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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 148, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/148\hfil An iteration method]
{An iteration method for controllability of semilinear parabolic
equations}

\author[B. Sun\hfil EJDE-2010/148\hfilneg]
{Bo Sun}

\address{Bo Sun \newline
College of Mathematics and Computers, 
Changsha University of Science and Technology,
Changsha, Hunan, China}
\email{sunbo52002@yahoo.com.cn}

\thanks{Submitted May 30, 2010. Published October 20, 2010.}
\subjclass[2000]{35K61, 93B05}
\keywords{Semilinear; parabolic equation; controllability;
iteration method; \hfill\break\indent coupled systems}

\begin{abstract}
 We present a method based on Picard's idea to construct a
 sequence of controls and a sequence of solutions of linearized
 systems such that their limits form a solution to the control
 problem.  By doing this, we simplified the works in the references,
 and deduced the controllability for semilinear coupled
 parabolic systems.
\end{abstract}

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

\section{Introduction}

The controllability for semilinear parabolic equations has been
intensively studied in previous decades; see for example
\cite{Fabre,Zuazua,Fernandez,Khapalov, Khapalov2,Khapalov3}. Some
new systems were studied recently, such as nonlinear heat equation
with memory effects, Lavanya \cite{Lavanya}, and some semilinear
parabolic equations arising in finance, Sakthivel \cite{Sakthivel}.
Tang and Zhang \cite{Tang} studied null
controllability for forward and backward linear stochastic parabolic
equations. It may not be easy to generalize their results to
nonlinear cases.

Generally speaking, all these works are based on the same idea:
the methods of approximating controllability for linear systems are
based on the unique continuation of solutions from an open set,
while the exact null-controllability depends on the inverse
estimates of dual observed systems. Furthermore, the controllability
are generalized to semilinear systems by Schauder fixed-point
arguments. All the arguments are based on the compact embedding of
$H_0^1$ into $L^2$. The Schauder fixed-point arguments are good
for semilinear parabolic systems with globally Lipschitz
nonlinearity, but difficult for those with superlinear terms. In
this paper, we try to do it in other ways, and give a method based
on iteration idea, by which we simplified some works concerning
the controllability of semilinear parabolic systems in literatures.
Moreover, we deduced the controllability of coupled parabolic systems,
which seems impossible to be obtained by previous arguments.

This paper is organized as follows. In Section~2 we recall two
lemmas which are necessary for our analysis. In Section~3 we prove
the exact null-controllability of semilinear parabolic systems with
mobile distributed control, to illustrate our method. We discuss the
controllability of coupled parabolic systems in Section~4.

\section{Preliminaries}

For convenience, we recall the following two propositions, from
\cite{Robinson}.

\begin{proposition}\label{Robison 1}
Let $X, H, Y$ be Banach spaces with $X\subset\subset
H\subseteq Y$ and $X$ reflexive. Suppose that
$\{u_n\}_{n=1}^{\infty}$ is a sequence that is bounded in
$L^2(0,T;X)$, and that $\{\frac{{\rm d}u_n}{{\rm
d}t}\}_{n=1}^{\infty}$ is bounded in $L^p(0,T;Y)$, for some $p>1$.
Then there is a subsequence of $\{u_n\}_{n=1}^{\infty}$ that
converges strongly in $L^2(0;T;H)$.
\end{proposition}


\begin{proposition}\label{Robison 2}
Let $\mathcal{O}$ be a bounded open set in $\mathbb{R}^m$, and let
$\{g_j\}_{j=1}^{\infty}$ be a sequence of functions in
$L^p(\mathcal{O})$ with
$$
\|g_j\|_{L^p(\mathcal{O})}\le C \quad \text{for all }
 j\in \mathbb{Z}^+.
$$
If $g_j\to g\in L^p(\mathcal{O})$
almost everywhere, then $g_j\rightharpoonup g$ in $L^p(\mathcal{O})$.
\end{proposition}

\section{Exact null-controllability for semilinear systems
with mobile distributed control}\label{sec2}

Consider the controllability for the Dirichlet problem
\begin{gather}
\frac{\partial u}{\partial t}=\Delta
u+g(u)+f(x,t)\chi_{\omega(t)}(x) \quad \text{in } Q_T,
\label{orignal1}
\\
u=0 \quad\text{in }  \Sigma_T, \quad u|_{t=0}=u_0,\quad u_0\in
L^2(\Omega), \label{orignal2}
\end{gather}
where $\Omega$ is an open bounded domain in $\mathbb{R}^n$ with
boundary $\partial \Omega$, $Q_T=\Omega\times (0,T)$,
 $\Sigma_T=\partial \Omega \times (0,T)$, $f$ is a distributed
control in $L^2(Q_T)$, $\omega(\cdot)$ is a mobile support, and
$\chi_{\omega(\cdot)}$denotes its indication function.

We assume growth conditions as follows:
\begin{gather}
 |g(u)|  \leq c_1+c_2|u|^p, \quad p=1+4/n,\quad \forall u\in R,\label{cond1}\\
 g(u)u  \leq c_3u^2, \quad \forall u\in R, \label{cond2} \\
 g(0)=0,\quad \lim_{u\to 0}\frac{g(u)}{u}=g'(0).\label{cond3}
\end{gather}
Conditions \eqref{cond1}-\eqref{cond3} ensure the existence of at
least one generalized solution of
\eqref{orignal1}-\eqref{orignal2} from
$C([0,T];L^2(\Omega))\bigcap H_0^{1,0}(Q_T)$ \cite[pp.
466-467]{Ladyzhenskaja}. Khapalov proved that
\eqref{orignal1}-\eqref{cond3} is exactly null-controllable
\cite{Khapalov2}, but his generalization to nonlinear systems is
somewhat complicated. So in this paper we put forward some
different methods.

The corresponding linear system is as follows:
\begin{gather}
\frac{\partial w}{\partial t}=\Delta w+a(x,t)w
+f(x,t)\chi_{\omega(t)}(x) \quad\text{in } Q_T, \label{linear1}\\
w=0 \quad\text{in} \quad \Sigma_T,\quad w|_{t=0}=w_0\in
L^2(\Omega). \label{linear2}
\end{gather}
The counterparts to \eqref{cond1}-\eqref{cond2} are
\begin{gather}
a(x,t)\leq c_3, \quad a\in L^{r_2}(0,T;L^{r_1}(\Omega)),\\
\frac{1}{r_2}+\frac{n}{2r_1}=1,\\
r_1\in (\frac{n}{2}, \infty],\quad r_2\in [1, \infty), \quad \text{for } n\geq 2,\\
r_1\in [1, \infty], \quad r_2\in [1, 2], \quad \text{for } n=1.
\label{linear conditions}
\end{gather}

Khapalov proved that \eqref{linear1}-\eqref{linear conditions} is exactly
null-controllable by a mobile internal control, while the support
may be chosen to be arbitrarily small. Moreover, he gives the
following estimates:
\begin{gather}
\|w\|_{L^q(Q_T)}, \|w\|_{\mathcal{B}}  \leq
c(T)(\|w_0\|_{L^2(\Omega)}+\|v\|_{L^2(\omega(\cdot))}),
\label{estimate1}\\
\|f\|_{L^2(\omega(\cdot))}  \leq M \|w_0\|_{L^2(\Omega)},
\label{estimate2}
\end{gather}
 where $q=2+4/n$,
$$
\|w\|_\mathcal{B}=\|w\|_{C([0,T];L^2(\Omega))}
+\Big(\int_0^T\int_\Omega \|\nabla
w\|^2_{R^n}\text{d}x\text{d}t\Big)^{1/2},
$$
$c(T)$ and $M$ do not depend on the choice of $a(x,t)$.
Then he deduced the exact
null-controllability for \eqref{orignal1}-\eqref{orignal2} by
Schauder fixed-point argument, with $p\le 1+4/n$. Now we do in a
different way.

As usual, we introduce the non-linearity
$$
h(s)=\begin{cases}
g(s)/s, & s\neq 0,\\
g'(0), & s=0,
\end{cases}
$$
and construct a ``linearized'' system
\begin{equation}
\frac{\partial u}{\partial t}  =\Delta u+h(z)u
+f(x,t)\chi_{\omega(t)}(x)\quad \text{in } Q_T \label{linearized}
\end{equation}
with the same initial and boundary conditions as \eqref{linear2}.

We construct a sequence in $L^2(Q_T)$ as follows: Take any
$z_1\in L^2(Q_T)$, substitute it for $z$ in \eqref{linearized},
then there exists a control $f_1\in L^2(Q_T)$ such that the
corresponding solution $u_1$ satisfies that $u_1(T)=0$; Take $z_2=u_1$,
substitute it for $z$ in \eqref{linearized}, then there exists a
control $f_2\in L^2(Q_T)$ such that the corresponding solution
$u_2$ satisfies that $u_2(T)=0$; Repeating this process yields two
sequences $\{z_n\}_{n=1}^{\infty}$(or $\{u_n\}_{n=1}^{\infty}$)
and $\{f_n\}_{n=1}^{\infty}$ in $L^2(Q_T)$. This process can be
illustrated as follows:
\begin{gather}
z_1\Rightarrow f_1\Rightarrow u_1=z_2\Rightarrow f_2\Rightarrow
u_2= \dots , \notag\\
\frac{\partial u_n}{\partial
t}=\Delta u_n+h(z_n)u_n+f_n\chi_{\omega(t)}.
\label{iterationeq}
\end{gather}

Now we prove that they converge in some topology, their limits
solve \eqref{orignal1}-\eqref{orignal2}, and the limit satisfies
that $u(T)=0$. To simplify notation, we denote
$\int_{Q_T}\cdot {\rm d}x{\rm d}t$ by $\int_{Q_T}\cdot $.

Condition \eqref{cond3} implies that there is $r>0$ such that
$$
|h(z)|\le c_4=|g'(0)|+1
$$
when $|z|<r$. On the other hand, it follows from \eqref{cond1}
that
\begin{align*}
|h(z)|
 & \le \frac{c_1}{|z|}+c_2|z|^{p-1}  \\
 & \le \frac{c_1}{|z|^p}|u|^{p-1}+c_2|z|^{p-1}  \\
 & \le \frac{c_1}{r^p}|z|^{p-1}+c_2|z|^{p-1}\\
 & \le c_5 |z|^{p-1}
\end{align*}
when $|z|\geq r$.
 Therefore,
\begin{equation}
\begin{aligned}
&\int_{Q_T}|h(z_n)u_n|^{2n/(n+2)}\\
&= \int_{|z_n|<r}|h(z_n)u_n|^{2n/(n+2)}
  +\int_{|z_n|\geq r}|h(z_n)u_n|^{2n/(n+2)}  \\
&\le c_4\int_{Q_T}|u_n|^{2n/(n+2)}
 +c_5\int_{Q_T}|z_n|^{\frac{8}{n+2}}|u_n|^{2n/(n+2)}.
\end{aligned}\label{ineq1}
\end{equation}
 It follows from the continuous embedding
of $L^{2+4/n}(Q_T)$ into $L^{2n/(n+2)}(Q_T)$ that
\begin{align}\int_{Q_T}|u_n|^{2n/(n+2)}\le c _4\int_{Q_T}|u_n|^{2+4/n}. \label{ineq2} \end{align}
 By H\"{o}lder inequality we have
\begin{equation}
\begin{aligned}
& \int_{Q_T}|z_n|^{\frac{8}{n+2}}|u_n|^{2n/(n+2)}\\
& \le c_5\Big(\int_{Q_T}|z_n|^{2+4/n}\Big)^{4(n+1)/(n+2)^2}
\Big(\int_{Q_T}|u_n|^{2+4/n}  \Big)^{n^2/(n+2)^2}.
\end{aligned}\label{ineq3}
\end{equation}
Combining \eqref{ineq1}, \eqref{ineq2} and \eqref{ineq3} yields
\begin{align*}
& \int_{Q_T}|h(z_n)u_n|^{2n/(n+2)}\\
&\le c _4\int_{Q_T}|u_n|^{2+4/n}
+c_5\Big(\int_{Q_T}|z_n|^{2+4/n}\Big)^{4(n+1)/(n+2)^2}
\Big(\int_{Q_T}|u_n|^{2+4/n} \Big)^{n^2/(n+2)^2}.
 \end{align*}

Due to \eqref{estimate1} and \eqref{estimate2}, the sequence
$\{u_n\}_{n=1}^{\infty}$ or he sequence $\{z_n\}_{n=1}^{\infty}$
is  bounded  in $L^{2+4/n}(Q_T)$. So $\{h(z_n)u_n\}_{n=1}^\infty$ is
bounded in $L^{2n/(n+2)}(Q_T)$, and it has a weak convergent
subsequence. Furthermore, it follows from \eqref{estimate1} and
\eqref{estimate2} that $\{u_n\}_{n=1}^{\infty}$ is bounded in
$L^2(0,T;H_0^1(\Omega))$, and $\{f_n\}_{n=1}^{\infty}$ is bounded
in $L^2(\omega(\cdot))$. By extracting subsequences (that we
denote by the index $j$ to simplify the notation) we have
\begin{gather*}
u_j\rightharpoonup u \quad \text{in } L^2(0,T;H_0^1(\Omega)),\\
\Delta u_j\rightharpoonup \Delta u \quad \text{in }
 L^2(0,T;H^{-1}(\Omega)),\\
f_j\rightharpoonup f \quad \text{in } L^2(\omega(\cdot)).
\end{gather*}
Moreover, it follows from the continuous embedding of
$H_0^1(\Omega)$ into $L^{2n/(n-2)}$ that $L^{2n/(n+2)}(\Omega)$ is
continuously embedded in $H^{-1}(\Omega)$. Therefore,
$L^{2n/(n+2)}(Q_T)$ is continuously embedded in
$L^{2n/(n+2)}(0,T;H^{-1}(\Omega))$, and thus
$\{h(z_n)u_n\}_{n=1}^\infty$ is bounded in
$L^{2n/(n+2)}(0,T;H^{-1}(\Omega))$. These facts imply that
$\frac{\partial u_n}{\partial t}$ is bounded in
$L^{2n/(n+2)}(0,T;H^{-1}(\Omega))$, so it has subsequence which
converges weakly to $\frac{\partial u}{\partial t}$. Next we will
show that the subsequence of $\{h(z_n)u_n\}_{n=1}^\infty$
converges weakly to $h(u)u$ in $L^{2n/(n+2)}(0,T;H^{-1}(\Omega))$.

\begin{remark} \label{rmk3.1} \rm
The constants $c_4$ and $c_5$ above are generic constants.
\end{remark}

Substituting $H_0^1(\Omega)$ for $X$, $L^2(\Omega)$ for $H$, and
$H^{-1}(\Omega)$ for $Y$ in Proposition \ref{Robison 1}, by
extracting subsequence we have that $\{u_j\}_{j=1}^{\infty}$
converges to $u$ strongly in $L^2(Q_T)$. A classical result in
real analysis tells that another subsequence
$\{u_j\}_{j=1}^{\infty}$ converges to $u$ almost everywhere in
$Q_T$. It follows, using the continuity of $g$, that $h(z_j)$
converges to $h(u)$ almost everywhere. Combining this fact with
Proposition \ref{Robison 2} leads to that
$\{h(z_j)u_j\}_{j=1}^{\infty}$ converges to $h(u)u$ weakly in
$L^{2n/(n+2)}(Q_T)$, and thus in
$L^{2n/(n+2)}(0,T;H^{-1}(\Omega))$. Taking limit of
\eqref{iterationeq} in $L^{2n/(n+2)}(0,T;H^{-1}(\Omega))$,
we see that $u$ and $v$ solve \eqref{orignal1}-\eqref{orignal2}.

It remains to show that $u(T)=0$. The basic idea is that
$\{u_j\}_{j=1}^{\infty}$ converges to $u$ strongly in
$L^2(0,T;L^2(\Omega))$ implies that a subsequence
$\{u_j(t)\}_{j=1}^{\infty}$ converges to $u(t)$ strongly in
$L^2(\Omega)$, for almost every $t$ in $[0,T]$. So we may think
that $\{u_j(T)\}_{j=1}^{\infty}$ converges to $u(T)$ strongly in
$L^2(\Omega)$. It follows from $u_n(T)=0$ that $u(T)=0$.

However, the a.e. convergence of $\{u_n\}$ to $u$ and that $u_n(T)=0$
do not always imply that $u(T)=0$, e.g., for a sequence $\{t^n\}$,
$t\in (0,1)$, $T=1$.
To avoid the strange case with steep curve near $t=T$,
we require that each control system as \eqref{iterationeq} vanishes
in advance, e.g.,
$$
u_n(t)\equiv 0
$$
for $t\in (\frac{T}{2},T)$ or $(\frac{3T}{4},T)$. Then it follows
from the a.e. convergence of $u_n$ to $u$ that
$u(t)=0$
in $(\frac{T}{2},T)$ or $(\frac{3T}{4},T)$.
On the other hand, it follows from the regularity of parabolic
systems that
 $$
u\in C([0,T];L^2(\Omega)).
$$
 So $u(T)=0$.
We summarize the analysis above as follows:

\begin{theorem} \label{thm3.2}
Suppose $ n\geq 3$ and that \eqref{cond1}, \eqref{cond2},
\eqref{cond3} hold. Then \eqref{orignal1}-\eqref{orignal2} is
exactly null-controllable by a mobile internal control, and that
the support can be chosen arbitrarily small.
\end{theorem}

\begin{remark} \label{rmk3.3} \rm
Schauder fixed-point argument requires that map $N: z\mapsto u$ is
continuous and compact, from a bounded closed convex set into
itself. These are difficult to verify for systems with superlinear
terms. Our method just requires that the iteration generates
bounded sequences of solutions and controls.
\end{remark}

\begin{remark} \label{rmk3.4} \rm
 Of course, our method is also fit for those systems with
globally Lipschitz nonlinear terms or point controls. We take a
system with superlinear term and a mobile internal control just to
show our method and its advantage.\end{remark}


\section{Controllability of coupled parabolic systems}\label{sec3}

Consider a coupled parabolic system
\begin{gather}
u_t-\Delta u=\Phi(u,v)+f\chi_{\omega_1} \quad \text{in }  Q_T,
\label{couple1} \\
v_t-\Delta v=\Psi(u,v)+g\chi_{\omega_2} \quad \text{in }  Q_T,
\label{couple2} \\
u(x,t)=v(x,t)=0 \quad \text{on }  \Sigma_T , \label{couple3} \\
u|_{t=0}=u_0, \quad v|_{t=0}=v_0, \quad u_0,v_0 \in L^2(\Omega),
\label{couple4}
\end{gather}
where $\omega_1$ and $\omega_2$ are proper subsets of $\Omega$.

The system above is a widely used mathematical modelling for many
chemical, physical, biological or ecological phenomena. Many
papers are devoted to study the existence and uniqueness of local
solution, global existence and blow-up of solutions (see
\cite{Escobedo,Henry}), but less is known about its
controllability. So we consider this problem. Suppose that $\Phi$
and $\Psi$ are continuous functions, $\Phi$ is globally Lipschitz
continuous and differentiable with respect to $u$, so is $\Psi$
with respect to $v$, and
$$
\Phi(0,v)=\Psi(u,0)=0, \quad \forall u,v\in\mathbb{R}.
$$
The solvability will be implied in our arguments on
controllability.

The globally approximate controllability and finite dimensional
exact controllability for \eqref{couple1}-\eqref{couple4} is
formulated as follows (see \cite{Zuazua} for single parabolic
equation): Given $u_T$, $v_T$ in $L^2(\Omega)$, $\varepsilon >0$
and finite dimensional subspaces $E_1$, $E_2$ of $L^2(\Omega)$,
there exist controls $f$ and $g$ in $L^2(Q_T)$ such that
\begin{gather}
\|u(T)-u_T\|_{L^2(\Omega)}\le \varepsilon, \quad
\Pi_1u(T)=\Pi_1u_T, \label{coupledcontro1} \\
|v(T)-v_T|_{L^2(\Omega)}\le \varepsilon, \quad
\Pi_2v(T)=\Pi_2v_T, \label{coupledcontro2}
\end{gather}
where $\Pi_1$ and $\Pi_2$ are the orthogonal projections from
$L^2(\Omega)$ into $E_1$ and $E_2$ respectively.

It seems difficult or impossible to construct a linearized systems
for \eqref{couple1}-\eqref{couple4}, as \eqref{linearized} for
\eqref{orignal1}-\eqref{orignal2}. We will deduce the
controllability without linearized system. At first, we construct
sequences of solutions and controls as follows: Take any $v_1\in
L^2(Q_T)$, substitute it for $v$ in \eqref{couple1}, then it
follows from the controllability results on single parabolic
systems (see \cite{Zuazua}) that there exists control $f_1\in
L^2(Q_T)$ such that the corresponding solution $u_1$ of
\eqref{couple1} satisfies
 \eqref{coupledcontro1}. Substitute $u_1$ for $u$ in
\eqref{couple2}, there exists control $g_1\in L^2(Q_T)$ such that
the corresponding solution $v_2$ satisfies
\eqref{coupledcontro2}. Then substitute $v_2$ for $v$
in \eqref{couple1}, and get
$f_2$ and $u_2$. Repeating this process yields sequences
$\{u_n\}_{n=1}^{\infty}$, $\{v_n\}_{n=1}^{\infty}$,
$\{f_n\}_{n=1}^{\infty}$ and $\{g_n\}_{n=1}^{\infty}$, which
satisfy \eqref{coupledcontro1} and \eqref{coupledcontro2}.
This process can be illustrated as follows:
\begin{gather}
v_1\Rightarrow f_1\Rightarrow u_1\Rightarrow g_1\Rightarrow
v_2\Rightarrow f_2\Rightarrow u_2\Rightarrow \dots , \notag \\
\frac{\partial u_n}{\partial t}-\Delta u_n=\Phi
(u_n,v_n)+f_n\chi_{\omega_1}, \label{coupleiteration1} \\
\frac{v_{n+1}}{\partial t}-\Delta v_{n+1}=\Psi
(u_n,v_{n+1})+g_n\chi_{\omega_2}. \label{coupleiteration2}
\end{gather}

We will prove that these sequences converge in some topology, and
that their limits solve \eqref{couple1}-\eqref{couple4}, which
satisfy \eqref{coupledcontro1} and \eqref{coupledcontro2}.

Due to the global Lipschitz continuity of $\Phi$ and $\Psi$,
sequences $\{f_n\}_{n=1}^{\infty}$, $\{g_n\}_{n=1}^{\infty}$,
$\{u_n\}_{n=1}^{\infty}$ and $\{v_n\}_{n=1}^{\infty}$ are bounded
in $L^2(Q_T)$. So they have subsequences which converge weakly.
Let $u_j\rightharpoonup u$, $v_j\rightharpoonup v$,
$f_j\rightharpoonup f$ and $g_j\rightharpoonup g$ in $L^2(Q_T)$.
By similar arguments to those in last section, we have that
\begin{gather*}
\Phi (u_n,v_n)\rightharpoonup \Phi (u,v),\\
\Psi (u_n,v_{n+1})\rightharpoonup \Psi (u,v)
\end{gather*}
in $L^2(Q_T)$. Taking limits of \eqref{coupleiteration1}
and \eqref{coupleiteration2},
one can easily verify that $u$, $v$, $f$ and $g$ solve
\eqref{couple1}-\eqref{couple4}, and satisfy
\eqref{coupledcontro1}-\eqref{coupledcontro2}. We summarize our analysis as
follows:

\begin{theorem} \label{couplethem} Suppose
that $\Phi$ and $\Psi$ are continuous functions, $\Phi$ is
globally Lipschitz continuous and differentiable with respect to
$u$, so is $\Psi$ with respect to $v$, and
$$
\Phi(0,v)=\Psi(u,0)=0, \quad \forall u,v\in\mathbb{R}.
$$
Then \eqref{couple1}-\eqref{couple4} is globally
approximately controllable and finite-dimensional exactly
controllable.
\end{theorem}


\begin{remark} \label{rmk4.2} \rm
By our method, all results concerning controllability of semilinear
parabolic systems can be extended to coupled parabolic systems.
For example, We may consider exact null-controllability of semilinear
coupled parabolic systems with superlinear terms, mobile internal
controls or point-controls.
\end{remark}

 From the point view of application, it is reasonable to consider
coupled parabolic system with a single control as follows:
\begin{gather}
u_t-\Delta u=\Phi(u,v)+f\chi_{\omega} \quad \text{in }  Q_T,
\label{single1} \\
v_t-\Delta v=\Psi(u,v) \quad \text{in } Q_T,
\label{single2} \\
u(x,t)=v(x,t)=0 \quad \text{on } \Sigma_T , \label{single3} \\
u|_{t=0}=u_0, \quad v|_{t=0}=v_0, \quad u_0,v_0 \in L^2(\Omega).
\label{single4}
\end{gather}

\begin{theorem} \label{onecontroltheorem}
Under the same conditions as in Theorem \ref{couplethem}, system
\eqref{single1}-\eqref{single4} is globally approximately
controllable and finite-dimensional exactly controllable.
\end{theorem}

The proof is similar to that of Theorem \ref{couplethem}, but
there is some difference. Let us give a sketch: Given $u_T\in
L^2(\Omega)$, $\varepsilon >0$ and finite-dimensional subspace $E$
of $L^2(\Omega)$. Take any $v_1\in L^2(Q_T)$, substitute it for
$v$ in \eqref{single1}, then there exist control $f_1$ and the
corresponding solution $u_1$ such that
$\|u_1(T)-u_T\|_{L^2(\Omega)}< \varepsilon $, and
$\Pi_Eu_1(T)=\Pi_Eu_T$. Substitute $u_1$ for $u$
in \eqref{single2}, then there is an unique solution $v_2$.
Substitute $v_2$ for
$v$ in \eqref{single1}, there exist control $f_2$ and the
corresponding solution $u_2$, which satisfy that
$\|u_2(T)-u_T\|_{L^2(\Omega)}< \varepsilon $, and
$\Pi_Eu_2(T)=\Pi_Eu_T$. Repeating this process yields sequences
$\{u_n\}_{n=1}^{\infty}$, $\{v_n\}_{n=1}^{\infty}$ and
$\{f_n\}_{n=1}^{\infty}$. Their limits solve
\eqref{single1}-\eqref{single4}, and satisfy
\begin{gather*}
\|u(T)-u_T\|_{L^2(\Omega)}<\varepsilon ,\\
\Pi_Eu(T)=\Pi_Eu_T,
\end{gather*}
where $\Pi_E$ represents the orthogonal projection from
$L^2(\Omega)$ into $E$.

Theorem \ref{couplethem} and Theorem \ref{onecontroltheorem} can
be generalized to multi-coupled parabolic systems as follows:
\begin{gather*}
\frac{\partial u_1}{\partial t}-\Delta u_1=
  \Phi_1(u_1,u_2,\dots,u_m)+f_1\chi_{\omega_1} \quad \text{in } Q_T,
 \\
\frac{\partial u_2}{\partial t}-\Delta u_2=
  \Phi_2(u_1,u_2,\dots,u_m)+f_2\chi_{\omega_2} \quad \text{in }
 Q_T, \\
    \dots  \\
 \frac{\partial u_m}{\partial t}-\Delta u_m=
  \Phi_m(u_1,u_2,\dots,u_m)+f_m\chi_{\omega_m} \quad \text{in } Q_T, \\
u_i|_{\Sigma_T}=0,\quad u_i|_{t=0}=u_{i0},\quad u_{i0}\in L^2(\Omega),
\quad i=1,2,\dots,m.
\end{gather*}


\section{Examples}

The following example, from combustion theory \cite{Temam},
illustrates our results.
\begin{gather}
\frac{\partial u_1}{\partial t}-d_1\Delta
u_1-(u_2)^ph(u_1)=f\chi_{\omega_1} \quad \text{in }  Q_T,
\label{examp1e1}
\\
\frac{\partial u_2}{\partial t}-d_2\Delta
u_2-(u_2)^ph(u_1)=g\chi_{\omega_2} \quad \text{in }  Q_T,
\label{examp1e2}
\\
u_1(x,t)=u_2(x,t)=0 \quad \text{on} \quad \Sigma_T, \label{example3}
\end{gather}
where $h(s)=|s|^\gamma\exp(-\alpha/|s|)$, and $p$, $\alpha$, $\gamma$
are positive constants. $u_1$ represents a temperature,
while $u_2$ represents a concentration. The nonlinearity vanishes
at $u_1=0$ and $u_2=0$. It is Lipschitz continuous, provided that
$p=\gamma=1$. It follows from section 4 that the coupled system
\eqref{examp1e1}-\eqref{example3} is globally approximately
controllable and exactly null-controllable.

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\end{document}