# Null controllability for a parabolic-elliptic coupled system

E. Fernández-Cara, J. Limaco , S. B. de Menezes Dpto. E.D.A.N., Universidad de
Sevilla, Aptdo. 1160, 41080 Sevilla, Spain<EMAIL_ADDRESS>Matemática,
Universidade Federal Fluminense, Valonguinho, 24020-140, Niterói, RJ, Brasil,
<EMAIL_ADDRESS>author: Dpto. Matemática, Universidade Federal
do Ceará, Campus do Pici - Bloco 914, 60455-760, Fortaleza, CE, Brasil,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we prove the null controllability of some parabolic-elliptic
systems. The control is distributed, locally supported in space and appears
only in one PDE. The arguments rely on fixed-point reformulation and suitable
Carleman estimates for the solutions to the adjoint system. Under appropriate
assumptions, we also prove that the solution can be obtained as the asymptotic
limit of some similar parabolic systems.

Mathematics Subject Classification 2000: 35B37, 35A05, 35B40.

Keywords: Null controllability, parabolic-elliptic systems, Carleman
inequalities.

##  1 Introduction and main results

Let ${\Omega}$ be a bounded domain of ${\mathbb{R}}^{N}$ ($N\geq 1$), with
boundary $\Gamma=\partial{\Omega}$ of class $C^{2}$. We fix $T>0$ and we
denote by $Q$ the cylinder $Q={\Omega}\times(0,T)$, with lateral boundary
$\Sigma=\Gamma\times(0,T)$. We also consider a non-empty (small) open set
$\mathcal{O}\subset{\Omega}$; as usual, $1_{\mathcal{O}}$ denotes the
characteristic function of $\mathcal{O}$.

Throughout this paper, $C$ (and sometimes $C_{0}$, $K$, $K_{0}$, …) denotes
various positive constants. Frequently, we will emphasize the fact that $C$
depends on (say) $f$ by writing $C=C(f)$. The inner product and norm in
$L^{2}({\Omega})$ will be denoted, respectively, by $(\cdot\,,\cdot)$ and
$\|\cdot\|$. On the other hand, $\|\cdot\|_{\infty}$ will stand for the norm
in $L^{\infty}(Q)$.

We will consider the following semilinear parabolic-elliptic coupled systems

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}\end{array}\right.$ (1.1)

and

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$ (1.2)

under some hypotheses for $F$ and $f$.

In (1.1) and (1.2), we have $y=y(x,t)$ and $z=z(x,t)$; $1_{\mathcal{O}}$ is
the characteristic function of $\mathcal{O}$ and $y^{0}=y^{0}(x)$ is the
initial state. We will assume that the possibly nonlinear functions
$F:{\mathbb{R}}\times{\mathbb{R}}\mapsto{\mathbb{R}}$ and
$f:{\mathbb{R}}\mapsto{\mathbb{R}}$ satisfy:

$\left\\{\begin{array}[]{l}\displaystyle\text{$F$ and $f$ are (globally)
Lipschitz-continuous,}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
F(0,0)=f(0,0)=0,\quad{\partial f\over\partial z}(y,z)\leq\mu<\mu_{1}\ \
\text{a.e.},\\\ \end{array}\right.$ (1.3)

where $\mu_{1}$ the first eigenvalue of the Dirichlet Laplacian in ${\Omega}$.

The analysis of systems of the kind (1.1) and (1.2) can be justified by
several applications. Let us indicate two of them:

  * •

Reaction-diffusion systems with origin in physics, chemistry, biology, etc.
where two scalar “populations” interact and the natural time scale of the
growth rate is much smaller for one of them than for the other one. Precise
examples can be found in the study of prey-predator interaction, chemical
heating, tumor growth therapy, etc.

  * •

Semiconductor modeling, where one of the state variables is (for example) the
density of holes and the other one is the electrical potential of the device;
see for instance [7]. Other problems with this motivation will be analyzed
with more detail by the authors in the next future.

The system (1.1) (resp. (1.2)) is well-posed in the sense that, for each
$y^{0}\in L^{2}({\Omega})$ and each $v\in L^{2}(\mathcal{O}\times(0,T))$
(resp. $w\in L^{2}(\mathcal{O}\times(0,T))$) possesses exactly one solution
$(y,z)$, with

$y\in L^{2}(0,T;H_{0}^{1}({\Omega})),\quad y_{t}\in
L^{2}(0,T;H^{-1}({\Omega})),\quad z\in L^{2}(0,T;D(-\Delta)).$

This statement is justified in Appendix A.

In this paper we will analyze some controllability properties of (1.1) and
(1.2).

It will be said that (1.1) (resp. (1.2)) is null-controllable at time $T$ if
the following holds: for any given $y^{0}\in L^{2}({\Omega})$, there exist
controls $v\in L^{2}(\mathcal{O}\times(0,T))$ (resp. controls $w\in
L^{2}(\mathcal{O}\times(0,T))$) and associated solutions satisfying

$z\in C^{0}([0,T];L^{2}({\Omega}))$

and

$y(x,T)=0\ \mbox{ in }\ {\Omega},\quad\limsup_{t\to T^{-}}\|z(\cdot\,,t)\|=0,$
(1.4)

with an estimate of the form

$\|v\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|\quad(\mbox{resp.\
}\|w\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|).$ (1.5)

This inequality indicates that the “null controls” can be chosen depending
continuously on the initial data.

The control of PDEs equations and systems has been the subject of a lot of
papers the last years. In particular, important progress has been made
recently in the controllability analysis of semi-linear parabolic equations.
We refer to the works [1, 2, 3, 5, 6, 9, 11, 12] and the references therein.
Consequently, it is natural to try to extend the known results to systems of
the kind (1.1) and (1.2).

The main results in this paper are the following:

###### Theorem 1.1

Assume that

$F(y,z)\equiv F_{0}(y)+bz$, with $F_{0}$ Lipschitz-continuous,
$b\in{\mathbb{R}}$, (1.6) $f(y,z)\equiv cy+dz$, with $c,d\in{\mathbb{R}}$,
$c\not=0$, $d<\mu_{1}$. (1.7)

Then (1.1) is null-controllable at any time $T>0$.

###### Theorem 1.2

Let us assume that (1.6) holds and

$f(y,z)\equiv f_{0}(y)+dz$, with $f_{0}$ Lipschitz-continuous,
$d\in{\mathbb{R}}$, $d<\mu_{1}$. (1.8)

Then (1.2) is null-controllable at any time $T>0$.

The proofs of these results rely on relatively well known arguments and some
new estimates.

More precisely, in a first step, we will first consider similar linearized
systems of the form

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=a(x,t)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}\end{array}\right.$ (1.9)

and

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=a(x,t)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$ (1.10)

We will establish null controllability results for (1.9) and (1.10) by
previously proving appropriate Carleman estimates for the solutions to the
associated adjoint systems. Then, in a second step, we will adapt a fixed-
point argument to get the null controllability results stated in Theorems 1.1
and 1.2.

In this paper, we will also consider systems of the form

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon z_{t}-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\,\,z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;\Omega.\end{array}\right.$
(1.11)

It will be shown that, under the assumptions of Theorem 5.1, (1.11) is
uniformly null-controllable as $\varepsilon\to 0$, i.e. null-controlable with
controls $w_{\varepsilon}$ satisfying the estimates (1.5) with $C$ independent
of $\varepsilon$. We will also see that the $w_{\varepsilon}$ can be chosen in
such a way that they converge weakly to a null control of (1.2) (see Theorem
(5.2) below).

This paper is organized as follows. In Section 2, we introduce some adjoint
(backwards in time) parabolic-elliptic systems and we prove that their
solutions satisfy suitable Carleman estimates. In Section 3, we deduce from
these estimates null controllability results for (1.9) and (1.10). Section 4
deals with the proofs of Theorems 1.1 and 1.2. The uniform null
controllability property of (1.11) and the convergence of the associated null
controls are established in Section 5. Finally, we give the proofs of some
technical results in Section 6 (Appendix A).

##  2 Some Carleman estimates

We will first consider the general linear backwards in time system

$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a(x,t)\varphi+c(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle-\Delta\psi=b(x,t)\varphi+d(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle\varphi=0,\ \psi=0\;\;\mbox{on}\;\;\Sigma,\\\
\displaystyle\varphi(x,T)=\varphi^{T}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$
(2.1)

where $\varphi^{T}\in L^{2}({\Omega})$ and we assume that

$a,b,c,d\in L^{\infty}(Q),\ \ d\leq\mu<\mu_{1}\ \text{a.e.}$ (2.2)

Also, it will be convenient to introduce a new non-empty open set
$\mathcal{O}_{0}$, with $\mathcal{O}_{0}\Subset\mathcal{O}$. We will need the
following result, due to Fursikov and Imanuvilov [6]:

###### Lemma 2.1

There exists a function $\alpha_{0}\in C^{2}(\overline{{\Omega}})$ satisfying:

$\left\\{\begin{array}[]{l}\alpha_{0}(x)>0\quad\forall
x\in{\Omega},\quad\alpha_{0}=0\quad\forall x\in\partial{\Omega},\\\
|\nabla\alpha_{0}(x)|>0\quad\forall
x\in\overline{{\Omega}}\setminus\mathcal{O}_{0}.\end{array}\right.$

Let us introduce the functions

$\beta(t):=t(T-t),\ \phi(x,t):=\frac{e^{\lambda\alpha_{0}(x)}}{\beta(t)}\,,\
\overline{\alpha}(x):=e^{k\lambda}-e^{\lambda\alpha_{0}(x)},\
\alpha(x,t):=\frac{\overline{\alpha}(x)}{\beta(t)}\,,$

where $k>\|\alpha_{0}\|_{L^{\infty}}+\log 2$ and $\lambda>0$. Also, let us set

$\begin{array}[]{l}\displaystyle\hat{\alpha}(t):=\min_{x\in\overline{\Omega}}\alpha(x,t),\quad\alpha^{*}(t):=\max_{x\in\overline{\Omega}}\alpha(x,t),\\\
\displaystyle\hat{\phi}(t):=\min_{x\in\overline{\Omega}}\phi(x,t),\quad\phi^{*}(t):=\max_{x\in\overline{\Omega}}\alpha(x,t).\end{array}$

Then the following Carleman estimates hold:

###### Proposition 2.1

Assume that (2.2) holds. There exist positive constants $\lambda_{0}$, $s_{0}$
and $C_{0}$ such that, for any $s\geq s_{0}$ and $\lambda\geq\lambda_{0}$ and
any $\varphi^{T}\in L^{2}({\Omega})$, the associated solution to (2.1)
satisfies

$\begin{array}[]{c}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!\left(|\varphi_{t}|^{2}\\!+\\!|\Delta\varphi|^{2}\right)\\!+\\!\lambda^{2}(s\phi)|\nabla\varphi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]dxdt\\\
\displaystyle\leq
C_{0}\left(\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}|\psi|^{2}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right)dxdt\end{array}$
(2.3)

and

$\begin{array}[]{c}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!|\Delta\psi|^{2}\\!+\\!\lambda^{2}(s\phi)|\nabla\psi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\psi|^{2}\right]dxdt\\!\\\
\displaystyle\leq
C_{0}\left(\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}|\varphi|^{2}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}\right)dxdt\end{array}$
(2.4)

Furthermore, $C_{0}$ and $\lambda_{0}$ only depend on ${\Omega}$ and
$\mathcal{O}$ and $s_{0}$ can be chosen of the form

$\displaystyle s_{0}=\sigma_{0}(T+T^{2}),$ (2.5)

where $\sigma_{0}$ only depends on ${\Omega}$, $\mathcal{O}$,
$\|a\|_{\infty}$, $\|b\|_{\infty}$, $\|c\|_{\infty}$ and $\|d\|_{\infty}$.

This result is proved in [6]. In fact, similar Carleman inequalities are
established there for more general linear parabolic equations. The explicit
dependence in time of the constants is not given in [6]. We refer to [4],
where the above formula for $s_{0}$ is obtained.

For further purpose, we introduce the following notation:

$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!\left(|\varphi_{t}|^{2}\\!+\\!|\Delta\varphi|^{2}\right)\\!+\\!\lambda^{2}(s\phi)|\nabla\varphi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]dxdt\end{array}$

and

$\begin{array}[]{l}\displaystyle\tilde{I}(s,\lambda;\psi)=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}|\Delta\psi|^{2}\\!+\\!\lambda^{2}(s\phi)|\nabla\psi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\psi|^{2}\right]dxdt.\end{array}$

Now, we will deduce several consequences from Proposition 2.1 under particular
hypotheses on the coefficients of (2.1). First, it will be assumed that $c$ is
a.e. equal to a non-zero constant and $b$ and $d$ do not depend of $t$:

$a\in L^{\infty}(Q),\ c\in{\mathbb{R}},\ c\not=0,\ b,d\in
L^{\infty}({\Omega}),\ d\leq\mu<\mu_{1}\ \text{a.e.}$ (2.6)

Accordingly, (2.1) reads:

$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a(x,t)\varphi+c\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle-\Delta\psi=b(x)\varphi+d(x)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle\varphi(x,t)=0,\ \psi(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\
\displaystyle\varphi(x,T)=\varphi^{T}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$
(2.7)

###### Proposition 2.2

Assume that (2.6) holds. There exist positive constants $\lambda_{0}$, $s_{0}$
and $C_{1}$ such that, for any $s\geq s_{0}$ and $\lambda\geq\lambda_{0}$ and
any $\varphi^{T}\in L^{2}({\Omega})$, the associated solution to (2.7)
satisfies

$I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq
C_{1}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.$
(2.8)

Furthermore, $C_{1}$ and $\lambda_{0}$ only depend on ${\Omega}$ and
$\mathcal{O}$ and $s_{0}$ can be chosen of the form

$\displaystyle s_{1}=\sigma_{1}(T+T^{2}),$ (2.9)

where $\sigma_{1}$ only depends on ${\Omega}$, $\mathcal{O}$,
$\|a\|_{\infty}$, $\|b\|_{L^{\infty}}$, $|c|$ and $\|d\|_{L^{\infty}}$.

Proof: Obviously, it will be sufficient to show that there exist
$\lambda_{0}$, $s_{0}$ and $C_{1}$ such that, for any small $\varepsilon>0$,
one has:

$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq C\varepsilon
I(s,\lambda;\varphi)+C\varepsilon\tilde{I}(s,\lambda;\psi)\\\
\displaystyle\qquad+\
C_{1}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.\end{array}$
(2.10)

We start from (2.3) and (2.4). After addition, by taking $\sigma_{1}$
sufficiently large and $s\geq\sigma_{1}(T+T^{2})$, we obtain:

$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\\\ \displaystyle\quad\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\left(|\varphi|^{2}+|\psi|^{2}\right)dxdt.\end{array}$
(2.11)

Let us now introduce a function $\xi\in\mathcal{D}(\mathcal{O})$ satisfying
$0<\xi\leq 1$ and $\xi\equiv 1$ in $\mathcal{O}_{0}$. Then

$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}dxdt\leq\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi|\psi|^{2}dxdt\\\
\displaystyle\quad=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi(x)\psi\left(-\frac{1}{c}(\varphi_{t}+\Delta\varphi+a(x,t)\varphi)\right)dxdt\\\
\displaystyle\quad=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\,\psi\,\varphi_{t}\;dxdt\\\
\displaystyle\phantom{\quad=}-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\,\psi\,\Delta\varphi\;dxdt\\\
\displaystyle\phantom{\quad=}-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\,a(x,t)\,\psi\,\varphi\;dxdt\\\
\displaystyle\quad:=M_{1}+M_{2}+M_{3}.\end{array}$ (2.12)

Let us compute and estimate the $M_{i}$.

First,

$\begin{array}[]{l}\displaystyle
M_{1}=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\frac{2\xi(x)}{c}\lambda^{4}s^{4}\phi^{3}\alpha_{t}\psi\varphi\;dxdt\\\
\displaystyle\phantom{M_{1}=}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\frac{3\xi(x)}{c}\lambda^{4}s^{3}\phi^{2}\phi_{t}\psi\varphi\;dxdt\\\
\displaystyle\phantom{M_{1}=}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\frac{\xi(x)}{c}\lambda^{4}(s\phi)^{3}\psi_{t}\varphi\;dxdt.\end{array}$

Using that $|\alpha_{t}|\leq C\phi^{2}$ and $|\phi_{t}|\leq C\phi^{2}$ for
some $C>0$, we get:

$\begin{array}[]{l}\displaystyle M_{1}\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}s^{4}\phi^{5}|\psi|\,|\varphi|\;dxdt\\\
\displaystyle\phantom{M_{1}\leq}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi_{t}|\,|\varphi|\;dxdt\\\
\displaystyle\phantom{M_{1}}\leq\varepsilon\tilde{I}(s,\lambda;\psi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}s^{5}\phi^{7}|\varphi|^{2}dxdt\\\
\displaystyle\phantom{M_{1}\leq}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi_{t}|\,|\varphi|\;dxdt.\end{array}$

The last integral in this inequality can be bounded as follows:

$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi_{t}|\,|\varphi|\;dxdt\\\
\displaystyle\quad\leq\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\hat{\alpha}}\lambda^{4}(s\phi^{*})^{3}|\psi_{t}|\,|\varphi|\;dxdt\\\
\displaystyle\quad=\displaystyle\int_{0}^{T}e^{-2s\hat{\alpha}(t)}\lambda^{4}(s\phi^{*}(t))^{3}\|\psi_{t}(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\|\varphi(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\;dt\\\
\displaystyle\quad\leq
C\displaystyle\int_{0}^{T}e^{-2s\hat{\alpha}(t)}\lambda^{4}(s\phi^{*}(t))^{3}\|\varphi_{t}(\cdot\,,t)\|\|\varphi(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\;dt\\\
\displaystyle\quad=C\displaystyle\int_{0}^{T}e^{-s\alpha^{*}}(s\phi^{*}(t))^{-1/2}\|\varphi_{t}(\cdot\,,t)\|\cdot
e^{-2s\hat{\alpha}+s\alpha^{*}}\lambda^{4}(s\phi^{*})^{7/2}\|\varphi(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\;dt\\\
\displaystyle\quad\leq\varepsilon
I(s,\lambda;\varphi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.\end{array}$

Here, we have used that

$\left\\{\begin{array}[]{l}\displaystyle-\Delta\psi_{t}=b\varphi_{t}+d\psi_{t}\
\text{ in }\ {\Omega},\\\ \displaystyle\psi_{t}=0\ \text{ on }\
\partial{\Omega},\end{array}\right.$

whence we obviously need $b$ and $d$ independent of $t$.

Thus, the following is found:

$\begin{array}[]{l}\displaystyle M_{1}\leq\varepsilon
I(s,\lambda;\varphi)+\varepsilon\tilde{I}(s,\lambda;\psi)\\\
\displaystyle\phantom{M_{1}}+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.\end{array}$
(2.13)

Secondly, we see that

$\begin{array}[]{l}\displaystyle
M_{2}=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}\Delta\left(e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\psi\right)\varphi\;dxdt\\\
\displaystyle\phantom{M_{2}\leq}C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\left[\lambda^{6}(s\phi)^{5}|\psi|+\lambda^{5}(s\phi)^{4}|\nabla\psi|+\lambda^{4}(s\phi)^{3}|\Delta\psi|\right]\varphi\;dxdt\\\
\displaystyle\phantom{M_{2}}\leq\varepsilon\tilde{I}(s,\lambda;\psi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{8}(s\phi)^{7}|\varphi|^{2}dxdt.\end{array}$
(2.14)

Here, we have used the identity

$\begin{array}[]{c}\displaystyle\Delta\left(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\psi\right)=\Delta\left(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\right)\psi\\\
\displaystyle+2\nabla\left(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\right)\cdot\nabla\psi+e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\Delta\psi\end{array}$

and the estimates

$|\Delta(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c})|\leq
Ce^{-2s\alpha}\lambda^{2}s^{2}\phi^{5}\ \text{ and }\
|\nabla(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c})|\leq Ce^{-2s\alpha}\lambda
s\phi^{4}.$

Finally, it is immediate that

$M_{3}\leq\varepsilon\tilde{I}(s,\lambda;\psi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\varphi|^{2}dxdt.$
(2.15)

From (2.11), (2.12) and (2.13)–(2.15), we directly obtain (2.10) for all small
$\varepsilon>0$. This ends the proof. $\Box$

An almost immediate consequence of Proposition 2.2 is the following
observability inequality:

###### Corollary 2.1

Under the assumptions of Proposition 2.2, there exists constants $M$ and $K$,
depending on ${\Omega}$, $\mathcal{O}$, $T$, $\|a\|_{\infty}$,
$\|b\|_{L^{\infty}}$, $|c|$ and $\|d\|_{L^{\infty}}$, such that every solution
$(\varphi,\psi)$ to (2.7) verifies:

$\begin{array}[]{l}\displaystyle\|\varphi(\cdot\,,0)\|^{2}+\|\psi(\cdot\,,0)\|^{2}\leq
M\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi|^{2}dxdt.\end{array}$
(2.16)

Proof: From the Carleman inequality in Proposition 2.2 with $s=s_{1}$ and
$\lambda=\lambda_{1}$, we see that there exists a constant $C>0$ such that

$\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}\phi^{3}\left(|\varphi|^{2}+|\psi|^{2}\right)dxdt\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}(t(T-t))^{-7}e^{-4s\hat{\alpha}+2s\alpha^{*}}|\varphi|^{2}dxdt.$

Since $2\hat{\alpha}-\alpha^{*}\equiv a^{*}/\beta(t)$ for some $a^{*}>0$ and
$e^{-2s\alpha}\phi^{3}$ is uniformly bounded from below in
${\Omega}\times[T/4,3T/4]$ we find that

$\displaystyle\int_{T/4}^{3T/4}\int_{\Omega}\left(|\varphi|^{2}+|\psi|^{2}\right)dxdt\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi|^{2}dxdt$
(2.17)

for some $C,K>0$.

On the other hand, we can easily get from (2.7) the standard (backwards)
energy inequalities

$-\frac{1}{2}\frac{d}{dt}\|\varphi\|^{2}+\|\varphi\|_{H_{0}^{1}}^{2}\leq
C\left(\|\varphi\|^{2}+\|\psi\|^{2}\right)\ \text{ and }\
\|\psi\|_{H_{0}^{1}}^{2}\leq C\|\varphi\|^{2}.$ (2.18)

This yields

$-\frac{1}{2}\frac{d}{dt}\|\varphi\|^{2}\leq C\|\varphi\|^{2},$ (2.19)

whence we deduce that

$\|\varphi(\cdot\,,0)\|^{2}\leq C\|\varphi(\cdot\,,t)\|^{2}\;\;\mbox{ for all
}t.$ (2.20)

Combining (2.17), (2.20) and the second part of (2.18), we obtain at once
(2.16). $\Box$

Now, we will assume that $b$ is a non-zero constant:

$a,c,d\in L^{\infty}(Q),\ b\in{\mathbb{R}},\ b\not=0,\ d\leq\mu<\mu_{1}\
\text{a.e.}$ (2.21)

The corresponding (2.1) becomes

$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a(x,t)\varphi+c(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle-\Delta\psi=b\varphi+d(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle\varphi(x,t)=0,\ \psi(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\
\displaystyle\varphi(x,T)=\varphi^{T}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$
(2.22)

###### Proposition 2.3

Assume that (2.21) holds. There exist positive constants $\lambda_{2}$,
$s_{2}$ and $C_{2}$ such that, for any $s\geq s_{2}$ and
$\lambda\geq\lambda_{2}$ and any $\varphi^{T}\in L^{2}({\Omega})$, the
associated solution to (2.22) satisfies

$I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq
C_{2}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{8}(s\phi)^{7}|\psi|^{2}dxdt.$

Furthermore, $C_{2}$ and $\lambda_{2}$ only depend on ${\Omega}$ and
$\mathcal{O}$ and $s_{2}$ can be chosen of the form

$\displaystyle s_{2}=\sigma_{2}(T+T^{2}),$

where $\sigma_{2}$ only depends on ${\Omega}$, $\mathcal{O}$,
$\|a\|_{\infty}$, $|b|$, $\|c\|_{\infty}$ and $\|d\|_{\infty}$.

Proof: We start again from (2.11). Recalling that
$\xi\in\mathcal{D}(\mathcal{O})$, $0<\xi\leq 1$ and $\xi\equiv 1$ in
$\mathcal{O}_{0}$, we see that

$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\varphi|^{2}dxdt\leq\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi|\varphi|^{2}dxdt\\\
\displaystyle\quad=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi(x)\varphi\left(-\frac{1}{b}(\Delta\psi+d(x,t)\psi)\right)dxdt\\\
\displaystyle\quad=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{b}\varphi\Delta\psi\;dxdt\\\
\displaystyle\phantom{\quad=}-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{b}d(x,t)\varphi\psi\;dxdt\\\
\displaystyle\quad:=M^{\prime}_{1}+M^{\prime}_{2}.\end{array}$ (2.23)

As in the proof of Proposition 2.2, it is not difficult compute and estimate
the $M^{\prime}_{i}$. Indeed,

$\begin{array}[]{l}\displaystyle
M^{\prime}_{1}=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}\Delta\left(e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{b}\varphi\right)\psi\;dxdt\\\
\displaystyle\phantom{M^{\prime}_{1}}\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\left[\lambda^{6}(s\phi)^{5}|\varphi|+\lambda^{5}(s\phi)^{4}|\nabla\varphi|+\lambda^{4}(s\phi)^{3}|\Delta\varphi|\right]|\psi|\;dxdt\\\
\displaystyle\phantom{M^{\prime}_{1}}\leq\varepsilon
I(s,\lambda;\varphi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{8}(s\phi)^{7}|\psi|^{2}dxdt.\end{array}$
(2.24)

On the other hand,

$M^{\prime}_{2}\leq\varepsilon
I(s,\lambda;\varphi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}dxdt.$
(2.25)

From (2.11), (2.23) and (2.24)–(2.25), we find that

$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq C\varepsilon
I(s,\lambda;\varphi)+C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}dxdt,\end{array}$

for all small $\varepsilon>0$.

This ends the proof. $\Box$

Arguing as in the proof of Corollary 2.1, the following can be easily
established:

###### Corollary 2.2

Under the assumptions of Proposition 2.3, there exists constants $M^{\prime}$
and $K^{\prime}$, depending on ${\Omega}$, $\mathcal{O}$, $T$,
$\|a\|_{L^{\infty}(Q)}$, $|b|$, $\|c\|_{L^{\infty}(Q)}$ and
$\|d\|_{L^{\infty}}$, such that every solution $(\varphi,\psi)$ to (2.7)
verifies:

$\|\varphi(\cdot\,,0)\|^{2}+\limsup_{t\to 0^{+}}\|\psi(\cdot\,,t)\|^{2}\leq
M^{\prime}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K^{\prime}/(T-t)}|\psi|^{2}dxdt.$
(2.26)

##  3 The null controllability of the linearized systems

In this Section, we will deduce from the observability estimates (2.16) and
(2.26) null controllability results for (1.9) and (1.10).

More precisely, we have:

###### Theorem 3.1

Assume that (2.6) holds. Then (1.9) is null-controllable at any time $T>0$.
That is to say, for any $y^{0}\in L^{2}({\Omega})$ there exist null controls
$v\in L^{2}(\mathcal{O}\times(0,T))$ for (1.9) satisfying

$\|v\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$ (3.1)

where $C$ only depends on ${\Omega}$, $\mathcal{O}$, $T$,
$\|a\|_{L^{\infty}(Q)}$, $\|b\|_{L^{\infty}}$, $|c|$ and $\|d\|_{L^{\infty}}$.

Proof: There are several ways to prove that the observability inequality
(2.16) implies the null controllability of (1.9). One of them is the
following.

For any $v\in L^{2}(\mathcal{O}\times(0,T))$ and any $\varepsilon>0$, let us
set

$J_{\varepsilon}(v)=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|v|^{2}dxdt+\frac{1}{\varepsilon}\|y(\cdot\,,T)\|^{2}.$
(3.2)

Here, $(y,z)$ is the solution to (1.9) associated to the initial data $y^{0}$.
It is not difficult to check that $v\mapsto J_{\varepsilon}(v)$ is lower semi-
continuous, strictly convex and coercive in $L^{2}(\mathcal{O}\times(0,T))$.
Hence, it possesses a unique minimizer $v_{\varepsilon}\in L^{2}(Q)$. We will
denote by $(y_{\varepsilon},z_{\varepsilon})$ the associated state.

We will show that, at least for a subsequence, $v_{\varepsilon}$ converges
weakly in $L^{2}(\mathcal{O}\times(0,T))$ towards a control $v\in
L^{2}(\mathcal{O}\times(0,T))$ and the associated $y_{\varepsilon}(\cdot\,,T)$
converges strongly in $L^{2}({\Omega})$ to zero. Obviously, this proves that
$v$ is a null control for (1.9), i.e. that the state associated to $v$
satisfies (1.4).

Notice that the unique minimizer of (3.11) is characterized by the following
optimality system:

$\left\\{\begin{array}[]{l}\displaystyle y_{\varepsilon,t}-\Delta
y_{\varepsilon}=ay_{\varepsilon}+bz_{\varepsilon}+v_{\varepsilon}1_{\mathcal{O}}\
\ \ \mbox{ in }\ Q,\\\ \displaystyle-\Delta
z_{\varepsilon}=cy_{\varepsilon}+dz_{\varepsilon}\ \ \ \mbox{ in }Q,\\\
\displaystyle y_{\varepsilon}=z_{\varepsilon}=0\ \ \ \mbox{ on }\ \Sigma,\\\
\displaystyle y_{\varepsilon}(x,0)=y^{0}(x)\ \ \ \mbox{ in }\
\Omega,\end{array}\right.$ (3.3)
$\left\\{\begin{array}[]{l}\displaystyle-\varphi_{\varepsilon,t}-\Delta\varphi_{\varepsilon}=a\varphi_{\varepsilon}+c\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\
\displaystyle-\Delta\psi_{\varepsilon}=b\varphi_{\varepsilon}+d\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\ \displaystyle\varphi_{\varepsilon}=\psi_{\varepsilon}=0\
\ \ \;\mbox{ on }\ \Sigma\\\
\displaystyle\varphi_{\varepsilon}(x,T)=-\frac{1}{\varepsilon}y_{\varepsilon}(x,T)\
\;\;\;\mbox{ in }\;\Omega,\end{array}\right.$ (3.4)
$v_{\varepsilon}=\bigl{.}e^{-2K/(T-t)}\varphi_{\varepsilon}\bigr{|}_{\mathcal{O}\times(0,T)}\,$
(3.5)

(see for instance [8]; see also [4]).

By multiplying both sides of $(\ref{asep})_{1}$ by $y_{\varepsilon}$ and both
sides of $(\ref{asep})_{2}$ by $z_{\varepsilon}$, integrating in time and
space and adding the resulting identities, we obtain:

$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt+\frac{1}{\varepsilon}\|y_{\varepsilon}(\cdot\,,T)\|^{2}\\\\[11.0pt]
\displaystyle\leq\|y^{0}\|\;\|\varphi_{\varepsilon}(\cdot\,,0)\|\leq
M^{1/2}\|y^{0}\|\left(\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt\right)^{1/2}\\\\[11.0pt]
\displaystyle\leq
C\|y^{0}\|^{2}+\frac{1}{2}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt,\end{array}$
(3.6)

where we have used the observability estimate (2.16).

From (3.6), we see that

$\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|v_{\varepsilon}|^{2}dxdt=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt\leq
C\|y^{0}\|^{2}.$ (3.7)

Consequently, at least for a subsequence, one has

$\begin{array}[]{l}\displaystyle v_{\varepsilon}\to v\ \text{ weakly in }\
L^{2}(\mathcal{O}\times(0,T)),\ \text{ with }\
\|v\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|.\end{array}$ (3.8)

We also have from (3.6) that

$\begin{array}[]{l}\displaystyle y_{\varepsilon}(\cdot\,,T)\to 0\ \text{
strongly in }\ L^{2}({\Omega})\ \text{ as }\ \varepsilon\to 0.\end{array}$
(3.9)

From the usual energy method, it is clear that a subsequence can be extracted
such that

$\begin{array}[]{l}\displaystyle y_{\varepsilon}\to y\ \text{ and }\
z_{\varepsilon}\to z\;\;\mbox{strongly in }\;L^{2}(Q)\;\;\mbox{ as
}\;\varepsilon\to 0\end{array}$ (3.10)

(see for instance [10]) and, consequently, the limit $v$ is such that the
solution $(y,z)$ to (1.9) satisfies (1.4).

This ends the proof of Theorem 3.1. $\Box$

We also deduce from Corollary 2.2 the following result:

###### Theorem 3.2

Assume that (2.21) holds. Then (1.10) is null-controllable at any time $T>0$.
In other words, for any $y^{0}\in L^{2}({\Omega})$ there exist null controls
$w\in L^{2}(\mathcal{O}\times(0,T))$ for (1.10) such that

$\|w\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$

where $C$ only depends on ${\Omega}$, $\mathcal{O}$, $T$, $\|a\|_{\infty}$,
$|b|$, $\|c\|_{\infty}$ and $\|d\|_{\infty}$.

Proof: It is very similar to the proof of Thoerem 3.1.

Indeed, we can introduce the functional $L_{\varepsilon}$, with

$L_{\varepsilon}(v):=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|w|^{2}\;dxdt+\frac{1}{\varepsilon}\|y(\cdot\,,T)\|^{2}$
(3.11)

and we can again check that $L_{\varepsilon}$ possesses exactly one minimizer
$w_{\varepsilon}\in L^{2}(Q)$. The optimality system is now

$\left\\{\begin{array}[]{l}\displaystyle y_{\varepsilon,t}-\Delta
y_{\varepsilon}=ay_{\varepsilon}+bz_{\varepsilon}\ \ \ \mbox{ in }\ Q,\\\
\displaystyle-\Delta
z_{\varepsilon}=cy_{\varepsilon}+dz_{\varepsilon}+w_{\varepsilon}1_{\mathcal{O}}\
\ \ \mbox{ in }Q,\\\ \displaystyle y_{\varepsilon}=z_{\varepsilon}=0\ \ \
\mbox{ on }\ \Sigma,\\\ \displaystyle y_{\varepsilon}(x,0)=y^{0}(x)\ \ \
\mbox{ in }\ \Omega,\end{array}\right.$
$\left\\{\begin{array}[]{l}\displaystyle-\varphi_{\varepsilon,t}-\Delta\varphi_{\varepsilon}=a\varphi_{\varepsilon}+c\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\
\displaystyle-\Delta\psi_{\varepsilon}=b\varphi_{\varepsilon}+d\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\ \displaystyle\varphi_{\varepsilon}=\psi_{\varepsilon}=0\
\ \ \;\mbox{ on }\ \Sigma\\\
\displaystyle\varphi_{\varepsilon}(x,T)=-\frac{1}{\varepsilon}y_{\varepsilon}(x,T)\
\;\;\;\mbox{ in }\;\Omega,\end{array}\right.$
$w_{\varepsilon}=\bigl{.}e^{-2K/(T-t)}\psi_{\varepsilon}\bigr{|}_{\mathcal{O}\times(0,T)}\,.$

We can argue as before and deduce that

$\begin{array}[]{l}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|w_{\varepsilon}|^{2}dxdt\\\\[5.0pt]
\displaystyle=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\psi_{\varepsilon}|^{2}dxdt\leq
C\|y^{0}\|^{2},\end{array}$ (3.12)

whence a weakly convergent sequence of control exists and, in the limit, we
get a null control for (1.10).

Notice that, by construction,

$\limsup_{t\to T^{-}}\|w(\cdot\,,t)\|_{L^{2}(\mathcal{O})}=0.$

This, together with the energy estimates

$\|z_{\varepsilon}(\cdot\,,t)\|_{H_{0}^{1}}\leq
C\left(\|y(\cdot\,,t)\|+\|w(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\right),$

ensures the second part of (1.4). $\Box$

##  4 The null controllability of the semilinear systems

In this Section, we present the proofs of the main results in this paper,
namely Theorems 1.1 and 1.2. They will be obtained by combining the linear
controllability results in the previous Section and a standard fixed-point
argument.

Proof of Theorem 1.1: Let us first assume that, in (1.6), $F_{0}$ is $C^{1}$.
In view of (1.6) and (1.7), we observe that (1.1) can be written as follows:

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(y)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$

with

$A_{0}(s)=\left\\{\begin{array}[]{ll}\displaystyle{F_{0}(s)\over s}&\ \text{
if }\ s\not=0,\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
F^{\prime}_{0}(0)&\ \text{ otherwise. }\end{array}\right.$ (4.1)

For any $k\in L^{2}(Q)$, let us consider the linear system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(k)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$ (4.2)

In view of Theorem 3.1, there exists controls $v\in
L^{2}(\mathcal{O}\times(0,T))$ such that the associated states $(y,z)$ satisfy
(1.4) and (3.1), where the constant $C$ only depends on ${\Omega}$,
$\mathcal{O}$, $T$, $\|F_{0}\|_{C^{1}({\mathbb{R}})}$, $|b|$, $|c|$ and $|d|$.

Let us introduce the mapping $\Phi:L^{2}(Q)\mapsto 2^{L^{2}(Q)}$, as follows:
for any $k\in L^{2}(Q)$, we set by definition

$\Phi_{0}(k)=\\{\,v\in L^{2}(\mathcal{O}\times(0,T)):\text{ the solution
to~{}\eqref{tk1} satisfies \eqref{eq2n} and~{}\eqref{eq3}}\,\\}$

and

$\Phi(k)=\\{\,y\in L^{2}(Q):\text{ $(y,z)$ solves \eqref{tk1} for some
$v\in\Phi_{0}(k)$}\,\\}.$

Then $\Phi$ satisfies the hypotheses of Kakutani’s Fixed-Point Theorem.

Indeed, the following holds:

  * •

For any $k\in L^{2}(Q)$, $\Phi(k)\subset L^{2}(Q)$ is a non-empty compact set.
Furthermore, there exists a fixed compact set $K\subset L^{2}(Q)$ such that
$\Phi(k)\subset K$ for all $k\in L^{2}(Q)$.

This is an obvious consequence of (3.1), the energy estimates

$\|y_{t}\|_{L^{2}(0,T;H^{-1}({\Omega}))}^{2}+\|y\|_{L^{2}(0,T;H_{0}^{1}({\Omega}))}^{2}\leq
C\left(\|v\|_{L^{2}(\mathcal{O}\times(0,T))}^{2}+\|y^{0}\|^{2}\right)$

(established in the Appendix) and the compactness of the embedding
$W\hookrightarrow L^{2}(Q)$, where

$W=\\{\,\xi\in L^{2}(0,T;H_{0}^{1}({\Omega})):\xi_{t}\in
L^{2}(0,T;H^{-1}({\Omega}))\,\\}.$

  * •

$\Phi$ is sequentially upper semicontinuous on $L^{2}(Q)$, i.e. if $k_{n},k\in
L^{2}(Q)$ and $k_{n}\to k$ in $L^{2}(Q)$, then

$\displaystyle\limsup_{n\to\infty}\left(\sup_{y\in\Phi(k_{n})}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}y\,\xi\;dxdt\right)\leq\sup_{y\in\Phi(k_{n})}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}y\,\xi\;dxdt\quad\forall\xi\in
L^{2}(Q).$

Therefore, $\Phi$ possesses at least one fixed-point $y$.

Obviously, $y$ solves, together with some $z$, the semilinear system (1.1) for
some $v\in L^{2}(\mathcal{O}\times(0,T))$ and (1.4) holds.

Let us now assume that the function $F_{0}$ in (1.6) is only globally
Lipschitz-continuous. Suppose that

$|F_{0}(s_{1})-F_{0}(s_{2})|\leq L|s_{1}-s_{2}|\quad\forall
s_{1},s_{2}\in{\mathbb{R}}.$

Then, we can find $C^{1}$ functions $F_{01},F_{02},\dots$ with the following
properties:

  1. 1.

$F_{0n}:{\mathbb{R}}\mapsto{\mathbb{R}}$ is $C^{1}$ and globally Lipschitz-
continuous., with Lpschitz constant $L$, i.e. Suppose that

$|F_{0n}(s_{1})-F_{0n}(s_{2})|\leq L|s_{1}-s_{2}|\quad\forall
s_{1},s_{2}\in{\mathbb{R}},\quad\forall n\geq 1.$

  2. 2.

$F_{0n}\to F_{0}$ uniformly on each compact interval $I\subset{\mathbb{R}}$.

For each $n\geq 1$, let us consider the system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F_{0n}(y)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$ (4.3)

Let $v_{n}$ be a null control for (4.3) satisfying

$\|v_{n}\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$

with $C$ independent of $n$. In view of the previous arguments and the
properties of $F_{0n}$, such a $v_{n}$ exists. It is clear that, at least for
a subsequence, one has

$v_{n}\to v\ \text{ weakly in }\ L^{2}(\mathcal{O}\times(0,T)),$

where $v$ is a null control for (1.1) again satisfying (1.4).

This ends the proof. $\Box$

Proof of Theorem 1.2: The proof is similar to the previous one, although a
little more intrincate.

Again, let us first assume that, in (1.6) and (1.8), the functions $F_{0}$ and
$f_{0}$ are $C^{1}$. Then, (1.3) can be written in the form

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(y)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=C_{0}(y)y+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$

where $A_{0}$ is given by (4.1) and

$C_{0}(s)=\left\\{\begin{array}[]{ll}\displaystyle{f_{0}(s)\over s}&\ \text{
if }\ s\not=0,\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
f^{\prime}_{0}(0)&\ \text{ otherwise. }\end{array}\right.$

For each $k\in L^{2}(Q)$, we can consider the linear system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(k)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=C_{0}(k)y+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$ (4.4)

As in the proof of Theorem 1.1, we can prove that there exist controls $w\in
L^{2}(\mathcal{O}\times(0,T))$ such that the associated solutions to (4.4)
satisfy (1.4) and

$\|w\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$

for some fixed $C$. We can again introduce a multi-valued mapping
$\Psi:L^{2}(Q)\mapsto 2^{L^{2}(Q)}$ (similar to $\Phi$) and we can show that
the assumptions of Kakutani’s Theorem are satisfied by $\Psi$, etc.

An easy adaptation of the remaining results leads to the desired
controllability result. We omit the details, that can be checked easily.
$\Box$

##  5 An asymptotic controllability property

In this Section, we prove that, under appropriate conditions on $F$ and $f$
(in fact the same in Theorem 1.2), the semilinear parabolic system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon\,z_{t}-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\,\,z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;\Omega,\end{array}\right.$
(5.1)

is uniformly null-controllable as $\varepsilon\to 0$. We also prove the
convergence of the null controls to a null control for the similar parabolic-
elliptic system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;\Omega.\end{array}\right.$ (5.2)

To this end, we will first consider the linear system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=ay+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon z_{t}-\Delta
z=cy+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\;\;\;z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$
(5.3)

and we will establish a uniform null controllability result.

More precisely, the following holds:

###### Theorem 5.1

Assume that (2.21) holds. Then, for any $\varepsilon>0$ and any
$y^{0},z^{0}\in L^{2}({\Omega})$, there exist controls $w_{\varepsilon}\in
L^{2}(\mathcal{O}\times(0,T))$ such that the corresponding solutions
$(y_{\varepsilon},z_{\varepsilon})$ to (5.3) satisfy

$y_{\varepsilon}(x,T)=0,\;\;z_{\varepsilon}(x,T)=0\;\;\;\mbox{in}\;\;{\Omega},$

with an estimate of the form

$\|w_{\varepsilon}\|_{L^{2}(\mathcal{O}\times(0,T))}\leq
C\left(\|y^{0}\|+\varepsilon\;\|z^{0}\|\right).$ (5.4)

where $C$ is independent of $\varepsilon$.

Sketch of the proof: Let us consider the adjoint system of (5.3), that is:

$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a\varphi+c\psi\;\;\mbox{in}\;\;Q,\\\\[5.0pt]
\displaystyle-\varepsilon\psi_{t}-\Delta\psi=b\varphi+d\psi\;\;\mbox{in}\;\;Q,\\\\[5.0pt]
\displaystyle\varphi=0,\;\psi=0\;\;\mbox{on}\;\;\Sigma,\\\\[5.0pt]
\displaystyle\varphi(x,T)=\varphi^{0}(x),\,\,\,\psi(x,T)=\psi^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$

with $\varphi^{0},\psi^{0}\in L^{2}({\Omega})$.

As in Section 2, we will use an abridged notation for the weighted integrals
concerning $\varphi$, $\psi$ and their derivatives: for any positive $\lambda$
and $s$, we set

$\begin{array}[]{l}\displaystyle
I(s,\lambda,\varphi)=\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!\left(|\varphi_{t}|^{2}\\!+\\!|\Delta\varphi|^{2}\right)\\!+\\!\lambda^{2}s\phi|\nabla\varphi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]\\!\,dx\,dt,\\\\[9.0pt]
\displaystyle
I_{\varepsilon}(s,\lambda,\psi)=\displaystyle\varepsilon^{2}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}(s\phi)^{-1}|\psi_{t}|^{2}\,dx\,dt\\\\[9.0pt]
\displaystyle\phantom{I_{\varepsilon}(s,\lambda,\psi)}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}|\Delta\varphi|^{2}\\!+\\!\lambda^{2}s\phi|\nabla\psi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]\\!\,dx\,dt\end{array}$

Then one has

$\displaystyle I(s,\lambda,\varphi)+I_{\varepsilon}(s,\lambda,\psi)\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}(s\phi)^{7}\lambda^{8}e^{-2s{\alpha}(x,t)}|\psi|^{2}\,dx\,dt,$
(5.5)

where $C$ is independent of $\varepsilon$.

The proof of (5.5) is very similar to the proof of Proposition 2.2 and will be
omitted. An almost immediate consequence is the following observability
inequality:

$\|\varphi(\cdot\,,0)\|^{2}+\varepsilon\;\|\psi(\cdot\,,0)\|^{2}\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2s\alpha}(s\phi)^{7}{\lambda}^{8}|\psi|^{2}\,dx\,dt.$
(5.6)

Now, arguing as in Section 3, it becomes clear that the uniform null
controllability property of (5.3) is implied by the observability estimate
(5.6). $\Box$

From this result, we get the following for the semilinear systems (5.1) and
(5.2):

###### Theorem 5.2

Under the assumptions of Theorem 1.2, for any $\varepsilon>0$ and any
$y^{0},z^{0}\in L^{2}(\Omega)$, there exist null controls $w_{\varepsilon}\in
L^{2}(\mathcal{O}\times(0,T))$ for (5.1). They can be chosen such that, at
least for a subsequence,

$w_{\varepsilon}\to w\;\;\;\mbox{ weakly in
}\;\;L^{2}(\mathcal{O}\times(0,T)),$

where $w$ is a null control for (5.2).

Sketch of the proof: For the proof of the first assertion, it suffices to
argue as in the proof of Theorem 1.2.

Indeed, for any $\varepsilon>0$ and any fixed $k\in L^{2}(Q)$, we can apply
Theorem 5.1 to the linearized system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(k)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon\,z_{t}-\Delta
z=C_{0}(k)y+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\,\,z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;\Omega.\end{array}\right.$
(5.7)

We deduce that, for any $\varepsilon>0$, (5.7) is null-controllable, with
controls $w_{\varepsilon}$ satisfying (5.4) (where $C$ is independent of
$\varepsilon$). Observe that, in fact, we can get a stronger estimate:

$\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|w_{\varepsilon}|^{2}dxdt\leq
C\|y^{0}\|^{2}.$

We can again introduce a multi-valued mapping $\Psi_{\varepsilon}$ (similar to
$\Psi$) and we can show that the assumptions of Kakutani’s Theorem are
satisfied by $\Psi_{\varepsilon}$. Therefore, there exist controls
$w_{\varepsilon}\in L^{2}(\mathcal{O}\times(0,T))$ satisfying (5.4) such that
the associated solutions to (5.1) satisfy

$y_{\varepsilon}(x,T)=0,\;\;z_{\varepsilon}(x,T)=0\;\;\;\mbox{in}\;\;{\Omega}.$

Let us multiply both sides of $(\ref{6epsn})_{1}$ (resp. $(\ref{6epsn})_{2}$)
by $\displaystyle y_{\varepsilon}$ (resp. $\displaystyle z_{\varepsilon}$) and
let us integrate in $Q$. We easily obtain the following for all $t$:

$\begin{array}[]{l}\displaystyle\|y_{\varepsilon}(\cdot\,,t)\|^{2}+\varepsilon\|z_{\varepsilon}(\cdot\,,t)\|^{2}+\displaystyle\int_{0}^{t}\left(\|y_{\varepsilon}(\cdot\,,s)\|_{H_{0}^{1}}^{2}+\|y_{\varepsilon}(\cdot\,,s)\|_{H_{0}^{1}}^{2}\right)\,ds\\\
\displaystyle\ \
\leq\|y^{0}\|^{2}+\varepsilon\|z^{0}\|^{2}+C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,t)}|w_{\varepsilon}|^{2}\,dx\,ds+C\displaystyle\int_{0}^{t}\|y_{\varepsilon}(\cdot\,,s)\|^{2}\,ds.\end{array}$

Using Gronwall’s inequality and extracting appropriate subsequences, we deduce
that, at least for a subsequence, $w_{\varepsilon}$, $y_{\varepsilon}$ and
$z_{\varepsilon}$ respectively converge to $w$, $y$ and $z$, where $(y,z)$
solves (5.2) and satisfies (1.4). $\Box$

##  6 Appendix A: Some technical results

In this Appendix, for completeness, we give a theoretical result for the
semilinear system

$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$ (6.1)

where $F$ and $f$ satisfy (1.6) and (1.8), respectively.

We have the following:

###### Theorem 6.1

For any given $y^{0}\in L^{2}({\Omega})$ and $v,w\in
L^{2}(\mathcal{O}\times(0,T))$, (6.1) possesses a unique solution $(y,z)$,
with

$y\in L^{2}(0,T;H_{0}^{1}({\Omega})),\quad y_{t}\in
L^{2}(0,T;H^{-1}({\Omega})),\quad z\in L^{2}(0,T;D(-\Delta)).$ (6.2)

Sketch of the proof: We will apply the Faedo-Galerkin method, see J.-L. Lions
[10].

Let $\\{h_{j}\\}$ be a special basis of $H_{0}^{1}({\Omega})$, more precisely,
the basis formed by the eigenfuncions of the Dirichlet Laplacian in
${\Omega}$. Let us introduce the finite-dimensional Galerkin approximations as
follows: find $y_{N},z_{N}$, with $y_{N}(t),z_{N}(t)\in V_{N}$ for all $t$,
such that

$\displaystyle(y_{N}^{\prime}(t),h)+(\nabla y_{N}(t),\nabla
h)=(F(y_{N},z_{N}),h)+(v1_{\mathcal{O}},h)\quad\forall h\in V_{N},$ (6.3)
$\displaystyle(\nabla z_{N}(t),\nabla
k)=(f(y_{N},z_{N}),k)+(w1_{\mathcal{O}},k)\quad\forall k\in V_{N},$ (6.4)
$\displaystyle\begin{array}[]{l}y_{N}(0)=y_{0N}.\end{array}$ (6.6)

Here, $V_{N}=[h_{1},h_{2},\dots,h_{N}]$ is the subspace of
$H_{0}^{1}({\Omega})$ spanned by the first $N$ eigenfunctions $h_{j}$ and
$y_{0N}=P_{N}y_{0}$, where $P_{N}:L^{2}({\Omega})\mapsto V_{N}$ is the
orthogonal projector.

This Cauchy problem has at least one local solution on $[0,t_{N})$. That the
maximal solution is defined in the whole interval $[0,T]$ is a consequence of
the estimates given below.

Estimates I : Let us set $h=y_{N}(t)$ in (6.3) and $k=z_{N}(t)$ in (6.4).
After some computations we obtain the following for any $t$ and for all small
$\delta>0$:

$\begin{array}[]{l}\displaystyle{1\over
2}\|y_{N}(t)\|^{2}+\displaystyle\int_{0}^{t}\|y_{N}(s)\|_{H_{0}^{1}}^{2}\,ds+\left(1-{\mu+\delta\over\mu_{1}}\right)\displaystyle\int_{0}^{t}\|z_{N}(s)\|_{H_{0}^{1}}^{2}\,ds\\\
\displaystyle\ \ \leq{1\over
2}\|y^{0}\|^{2}+C_{\delta}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,t)}\left(|v|^{2}+|w|^{2}\right)\,dx\,ds+C_{\delta}\displaystyle\int_{0}^{t}\|y_{N}(s)\|^{2}\,ds\end{array}$
(6.7)

and

$\begin{array}[]{l}\displaystyle\left(1-{\mu+\delta\over\mu_{1}}\right)\|z_{N}(t)\|_{H_{0}^{1}}\leq
C_{\delta}\left(\|y_{N}(t)\|+\|w\|_{L^{2}(\mathcal{O})}\right).\end{array}$
(6.8)

Estimates II  Since the $h_{j}$ are the eigenfunctions of $-\Delta$ in
$H_{0}^{1}({\Omega})$, one has

$\|y^{\prime}_{N}(t)\|_{H^{-1}(\Omega)}\leq\|(\Delta
y_{N}+F(y_{N},z_{N})+v1_{\mathcal{O}})(t)\|_{H^{-1}(\Omega)}$ (6.9)

for all $t$. Therefore, we get the following for some $C>0$:

$\|y^{\prime}_{N}(t)\|_{H^{-1}(\Omega)}\leq
C\left(\|y_{N}(t)\|_{H_{0}^{1}}+\|(v1_{\mathcal{O}})(\cdot\,,t)\|+\|(w1_{\mathcal{O}})(\cdot\,,t)\|\right).$
(6.10)

From (6.7)–(6.10), it is standard to deduce that, at least for a subsequence,
the $y_{N}$ and $z_{N}$ converge to a solution to (6.1). This solution must
satisfy

$y\in L^{2}(0,T;H_{0}^{1}({\Omega})),\quad y_{t}\in
L^{2}(0,T;H^{-1}({\Omega})),\quad z\in L^{2}(0,T;H_{0}^{1}({\Omega})).$

Furthermore, from the usual elliptic estimates, we also obtain that $z\in
L^{2}(0,T;D(\Delta))$. This yields (6.2).

The uniqueness of the solution is also a standard consequence of the previous
estimates (written for $y:=y^{1}-y^{2}$ and $z:=z^{1}-z^{2}$ where
$(y^{1},z^{1})$ and $(y^{2},z^{2})$ are assumed to solve the system) and the
global Lipschitz-continuity of $F$ and $f$. $\Box$

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