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\markboth{ A semilinear control problem involving homogenization }
{ C. Conca, A. Osses \&  J. Saint Jean Paulin }

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 109--122.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 A semilinear control problem involving homogenization
% 
\thanks{ {\em Mathematics Subject Classifications: 35B37, 35B27, 35J60.} 
 \hfil\break\indent 
{\em Key words:} control, homogenization, semilinear elliptic equation.
 \hfil\break\indent
\copyright 2001 Southwest Texas State University. 
\hfil\break\indent Published January 8, 2001 \hfil\break\indent
Partially supported by the Mathematical-Mechanics program of FONDAP. } } 

\date{}
\author{ Carlos Conca, Axel Osses, \&  Jeannine Saint Jean Paulin }
\maketitle
\begin{abstract} 
We consider a  control problem involving a semilinear elliptic equation 
with a uniformly Lipschitz non-linearity and rapidly oscillating 
coefficients in a bounded domain of $\mathbb{R}^N$. The control is
distributed on a compact subset interior to the domain. Given an $N-1$
dimensional hypersurface at the interior of the domain not
intersecting the control zone, the trace of the solution on the curve
has to be controlled. We prove that there exists a limit control as
the homogenization parameter converges to zero, which results as the 
limit of fixed points for controllability problems.
We link this limit control with the corresponding homogenized problem.
\end{abstract}

\def\norm #1{\left\|{#1}\right\|}
\def\mod #1{\left|{#1}\right|}
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\newtheorem{theorem}{Theorem}[section] 
\newtheorem{lemma}{Lemma}[section] 
\newtheorem{remark}[theorem]{Remark} 

\section{Introduction}
Let $\Omega$ be a connected and open subset of $\mathbb{R}^N$ with 
smooth boundary $\Gamma$.  
Let $\omega\subset\subset\Omega$ be a non-empty open subset
with indicatrix set $1_\omega$ and let $S$ be a $N-1$ dimensional manifold
strictly included in $\Omega$ and not intersecting $\omega$. 
Consider the following control problem. Given $\varepsilon>0$, $\alpha>0$ 
and $y_1\in L^2(S)^N$ find a control function $v^\varepsilon$ with support 
in $\omega$ such that
\begin{equation} \label{primal-system}
\gathered
-\mathop{\rm div}(A^\varepsilon\nabla y^\varepsilon) 
+ f(y^\varepsilon)=1_{\omega}v^\varepsilon\quad\mbox{in }\Omega\\
y^\varepsilon=0 \quad\mbox{on }\Gamma
\endgathered
\end{equation}
and
\begin{equation}\label{error-condition}
\|{y^\varepsilon}_{\vert_S}-y_1\|_{0,S}\le\alpha,
\end{equation}
where ${y^\varepsilon}_{\vert_S}$ is the trace of $y^\varepsilon$ on
$S$ and $\norm{\ }_{0,S}$ denotes the standard $L^2$-norm on~$S$. The
nonlinear function $f$ is such that
\begin{equation}\label{f-hypothesis-a}
f\in C^0,\ f(0)=0,
\end{equation} 
and uniformly Lipschitz, that is
\begin{equation}\label{f-hypothesis-b}
\exists \gamma>0\hbox{ such that }\forall s\in\mathbb{R}\setminus\{0\},\ 
0\le{f(s)\over s}\le\gamma.
\end{equation}
The coefficients of the symmetric matrix $A^\varepsilon$ are real 
and piecewise $C^1$ in $\overline\Omega$. We assume the condition
\begin{equation}\label{A-elliptic}
\exists\alpha_m,\alpha_M>0\hbox{ such that }\forall\xi\in\mathbb{R}^N,\ \mod{\xi}=1,\ 
\alpha_m\le\sum_{i,j=1}^N A^\varepsilon_{ij}(x)\xi_i\xi_j\le\alpha_M,
\end{equation}
for a.e. $x\in\Omega$. 
The following result can be established as in \cite{OsPu98a,OsPu98b}
using a fixed point technique introduced in \cite{FaPuZu3}.

\begin{theorem}\label{thm:1.1}
Assume that each point $x_0$ on $S$ can be connected by an arc
included in $\Omega$ to some point in $\omega$ without intersecting
$S\setminus\{x_0\}$. Then, under the hypotheses
$(\ref{f-hypothesis-a})$, $(\ref{f-hypothesis-b})$ and
$(\ref{A-elliptic})$, there exists a control $v^\varepsilon\in
L^2(\omega)^N$ satisfying $(\ref{primal-system})$ and
$(\ref{error-condition})$.
\end{theorem}
Moreover a control $v^\varepsilon_*$ of minimal norm and solution of
$(\ref{primal-system})$-$(\ref{error-condition})$ can be
constructed as follows. Using a density argument, we can assume $f\in
C^1$. Define the real function
$$ g(s)=\left\{
\begin{array}{ll}
{\displaystyle f(s)/s}&\hbox{if }s\not=0\\[5pt]
f'(0)&\hbox{if }s=0.
\end{array}\right.
$$
For each $z\in L^2(\Omega)^N$ consider the following 
auxiliary control problem. Given $\varepsilon>0$, $\alpha>0$ and
$y_1\in L^2(S)^N$ find a control function $v^\varepsilon$ supported in $\omega$
such that
\begin{equation}\label{aux-primal-system}
\gathered
-\mathop{\rm div}(A^\varepsilon\nabla y^\varepsilon) 
+ g(z)y^\varepsilon=1_{\omega}v^\varepsilon\quad\mbox{in }\Omega\\
y^\varepsilon=0 \quad\mbox{on }\Gamma
\endgathered 
\end{equation}
and
\begin{equation}\label{aux-error-condition}
\norm{y^\varepsilon(z)_{\vert_S}-y_1}_{0,S}\le\alpha.
\end{equation}
For the existence of these controls see \cite{OsPu98a}. 
Among the controls satisfying $(\ref{aux-primal-system})$ and 
$(\ref{aux-error-condition})$  we choose as an optimal the
minimizer of the functional (see \cite{Lions90,Lions92})
\begin{equation}\label{primal-functional}
I^\varepsilon_z(v)= \left\{\begin{array}{ll}
 {1\over 2}\norm{v}^2_{0,\omega}&
\hbox{if $(\ref{aux-error-condition})$ is satisfied}\\[2pt]
+\infty&\hbox{otherwise.}
\end{array}
\right.
\end{equation}
We denote by $v^\varepsilon_*(z)$ the point of minimum value,
which depends on $z$ and $\varepsilon$ of course. Associated to this 
control we have the solution of $(\ref{aux-primal-system})$ that we 
denote by $y^\varepsilon_*(z)$.
Now we define the mapping
\begin{equation}\label{F-operator}
{\cal F}^\varepsilon:z\in L^2(\Omega)^N\rightarrow y^\varepsilon_*(z)\in L^2(\Omega)^N.
\end{equation} 
We will show that it has a fixed point $\overline z^\varepsilon$,
that is to say
\begin{equation}\label{fixed-point}
{\cal F}^\varepsilon(\overline z^\varepsilon)=\overline z^\varepsilon.
\end{equation}
An admissible control for the semilinear control problem 
$(\ref{primal-system})$ and $(\ref{error-condition})$ is simply
\begin{equation}\label{optimal-control}
v^\varepsilon_*=v^\varepsilon_*(\overline z^\varepsilon).
\end{equation}
Our main goal is to study the behavior of $v^\varepsilon_*$ as
$\varepsilon\rightarrow 0$.

\paragraph{Notation.} We will denote by $y^\varepsilon$ (or
$y^\varepsilon(v^\varepsilon)$) the solution of the original problem
(\ref{primal-system}) and by $y^\varepsilon(z)$ (or $y^\varepsilon(z,v^\varepsilon)$) the
solution of the auxiliary problem (\ref{aux-primal-system}).

\section{Dual context}
For each $z\in L^2(\Omega)^N$, $\varepsilon>0$, $\alpha>0$ and $y_1\in
L^2(S)^N$ the optimal control $v^\varepsilon_*(z)$ minimizing 
$(\ref{primal-functional})$ and satisfying simultaneously 
$(\ref{aux-primal-system})$ and $(\ref{aux-error-condition})$
can be expressed in a dual context. Indeed, we have the relationship
\cite{OsPu98a}
\begin{equation}\label{dual-relationship}
v^\varepsilon_*(z)=\varphi^\varepsilon_*(z)_{\vert_\omega},
\end{equation} 
where $\varphi^\varepsilon_*(z)$ is the solution of the following 
dual problem associated to $(\ref{aux-primal-system})$
($\delta_S$ is a Dirac mass concentrated on $S$)
\begin{equation}\label{aux-dual-system}
\gathered
-\mathop{\rm div}(\,{}^t\!A^\varepsilon\nabla\varphi^\varepsilon) + 
g(z)\varphi^\varepsilon=\delta_S\varphi_1\quad\mbox{in }\Omega \\
\varphi^\varepsilon=0 \quad\mbox{on }\Gamma
\endgathered
\end{equation}
for 
\begin{equation}\label{optimal-choice}
\varphi_1=\varphi_{1*}^\varepsilon(z),
\end{equation} 
where $\varphi_{1*}^\varepsilon(z)$ is the point of minimum in $L^2(S)^N$
of the following dual functional of $(\ref{primal-functional})$
\begin{equation}\label{dual-functional}
J^\varepsilon_z(\varphi_1)=\frac{1}{2}\int_\omega\mod{\varphi^\varepsilon}^2\,dx
+\alpha\norm{\varphi_1}_{0,S}-\int_S y_1\varphi_1\, ds
\end{equation} 
in the sense of Fenchel-Rockafellar \cite{Ekeland-Temam,Lions92}. 
Note that in order to evaluate this
dual functional we have to solve the dual problem
$(\ref{aux-dual-system})$ for each $\varphi_1\in L^2(S)^N$.

\paragraph{Notation.} We will denote by $\varphi^\varepsilon(z)$ 
(or $\varphi^\varepsilon(z,\varphi_1)$) the solution of the
auxiliary dual problem (\ref{aux-dual-system}).

\section{Main result}

Our main result can be summarized as follows (the definition of
$H$-convergence can be found in \cite{Murat_Tartar}).
\begin{theorem}\label{thm:H-control}
Assume that $A^\varepsilon$ H-converges to $A^0$ and that the
hypotheses of Theorem~\ref{thm:1.1} are satisfied, then up to a
subsequence
$$ v^\varepsilon_*\rightharpoonup v^0_*\hbox{ in }
L^2(\omega)^N-\hbox{weakly}\quad
\hbox{and}\quad
y(v^\varepsilon_*)\rightharpoonup y^0_*\hbox{ in }H_0^1(\Omega)
-\hbox{weakly}\quad\mbox{as }\varepsilon\rightarrow 0,\nonumber$$
where { $v^0_*$} has minimal norm among all controls { $v$} satisfying
$$ \norm{y^0_*(v)_{\vert_S}-y^1}_{0,S}\le\alpha.$$
Moreover { $y^0_*$} is solution of the system
\begin{equation}\label{optimality-nonlinear-system}
\gathered
-\mathop{\rm div}(A^0\nabla y^0_*)+f(y^0_*)=1_\omega
\varphi^0_*\quad\mbox{in }\Omega\\
y_*^0=0\quad\mbox{on }\partial\Omega\\
-\mathop{\rm div}(\,{}^t\!A^0\nabla\varphi^0)+g(y_*^0)\varphi^0
=\delta_S\varphi_1\quad\mbox{in }\Omega\\
\varphi^0=0\quad\mbox{on }\partial\Omega\\
\varphi_{1*}=\mathop{\rm argmin}
\left(\frac{1}{2}\int_\omega\mod{\varphi^0}^2\,dx
+\alpha\norm{\varphi_1}_{0,S}-\int_S y_1\varphi_1\, 
ds\right)\,, \endgathered
\end{equation}
where $\varphi^0_*$ is the solution of 
$(\ref{optimality-nonlinear-system}\hbox{\rm c,d})$ 
associated to
$\varphi_{1*}$. In terms of this dual variable,
{\begin{equation}v^0_*={\varphi^0_*}_{\vert_\omega}.\end{equation}}
\end{theorem}
The proof of this theorem is developed in the rest of the paper and
uses the following Lemma.  The proof of this Lemma is similar to the
one in \cite{FaPuZu3} (see also \cite{OsPu98a}) taking care of
the $\varepsilon$ dependence in bounds and the regularity of
$A^\varepsilon$.

\begin{lemma}\label{lem:J-coercivity}
Assume that the coefficients of $A^\varepsilon$ are piecewise $C^1$ 
in $\overline\Omega$. Then, under the hypotheses of
Theorem~\ref{thm:1.1}, we have
\begin{equation}\label{J-coercivity}
\liminf_{\norm{\varphi_1}_{0,S}\to\infty}
\frac{J^\varepsilon_z(\varphi_1)}
{\norm{\varphi_1}_{0,S}}\ge\alpha>0.
\end{equation}
\end{lemma}

\paragraph{Proof.} We have
$$
\frac{J^\varepsilon_z(\varphi_1)}{\norm{\varphi_1}_{0,S}}=\frac{1}{2}\int_\omega
\frac{1}{\norm{\varphi_1}_{0,S}}\mod{\varphi^\varepsilon}^2\,dx+
\alpha-\int_S y_1\frac{\varphi_1}{\norm{\varphi_1}_{0,S}}\,ds.$$
Let 
$$
\widehat\varphi^\varepsilon=\frac{\varphi^\varepsilon}{\norm{\varphi_1}_{0,S}}
\quad\mbox{and}\quad
\widehat\varphi_1=\frac{\varphi_1^\varepsilon}{\norm{\varphi_1}_{0,S}}\,.
$$
Then 
\begin{equation}\label{J-proof}
\frac{J^\varepsilon_z(\varphi_1)}{\norm{\varphi_1}_{0,S}}
=\frac{\norm{\varphi_1}_{0,S}}{2}
\int_\omega\mod{\widehat\varphi^\varepsilon}^2\,dx
+\alpha-\int_S y_1\,\widehat\varphi_1\,ds\,.
\end{equation}
We write that for a sequence $\varphi_{1,n}$ such that
$\norm{\varphi_{1,n}}_{0,S}\to\infty$ as $n\to\infty$. Since 
$\norm{\widehat\varphi_{1,n}}_{0,S}=1$ it is easy to see using 
(\ref {f-hypothesis-b}) and (\ref {A-elliptic}) that the
associated solutions of (\ref{aux-dual-system}) satisfy
$$
\norm{\widehat\varphi^\varepsilon_n}_{1,\Omega}\le C
$$
where the constant $C$ does not depend on $n$ nor $\varepsilon$ and only
depends on $\alpha_m$, $\gamma$ and the norm of the
trace operator from $H^1(\Omega)$ into $L^2(S)$. 
For a fixed $\varepsilon$ up to a sequence (in $n$), we have
\begin{gather*}
\widehat\varphi_{1,n}\rightharpoonup
\widetilde\varphi_1\quad\mbox{in }L^2(S)-\hbox{weakly}\\
\widehat\varphi^\varepsilon_n\rightharpoonup
\widetilde\varphi^\varepsilon\quad\mbox{in }H^1(\Omega)-\hbox{weakly}.
\end{gather*}
Then
$$
\liminf_{\norm{\varphi_1}_{0,S}\to\infty}
\frac{J^\varepsilon_z(\varphi_1)}
{\norm{\varphi_1}_{0,S}}=\liminf_{n\to\infty}
\frac{J^\varepsilon_z(\varphi_{1,n})}
{\norm{\varphi_{1,n}}_{0,S}}.
$$
We consider two cases. Firstly, if 
$$
\lim_n\int_\omega\mod{\widehat\varphi^\varepsilon_n}^2\,dx=\int_\omega
\mod{\widetilde\varphi^\varepsilon}^2\,dx>0\,,
$$
then
$$
\norm{\varphi_{1,n}}_{0,S}\int_\omega\mod{\widehat\varphi^\varepsilon_n}^2\,
dx\to +\infty
$$
and since $\int_S y_1\widehat\varphi_{1,n}\to\int_S
y_1\widetilde\varphi_1$, from (\ref{J-proof}) we obtain 
(\ref{J-coercivity}).
Secondly, if 
$$
\lim_n\int_\omega\mod{\widehat\varphi^\varepsilon_n}^2\,dx=\int_\omega
\mod{\widetilde\varphi^\varepsilon}^2\,dx=0
$$
then $\widetilde\varphi^\varepsilon=0$ in $\omega$. Next, our aim is to
prove that $\widetilde\varphi^\varepsilon=0$ in the whole of $\Omega$.
The fact that we have supposed the coefficients of $A^\varepsilon$
piecewise $C^1$, implies that $\widetilde\varphi^\varepsilon=0$ till
$S$. Indeed, the classical Holmgren's unique continuation property
\cite{Hormander} shows that $\widetilde\varphi^\varepsilon$ is zero in
the regions intersecting $\omega$ where $A^\varepsilon$ is regular and
the transmission conditions allow to extend
$\widetilde\varphi^\varepsilon$ by zero to the contiguous regions till
$S$. This gives the desired result if $S$ is an open
curve. Conversely, if $S$ is closed, the geometrical hypothesis on $S$
and $\omega$ introduced in Theorem~\ref{thm:1.1} implies that
$\widetilde\varphi^\varepsilon$ is zero in the whole $\Omega$.  This
implies that $\widetilde\varphi_1=0$ on $S$, therefore
$$
\liminf_{n}
\frac{J^\varepsilon_z(\varphi_{1,n})}
{\norm{\varphi_{1,n}}_{0,S}}\ge
\alpha+\liminf_n \left(\norm{\varphi_{1,n}}_{0,S}
\int_\omega\mod{\widehat\varphi^\varepsilon_n}^2\,dx\right)-0\ge\alpha,
$$
which completes the proof of the lemma.

\section{Step 1. Fixed point}
We will establish that the operator ${\cal F^\varepsilon}$ defined in
$(\ref{F-operator})$ has a fixed point using Schauder's theorem. We
follow the ideas in \cite{FaPuZu3} and \cite{OsPu98a}, taking care of
the $\varepsilon$ dependence.

Let us prove that ${\cal F^\varepsilon}$ is continuous and maps
$L^2(\Omega)^N$ into a relatively compact subset of $L^2(\Omega)^N$.
Take
$$
z_n\rightarrow z_0\quad\mbox{in }L^2(\Omega)^N
$$
and in order to simplify notations let us set
$$
\varphi^\varepsilon_n=\varphi^\varepsilon(z_n)
$$
the solution of $(\ref{aux-dual-system})$ associated to $z_n$ and to a
fixed $\varphi_1\in L^2(S)^N$. Now, taking $\varphi^\varepsilon_n$ as a
function test in $(\ref{aux-dual-system})$ the following estimate is
easily obtained
\begin{equation}\label{a-priori-bound}
\norm{\varphi^\varepsilon_n}_{1,\Omega}\le C\norm{\varphi_1}_{0,S},
\end{equation}
where the constant $C$ depends only on the $A^\varepsilon$-ellipticity
constant $\alpha_m$, and on trace and Poincar\'e constants, but is
independent on $\varepsilon$ (we also use hypothesis $(\ref{f-hypothesis-b})$
about $f$). Thanks to $(\ref{a-priori-bound})$ we have 
up to a subsequence
$$
\varphi_n^\varepsilon\rightharpoonup\varphi_0^\varepsilon\quad\mbox{in }H^1_0(\Omega)-\hbox{weakly}.
$$
In order to pass to the limit in a variational formulation of 
$(\ref{aux-dual-system})$, note that
\begin{eqnarray*}
\lefteqn{ \int_\Omega g(z_n)\varphi_n^\varepsilon\varphi\,dx
-\int_\Omega g(z_0)\varphi_0^\varepsilon\varphi\,dx}\\
&=& \int_\Omega g(z_n)(\varphi_n^\varepsilon-\varphi_0^\varepsilon)
\varphi\,dx+\int_\Omega (g(z_n)-g(z_0))\varphi_0^\varepsilon\varphi\,dx,
\end{eqnarray*}
but $g(z_n)$ is bounded in $L^\infty(\Omega)$ and since $z_n$ converges to
$z_0$ a.e. then
\begin{equation}\label{weak-star-convergence}
g(z_n)\rightharpoonup g(z_0)\quad\mbox{in }L^\infty(\Omega)-\hbox{weakly*}.
\end{equation}
Therefore
$$
\int_\Omega g(z_n)\varphi_n^\varepsilon\varphi\,dx\to\int_\Omega
g(z_0)\varphi_0^\varepsilon\varphi\,dx\quad\forall\varphi\in H^1_0(\Omega).
$$

\begin{remark}\label{rem:Hminusone} \rm
Convergence $(\ref{weak-star-convergence})$ implies weak but
not strong convergence in $H^{-1}(\Omega)$. 
\end{remark}
Nevertheless, a technical
argument allows to obtain the strong convergence in $H^{-1}(\Omega)$. 
Indeed, for all $\varphi\in H^1_0(\Omega)$, we have
\begin{eqnarray*}
\lefteqn{\Big|\int_\Omega g(z_n)\varphi_n^\varepsilon\varphi\,dx
-g(z_0)\varphi_0^\varepsilon\varphi\,dx\Big|\le }\\
&\le& \Big|\int_\Omega g(z_n)(\varphi_n^\varepsilon
-\varphi_0^\varepsilon)\varphi\,dx\Big|+
\Big|\int_\Omega (g(z_n)-g(z_0))\varphi_0^\varepsilon\varphi\,dx\Big|\,.
\end{eqnarray*}
On the one hand
$$
\mod{\int_\Omega g(z_n)(\varphi_n^\varepsilon-\varphi_0^\varepsilon)\varphi\,dx}\le
\norm{g(z_n)}_{L^{p_1}}\norm{\varphi^\varepsilon_n-\varphi_0^\varepsilon}_{L^{p_2}}
\norm{\varphi}_{L^{p_3}}.
$$
Choosing $p_1=N$, $p_2=p_3=\frac{2N}{N-1}$ if $N\ge 2$ otherwise
$p_1=p_3=4$ and $p_2=2$, thanks to this choice of $p_2$, the injection
from $H^1(\Omega)$ to $L^{p_2}(\Omega)$ is compact and then
$$
\norm{\varphi_n^\varepsilon-\varphi_0^\varepsilon}_{L^{p_2}}\to 0\quad\mbox{as}\quad n\to\infty.
$$
Note that $g$ is bounded and
$$
\norm{g(z_n)}_{L^{p_1}}\le\gamma\mathop{\rm meas}(\Omega)^{1/p_1}.
$$
Finally, the injection from $H^1_0(\Omega)$ to $L^{p_3}(\Omega)$ is
continuous so
\begin{equation}\label{p-bound}
\norm{\varphi}_{L^{p_3}}\le\norm{i}_{{\cal L}
(H_0^1(\Omega);L^{p_3}(\Omega))}\norm{\varphi}_{1,\Omega}.
\end{equation}
On the other hand
$$\Big\| \int_\Omega (g(z_n)-g(z_0))\varphi_0^\varepsilon\varphi\,dx
\Big\| \le
\norm{g(z_n)-g(z_0)}_{L^{q_1}}\norm{\varphi_0^\varepsilon}_{L^{q_2}}
\norm{\varphi}_{L^{q_3}}$$ with $q_1=\frac{N}{2}$,
$q_2=q_3=\frac{2N}{N-2}$ for $N\ge 3$, otherwise $q_1=2$,
$q_2=q_3=4$. Thanks to this choice the injection from $H^1(\Omega)$
into $L^{q_3}(\Omega)$ is continuous and a bound can be obtained as
in $(\ref{p-bound})$. In virtue of dominated convergence theorem and
bounds on $g(z_n)$ we have
$$
\norm{g(z_n)-g(z_0)}_{L^{q_1}}\rightarrow 0\quad\mbox{as}\quad n\to\infty.
$$
{}From the above convergences we see that
$$
g(z_n)\varphi_n^\varepsilon\to
g(z_0)\varphi_0^\varepsilon\quad\mbox{in }H^1_0(\Omega)-\hbox{strongly}
\quad\mbox{as}\quad n\to\infty.
$$
Let us continue with our problem. Multiplying
$(\ref{aux-dual-system})$ by $\phi\in H^1_0(\Omega)$ and integrating
by parts we obtain
$$
\int_{\Omega}A^\varepsilon\nabla\varphi^\varepsilon_n\cdot\nabla\phi\,dx+
\int_{\Omega}g(z_n)\varphi_n^\varepsilon\phi\, dx=\int_S\varphi_1\phi\,d\sigma,
$$
for a fixed $\varepsilon$ and we take $n\to\infty$ to obtain
$$
\int_\Omega A^\varepsilon\nabla\varphi_0^\varepsilon\cdot\nabla\phi\,dx+\int_\Omega
g(z_0)\varphi_0^\varepsilon\phi\, dx=\int_S\varphi_1\phi\,d\sigma,
$$
and this shows that
$\varphi_0^\varepsilon=\varphi^\varepsilon(z_0)$.
Let us now show that 
\begin{equation}\label{phi-strong-convergence}
\varphi_n^\varepsilon\to \varphi^\varepsilon_0\quad
\mbox{in }H^1_0(\Omega)-\hbox{strongly}.
\end{equation}
Take $\varphi_n^\varepsilon$ as a test function in the problem
\begin{gather*}
-\mathop{\rm div}(\,{}^t\!A^\varepsilon\nabla\varphi_n^\varepsilon)
+g(z_n)\varphi_n^\varepsilon
=\delta_S\varphi_1\quad\mbox{in }\Omega\\
\varphi_n^\varepsilon=0\quad\mbox{on }\Gamma\,.
\end{gather*}
Passing to the limit, we obtain
$$
\lim_{n\to\infty}\int_\Omega
{}^tA^\varepsilon\nabla\varphi_n^\varepsilon\cdot\nabla\varphi_n^\varepsilon\,dx
=\int_S\varphi_1\varphi_0^\varepsilon\,d\sigma-\int_\Omega g(z_0)\varphi_0^\varepsilon\varphi_0^\varepsilon\,dx\,.
$$
Now, taking $\varphi_0^\varepsilon$ as a test function in
\begin{gather*}
-\mathop{\rm div}(\,{}^t\!A^\varepsilon\nabla\varphi_0^\varepsilon)+g(z_0)\varphi_0^\varepsilon
=\delta_S\varphi_1\quad\mbox{in }\Omega\\
\varphi_0^\varepsilon=0\quad\mbox{on }\Gamma\,,
\end{gather*}
we obtain
$$
\int_\Omega
A^\varepsilon\nabla\varphi_0^\varepsilon\cdot\nabla\varphi_0^\varepsilon\,dx=
\int_S\varphi_1\varphi_0^\varepsilon\,d\sigma-\int_\Omega
g(z_0)\varphi_0^\varepsilon\varphi_0^\varepsilon\,dx\,.
$$
By comparison
$$
\lim_{n\to\infty}\int_\Omega {}^t
A^\varepsilon\nabla\varphi_n^\varepsilon\cdot\nabla\varphi_n^\varepsilon\,dx=
\int_\Omega \,{}^t\!A^\varepsilon\nabla\varphi_0^\varepsilon\cdot\nabla\varphi_0^\varepsilon\,dx.
$$
We conclude $(\ref{phi-strong-convergence})$ since $\left(\int_\Omega
{}^t\!A^\varepsilon\nabla v\cdot\nabla v\,dx\right)^{1/2}$ is equivalent to the
standard norm in $H^1_0(\Omega)$.

By a method analogous to the one that yields
$(\ref{phi-strong-convergence})$ from $(\ref{a-priori-bound})$, 
we show that
\begin{equation}\label{second-a-priori-bound}
\norm{\varphi^\varepsilon_{1*}}_{0,S}\le C
\end{equation}
with $C$ independent of $n$ and of $\varepsilon$, that is
$$
\varphi_{1*}^\varepsilon(z_n)\rightharpoonup\xi^\varepsilon
\quad\mbox{in } L^2(S)-\hbox{weakly}
$$
and
\begin{equation}\label{varphi-strong}
\varphi^\varepsilon(z_n,\varphi_{1*}^\varepsilon(z_n))\to
\varphi^\varepsilon(z_0,\xi^\varepsilon)
\quad\mbox{in }H^1_0(\Omega)-\hbox{strongly}.
\end{equation}
Let us show by contradiction that $(\ref{second-a-priori-bound})$
holds. Otherwise, there exists a sequence
$\{\varphi_{1*}^\varepsilon(z_n)\}_{n\ge 0}$ such that
\begin{equation}\label{first-sequence-not-bounded}
\norm{\varphi_{1*}^\varepsilon(z_n)}_{0,S}\to +\infty\quad\mbox{as}\quad n\to\infty,
\end{equation}
but for each $z_n$, the function $\varphi_{1*}^\varepsilon(z_n)$ minimizes
$J^\varepsilon_{z_n}$ and consequently
\begin{equation}\label{bound-for-J}
J_{z_n}^\varepsilon(\varphi_{1*}^\varepsilon(z_n))\le
J_{z_n}^\varepsilon(\varphi_1)\quad\forall\varphi_1\in L^2(S).
\end{equation}
At the same time, we see that
$$J_{z_n}^\varepsilon(\varphi_1)=
\frac{1}{2}\int_\omega\mod{\varphi^\varepsilon(z_n)}^2\,dx+
\alpha\norm{\varphi_1}_{0,S}-\int_S y_1\varphi_1\,d\sigma$$
converges as $n\to\infty$ to
$$J_{z_0}^\varepsilon(\varphi_1)=
\frac{1}{2}\int_\omega\mod{\varphi^\varepsilon(z_0)}^2\,dx+
\alpha\norm{\varphi_1}_{0,S}-\int_S y_1\varphi_1\,d\sigma.$$ 
Therefore from $(\ref{bound-for-J})$, for each fixed $\varphi_1$
$$J_{z_n}^\varepsilon(\varphi_{1*}^\varepsilon(z_n))\le C$$
with $C$ independent of $n$ (and of $\varepsilon$). 
This last upper bound contradicts 
$(\ref{first-sequence-not-bounded})$ since
\begin{equation}\label{aux-J-coercivity}
\liminf_{\norm{\varphi_{1*}^\varepsilon(z_n)}_{0,S}\to\infty}
\frac{J_{z_n}^\varepsilon(\varphi_{1*}^\varepsilon(z_n))}
{\norm{\varphi_{1*}^\varepsilon(z_n)}_{0,S}}\ge\alpha>0.
\end{equation}
Proof of $(\ref{aux-J-coercivity})$ is similar to the proof of Lemma
\ref{lem:J-coercivity} since
$$
\norm{\varphi_{*}^\varepsilon(z_n,\varphi_{1*}^\varepsilon(z_n))}_{1,\Omega}\le
C\norm{\varphi_{1*}^\varepsilon(z_n)}_{0,S}
$$
with a constant $C$ independent of $n$ (and of $\varepsilon$).
Since $(\ref{first-sequence-not-bounded})$ does not hold, we have up to a
subsequence 
\begin{equation}\label{xi-weak-convergence}
\varphi_{1*}^\varepsilon(z_n)\rightharpoonup \xi^\varepsilon\quad\mbox{in }
L^2(S)-\hbox{weakly}\quad\mbox{as}\quad n\to\infty.
\end{equation}
It remains to identify the limit. Let us show that $\xi^\varepsilon$
minimizes $J^\varepsilon_{z_0}$, that is to say
\begin{equation}\label{second-bound-for-J}
J_{z_0}^\varepsilon(\xi^\varepsilon)\le
J_{z_0}^\varepsilon(\varphi_1)\quad\forall\varphi_1\in L^2(S)^N.
\end{equation}
First, note that $\varphi^\varepsilon_{1*}(z_n)$ is optimal 
for $J^\varepsilon_{z_n}$, that is
$$
J_{z_n}^\varepsilon(\varphi_{1*}^\varepsilon(z_n))\le
J^\varepsilon_{z_n}(\varphi_1)\quad\forall\varphi_1\in L^2(S)^N
$$
hence
$$
\liminf_n J_{z_n}^\varepsilon(\varphi_{1*}^\varepsilon(z_n))\le\liminf_n
J^\varepsilon_{z_n}(\varphi_1)=J_{z_0}^\varepsilon(\varphi_1)\quad\forall\varphi_1\in L^2(S)^N.
$$
In order to get $(\ref{second-bound-for-J})$
it remains to proof that
\begin{equation}\label{thirth-bound-for-J}
J_{z_0}^\varepsilon(\xi^\varepsilon)\le\liminf_n J^\varepsilon_{z_n}(\varphi_{1*}^\varepsilon(z_n)).
\end{equation}
Let us recall that
$$
J_{z_n}^\varepsilon(\varphi_{1*}^\varepsilon(z_n))=\frac{1}{2}\int_\omega
\mod{\varphi^\varepsilon(z_n,\varphi_{1*}^\varepsilon(z_n))}^2\,dx +
\alpha\norm{\varphi_{1*}^\varepsilon(z_n)}_{0,S}
-\int_S y_1\varphi_{1*}^\varepsilon(z_n)\,d\sigma,
$$
so from $(\ref{xi-weak-convergence})$ we have
$$
\liminf_n\alpha\norm{\varphi_{1*}^\varepsilon(z_n)}_{0,S}
-\int_S y_1\varphi_{1*}^\varepsilon(z_n)\,d\sigma\ge
\liminf_n\alpha\norm{\xi^\varepsilon}_{0,S}-\int_S y_1\xi^\varepsilon\,d\sigma
$$
and from $(\ref{varphi-strong})$
$$
\liminf_n\int_\omega\mod{\varphi^\varepsilon(z_n,\varphi_{1*}^\varepsilon(z_n))}^2\,dx\ge
\int_\omega\mod{\varphi^\varepsilon(z_0,\xi^\varepsilon)}^2\,dx.
$$
In this way, we obtain $(\ref{thirth-bound-for-J})$ and consequently
$(\ref{second-bound-for-J})$, in other words
$$
\xi^\varepsilon=\varphi_{1*}^\varepsilon(z_0).
$$
With this relation, convergence in $(\ref{varphi-strong})$
becomes
\begin{equation}\label{new-varphi-strong}
\varphi^\varepsilon(z_n,\varphi_{1*}^\varepsilon(z_n))\to
\varphi^\varepsilon(z_0,\varphi_{1*}^\varepsilon(z_0))\quad\mbox{in }H^1_0(\Omega)-\hbox{strongly}.
\end{equation}
The rest of the proof is straightforward since
\begin{gather*}
v_*^\varepsilon(z_n)=\varphi^\varepsilon(z_n,
\varphi_{1*}^\varepsilon(z_n))_{\vert_\omega}\cr
v_*^\varepsilon(z_0)=\varphi^\varepsilon(z_0,
\varphi_{1*}^\varepsilon(z_0))_{\vert_\omega}
\end{gather*}
and it is clear from ($\ref{new-varphi-strong}$) that
$v_*^\varepsilon(z_n)\rightarrow v_*^\varepsilon(z_0)$
in $H^1(\omega)$ - strongly.
An analogous proof as for the adjoint problem shows that
$$
y^\varepsilon(z_n,v^\varepsilon_*(z_n))\rightarrow
y^\varepsilon(z_0,v^\varepsilon_*(z_0))\quad\mbox{in } H^1(\Omega)-\hbox{strongly},
$$
proving the continuity of the map ${\cal F}^\varepsilon$ for a fixed $\varepsilon>0$.

Next we show that ${\cal F}^\varepsilon$ is compact
(uniformly in $\varepsilon$). Let $z\in
L^2(\Omega)^N$ since
$$
\norm{g(z)}_{\infty,\Omega}\le\gamma
$$
then
$$
\norm{\varphi^\varepsilon(z,\varphi_1)}_{1,\Omega}\le C\norm{\varphi_1}_{0,S}
$$
with $C$ independent of $z$ (and of $\varepsilon$). This implies that
$\mod{J_z^\varepsilon(\varphi_1)}\le C(\varphi_1)$, therefore
$$
\mod{J_z^\varepsilon(\varphi_{1*}^\varepsilon(z))}\le C(\varphi_1).
$$
Using again the coercitivity of $J_z^\varepsilon$ we see that
$\norm{\varphi_{1*}^\varepsilon}_{0,S}$ is bounded independently of
$z$ (and $\varepsilon$). Then
$\norm{\varphi^\varepsilon(z,\varphi_{1*}^\varepsilon)}_{1,\Omega}$ is
bounded independently of $z$ (and $\varepsilon$) and consequently the
same is true for
$v^\varepsilon_*(z)=\varphi^\varepsilon(z,\varphi_{1*}^\varepsilon)_{\vert_\omega}$
and $y^\varepsilon(z,v^\varepsilon_*(z))$. \hfill$\diamondsuit$

\section{Step 2. H-convergence}
We first consider the $H$-convergence in the original problem
(\ref{primal-system}) with fixed control $v\in L^2(\omega)^N$,
that is the $H$-convergence in the problem
\begin{equation}\label{primal-system-bis}
\gathered
-\mathop{\rm div}(A^\varepsilon\nabla y^\varepsilon) 
+ f(y^\varepsilon)=1_{\omega}v\quad\mbox{in }\Omega\\
y^\varepsilon=0 \quad\mbox{on }\Gamma, \endgathered
\end{equation}
under the hypotheses (\ref{f-hypothesis-a}) and (\ref{f-hypothesis-b}) 
on $f$.
To have {\it a priori} estimates, we multiply
(\ref{primal-system-bis}) by $y^\varepsilon$ and we integrate in $\Omega$ to
obtain
$$
\int_\Omega A^\varepsilon\nabla y^\varepsilon\cdot\nabla y^\varepsilon\, dx+
\int_\Omega f(y^\varepsilon)\,y^\varepsilon\, dx=\int_\omega v\, y^\varepsilon\, dx,
$$ 
but from (\ref{f-hypothesis-a})
$$
f(y^\varepsilon)y^\varepsilon=\frac{f(y^\varepsilon)}{y^\varepsilon}\mod{y^\varepsilon}^2\ge 0
$$
and it is true also in the case $y^\varepsilon(x)=0$. Hence
$\norm{y^\varepsilon}_{1,\Omega}\le C\norm{v}_{0,\omega}$
with $C$ independent of $\varepsilon$. Up to a subsequence
$$
y^\varepsilon\rightharpoonup y^0\quad\mbox{in } H_0^1(\Omega)-\hbox{weakly}.
$$
Now let us see which is the limit of $f(y^\varepsilon)$. Take $\varphi\in
H^1_0(\Omega)$, then
\begin{eqnarray}
&\displaystyle{{\mod{\int_\Omega(f(y^\varepsilon)\varphi-f(y^0)\varphi)\,dx}}
=\mod{\int_\Omega(g(y^\varepsilon)y^\varepsilon\varphi-g(y^0)y^0\varphi)
\,dx}\le}&\\
&\displaystyle{\leq \mod{\int_\Omega(g(y^\varepsilon)(y^\varepsilon-y^0)\varphi)\,dx}
+\mod{\int_\Omega((g(y^\varepsilon)-g(y^0))y^0\varphi)\,dx}.}&\nonumber
\end{eqnarray}
Starting from this and reasoning as in Remark~\ref{rem:Hminusone} 
we can show that
$$
f(y^\varepsilon)=g(y^\varepsilon)y^\varepsilon\to g(y^0)y^0=f(y^0)\quad\mbox{in }H^{-1}
(\Omega)-\hbox{strongly}.
$$
Thanks to the $H$-convergence definition, we immediately deduce that
\begin{equation}\label{H-primal-system-bis}
\gathered
-\mathop{\rm div}(A^0\nabla y^0) + f(y^0)=1_{\omega}v\quad\mbox{in }
\Omega\\
y^0=0 \quad\mbox{on }\Gamma,\endgathered
\end{equation}
where $A^0$ is the $H$-limit of $A^\varepsilon$ 
(and $A^\varepsilon\nabla y^\varepsilon\rightharpoonup A^0\nabla y^0$ in $L^2(\Omega)^N$-weakly). 

Consider now the $H$-convergence with the optimal control
$v^\varepsilon_*=v^\varepsilon_*(\overline z^\varepsilon)$ satisfying
(\ref{aux-primal-system})-(\ref{optimal-control}) where $\overline
z^\varepsilon$ is the fixed point of ${\cal F}^\varepsilon$. We have already seen at
the end of the previous section that $\norm{v^\varepsilon_*(z)}_{0,\omega}$
is bounded independently of $z\in L^2(\Omega)$ and $\varepsilon$. In
particular
$$
\norm{v^\varepsilon_*}_{0,\omega}\le C
$$
with $C$ independent of $\varepsilon$. Hence there exists $v^0\in
L^2(\omega)^N$ such that 
\begin{equation}\label{weak-limit-control}
\gathered
v^\varepsilon_*\rightharpoonup v^0\quad\mbox{in }L^2(\omega)
-\hbox{weakly}\\
1_\omega v^\varepsilon_*\to1_\omega v^0\quad\mbox{in }H^{-1}(\Omega)
-\hbox{strongly}. \endgathered
\end{equation}
The same proof as in the case of a fixed $v$ shows that the solution
 $y^\varepsilon_*$ of  
\begin{gather*}
-\mathop{\rm div}(A^\varepsilon\nabla y^\varepsilon_*) 
+ f(y^\varepsilon_*)=1_{\omega}v^\varepsilon_*\quad\mbox{in }\Omega\\
y^\varepsilon_*=0 \quad\mbox{on }\Gamma\,
\end{gather*}
converges weakly to $y_0$, i.e.,
\begin{equation}\label{weak-limit-solution}
y^\varepsilon_*\rightharpoonup y^0\quad\mbox{in }H^1_0(\Omega)-\hbox{weakly}
\end{equation}
where $y^0$ is a solution of
\begin{equation}\label{H-primal-system}
\gathered
-\mathop{\rm div}(A^0\nabla y^0) + f(y^0)=1_{\omega}v^0\quad\mbox{in }\Omega\\
y^0=0 \quad\mbox{on }\Gamma,
\endgathered
\end{equation}
and $A^0$ is the $H$-limit of $A^\varepsilon$.

\paragraph{Notation.} In the following sections $v^0$ stands for
the $L^2$-weak limit of the control in (\ref{weak-limit-control}) and
$y^0$ (or $y^0(v^0)$) stands for the weak $H^1$-limit of the solution
in (\ref{weak-limit-solution}), which is solution of the limit problem
(\ref{H-primal-system}).

\section{Step 3. Limit of optimal controls}
The objective is now to identify $v^0$. Is it an optimal solution?
First at all, note that 
$$
\norm{y^\varepsilon_*(v^\varepsilon_*)_{\vert_S}-y_1}_{0,S}\le\alpha\,.
$$
Since weak convergence in (\ref{weak-limit-solution}) implies
$$
{y^\varepsilon_*}_{\vert_S}\rightharpoonup y^0_{\vert_S}\quad\mbox{in }H^{1/2}(S)^N-\hbox{weakly}
\ \hbox{($L^2(S)^N$-strongly)},
$$
we conclude that $v^0$ satisfies the approximate
controllability inequality
$$
\norm{y^0(v^0)_{\vert_S}-y_1}_{0,S}\le\alpha.
$$
Also from 
(\ref{weak-limit-solution}), since $\overline z^\varepsilon$ is a fixed
point (see (\ref{F-operator}), (\ref{fixed-point})), we have
$$ \overline z^\varepsilon=y^\varepsilon_*\rightharpoonup
y^0\quad\mbox{in }H^1_0(\Omega)-\hbox{weakly}.$$

Let $v^0_*$ be the minimizer in $L^2(\omega)^N$ of the functional
\begin{equation}\label{primal-H-functional}
I(v)=\left\{\begin{array}{ll}
{1\over 2}\norm{v}^2_{0,\omega} 
  &\hbox{if $\norm{y^0(v)_{\vert_S}-y^1}_{0,S}\le\alpha$}\\[2pt]
+\infty &\hbox{otherwise,}
\end{array}
\right.
\end{equation}
where for each $v\in L^2(S)^N$, we denote $y^0(v)$ the solution
of
\begin{equation}\label{H-primal-system-bis-bis}
\gathered
-\mathop{\rm div}(A^0\nabla y^0) + f(y^0)=1_{\omega}v\quad\mbox{in }
\Omega\\
y^0=0 \quad\mbox{on }\Gamma. \endgathered
\end{equation}
We will establish that
\begin{equation}\label{H-control-is-optimal}
v^0=v^0_*.
\end{equation}

In virtue of Fenchel-Rockafellar duality, the minimum $v^0_*$ can be
characterized as follows. Let us consider the dual problem associated
to (\ref{H-primal-system-bis-bis}), that is, for each $\varphi_1\in
L^2(S)$, find $\varphi^0\in L^2(\Omega)^N$ such that
\begin{equation}\label{H-limit-dual-system}
\gathered
-\mathop{\rm div}(\,{}^t\!A^0\nabla\varphi^0) + 
g(y^0)\varphi^0=\delta_S\varphi_1\quad\mbox{in }\Omega\\
\varphi^0=0 \quad\mbox{on }\Gamma \endgathered
\end{equation}
and let us define the respective dual functional of 
(\ref{primal-H-functional}) as
\begin{equation}\label{dual-H-functional}
J^0(\varphi_1)=\frac{1}{2}\int_\omega\mod{\varphi^0}^2\,dx
+\alpha\norm{\varphi_1}_{0,S}-\int_S y_1\varphi_1\,ds.
\end{equation} 
If $\varphi^0_{1*}$ is the point of minimum of $J^0$ in $L^2(S)^N$,
and if $\varphi^0_*$ is the solution of (\ref{H-limit-dual-system})
associated to it, then the duality theory gives the relationship
\begin{equation}\label{limit-dual-relationship}
v^0_*={\varphi^0_*}_{\vert_\omega}.
\end{equation}
We will pass to the limit in (\ref{dual-relationship}),
(\ref{aux-dual-system}), (\ref{optimal-choice}),
(\ref{dual-functional}) with $z=\overline z^\varepsilon$ as
$\varepsilon\to\ 0$.  An argument similar to the one used for
obtaining (\ref{H-primal-system-bis}) shows that if we pass to the
limit in (\ref{aux-dual-system}) with $z=\overline z^\varepsilon$ as
$\varepsilon\to\ 0$ then
$$
\varphi^\varepsilon(\overline z^\varepsilon)\rightharpoonup\varphi^0\quad\mbox{in }H_0^1(\Omega)-\hbox{weakly}
$$
where $\varphi^0$ is the solution of 
\begin{equation}\label{optimality-nonlinear-system-sec}
\gathered
-\mathop{\rm div}(\,{}^t\!A^0\nabla\varphi^0)+g(y^0)\varphi^0
=\delta_S\varphi_1\quad\mbox{in }\Omega\\
\varphi^0=0\quad\mbox{on }\partial\Omega. \endgathered
\end{equation}
Taking the limit in (\ref{dual-functional}) with $\varphi_1$ fixed,
$$
J^\varepsilon_{\overline z^\varepsilon}(\varphi_1)\to J^0(\varphi_1).
$$
Let us consider now the sequence $\varphi^\varepsilon_{1*}(\overline
z^\varepsilon)$. From the uniform coecivity of $J^\varepsilon_{\overline z^\varepsilon}$
with respect to $\varepsilon$ (an analogous to Lemma \ref{lem:J-coercivity}
with $z=\overline z^\varepsilon$), we deduce that
$\varphi^\varepsilon_{1*}(\overline z^\varepsilon)$ is bounded in $L^2(S)^N$ then,
up to a subsequence
$$
\varphi^\varepsilon_{1*}(\overline z^\varepsilon)\rightharpoonup\varphi_1^0\quad\mbox{in }L^2(S)^N-\hbox{weakly}.
$$
Then (see the proof of (\ref{thirth-bound-for-J})) for each
$\varphi_1\in L^2(S)$
$$
J^0(\varphi_1^0)\le\liminf_\varepsilon J^\varepsilon_{\overline
z^\varepsilon}(\varphi^\varepsilon_{1*}(\overline z^\varepsilon))\le\liminf_\varepsilon J^\varepsilon_{\overline
z^\varepsilon}(\varphi_1)=J^0(\varphi_1).
$$
Therefore $\varphi_1^0=\varphi_{1*}^0$ and consequently
$\varphi^0=\varphi^0_*$.  Finally, passing to the limit as
$\varepsilon\to 0$ in (\ref{dual-relationship}), we obtain
$$
v^\varepsilon_*={\varphi^\varepsilon_*}_{\vert_\omega}\to{\varphi^0_*}_{\vert_\omega}=v^0_*
\quad\mbox{in }L^2(\omega)-\hbox{strongly}.
$$
This, together with (\ref{weak-limit-control}), implies 
(\ref{H-control-is-optimal}).

\paragraph{Acknowledgments.}
This research was carried out when J.~Saint Jean Paulin visited the
Department of Mathematical Engineering at the University of Chile.
The authors thank warmly this institution for
supporting the Fondap's Program on Mathematical-Mechanics.
The authors also want to thank the Chilean and French Governments 
for supporting the Scientific Committee {\sc Ecos-Conicyt}.


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\noindent{\sc Carlos Conca \& Axel Osses} \\
Departamento de Ingenier\'{\i}a Matem\'atica,\\
CMM, UMR 2071 CNRS-Uchile,\\
Universidad de Chile, Casilla 170/3 - Correo 3,\\
Santiago, Chile \\
e-mail: cconca@dim.uchile.cl, axosses@dim.uchile.cl \smallskip

\noindent{\sc Jeannine Saint Jean Paulin} \\
D\'epartement de Math\'ematiques, Universit\'e de Metz, \\
Ile du Saulcy, 57045 Metz Cedex 01, France\\
e-mail: sjpaulin@poncelet.sciences.univ-metz.fr 

\end{document}