+Y_i^TB_i^T+\lambda X_i$.
Next, we show a result which can be obtained using matlab software.

\begin{theorem} \label{thm4.2}
For given scalars $\lambda>0$ and $\gamma>0$, if there exist diagonal 
matrices $X_i>0$ and matrices $Y_i>0$, such that the LMIs \eqref{e26} are feasible. 
Then the system \eqref{e1} can be exponentially stabilized by the state
feedback control \eqref{e2} with  $K_i=Y_iX_i^{-1}$ with the $H_{\infty}$ 
disturbance level $\gamma>0$ for the arbitrary switching signal $\sigma(t)$ 
with the ADT $\tau_{a}>\frac{ln\mu}{\lambda}$, where $\mu$ is determined 
by \eqref{e11}.
\end{theorem}

\begin{example} \label{exam4.1}
Consider the switched parabolic equations \eqref{e1} under the state 
feedback \eqref{e2}. Suppose there are two subsystems with parameters
\begin{gather*}
D_1=[1\; 0;0 \;2], \quad
D_2=[2\; 0;0\; 1], \quad
A_1=[1\; 3;2\; 3],\quad
A_2=[1\; 2;3\; 1], \\
B_1=[6\; 7;5\; 1], \quad
B_2=[5\; 2;3\; 1],\quad
C_1=[3\; 5;4\; 2], \quad
C_2=[5\; 2;3\; 2],\\
E_1=[1\; 0;0\; 1], \quad
E_2=[1\; 0;0\; 1], \quad
F_1=[1\; 2;2\; 1],\quad
F_2=[2\; 1;1\; 2].
\end{gather*}
Set  $\lambda=0.6$, $\gamma=0.8$, using Theorem \ref{thm4.2}, 
by resolving LMIs \eqref{e26},
we obtain
\[
X_1=[489.9209\;0;\; 0\;490.7482], \quad
X_2=[ 195.4787\;0;\; 0\; 193.5938].
\]
The state feedback matrices are
\[
K_1=\begin{bmatrix}
 -0.1487  & -6.4763\\
  -4.0296 &3.3322
\end{bmatrix}, \quad 
K_2=\begin{bmatrix}
 1.8293 & -3.8292\\
-5.6515 & 9.3622 
\end{bmatrix}
\]
From \eqref{e11} we obtain that $\mu= 2.5349$ and 
$\tau_{a}> \frac{ln\mu}{\lambda}=1.5503$. So system \eqref{e1} can be 
exponentially stabilized.
\end{example}

\subsection*{Conclusion}
By using multiple Lyapunov function and ADT method, we establish some new 
criteria for the exponential stabilization and the $H_{\infty}$ control synthesis
of switched linear parabolic systems via the state feedback. 
All the results are given in terms of LMIs and a class of signals which can
 be easily tested by matlab software. 

\subsection*{Acknowledgments}
This work is supported by the National Science Foundation of 
China (No. 61273119, 61374038), and by the Natural Science Foundation 
of Jiangsu Province (BK2011253).


\begin{thebibliography}{00}

\bibitem{a1}S. Amin, F. M. Hante , A. Bayen;
\emph{Exponential stability of switched linear hyperbolic initial-boundary
value problems}, IEEE Transactions on Automatic Control 57 (2012) 291-301.

\bibitem{b1} S. Boyd, L. Ghaoui, E. Feron, V. Balakrishnan;
\emph{Linear Matrix Inequalities in System and Control Theory},
SIAM, Philadelphia, PA, 1994.

\bibitem{c1} D. Cheng, L. Guo;
\emph{Stabilization of  switched linear systems},
IEEE Transactions on Automatic Control, 50(5)(2005) 661-666.

\bibitem{c2} Z. C. Chen;
\emph{Partial Differential Equations} (second edition). University
of Science and Technology of China Press, 2002.

\bibitem{c3} R. F. Curtain, H. Zwart;
\emph{An introduction to infinite dimensional linear system theory},
New York, MA: Springer, 1995.

\bibitem{d1} G. Deaecto, J. Geromel, J. Daafouz;
\emph{Dynamic output feedback $H_{\infty}$ control of switched linear systems},
Automatica 47 (2011) 1713-1720.

\bibitem{d2} X. Dong, R. Wen, H. Liu;
\emph{Feedback stabilization for a class of distributed parameter
switched systems with time delay},
 Journal of applied sciences - Electronics and information engineering,
29(1) (2011) 92-96.

\bibitem{f1} E. Fridman, Y. Orlov;
\emph{An LMI approach to $H_{\infty}$ boundary control of semilinear parabolic
and hyperbolic systems}, Automatica 45 (2009) 2060-2066.

\bibitem{h1} F. M. Hante, M. Sigalotti;
\emph{Converse Lyapunov theorems for switched systems in Banach and Hilbert spaces},
SIAM Journal on Control and Optimization 49(2)(2011) 752-770.

\bibitem{l1} D. Liberzon;
\emph{Switching in systems and control. Volume in series
Systems and Control: Foundations and Applications}, Birkhauser, 2003.

\bibitem{l2} H. Lin, P. J. Antsaklis;
\emph{Stability and stabilizability of switched linear
systems: A survey of recent results}, IEEE Transactions on Automatic
Control 54 (2009) 308-322.

\bibitem{l3} F. Long, S. Fei, Z. Fu, S. Zheng, W. We;
\emph{$H_{\infty}$ control and quadratic stabilization of switched linear systems
with linear fractional uncertainties via output feedback},
Nonlinear Analysis: Hybrid Systems 2 (2008) 18-27.

\bibitem{l4} Z. H. Luo, B. Z. Guo, O. Morgul;
\emph{Stability and stabilization for infinite
dimensional systems with applications}, London, MA: Springer, 1999.

\bibitem{m1} A. Michel, Y. Sun;
\emph{Stability of discontinuous cauchy problems in Banach space},
Nonlinear Analysis, 65 (2006) 1805-1832.

\bibitem{n1} K. S. Narendra, J. Balakrishnan;
\emph{A common Lyapunov function for stable LTI systems with commuting A-matrices}.
IEEE Transactions on Automatic Control, 39 (1994) 2469-2471.

\bibitem{o1} M. Ouzahra;
\emph{Global stsbilization of semilinear systems using switching controls}.
Automatica, 48 (2012) 837-843.

\bibitem{p1} A. Pazy;
\emph{Semigroup of linear operators and applications to partial differential
equations}, New York, MA: Springer-Verlag, 1983.

\bibitem{p2} V. Phat, S. Pairote;
\emph{Exponential stability of switched linear systems with time-varying delay}, 
Electronic Journal of Differential Equations, 159 (2007) 1-10.

\bibitem{p3} C. Prieur, A. Girard, E. Witrant;
\emph{Lyapunov functions for switched linear hyperbolic systems}. 51th CDC, 2012.

\bibitem{q1} J. Qi, Y. Sun;
\emph{Global exponential stability of certain switched systems with
time-varying delays}, Applied Mathematics Letters 26 (2013) 760-765.

\bibitem{s1} A. Sasane;
\emph{Stability of switching infinite-dimensional systems}, 
Automatica, 41 (2005) 75-78.

\bibitem{z1} G. Zhai, B. Hu, K. Yasuda, A. N. Michel;
\emph{Disturbance attenuation properties of time-controlled switched system},
Journal of the franklin institute 338 (2001) 765-77.

\bibitem{z2} J. Zhao, J. H. David;
\emph{On stability, $L_2$-gain and $H_{\infty}$ control for switched systems},
 Automatica 44 (2008) 1220-1232.

\end{thebibliography}

\end{document}