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\begin{document}
\title{{\bf $\Gamma$-convergence for infinite dimensional optimal control 
problems}}
\author{Paolo Acquistapace \qquad  Ariela Briani \\
{\small Dipartimento di Matematica - Universit\`a di Pisa}\vspace{-2mm} \\
{\small via Filippo Buonarroti, 2 - I-56127 Pisa}}
\date{}
\maketitle 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-------------------------------------------------------
\ssection{Introduction}
%-------------------------------------------------------
Among the general goals in the various theories of variational
convergence, an important one consists in singling out a notion of
$\Gamma$-limit problem $({\cal P})$ for a sequence of minimum problems
$({\cal P}_n)$ of the form 
$$\min \left\{F_n(y,u): (y,u) \in Y \times \spc \right\}.
\leqno({\cal P}_n)$$ 
Loosely speaking, a good definition of $\Gamma$-limit problem 
should guarantee the following property:  \vspace{2mm}\\
``{\it If $(y_n,u_n)$ is an optimal pair for problem $({\cal P}_n)$,
or simply a minimizing sequence, and if $(y_n,u_n) \to (y,u)$ in $Y
\times {\cal U}$, then $(y,u)$ is an optimal pair for the $\Gamma$-limit
problem $({\cal P})$.}''\vspace{2mm} \\ 
The next Theorem indicates which are the relevant features that the
$\Gamma$-limit problem should possess in order to satisfy this property
(the theorem is proved in \cite{BDM82}, Proposition 2.1).
\begin{THEO} \label{minglim}
Let $Y$ and ${\cal U}$ be topological spaces and let $F_n:Y\times
{\cal U} \to \overline{\R}$ be a sequence of functions. Let
$(y_n,u_n)$ be a minimum point for $F_n$, or simply a pair such that 
\beqsn 
\lim_{n \to \infty} F_n(y_n,u_n)= \lim_{n \to \infty} \left[ \inf_{Y 
\times {\cal U}}F_n\right]. 
\eeqsn
Assume further that $(y_n,u_n)\to (y,u)$ in $Y \times {\cal U}$,
and there exists 
\beqs\label{absglim}
F(y,u)= \Gamma(\N,Y^-,{\cal U}^-)\lim_{n \to \infty} F_n(y,u).
\eeqs
Then $(y,u)$ is a minimum point for $F$ on $Y \times {\cal U}$, and
$$\lim_{n \to \infty} \left[ \inf_{Y \times {\cal U}} F_n\right]=\min_{Y
\times {\cal U}} F.$$
\end{THEO}  
The definition of the $\Gamma$-limit (\ref{absglim}) is given in
Section 4 (Definition \ref{multiglim}).\\
The  above theorem motivates the following definition of $\Gamma$-limit 
problem.
\begin{DEF} \label{defglim}
When (\ref{absglim}) is satisfied, we say that the problem 
$$\min \left\{F(y,u): (y,u) \in Y \times \spc \right\},
\leqno({\cal P})$$ 
is the {\em $\Gamma$-limit} of problems $({\cal P}_n)$. 
\end{DEF}
See \cite{BDM82}, \cite{BF95} for the explicit calculation of 
$\Gamma$-limits in various situations. \\
In this paper we consider the following family of optimal control
problems: for fixed (small) $\varepsilon>0$, we look for 
\beq \label{minapprox}
\min_{u\in L^2(0,T;V)} \Jee(u),
\eeq 
where
\beq \label{costoee}
\Jee(u)=\int_0^T \apt \| u(s,\cdot) \|_V^2 + \| \yee(s,\cdot) \|_H^2 \cht ds; 
\eeq
here $V$ is the space $H^{1/2}_{00}(\gao,\C^N)\times
H^{-1/2}(\gai,\C^N)$, and for each control $u\in L^2(0,T;V)$ the state
function $\yee$ is the solution of the second order regular parabo-lic
initial-boundary value problem
\beq \label{pbapprox}
\left\{ \begin{array}{rcll} \frac{\de y}{\de t}(t,x)-A(x,D)y(t,x) & = & 0 &
\quad \textrm{in } [0,T] \times \cho \vspace{1mm} \\ 
\teee(x)\frac{\de y}{\de \nu_A}(t,x) + (1-\teee(x))y(t,x) & = & u(t,x) &
\quad \textrm{in } [0,T] \times \de \Omega \vspace{1mm}\\ 
y(0,x) & = & y_0(x) & \quad \textrm{in } \cho, \end{array} \right.
\eeq
where the scalar function $\teee$ is defined as follows:
\beq \label{tetaeps}
\teee(x) = \apg \begin{array}{ll}
1-\ee & \textrm{if } x \in \gai \vspace{1mm}\\
1-\ee - \frac{1-2\ee}{\ee}\ d(x,\gai) \quad & \textrm{if } x \in \gao
\textrm{ and } d(x,\gai) < \ee \vspace{1mm}\\
\ee & \textrm{if } x \in \gao \textrm{ and } d(x,\gai) \geq \ee. \\
\end{array} \ch
\eeq
Here $\gao$ and $\gai$ are disjoint smooth submanifolds of $\de
\Omega$, such that $\overline{\gao} \cup \overline{\gai} = \de
\Omega$.\\ 
Our main result is that the $\Gamma$-limit of this family as
$\varepsilon \to 0^+$  is the following optimal control problem: 
look for 
\beq \label{minmisto} 
\min_{u\in L^2(0,T;U)} J(u)
\eeq 
where
\beq \label{costo}
J(u)=\int_0^T \apt \| u(s,\cdot) \|_U^2 + \| y(s,\cdot) \|_H^2 \cht ds;
\eeq
the space $U$ is $H^{1/2}(\gao,\C^N)\times [H^{1/2}_{00}(\gai,\C^N)]^*$, and
for each control $u=(u_0,u_1)\in L^2(0,T;U)$ the state function $y$ is the
solution of the second order parabolic mixed initial-boundary value problem 
\beq \label{pbmisto}
\left\{ \begin{array}{rll}
\frac{\de y}{\de t}(t,x)-A(x,D)y(t,x)= & 0 & \quad \textrm{in } [0,T] \times
\Omega \vspace{1mm}\\ 
y(t,x)= & u_0(t,x) & \quad \textrm{in } [0,T] \times \Gamma_0 \vspace{1mm}\\ 
\frac{\de y}{\de \nu_A}(t,x)= & u_1(t,x) & \quad \textrm{in } [0,T]
\times \Gamma_1 \vspace{1mm}\\ 
y(0,x)= & y_0(x) & \quad \textrm{in } \Omega. \end{array} \right.
\eeq
We note that the space of controls for problem (\ref{minmisto}) is
larger than that of problems (\ref{minapprox}), and in the framework of
$\Gamma$-convergence we are forced to choose the smaller one: thus we
can only approximate the solutions of (\ref{minmisto}) which belong to the
smaller space, with a little loss of generality (compare with Remark
\ref{restrspazi} in Section 2). \\
A special case of the above situation occurs when $\Gamma_1=\emptyset$, so
that the approximating boundary condition has the form  
$$\ee \frac{\de y}{\de \nu_A}(t,x) + y(t,x) = u(t,x) \qquad \textrm{in }
[0,T]\times \de \Omega;$$
in this case the $\Gamma$-limit problem contains the Dirichlet
boundary condition $y=u$ in $[0,T]\times \de \Omega$, and the
natural space of controls is just $L^2(0,T;L^2(\de \Omega, \C^N))$ (see
\cite{Br} for details). \\
This kind of result is interesting also from the numerical point of view. 
Indeed, in many programs that calculate numerically the solution of 
partial differential equations, the Neumann and Dirichlet conditions are 
treated by considering a unique boundary conditions of type 
$$a \ \frac{\de y}{\de \nu_A} + b y = u \qquad \textrm{in }
[0,T]\times \de \Omega,$$
where the biggest constant appears near the condition one wants to
consider (an example is the program \cite{PP}). \vspace{1mm} \\
Let us shortly describe the content of the following sections. In
order to apply the general result given in \cite{BF95}, we need to
find a representation formula for the solutions of equations
(\ref{pbapprox}) and (\ref{pbmisto}). Therefore, in Section 2 we
analyze the mixed problem and we prove that the solution of
(\ref{pbmisto}) can be written as 
\beqs
y(t,\p)= \etam y_0(\p) - \int_0^t A e^{(t-s)A} Gu(s,\p) \ ds, 
\eeqs 
where $A$:$D(A)\to L^2(\Omega,\C^N)$ and $G$:$U \to D([-A]^{\te})$ 
(for $0 \leq \te < 1/4$) are suitable operators. Next, in Section 3
we prove the same kind of formula for the solution of
(\ref{pbapprox}), i.e. 
\beqs 
\yee(t,\p)= e^{t\aee}y_0(\p)+\int_0^t \aee e^{(t-s)\aee} \Gee u(s,\p) \ ds,
\eeqs
where now $\aee$: $D(\aee) \to L^2(\Omega,\C^N)$ and $\Gee$:$V \to
D([-\aee]^\te)$ (for $0 \leq \te < 1/2$). In Section 4 we finally
prove our $\Gamma$-convergence result. \\
Finally we set some notations. \\
If $T>0$ and $X$ is a Banach space, we will use the standard spaces
$L^p(0,T;X)$, $1\le p \le \infty$, and $C^0([0,T];X)$, with their usual
norms. As a rule, the space $X$ will be a Sobolev space $H^m(A)$ of
functions defined on some subset $A\subseteq \Rn$, with values in $\R^N$ or
$\C^N$. We will simply write $H^m(A)$ instead of $H^m(A;\R^N)$ or
$H^m(A;\C^N)$; here the number $m>0$ may be integer or not. In particular,
we denote by $H^m_0(A)$ the closure of the space $C^\infty_0(A)$ in $H^m(A)$, 
where $C^\infty_0(A)$ is the set of infinitely differentiable
functions with compact support contained in $A$; the set $A$ may be an
open set of $\R^N$, or a submanifold of the boundary of a smooth open
set $\Omega \subset \R^N$. \\
In the latter case, we also define $H^{1/2}_{00}(A)$ as the Hilbert
space of functions belonging to $H^{1/2}(A)$ whose trivial
extension to $\de \Omega$ is an element of $H^{1/2}(\de \Omega)$. We recall
that $H^{1/2}_{00}(A)$ is a proper, closed subspace of $H^{1/2}(A)$.\\ 
We denote by $(\p,\p)$ and $|\p|$ the scalar product and the norm in
$\Cn$, whereas $\langle\p,\p\rangle_X$ denotes the duality pairing between a
space $X$ and its dual space $X^*$. In our estimates we will write $C$ for a
generic constant possibly varying from line to line. 
%-------------------------------------------------------
\ssection{The mixed problem}
%-------------------------------------------------------
We start with describing in a more precise way the optimal control
problem we are going to approximate. \\
Let $\Omega$ be an open set of $\R^N$ with $\partial \Omega = \Gamma_0 \cup
\Gamma_1$. For fixed $T>0$, we consider the state space $L^2(0,T;H)$
and the control space $L^2(0,T;U)$, where $H$:$=\ldue$ and 
$U$:$=\humgo\times [\humoogi]^*$. We look for 
\beqs 
\min_{u\in L^2(0,T;U)} J(u)
\eeqs 
where
\beqs 
J(u)=\int_0^T \apt \| u(s,\cdot) \|_U^2 + \| y(s,\cdot) \|_H^2 \cht ds 
\eeqs
and for each control $u=(u_0,u_1)\in L^2(0,T;U)$ the state function $y$
is the solution of the parabolic mixed initial-boundary value problem 
\beqs 
\left\{ \begin{array}{rcll}
\frac{\de y}{\de t}(t,x)-A(x,D)y(t,x)& = & 0 & \quad \textrm{in } [0,T] \times
\Omega \vspace{1mm}\\ 
y(t,x) & = & u_0(t,x) & \quad \textrm{in } [0,T] \times \Gamma_0 \vspace{1mm}\\ 
\frac{\de y}{\de \nu_A}(t,x) & = & u_1(t,x) & \quad \textrm{in } [0,T]
\times \Gamma_1 \vspace{1mm}\\ 
y(0,x) & = & y_0(x) & \quad \textrm{in } \Omega. \end{array} \right.
\eeqs
Here
$$A(x,D) = \sij D_i (a_{ij}(x)\cdot D_j)\ ,\qquad \frac{\de }{\de \nu_A}=\sij
\nu_i(x) a_{ij}(x)\cdot D_j\ ,$$
and $\nu(x)$ is the unit outward normal vector at $x\in \de \Omega$; $\gao$
and $\gai$ are suitable subsets of $\de \Omega$. We interpret this problem
in the variational sense: namely, we set $\hunog=\{u\in \huno: u=0 
\textrm{ on } \gao\}$, and introduce the bilinear form 
\beq \label{formabili}
a(u,v) = \sij \into (a_{ij}(x)\cdot D_j u, D_i v)\ dx, \qquad
u,v\in H^1_{\gao}(\Omega);
\eeq
finally we fix a function $g_0\in L^2(0,T;\huno)$ such that
$g_0|_{[0,T]\times \gao}=u_0$. Then, following \cite{Li61}, the variational
version of problem (\ref{pbmisto}) consists in looking for a function $y$
such that $v:=y-g_0 \in L^2(0,T;\hunog)$ and 
\beq \label{mistovariazA} 
\int_0^T [- \langle v(t),\varphi'(t)\rangle_{\ldue} + a(v(t),\varphi(t))]\ dt 
= \langle y_0,\varphi(0)\rangle_{\ldue} + F(\varphi) \  
\eeq
for all test functions $\varphi$ such that 
\beq \label{mistovariazB}
\varphi \in L^2(0,T;\hunog)\cap H^1(0,T;\ldue),\qquad \varphi(T)=0,
\eeq
where the functional $F$ is defined, for all functions $\varphi$ of the above 
type, by 
\beq 
F(\varphi) = \int_0^T \langle g_0(t),\varphi'(t)\rangle_{\ldue}\ dt -
a(g_0,\varphi) + \langle u_1,\varphi|_{\gai}\rangle_{\humoogi}. 
\eeq
Of course, any solution of system (\ref{pbmisto}) solves equation
(\ref{mistovariazA}) too, and any sufficiently smooth solution of equation
(\ref{mistovariazA}) is also a solution of system (\ref{pbmisto}).\\ 
We list now our basic assumptions.
\begin{enumerate}
\item[(H0)] $\Omega$ is a bounded connected open set of $\R^N$ with boundary 
 $\de \Omega$ of class $C^{1,1}$. 
\item[(H1)] $y_0 \in L^2(\Omega)$.
\item[(H2)] $a_{ij} \in L^{\infty}(\Omega;\C^{N^2})$, with $M=\sij
|\aij(\p)|_{L^{\infty}(\Omega;\C^{N^2})}$, and there exists $\nu >0$
such that 
$$Re \sij (\aij \p \eta_j, \eta_i ) \geq \nu \si |\eta_i|^2$$
for all $\eta_1,{\ldots},\eta_n \in \Cn$, and for all $x \in \Omega$.
\item[(H3)] $\gao$ and $\gai$ are $(n-1)$-manifolds of class $C^2$, such 
that $\gao \subset \de \Omega$, $\gai=\de \Omega \setminus
\overline{\gao}$ and $\overline{\gao} \cap \overline{\gai} \neq
\emptyset$. 
\end{enumerate}
For the solution of system (\ref{pbmisto}) in the variational form
(\ref{mistovariazA}) we have the following existence and uniqueness result. 
\begin{THEO} \label{eupbmisto}
We assume {\em (H0), (H1), (H2), (H3)}. If $u \in L^2(0,T;U)$, then equation
(\ref{mistovariazA}) has a unique solution $y$ in the class $L^2(0,T;
\huno)$; moreover $y \in C^0([0,T]; \ldue) \cap H^{1/2}(0,T;\ldue)$ and
\beqsn
\lefteqn{\|y\|_{C^0([0,T];\ldue)} + \|y\|_{L^2(0,T;\huno)} +
\|y\|_{H^{1/2}(0,T;\ldue)} \le} \\
& & \le C \Big[\|y_0\|_{\ldue} + \|u\|_{L^2(0,T;\humgo)} +
\|u\|_{L^2(0,T;[\humoogi]^*)}\Big]. 
\eeqsn 
\end{THEO}
{\bf Proof.} For each $\lambda>0$ the bilinear form $a(u,v)+\lambda\langle
u,v\rangle_{\ldue}$ is coercive on the space $\huno$; hence the same holds
for the space $\hunog$. Thus the result follows by Theorems 1.1 and 2.2,
Chapter IV, in \cite{Li61}. \vspace{2mm} \quad \qed \\ 
In order to obtain a representation for the solution of equation
(\ref{mistovariazA}), we define the following operators $A$ and $G$: 
\beq \label{definA}
\left\{ \begin{array}{l}
\begin{array}{ll} D(A):= \{v \in \huno: & A(\p,D)v \in \ldue, \quad
\vspace{1mm} \\
& v|_{\gao}=0, \ \frac{\de v}{\de \nu_A}|_{\gai}=0 \} \end{array}
\vspace{2mm} \\ 
Av:=A(\p,D)v, \end{array} \right.
\eeq
\beq \label{definG}
v:=G(u_0,u_1) \qquad \iff \qquad \left\{ \begin{array}{rcll}
A(\p,D)v & = & 0 & \quad \textrm{in }\Omega \\
v & = & u_0 & \quad \textrm{on } \gao \\ 
\frac{\de v}{\de \nu_A} & = & u_1 & \quad \textrm{on } \gai. 
\end{array} \right. 
\eeq
We prove now that $A$ is the generator of an analytic semigroup in $\ldue$. 
\begin{PROP} \label{Agisga}
Under assumptions {\em (H0), (H2), (H3)}, let the operator $A$ be defined by
(\ref{definA}). Then $A$ is densely defined and generates an analytic
semigroup in $H=\ldue$. Moreover for any $\delta>0$ we have the estimate 
\beq \label{stispe}
\|R(\lambda,A)f\|_H \le \frac{C(\delta)}{|\lambda|} \ \|f\|_H
\qquad \forall \lambda\in S_{\te_0}, 
\eeq
where $S_{\te_0}:=\{ \lambda \in \C: |\arg(\lambda)|\le \te_0\}$, and 
$\te_0 = \pi - \arctan \delta$.
\end{PROP}
{\bf Proof.} Obviously, $C_0^{\infty}(\Omega) \subset D(A)$ so that
$D(A)$ is dense in $H$. Next, we show that the resolvent set $\rho(A)$
contains the positive real half-line: indeed, by the coerciveness in $\huno$
of the form $a(u,\varphi)+\lambda \langle u,\varphi\rangle_H$ for all
$\lambda>0$, we get that for all $f\in H$ the problem
\beq \label{ellitt}
\left\{ \begin{array}{l} u\in \hunog \vspace{2mm}\\
a(u,\varphi) + \lambda \langle u,\varphi\rangle_H = \langle f,\varphi
\rangle_H \qquad \forall \varphi \in \hunog
\end{array} \right.
\eeq
is uniquely solvable. Choosing $\varphi \in H^1_0(\Omega)$ we easily
find $A(\p, D)u = f-\lambda u \in H$, i.e. $u\in D(A)$ and $\lambda u -
Au=f$. This shows in particular that the solution $u$ of (\ref{ellitt})
solves the problem
\beq \label{ellittmisto}
\left\{ \begin{array}{rcl} 
\lambda u - A(\p,D)u & = & f \qquad \textrm{in } \Omega
\vspace{2mm}\\
u & = & 0 \qquad \textrm{on } \gao \vspace{2mm}\\
\frac{\de u}{\de \nu_A} & = & 0 \qquad \textrm{on } \gai. 
\end{array} \right.
\eeq
Thus we only need to prove (\ref{stispe}). Fix $\delta>0$ and take $\lambda
\in S_{\te_0}$, with $\te_0=\pi - \arctan{\delta}$. Let $v\in D(A)$ and set 
$f = \lambda v - Av$. 
Multiplying this equation by $v$ in $H$ and integrating by parts we get 
\beq \label{stispe2}
\into \lambda |v|^2 \ dx+ \into \sij (\aij\p D_j v, D_i v) \ dx =
\into (f,v) \ dx.
\eeq
By taking the real part and recalling hypothesis (H2) we obtain
$$Re \lambda \into |v|^2 \ dx + \nu \into |Dv|^2 \ dx \leq \| f \|_H
\| v \|_H \ ;$$
by taking the imaginary part we get
$$|Im \lambda | \into |v|^2 \ dx \le \| f \|_H \ \| v \|_H
+M\into |Dv|^2 \ dx.$$
Now if $|Re \lambda| \leq \delta |Im\lambda|$, then
$$\begin{array}{rcl} |\lambda| \into |v|^2 \ dx & \le & 
\sqrt{1 + \delta^2} \ |Im \lambda| \into |v|^2 \ dx \le \vspace{2mm} \\
& \le & \sqrt{1 + \delta^2} \apt 1+ \frac{M}{\nu} \cht \| f \|_H \| v
\|_H \ , \end{array}$$
whereas if $Re \lambda >0$ and $\delta |Im \lambda| < Re \lambda$ we have
$$\begin{array}{rcl} |\lambda| \into |v|^2 \ dx &\le &
\sqrt{1+\frac{1}{\delta^2}} \ Re \lambda \ \into |v|^2 \ dx \leq \vspace{2mm}
\\
&\le & \sqrt{1+\frac{1}{\delta^2}}\ \| f \|_H \|v\|_H \ .\end{array} 
$$
These two estimates show that for any $\delta>0$ we have
\beq \label{stispe3}
|\lambda| \|v\|_H \le \sqrt{1+\delta^2} \apq \apt 1+
\frac{M}{\nu} \cht +\frac{1}{\delta}\chq \|f\|_H \quad \forall
\lambda \in S_{\te_0}\ . 
\eeq
Note that the smaller is $\delta$ the larger is the sector $S_{\te_0}$ but
also the larger is the constant in the estimate.\\
As $\rho(A) \supseteq ]0,\infty[$, by estimate (\ref{stispe3}) and standard
arguments we deduce that $\rho(A)$ also contains the right half-plane; hence
Proposition 2.1.11 in \cite{Lu95} implies that $A$ is sectorial. 
Therefore $A$ is the infinitesimal generator of an analytic semigroup;
moreover, by (\ref{stispe3}) we immediately deduce (\ref{stispe}). This
completes the proof. \vspace{2mm} \quad \qed \\ 
The adjoint operator of $A$ is defined as follows:
\beq \label{definAstar}
\left\{ \begin{array}{l}
\begin{array}{ll} D(A^*):= \{v \in \huno: & \overline{A(\p,D)}v \in
\ldue, \vspace{1mm}\\ 
& v|_{\gao}=0, \ \frac{\de v}{\de \nu_{A^*}}|_{\gai}=0 \} \end{array}
\vspace{2mm}\\
A^*v:=\overline{A(\p,D)}v, \end{array} \right.
\eeq
where $\overline{A(x,D)}v:= \sij D_i [\overline{\aij^t} \p D_j v]$ and
$\overline{\aij^t}$ is the matrix whose elements are the 
conjugates of the elements of the transposed $\aij^t$ of $\aij$.  
Consequently it is clear that the following statement holds:
\begin{PROP} \label{Astargisga}
Under assumptions {\em (H0), (H2), (H3)}, let the operator $A^*$ be given by
(\ref{definAstar}). Then $A^*$ is densely defined and generates an analytic
semigroup in $H=\ldue$. Moreover for any $\delta>0$ we have the estimate 
\beq \label{stispe*}
\|R(\lambda,A^*)f\|_H \le \frac{C(\delta)}{|\lambda|} \ \|f\|_H
\qquad \forall \lambda\in S_{\te_0}. \quad \qed
\eeq
\end{PROP}
Concerning the operator $G$, we have the following result: 
\begin{PROP} \label{regolG}
Let {\em (H0), (H1), (H2), (H3)} be fulfilled. If $A$ is defined in
(\ref{definA}) then the operator $G$, given by (\ref{definG}), is well
defined and continuous from $U = \humgo \times [\humoogi]^*$ into $D(\apq -A 
\chq^\te)$ for $0 \leq \te < \frac{1}{4}$. 
\end{PROP}
{\bf Proof.} Let $(\uo,\ui) \in U$ and fix a lifting $g_0\in \huno$ of the
datum $u_0$, with $\|g_0\|_{\huno} \le C\|u_0\|_{\humgo}$. Then the problem
$$\left\{ \begin{array}{l} u-g_0 \in \hunog \vspace{2mm}\\
a(u,\varphi) = \langle u_1,\varphi \rangle_{\humoogi} \qquad \forall \varphi
\in \hunog \end{array} \right.$$
has a unique solution $u\in \huno$. Hence $u=G(\uo,\ui)$ by definition
of $G$, and, in particular, we have $G(\uo,\ui) \in
H^{2\te}(\Omega)$ for all $\te \in [0,\frac{1}{2}]$. \\  
On the other hand, by Theorem 3.1 of \cite{Ban94} we know that $D(\apq
-A \chq^\te)$ coincides with $H^{2\te}(\Omega)$ if and only if
$\te \in [0,\frac{1}{4}[$. This completes the proof. \vspace{2mm}
\quad \qed \\ 
We are finally ready to prove the representation formula for the
solution of system (\ref{pbmisto}) in the variational form
(\ref{mistovariazA}). 
\begin{THEO} \label{formrappr}
Under assumptions {\em (H0), (H1), (H2), (H3)}, let $u=(\uo,\ui)$ be an
element of $L^2(0,T;U)$, where $U = \humgo \times [\humoogi]^*$. Then the
solution of (\ref{mistovariazA}) is given for each $t\in [0,T]$ by 
\beq \label{rapprsol}  
y(t,\p)= \etam y_0(\p) - \int_0^t A e^{(t-s)A} Gu(s,\p) \ ds. 
\eeq  
\end{THEO}
{\bf Proof.} By Theorem \ref{eupbmisto} we know that equation
(\ref{mistovariazA}) has a unique solution $y \in C^0([0,T];H) \cap
H^{1/2}(0,T;H) \cap L^2(0,T;\huno)$, where $H=\ldue$. Set now
\beq \label{formularap}
v(t) = \etam y_0 - \int_0^t A e^{(t-s)A} Gu(s) \ ds,\qquad t\in [0,T];
\eeq
by the standard properties of analytic semigroups and by Proposition
\ref{regolG} it is easily seen that $v\in C^0([0,T];H)\cap
L^2(0,T;H^{2\te}(\Omega))$ for any $\te\in [0,\frac{1}{4}[$. We have to
show that $y\equiv v$. \\
We suppose first that 
\beq \label{datireg}
\left\{ \begin{array}{l} y_0\in H^2(\Omega),\quad u=(u_0,u_1)\in
C^{\infty}([0,T],U),\vspace{1mm}\\ 
y_0-Gu(0)\in D(A). \end{array} \right.
\eeq
Under assumption (\ref{datireg}) we can integrate by parts in
(\ref{formularap}):
\beq \label{formulabis}
\begin{array}{rcl} v(t) & = & e^{tA}y_0 + \left[ e^{(t-s)A}Gu(s)\right]^t_0
- \int_0^t e^{(t-s)A} Gu'(s)\ ds = \vspace{2mm} \\
& = & e^{tA}y_0 + Gu(t) - e^{tA}Gu(0) - \int_0^t e^{(t-s)A} Gu'(s)\ ds.
\end{array}
\eeq
Hence we can compute
\beq \label{derivformula} 
v'(t) = Ae^{tA}[y_0-Gu(0)] - \int_0^t Ae^{(t-s)A} Gu'(s)\ ds \qquad \forall
t\in [0,T]; 
\eeq
thus we see that 
\beq \label{regollaterale}
v(t)-Gu(t) \in D(A) \qquad \forall t\in [0,T],
\eeq
and in addition, by (\ref{definA}) and (\ref{definG}),
\beq \label{rapprsol2}   
v'(t) = A[v(t)-Gu(t)] = A(\p,D)v(t,\p) \qquad \forall t\in [0,T].
\eeq
In particular, $v'\in C^0([0,T];H)$. Moreover, $v(0)=y_0$ and, by
(\ref{regollaterale}), $v(t)$ behaves at $\de \Omega$ just like $Gu(t)$, i.e.
$$v(t)|_{\gao} = u_0(t) \in \humgo, \qquad \frac{\de v(t)}{\de
\nu_A}|_{\gai} = u_1(t) \in [\humoogi]^*.$$
This shows that $v$ solves problem (\ref{pbmisto}). In particular, $v$
solves equation (\ref{mistovariazA}) too; by uniqueness, this implies
$v\equiv y$, provided (\ref{datireg}) holds.\\
Consider now the general case, i.e. $y_0\in H$ and $u=(u_0,u_1) \in
L^2(0,T;U)$; let $y$ be the solution of equation (\ref{mistovariazA}) and
let $v$ be the function (\ref{formularap}). There exist sequences
$\{y_{0,n}\}$ and $\{u_n\}$, satisfying assumption (\ref{datireg}) for all
$n\in \N$, such that $y_{0,n}\to y_0$ in $H$ and $u_n\to u$ in $L^2(0,T;U)$.
Denoting by $y_n$ the corresponding solution of (\ref{mistovariazA}) and by
$v_n$ the corresponding function (\ref{formularap}), we have by the above
argument $v_n\equiv y_n$ for all $n\in \N$. But since the $C^0([0,T];H)$-norm
of both $y_n$ and $v_n$ depends continuosly on the $H$-norm of $y_{0,n}$ and
on the $L^2(0,T;U)$-norm of $g_n$, as $n\to \infty$ we immediately deduce
that $v\equiv y$. This proves the result. \vspace{2mm} \quad \qed \\
After the above preparations, we see that the control problem (\ref{minmisto})
fits in the abstract setting described in \cite{LT92}: thus, by the
results of \cite{LT92} (see also Theorem 3.14 in \cite{AFT91}, and
Theorem 8.2 in \cite{AT99}) we can characterize the optimal control
through a feedback formula involving the Riccati operator. The
synthesis of the optimal control problem is summarized in the
following statement. 
\begin{THEO} \label{sintesimisto} Let {\em (H0), (H1), (H2), (H3)} be
fulfilled. Then:
\begin{enumerate}
\item[{\em(i)}] There exists a unique optimal pair $(\hat{u},\hat{y}) \in
\ldueotu \times \ldueoth)$ for problem (\ref{minmisto}), where $H=\ldue$ and 
$U=\humgo \times [\humoogi]^*$. 
\item[{\em(ii)}] The Riccati equation in integral form, i.e.
$$P(t)=\int_t^T e^{(s-t)A^*} \apq I- P(s) A G G^* A^* P(s) \chq e^{(s-t)A} \
ds,$$
has a unique solution $P \in C^1([0,T[;{\cal L}(H))\cap C^0([0,T];{\cal
L}(H))$. \\ 
Moreover, 
$P(t)=P(t)^*\ge 0$ and $P(t) \in D([-A^*]^{1-\te})$ for each $\te\in ]0,1]$,
with $\|[-A^*]^{1-\te} P(t)\|_{{\cal L}(H)} \le C(T-t)^{-(1-\te)}$ for all
$t\in [0,T[$; in addition it holds
$$J(\hat{u})=\langle P(0)y_0,y_0\rangle_H \ .$$
\item[{\em(iii)}]  We have the feedback formula for $\hat{u}$:
$$\hat{u}(t,\p)=G^* A^* P(t) \hat{y}(t,\p), \qquad t \in [0,T[ .$$
\item[{\em(iv)}] The optimal trajectory $\hat{y}$ is expressed by
$\hat{y}(t,\cdot)=\Phi(t,0)y_0(\p)$, where $\Phi(t,s)$ is defined by the
integral equation
$$\Phi(t,s) = e^{(t-s)A} - \int_s^t Ae^{(t-r)A}GG^* A^* P(r) \Phi(r,s) \
dr, \quad t \in [s,T].$$
\end{enumerate}
The expressions $P(s)AG$ and $G^*A^*P(s)$ are shorter forms relative to the
well defined operator $\left[-[(-A^*)^{1-\te} P(s)]^*(-A)^{\te} G\right]$
and to its adjoint, with fixed $\te \in ]0,\frac{1}{4}[$. \quad \qed
\end{THEO}
\begin{REM} \label{restrspazi} {\em As noted in the Introduction,
in order to obtain our $\Gamma$-convergence result we will need to
restrict somewhat the control space $U$, replacing $\humgo \times
[\humoogi]^*$ by its closed subspace $V=\humoogo \times \hmumgi$. Of
course, all results proved in this Section still hold if we use the
restricted control space.} 
\end{REM}
%-------------------------------------------------------------------
\ssection{The approximating problems}
%-------------------------------------------------------------------
Our goal is to approach problem (\ref{minmisto}) by a family of more
regular problems which we now describe.\\
Fix $\varepsilon>0$. In the same open set $\Omega$ as in the preceding
section, consider again the state space $L^2(0,T;H)$, with $H=\ldue$, and
take as control space $L^2(0,T;V)$, with 
\beq \label{spcapprox}
V = \{u \in {\cal D}'(\de \Omega): u|_{\gao} \in \humoogo, \ u|_{\gai} \in
\hmumgi \}, 
\eeq
endowed with its natural norm
$$\|u\|_V = \|u|_{\gao}\|_{\humoogo} + \|u|_{\gai}\|_{\hmumgi}\ .$$
We remind that a distribution $u\in {\cal D}'(\de \Omega)$ belongs
to $V$ if the restrictions of the functional $u$ to the subspaces ${\cal
D}(\gao)$ and ${\cal D}(\gai)$ verify respectively
\beq
\begin{array}{c} |\langle u, \varphi \rangle| \le C_0
\|\varphi\|_{\humgi} \quad \forall \varphi \in {\cal
D}(\gai),\vspace{2mm} \\ 
|\langle u, \varphi \rangle| \le C_1 \|\varphi\|_{[\humoogo]^*} \quad
\forall \varphi \in {\cal D}(\gao); \end{array} 
\eeq
by density, the above equalities are true for all $\varphi\in \humgi$
and for all $\varphi\in [\humoogo]^*$ respectively. 
\begin{LEMMA} \label{Visom} The space $V$, difined in (\ref{spcapprox}), is
isomorphic to $\humoogo \times \hmumgi$ by the map $j(u) =
(u|_{\gao},u|_{\gai})$.
\end{LEMMA}
{\bf Proof.} We just verify that the map $j$ is onto, the other
properties being quite easy. Fix $(u_0,u_1)\in \humoogo \times
\hmumgi$, and set $u=U+V$, where $U$ is the trivial extension of $u_0$
to $\de \Omega$ and $V$ is the element of $\hmunmeb$ defined by
$V(\varphi) = u_1(\varphi)$ for all $\varphi \in \hunmeb \subset
\humgi$. Then it is straightforward to check that that $u|_{\gao}=u_0$
and $u|_{\gai}=u_1$, i.e. $j(u) = (u_0,u_1) $. \vspace{2mm} \quad \qed \\ 
We recall here  our approximating control problem: in view of Lemma
\ref{Visom}, we set 
\beq \label{spcgamma} 
V=\humoogo \times \hmumgi,
\eeq
and we look for
\beqs 
\min_{u\in L^2(0,T;V)} \Jee(u),
\eeqs 
where
\beqs 
\Jee(u)=\int_0^T \apt \| u(s,\cdot) \|_V^2 + \| \yee(s,\cdot) \|_H^2 \cht ds 
\eeqs
and for each control $u\in L^2(0,T;V)$ the state function $\yee$ is the
solution of the regular parabolic initial-boundary value problem 
\beqs 
\left\{ \begin{array}{rcll} \frac{\de y}{\de t}(t,x)-A(x,D)y(t,x) & = & 0 &
\quad \textrm{in } [0,T] \times \cho \vspace{1mm} \\ 
\teee(x)\frac{\de y}{\de \nu_A}(t,x) + (1-\teee(x))y(t,x) & = & u(t,x) &
\quad \textrm{in } [0,T] \times \de \Omega \vspace{1mm}\\ 
y(0,x) & = & y_0(x) & \quad \textrm{in } \cho. \end{array} \right.
\eeqs
Here the scalar function $\teee$ is defined as follows:
\beqs 
\teee(x) = \apg \begin{array}{ll}
1-\ee & \textrm{if } x \in \gai \vspace{1mm}\\
1-\ee - \frac{1-2\ee}{\ee}\ d(x,\gai) \quad & \textrm{if } x \in \gao
\textrm{ and } d(x,\gai) < \ee \vspace{1mm}\\
\ee & \textrm{if } x \in \gao \textrm{ and } d(x,\gai) \geq \ee. \\
\end{array} \ch
\eeqs
The classical results on regular elliptic and parabolic problems
guarantee that, at least for smooth data, a unique solution of the
state equation (\ref{pbapprox}) exists; however the standard estimates
(see e.g. \cite{ADN64}) depend on $\varepsilon$, so that the solution might
be unbounded in certain spaces with respect to $\varepsilon$ as $\varepsilon
\to 0$.\\ 
On the other hand, problem (\ref{pbapprox}) has also a variational
formulation: for fixed $u\in L^2(0,T;V)$, problem (\ref{pbapprox}) consists
in looking for a function $y\in L^2(0,T;\huno)$ such that
\beq \label{approxvariazA}
\begin{array}{l} \int_0^T \big[- \langle y(t),\varphi'(t)\rangle_{\ldue} +
a(y(t),\varphi(t)) + \vspace{2mm} \\
\qquad \qquad \qquad \qquad + \langle \frac{1-\teee}{\teee} \
y(t),\varphi(t)\rangle_{\hunmeb}\big]\ dt = \vspace{2mm} \\ 
\qquad \qquad \qquad \qquad = \langle y_0,\varphi(0)\rangle_{\ldue} +
\int_0^T \langle u(t),\varphi(t)\rangle_{\hunmeb} \ dt \end{array} 
\eeq
for all test functions $\varphi$ such that 
\beq \label{approxvariazB}
\varphi \in L^2(0,T;\huno)\cap H^1(0,T;\ldue),\qquad \varphi(T)=0.
\eeq
Our first task is to give an existence and uniqueness result for the
variational problem (\ref{approxvariazA}), with an estimate not depending on
$\varepsilon$. 
\begin{THEO} \label{eupbapprox}
We assume {\em (H0), (H1), (H2), (H3)}. If $u\in L^2(0,T;V)$, with $V$
defined in (\ref{spcgamma}), then equation (\ref{approxvariazA}) has
a unique solution $\yee$ in the class $C^0([0,T];\ldue)\cap
L^2(0,T;\huno)\cap H^{1/2}(0,T;\ldue)$. Moreover there exists $C\ge
0$, independent of $\varepsilon$, such that
\beqsn
\lefteqn{\|\yee\|_{C^0([0,T];\ldue)} + \|\yee\|_{L^2(0,T;\huno)} +
\|\yee\|_{H^{1/2}(0,T;\ldue)} \le} \\
& & \le C \Big[\|y_0\|_{\ldue} + \|u\|_{L^2(0,T;\humoogo)} +
\|u\|_{L^2(0,T;\hmumgi)}\Big]. 
\eeqsn 
\end{THEO}
{\bf Proof.} The bilinear form $a(y,v) + \langle \frac{1-\teee}{\teee} \  
y,v\rangle_{\hunmeb}$ is clearly weakly coercive on $\huno$; in fact
it is even coercive, since the norm $\|u\|_{L^2(\de \Omega)} +
\|Du\|_{\ldue}$ is equivalent to the usual norm of $\huno$, as shown
e.g. by Lemma 4.4 in \cite{AT96}. Hence the result follows by Theorems
1.1 and 2.2, Chapter IV in \cite{Li61}. \vspace{2mm} \quad \qed \\
As we did for the case of the mixed problem, we want to find now a
representation formula for the solution $y_\ee$ of problem
(\ref{pbapprox}). We start with defining the operators 
\beq \label{defAe} 
\left\{ \begin{array}{l}
 D(A_\ee):= \left\{ v \in H^2(\Omega) :\ \teee \frac{\partial
v}{\partial \nu_A} +(1-\teee) v=0 \textrm{ on } \partial \Omega \right\}
\vspace{2mm}\\ 
\aee v := A(\p,D)v, \end{array} \right.
\eeq
and
\beq \label{defGe}
w:=\Gee h \qquad  \iff \qquad \left\{ \begin{array}{rl}
A(\p,D)w=0 & \textrm{ in } \Omega \\
\teee \frac{\partial w}{\partial \nu_A} + (1-\teee)w =h & \textrm{ on } 
\partial \Omega. \end{array} \right.
\eeq 
As (\ref{defAe}) is a regular elliptic operator, it is clear that
$\aee$ is the infinitesimal generator of an analytic semigroup; we
show now that this property holds ``uniformly'' with respect to
$\ee$. 
\begin{PROP} \label{Aegisga}
Let {\em (H0), (H1), (H2)} be fulfilled. Then the operator $\aee$
defined by (\ref{defAe}) is densely defined and generates an analytic
semigroup in $\ldue$. Moreover for any $\delta>0$ we have the estimate 
\beq \label{strAe}
\|R(\lambda,\aee)f\|_{\ldue} \leq \frac{C(\delta)}{|\lambda|} \ 
\|f\|_{\ldue} \qquad \forall \lambda\in S_{\te_0}\ ,
\eeq
where $S_{\te_0}:=\{ \lambda \in \C: |\arg(\lambda)|\leq \te_0\}$, and 
$\te_0=\pi - \arctan \delta$. In particular, the constant $C(\delta)$
does not depend on $\ee$.
\end{PROP}
{\bf Proof.} We just need to prove (\ref{strAe}).\\
Let $f\in \ldue$ and set $v_{\ee}=R(\lambda,\aee)f$: then $v_{\ee}$ solves
\beq \label{ellittapprox}
\left\{ \begin{array}{rcl} 
\lambda v_{\ee} - A(\p,D)v_{\ee} & = & f \qquad \textrm{in } \Omega
\vspace{1mm}\\
\teee\frac{\de v_{\ee}}{\de \nu_A} + (1-\teee)v_{\ee} & = & 0 \qquad
\textrm{on } \de \Omega. \end{array} \right.
\eeq
Let us multiply by $v_{\ee}$, integrate over $\Omega$ and use the
boundary condition: we get
\beq \label{relaziniz}
\begin{array}{l} \lambda \into |v_{\ee}|^2 \ dx + a(v_{\ee},v_{\ee}) +
\vspace{2mm} \\ 
\qquad + \int_{\Gamma_0} \frac{\teee}{1-\teee}|\frac{\de v_{\ee}}{\de
\nu_A}|^2 \ d\sigma + \int_{\Gamma_1} \frac{1-\teee}{\teee} |v_{\ee}|^2 \
d\sigma = \into |(f,v_{\ee})|\ dx. \end{array}
\eeq
This estimate plays the role of (\ref{stispe2}) in the proof of Proposition
\ref{Agisga}: thus, just repeating the argument used in that proof, we get
for any $\delta>0$
$$\|v_{\ee}\|_{\ldue} \le \frac{C(\delta)}{|\lambda|}\ \|f\|_{\ldue} \qquad
\forall \lambda \in S_{\te_0}\ ,$$
where $\te_0=\pi-\arctan \delta$. This proves the result. \vspace{2mm}
\quad \qed 
\begin{REM} \label{stimagrad} {\em By (\ref{relaziniz}) it follows in
particular the useful estimate, which holds true for all $\lambda \in
S_{\te_0}$\ :}
$$\|v_{\ee}\|_{\huno} + \int_{\Gamma_0} \frac{\teee}{1-\teee}\left|\frac{\de
v_{\ee}}{\de \nu_A}\right|^2 d\sigma + \int_{\Gamma_1}
\frac{1-\teee}{\teee} |v_{\ee}|^2\ d\sigma \le
\frac{C(\delta)}{|\lambda|^{1/2}}\ \|f\|_{\ldue}\ .$$
\end{REM}
The adjoint operator of $\aee$ is defined by:
\beq \label{defAestar} 
\left\{ \begin{array}{l}
 D(A_{\ee}^*):= \left\{ v \in H^2(\Omega):\ \teee \frac{\partial
v}{\partial \nu_{A^*}} +(1-\teee) v=0 \textrm{ on } \partial \Omega \right\}
\vspace{2mm}\\ 
\aee^* v := \overline{A(\p,D)}v. \end{array} \right.
\eeq 
Hence it is clear that the following result, parallel to that of Proposition
\ref{Aegisga}, holds:
\begin{PROP} \label{Aestargisga}
Let {\em (H0), (H1), (H2)} be fulfilled and fix $\ee>0$. Then the operator
$\aee^*$ given by (\ref{defAestar}) is densely defined and generates an
analytic semigroup in $\ldue$. Moreover for any $\delta>0$ we have the
estimate 
\beq \label{strAestar}
\|R(\lambda,\aee^*)f\|_{\ldue} \leq \frac{C(\delta)}{|\lambda|} \ 
\|f\|_{\ldue} \qquad \forall \lambda\in S_{\te_0}\ ,
\eeq
where $S_{\te_0}:=\{ \lambda \in \C: |\arg(\lambda)|\leq \te_0\}$, and 
$\te_0=\pi - \arctan \delta$. In particular, the constant $C(\delta)$
does not depend on $\ee$. \quad \qed
\end{PROP}
Concerning the operator $\Gee$, we have:
\begin{PROP} \label{regolGe}
Let {\em (H0), (H1), (H2), (H3)} be fulfilled; fix $\ee >0$ and let $\aee$ be
defined by (\ref{defAe}). The operator $\Gee$\ , given by (\ref{defGe}),
is well defined from the space $V$, defined by (\ref{spcgamma}), into the
space $D([-\aee]^\te)$ for each $\te \in ]0, \frac{1}{2}[$. In addition,
there exists $C>0$, independent of $\ee$, such that
\beqsn
\lefteqn{C(\ee)^{-1} \|[-\aee]^\te G_{\ee}h\|^2_{\ldue} \le
\|DG_{\ee}h\|^2_{\ldue} \le} \\
& & \le C \left[ \|h|_{\gao}\|^2_{\humoogo} +
\|h|_{\gai}\|^2_{\hmumgi} \right] \qquad \forall h\in V.
\eeqsn
\end{PROP}
{\bf Proof.} Let first $w=\Gee h$, where $h\in C^{\infty}(\de \Omega)$
and $h$ vanishes in a neighbourhood of $\overline{\gao}\cap \overline{\gai}$; 
then $w$ is smooth and solves problem (\ref{defGe}). We repeat the argument
used in the proof of Proposition \ref{Aegisga}. Multiply by $w(x)$ the
equation $A(x,D)w=0$ and integrate by parts: using Hypothesis (H2) and the
boundary condition, we get 
\beq \label{stima0approx} 
\begin{array}{l} \nu \into |Dw|^2\ dx + \int_{\gao}
\frac{\teee}{1-\teee}\ |\frac{\de w}{\de \nu_A}|^2\ d\sigma +
\int_{\gai} \frac{1-\teee}{\teee}\ |w|^2 \ d\sigma \le \vspace{2mm}\\
\qquad \qquad \qquad \le \left| \int_{\gao} (h,\frac{\de w}{\de
\nu_A})\ \frac{1}{1-\teee}\ d\sigma\right| + \left| \int_{\gai} (h,w) \
\frac{1}{\teee}\ d\sigma \right|. \end{array} 
\eeq
Let us estimate separately the addenda of the last term in
(\ref{stima0approx}). We start with the second one, which is easier to
handle. As $\teee(x)\equiv 1-\ee$ on $\gai$ we have 
\beq \label{pezzofacile} 
\begin{array}{rcl} \left|\int_{\gai} (h,w) \frac{1}{\teee} \ d\sigma \right|
& = & \frac{1}{1-\ee}\ \left|\langle h,w \rangle_{\humgi} \right| \le
\vspace{2mm} \\ 
& \le & \frac{1}{1-\ee}\ \|h\|_{\hmumgi} \ \|w\|_{\humgi}\ \le \vspace{2mm} \\ 
& \le & \frac{1}{1-\ee}\ \|h\|_{\hmumgi}\ \|w\|_{\hunmeb}\ \le
\vspace{2mm} \\
& \le & \frac{\nu}{2}\ \|h\|_{\hmumgi}^2 + C\ \|w\|_{\huno}^2 . \end{array} 
\eeq 
Concerning the first term on the last member of (\ref{stima0approx}), set
\beq \label{gaee}
\Gamma_{\ee} = \{x\in \gao : d(x,\gai) \le \ee\},
\eeq
and denote by $\overline{h}$ the function which coincides with $h$ on
$\gao$ and vanishes on $\gai$: then $\overline{h} \in \hunmeb$ and we can
write 
\beq \label{pezzodiff} 
\begin{array}{l} \left| \int_{\gao} (h,\frac{\de w}{\de \nu_A}) \ 
\frac{1}{1-\teee} \ d\sigma \right| \le \vspace{2mm} \\ 
\quad = \left| \int_{\gao} (h,\frac{\de w}{\de \nu_A})\ \frac{1}{1-\ee} \
d\sigma \right| + \int_{\Gamma_{\ee}} |(h,\frac{\de w}{\de \nu_A})| \
(\frac{1}{1-\teee} - \frac{1}{1-\ee})\ d\sigma \le \vspace{2mm} \\
\quad \le \left| \int_{\de \Omega} (\overline{h},\frac{\de w}{\de \nu_A})\
\frac{1}{1-\ee} \ d\sigma \right| + \int_{\Gamma_{\ee}} |(h,\frac{\de w}{\de 
\nu_A})|\ \frac{1}{1-\teee}\ d\sigma \vspace{2mm} \\
\quad = I + II.
\end{array} 
\eeq
On one hand, we get for all $\eta>0$ 
\beq \label{stima1approx}
\begin{array}{rcl} I & = & \frac{1}{1-\ee}\ \left|\int_{\de \Omega}
(\overline{h},\frac{\de w}{\de \nu_A}) \ d\sigma \right| \le \vspace{2mm} \\ 
& \le & C\ \|\frac{\de w}{\de \nu_A}\|_{\hmunmeb}\ \|\overline{h}\|_{\hunmeb}\
\le \vspace{2mm}\\ 
& \le & \eta\ \|w\|_{\huno}^2 + C(\eta)\ \|h|_{\gao}\|_{\humoogo}^2;
\end{array}
\eeq
on the other hand, we have 
\beq \label{stima2approx}
\begin{array}{rcl} II & = & \int_{\Gamma_{\ee}} |(h,\frac{\de w 
}{\de \nu_A})|\ \frac{1}{1-\teee}\ d\sigma \le \vspace{2mm} \\
& \le & \int_{\Gamma_{\ee}} \frac{1}{2}\ 
\frac{\teee}{1-\teee}\ |\frac{\de w}{\de \nu_A}|^2 \ d\sigma +
\int_{\Gamma_{\ee}} \frac{2}{\teee(1-\teee)}\ |h|^2 \ d\sigma ,
\end{array}
\eeq
and the second addendum in the last member of (\ref{stima2approx}) is
estimated by 
\beq \label{stima3approx}
\begin{array}{l} \int_{\Gamma_{\ee}} \frac{2}{\teee(1-\teee)}\ |h|^2\ d\sigma 
\le C \int_{\Gamma_{\ee}} \frac{1}{\ee}\ |h|^2 \ d\sigma \le \vspace{2mm} \\
\qquad \le C \int_{\gao} |h|^2 \ \frac{1}{d(x,\gai)}\ d\sigma \le
C \|h|_{\gao}\|_{\humoogo}^2 . \end{array}
\eeq 
By (\ref{stima0approx}), (\ref{pezzofacile}), (\ref{pezzodiff}), 
(\ref{stima1approx}), (\ref{stima2approx}) and (\ref{stima3approx}), 
we deduce for a sufficiently small $\eta$:
\beq \label{stima4approx} 
\begin{array}{l} \int_{\Omega} |Dw|^2 \ dx + \int_{\gao}
\frac{\teee}{1-\teee}\ |\frac{\de w}{\de \nu_A}|^2\ d\sigma +
\int_{\gai} \frac{1-\teee}{\teee}\ |w|^2\ d\sigma \le \vspace{2mm}\\ 
\qquad \qquad \qquad \qquad \le C \left[ \|h|_{\gao}\|_{\humoogo}^2 +
\|h|_{\gai}\|_{\hmumgi}^2 \right] . \end{array} 
\eeq
Now the general case, i.e. the case $h\in V$, follows by a density argument,
since we may approximate $h$ by a sequence $\{h_n\} \subset
C^{\infty}(\de \Omega)$ of functions vanishing in a neighbourhood of
$\overline{\gao} \cap \overline{\gai}$. Thus estimate
(\ref{stima4approx}) holds for $w=\Gee h$ too; this means that 
the map $\Gee$ is bounded, uniformly with respect to $\ee$, from
$\humoogo \times \hmumgi$ to $\huno$. \\
We invoke finally a result due to Fujiwara (Theorem 2 in \cite{Fu67}),
according to which we have $D([-\aee]^\te) = H^{2\te}(\Omega)$ for all
$\te \in [0,3/4[$ and
\beq
\|f\|_{D([-\aee]^\te)} \le C(\ee) \|f\|_{H^{2\te}(\Omega)} \qquad
\forall f\in H^{2\te}(\Omega),
\eeq
with $C(\ee)$ possibly depending on $\ee$. This estimate, together with
(\ref{stima4approx}), implies our result. \quad \qed 
\begin{REM} \label{stimebordo} {\em In the proof of Proposition
\ref{regolGe} we have proved in particular the following useful
estimate for $w=\Gee h$:}
\beq \label{stimage}
\begin{array}{l} \int_{\gao} \frac{\teee}{1-\teee}\ |\frac{\de w}{\de
\nu_A}|^2\ d\sigma + \int_{\gai} \frac{1-\teee}{\teee}\ |w|^2\ d\sigma \le
\vspace{2mm} \\ 
\qquad \le C \left[ \|h|_{\gao}\|_{\humoogo}^2 + \|h|_{\gai}\|_{\hmumgi}^2
\right] . \end{array} 
\eeq 
\end{REM}
\begin{REM} \label{stimepbapprox} {\em An estimate similar to (\ref{stimage}) 
is valid for the solution $\yee$ of the parabolic problem
(\ref{approxvariazA}), namely 
\beq 
\begin{array}{l} \int_0^T \int_{\gao} \frac{\teee}{1-\teee}\left| \frac{\de
\yee}{\de \nu_A}\right|^2 d\sigma dt + \int_0^T\int_{\gai}
\frac{1-\teee}{\teee} |y|^2\ d\sigma dt \le \vspace{2mm}\\ 
\qquad \le C \int_0^T \left[\|u_0(t,\p)\|_{\humgo}^2 +
\|u_1(t,\p)\|_{\hmumgi}^2 \right] dt. \end{array} 
\eeq
The proof requires the same argument used in the proof of Proposition
\ref{regolGe}, and we can omit it.} 
\end{REM}
We are finally ready to prove the representation formula for the
solution of problem (\ref{pbapprox}). 
\begin{THEO} \label{fdrapprox}
Assume {\em (H0), (H1), (H2), (H3)}. If $u \in L^2(0,T;V)$, with $V$
defined by (\ref{spcgamma}), then the solution $\yee$ of
(\ref{pbapprox}) is given for each $t\in [0,T]$ by 
\beq \label{frapprox} 
\yee(t,\p)= e^{t\aee}y_0(\p)+\int_0^t \aee e^{(t-s)\aee} \Gee u(s,\p) \ ds.
\eeq  
\end{THEO}
{\bf Proof.} As (\ref{pbapprox}) is a regular parabolic initial-boundary
value problem, the result follows by adapting the proof of Proposition
2.13 in \cite{AFT91}; otherwise, one may repeat the same argument used
in proving Theorem \ref{formrappr} above. We omit the
details. \vspace{2mm} \quad \qed \\  
Finally we state the result on the synthesis of the optimal control problem
(\ref{minapprox}), whose proof is parallel to that of Theorem
\ref{sintesimisto} (compare again with \cite{LT92}, \cite{AFT91}, and 
\cite{AT99}). 
\begin{THEO} \label{sintesiapprox} Let {\em (H0), (H1), (H2), (H3)} be
fulfilled. Then:
\begin{enumerate}
\item[{\em(i)}] There exists a unique optimal pair
$(\hat{u}_{\ee},\hat{y}_{\ee}) \in L^2(0,T;V) \times \ldueoth)$ for problem
(\ref{minmisto}), where $H=\ldue$ and $V$ is defined in (\ref{spcgamma}). 
\item[{\em(ii)}] The Riccati equation in integral form, i.e.
$$P_{\ee}(t)=\int_t^T e^{(s-t)\aee^*} \apq I- P_{\ee}(s)\aee \Gee \Gee^*
\aee^* P_{\ee}(s) \chq e^{(s-t)\aee} \ ds,$$
has a unique solution $P_{\ee} \in C^1([0,T[;{\cal L}(H))\cap C^0([0,T];{\cal
L}(H))$, such that $P_{\ee}(t)=P_{\ee}(t)^*\ge 0$ and $P_{\ee}(t) \in
D([-A^*]^{1-\te})$ for each $\te\in ]0,1]$, with $\|[-\aee^*]^{1-\te}
P_{\ee}(t)\|_{{\cal L}(H)} \le C(T-t)^{-(1-\te)}$ for all $t\in [0,T[$; in
addition it holds 
$$J(\hat{u}_{\ee})=\langle P_{\ee}(0)y_0,y_0\rangle_H \ .$$
\item[{\em(iii)}]  We have the feedback formula for $\hat{u}_{\ee}$:
$$\hat{u}_{\ee}(t,\p)=\Gee^* \aee^* P_{\ee}(t) \hat{y}_{\ee}(t,\p), \qquad t 
\in [0,T[ .$$ 
\item[{\em(iv)}] The optimal trajectory $\hat{y}_{\ee}$ is given by
$\hat{y}_{\ee}(t,\cdot)=\Phi_{\ee}(t,0)y_0(\p)$, where $\Phi_{\ee}(t,s)$ is
defined by the integral equation
$$\Phi_{\ee}(t,s) = e^{(t-s)\aee} - \int_s^t \aee e^{(t-r)\aee}\Gee\Gee^*
\aee^* P_{\ee}(r) \Phi_{\ee}(r,s) \ dr, \quad t \in [s,T].$$
\end{enumerate}
The expressions $P_{\ee}(s)\aee \Gee$ and $\Gee^*\aee^*P_{\ee}(s)$ are
shorter forms relative to the well defined operator
$\left[-[(-\aee^*)^{1-\te} P_{\ee}(s)]^*(-\aee)^{\te} \Gee\right]$ 
and to its adjoint, with fixed $\te \in ]0,\frac{1}{2}[$. \quad \qed
\end{THEO}
%-------------------------------------------------------------------
\ssection{$\Gamma$-convergence}
%-------------------------------------------------------------------
First of all, we observe that all $\Gamma$-convergence results in the
literature deal with a discrete parameter $n$ tending to $+\infty$. So, from
now on we will consider a fixed subsequence of our approximating problems
with parameter $\ee = \ee_n$ such that $\ee_n \to 0^+$ for $n \to \infty$. \\ 
We remark that, by (\ref{tetaeps}),
\beq \label{limtetaepsn}
\lim_{n \ds \infty} \teen(x) = \apg \begin{array}{ll}
1 & \textrm{if } x \in \gai\\
0 & \textrm{if } x \in \gao.
\end{array} \ch
\eeq
In order to apply the abstract result of \cite{BF95}, we need to rewrite
the optimal control problems in a different way. Therefore, we set
\beq \label{spazi}
H=\ldue, \qquad V=\humoogo \times \hmumgi,
\eeq
and
\beq \label{evspazi}
Y=\ldueoth, \qquad  \spc=\ldueotv. 
\eeq
Define the operators $M_{\enp},M:Y\to Y$ and $B_{\enp},B:\spc \to Y$ as
follows: 
\beq \label{defopstato}
\left\{ \begin{array}{rcl} M_{\enp}(y) & := & y -\etaen y_0\ \vspace{2mm} \\
M(y) & := & y -\etao y_0 \end{array} \right. \qquad \forall y\in Y, 
\eeq 
\beq \label{defopcosto}
\left\{ \begin{array}{rcl} B_{\enp}(u) & := &\int_0^t \aen \etsaen \Gen(s)
u(s) \ ds, \vspace{2mm} \\
B(u) & := & \int_0^t \ao \etsao \go(s) u(s) \ ds \end{array} \right. \qquad
\forall u\in \spc.
\eeq
As we already noted, the space of controls for problems 
(\ref{minapprox}) is smaller than that of problem (\ref{minmisto}). 
See the remark in the Introduction and compare with Remark
\ref{restrspazi}. \\
Due to Theorems \ref{formrappr} and \ref{fdrapprox} above, the state
equations of the approximating problems and of the mixed problem can be
written as
\beq \label{stateeq}
M_{\enp} (y)= B_{\enp}(u) \qquad \textrm{and} \qquad M(y)= B(u),
\eeq
respectively. We also observe that the corresponding cost functionals
$J_{\enp}$ and $J$, defined by (\ref{costoee}) and (\ref{costo}), are
in fact the same (they differed only in the choice of the space of
controls, which are equal to $V$ now). In order to stress their
dependence on $y$, we relabel both $J_{\enp}(u)$ and $J(u)$ as 
$J(y,u)$, i.e. we set
\beq\label{costi}
J(y,u):= \int_0^T \apt \| u(s,\cdot) \|_V^2 + \| y(s,\cdot) \|_H^2
\cht ds.
\eeq
Now our approximating problem (\ref{minapprox}) has the following
equivalent formulation: 
$$\min \apg J(y,u)+ \chi_{\{M_{\enp}(y)=B_{\enp}(u)\}}:\ (y,u) \in
Y \times \spc \chg , \leqno({\cal P}_n)$$
where for each set $A$ the function $\chi_A$ is given by
$$\chi_A(x) = \left\{ \begin{array}{ll} 0 & \qquad \textrm{if } x\in A \\
+\infty & \qquad \textrm{if } x\notin A. \end{array} \right.$$
Similarly, we rewrite the mixed problem (\ref{minmisto}) (with $U$
replaced by $V$) as
$$\min \apg J(y,u)+ \chi_{\{M(y)=B(u)\}}:\ (y,u) \in Y \times \spc
\chg . \leqno({\cal P})$$ 
Thus, setting
\beqs\label{funzapprox}
F_n(y,u):=J(y,u)+ \chi_{\{M_{\enp}(y)=B_{\enp}(u)\}}(y,u),
\eeqs
the sequence of optimal control problems we consider is 
$$\min \left\{F_n(y,u): (y,u) \in Y \times \spc \right\},
\leqno({\cal P}_n)$$ 
with $Y$ and $\spc$ given by (\ref{evspazi}). \\
Of course, a point $(y_n,u_n)\in Y\times \spc$ is an optimal pair for
problem $({\cal P}_n)$ if
$$F_n(y_n,u_n)=\min_{Y \times \spc} F_n(y,u).$$
We recall now the general definition of multiple $\Gamma$-limits.
\begin{DEF} \label{multiglim}
Let $X$ and $W$ be topological spaces and let $\{F_n\}_{n\in \N}$ be a
sequence of functions from $X \times W$ to $\overline{\R}$; we denote by
$Z(+)$ and $Z(-)$ the $\sup$ and $\inf$ operators respectively. For
every $x \in X$ and $w \in W$ we set
\beqsn
\Gamma(\N^\alpha,X^\beta,W^\gamma)\lim_{n \to \infty} F_n(x,w)=
\Zgamma_{\{w_n\} \in S(w)} \Zbeta_{\{x_n\} \in S(x)} \Zmenalpha_{k \in
\N} \Zalpha_{n \geq k} F_n(x_n,w_n)
\eeqsn
where $\alpha,\beta,\gamma \in \{+,-\}$, and  $S(x)$, $S(w)$
denote the set of all sequences $ x_n \rightarrow x$ in $X$
and $w_n \rightarrow w$ in $W$ respectively. Note that when the
$\Gamma$-limit does not depend on the signs $+$ or $-$ in one (or more)
of its variables, the corresponding sign is customarily omitted. 
\end{DEF}
Our aim is now to apply Proposition 2.1 in \cite{BDM82} 
(see also Theorem \ref{minglim} in the Introduction). 
Therefore we take as $F_n$
the functionals defined in (\ref{funzapprox}). Our goal is to prove that
the functional $F$ corresponding to problem (${\cal P}$), i.e.
\beqs\label{funzlim}
F(y,u) = J(y,u)+ \chi_{\{M(y)=B(u)\}}\ ,
\eeqs
coincides precisely with the multiple $\Gamma$-limit (\ref{absglim}):
\beqs\label{concglim}
F(y,u) = \Gamma(\N,Y^-,{\spc}^-)\lim_{n \to \infty} F_n(y,u).
\eeqs
This will mean that in fact problem (${\cal P}$) is the {\it $\Gamma$-limit
problem} of the sequence of problems $\{({\cal P}_n)\}$, in the sense 
given by Definition \ref{defglim} in the Introduction. \\
In order to prove (\ref{concglim}), we are going to apply an abstract 
result in \cite{BF95}. We first recall the definition of
$G$-convergence of operators. 
\begin{DEF} \label{defgconv}
Let $Z$ and $W$ be topological spaces and for all $n\in \N$ let $D_n$ 
be operators from $Z$ to $W$. We say that the sequence $\{D_n \}$ {\em
$G$-converges} to the operator $D:Z\to W$ if 
$$\Gamma(\N,W,Z^-)\lim_{n \to \infty} \chi_{\{D_n(z)=w\}}(w,z) =
\chi_{\{D(z)=w\}}(w,z) \qquad \forall w\in W, \ \forall z\in Z,$$
that is, if the following conditions are satisfied:
\begin{description}
\item[(i)] if $z_n \to z$ in $Z$, $w_n \to w$ in $W$ and $D_n(z_n)=w_n$ for
infinitely many $n \in \N$, then $D(z)=w$;  
\item[(ii)] if $z \in Z$ and $w \in W$ are such that $D(z)=w$ and
$w_n \to w$ in $W$, then there exists $\{z_n\}\subseteq Z$ such that
$z_n \to z$ in $Z$ and $D_n(z_n)=w_n$ for all sufficiently large $n
\in \N$. 
\end{description}
\end{DEF}
The following general result holds true (it is a special case of a
result proved in \cite{BF95}, Proposition 3.3 and Section 3). 
\begin{THEO} \label{condsuffglim} Let $Z,W$ be topological spaces, let
$G_n:W \times Z \to \R$ be a functional and let $D, D_n:Z\to Z$, $K,
K_n:W\to Z$ be operators. Consider a sequence of control
problems of the following form:
$$\min_{Z\times W} \{ G_n(z,w) + \chi_{\{D_n(z)=K_n(w)\}} \}.$$
Assume that:
\begin{description}
\item[(i)] \ the sequence $\{D_n\}$ $G$-converges to $D$; 
\item[(ii)] \ if $w_n \to w$ in $W$, then $K_n(w_n) \to K(w)$ in $W$;
\item[(iii)] \ there exist a function $\Psi:W \to \R$, bounded on
bounded sets of $W$, and a function $\omega : Z \times Z \to \R$,
with $\lim_{v \to z} \omega(z,v)=0$ for all $z \in Z$, such that
$$G_n(z,w) \leq G_n(v,w)+\Psi(w)\omega(z,v). \qquad \forall
w\in W, \ \forall z,v\in Z, \ \forall n\in \N.$$
\end{description}
Then
$$\Gamma(\N,Z^-,W^-)\lim_{n \to \infty} [G_n +
\chi_{\{D_n(z)=K_n(w)\}}](z.w) = [G + \chi_{\{ D(z)=K(w)\}}](z,w),$$
where
\beq \label{glimcosto}
G(z,w) = \Gamma(\N, W^-) \lim_{n\to \infty} G_n (z,w).
\eeq
\end{THEO}
We will apply this theorem with $Z=Y$, $W=\spc$ (defined in
(\ref{evspazi})), $G_n=G=J$ (given by (\ref{costi})), and 
$D_n=M_{\enp}$, $D=M$, $K_n=B_{\enp}$, $K=B$ (see (\ref{defopstato}) and
(\ref{defopcosto})). Thus what we have to do is to verify that
assumptions {\bf (i)-(iii)} of Theorem \ref{condsuffglim} are in fact
satisfied. \\ 
We start with proving hypothesis {\bf (i)}.
\begin{LEMMA} \label{lemgch}
Let {\em (H0), (H1), (H2), (H3)} be fulfilled. If $M_{\enp}$ and $M$
are defined by (\ref{defopstato}), then the sequence $\{M_{\enp}\}$
$G$-converges to $M$ as $n \to \infty$.
\end{LEMMA}
{\bf Proof.} Suppose we have shown that
\beq \label{convsg}
\etaen y_0 \to \etao y_0 \quad \textrm{in } Y \qquad \forall y_0\in H;
\eeq
then we easily deduce the following properties. Firstly, if $\{y_n\},\{v_n\}
\subset Y$ are such that $y_n \to y$ in $Y$ and $v_n = M_{\enp}(y_n) =
\yeen - \etaen y_0$ for infinitely many $n\in \N$, then letting $n \to
\infty$ we get $v=y-\etao y_0=M(y)$.\\ 
Secondly, if $v=M(y)=y-\etao y_0$, and $\{v_n\}\subset Y$ is a
sequence such that $v_n \to v$ in $Y$, then setting $y_n := v_n +
\etaen y_0$ we have $M_{\enp}(y_n)=v_n$ for infinitely many $n$ and
$y_n \to y$ in $Y$. \\
Recalling Definition \ref{defgconv}, these facts imply that $M_{\enp}$
$G$-converges to $M$; hence the proof of Lemma \ref{lemgch} is
achieved provided we show (\ref{convsg}).\\
Let us prove now (\ref{convsg}).\\
To begin with, fix $f \in \ldue$ and $\lambda \in \rho(A_{\enp}) \cap
\rho(A)$. If we set 
\beq \label{risolv}
\ven:=R(\lambda,A_{\enp}), \qquad v:= R(\lambda,A),
\eeq
then by (\ref{ellittapprox}) and (\ref{ellittmisto}) the function
$v-\ven$ solves the equation 
\beq \label{convris0}
\lambda(v-\ven)-A(\cdot,D)(v-\ven) = 0 \quad \textrm{in }
\Omega,
\eeq
whereas at the boundary $v$ and $\ven$ verify
\beq \label{convris0b}
\left\{ \begin{array}{rcll}
\teen \frac{ \de \ven}{\de \nu_A} + (1-\teen) \ven & = & 0 &\quad
\textrm{in } \de \Omega \vspace{2mm}\\ 
v & = & 0 & \quad \textrm{in } \Gamma_0 \vspace{2mm}\\
\frac{\de v}{\de \nu_A} & = & 0 & \quad \textrm{in } \Gamma_1.
\end{array} \right. 
\eeq
Hence multiplying by $v-\ven$ and integrating by parts we easily get
\beq \label{convris1}
\begin{array}{l}
|\lambda| \|v-\ven \|_{\ldue}^2 + \nu \|D(v-\ven)\|_{\ldue}^2 \le
\vspace{2mm}\\ 
\qquad \le \intbo (v-\ven) \frac{\de}{\de \nu_A}(v-\ven) d\sigma =
\vspace{2mm} \\ 
\qquad = - \int_{\gao} \ven \frac{\de}{\de \nu_A}(v-\ven) d\sigma 
- \int_{\gai} (v-\ven) \frac{\de \ven}{\de \nu_A} d\sigma =
\vspace{2mm}\\  
\qquad = - \intbo \ven \frac{\de}{\de \nu_A}(v-\ven) d\sigma 
- \intbo (v-\ven) \frac{\de \ven}{\de \nu_A} d\sigma - \vspace{2mm}\\ 
\qquad - \intbo \ven \frac{\de \ven}{\de \nu_A} d\sigma = I + II + III.
\end{array}
\eeq 
Let us estimate separately each term. We have
$$I \le c\|\ven\|_{\hunmeb}
\left\|\frac{\de}{\de \nu_A}(v-\ven)\right\|_{\hmunmeb} \le C
\|\ven\|_{\huno} \|v-\ven\|_{\huno}\ ,$$
and similarly
$$II \le C\left\|\frac{\de \ven}{\de \nu_A}\right\|_{\hmunmeb}
\|v-\ven\|_{\hunmeb} \le C \|\ven\|_{\huno} \|v-\ven\|_{\huno}\ ;$$
hence
$$I+II \le C \|\ven\|_{\huno} \|v-\ven\|_{\huno}
\le \frac{\nu}{4} \|v-\ven\|_{\huno}^2 + \frac{C}{|\lambda|}
\|\ven\|_{\huno}^2$$
and by Remark \ref{stimagrad} we obtain 
\beq \label{convris1a} I+II \le \frac{\nu}{4} \|v-\ven\|_{\huno}^2 +
\frac{C}{|\lambda|} \|f\|_{\ldue}^2 \ .
\eeq 
Concerning the third term, we get as before
\beq \label{convris1b} III \le C\|\ven\|_{\hunmeb} \left\|\frac{\de
\ven}{\de \nu_A}\right\|_{\hmunmeb} \le C \|\ven\|_{\huno}^2 \le 
\frac{C}{|\lambda|} \|f\|_{\ldue}^2\ .
\eeq 
Thus, by (\ref{convris1}), (\ref{convris1a}) and (\ref{convris1b}) it
follows that 
\beq \label{convris2}
|\lambda| \| v-\ven \|_{\ldue}^2 + \nu \|D(v-\ven)\|_{\ldue}^2
\leq \frac{C}{|\lambda|} \|f\|^2_{\ldue}.
\eeq
By compactness there exist a subsequence, that we still call $v-\ven$,
and a function $w \in \huno$ such that
\beq \label{convris2a}
v-\ven \rightharpoonup w \quad \textrm{ in } \huno \quad \textrm{as } n\to
\infty 
\eeq 
and
\beq \label{convris2b}
v-\ven \to w \quad \textrm{ in } \ldue \quad \textrm{as } n\to \infty. 
\eeq 
Since the operator $A(\p,D)$ is closed on $\ldue$ and $v-\ven$ solves
(\ref{convris0}), we deduce that
\beq \label{convris3}
\lambda w -A(\p,D)w=0 \qquad  \textrm{in } \Omega.
\eeq 
Now, using Remark \ref{stimagrad}, we can rewrite (\ref{convris2}) as follows 
\beq \label{convris4}  
\begin{array}{l}
|\lambda| \|v-\ven\|_{\ldue}^2 + \nu \|D(v-\ven)\|_{\ldue}^2 +
\vspace{2mm} \\
\qquad + \int_{\Gamma_0} \frac{\teee}{1-\teee} |\frac{\de v_{\ee}}{\de
\nu_A}|^2 d\sigma + \int_{\Gamma_1} \frac{1-\teee}{\teee} |v_{\ee}|^2\
d\sigma \le \frac{C}{|\lambda|} \|f\|_{\ldue}^2. 
\end{array}
\eeq
Using again compactness, passing possibly to another subsequence, we get the
existence of two functions $\gamma \in L^2(\Gamma_0)$ and
$\mu \in  L^2(\Gamma_1)$ such that
\beq \label{convris5}
\begin{array}{cc} z_{\enp}:=\sqrt{\frac{1-\teen}{\teen}}\ \ven
\rightharpoonup \gamma \textrm{ in } L^2(\Gamma_0),\ & \ q_{\enp} :=\sqrt{
\frac{\teen}{(1-\teen)}}\ \frac{\de\ven}{\de \nu_A} \rightharpoonup \mu 
\textrm{ in } L^2(\Gamma_1). \end{array} 
\eeq
Now, since $\Omega$ is bounded, the $(N-1)$-dimensional measure of the
set $\Gamma_{\varepsilon}$ introduced in (\ref{gaee}) is bounded by $C
\varepsilon$. Consequently, by Remark \ref{stimagrad} we get 
\beq
\|\ven\|_{L^1(\Gamma_{\varepsilon_n})} \le
\|1\|_{L^2(\Gamma_{\varepsilon_n})} \|\ven\|_{L^2(\partial \Omega)} \le
\frac{C}{|\lambda|^{1/2}} \ \|f\|_{\ldue} \ \sqrt{\varepsilon_n}
\eeq 
Thus, recalling (\ref{tetaeps}), we deduce
\beq \begin{array}{rcl}
\|\ven\|_{L^1(\gao)} & = & \|\ven\|_{L^1(\Gamma_{\varepsilon_n})} +
\|\ven\|_{L^1(\gao \setminus \Gamma_{\varepsilon_n})} \le \vspace{2mm}
\\  
& \le & C \sqrt{\varepsilon_n} + \|\frac{\teen}{1-\teen}\|_{L^2(\gao\setminus
\Gamma_{\varepsilon_n})} \|z_{\enp}\|_{L^2(\gao)} \le \vspace{2mm} \\
& \le & C \sqrt{\varepsilon_n} + C
\sqrt{\frac{\varepsilon_n}{1-\varepsilon_n}} \ , \end{array}
\eeq 
so that
\beq \label{convL1v}    
\ven \to 0 \qquad \textrm{in } L^1(\gao).
\eeq
Similarly, as
\beq 
\left\|\frac{\de\ven}{\de \nu_A}\right\|_{L^1(\gai)} \le
\sqrt{\frac{\varepsilon_n}{1-\varepsilon_n}} \
\|q_{\enp}\|_{L^2(\gai)} \le C
\sqrt{\frac{\varepsilon_n}{1-\varepsilon_n}} \ , 
\eeq
we get
\beq \label{convL1dv}
\frac{\de\ven}{\de \nu_A} \to 0 \qquad \textrm{in } L^1(\gai).
\eeq
Comparing (\ref{convris2a}), (\ref{convL1v}) and (\ref{convris0b}), we
see that
\beq \label{convris8}
w=0 \qquad \textrm{in } \gao.
\eeq
Similarly, by the closedness of the operator $\frac{\de}{\de
\nu_A}$ from $\huno$ into $\hmunmeb$, (\ref{convris2a}) implies that
$$\frac{\de (v-\ven)}{\de \nu_A} \rightharpoonup \frac{\de w}{\de
\nu_A} \qquad \textrm{in } \hmunmeb,$$ 
so that by (\ref{convL1dv}) and (\ref{convris0b}) we obtain
\beq \label{convris9}
\frac{\de w}{\de \nu_A}=0 \qquad \textrm{in } \gai.
\eeq
By (\ref{convris3}), (\ref{convris8}) and (\ref{convris9}) we deduce
that $w$ solves the mixed problem (\ref{ellittmisto}) with homogeneous 
data, so that Theorem \ref{eupbmisto} allows us to conclude that $w=0$. \\
As a result, by (\ref{convris2b}), ve get $\ven \to v$ in $\ldue$. 
Now an easy argument by contradiction proves that in fact the whole sequence
$\{\ven\}$ converges to $v$ in $\ldue$. Thus, recalling (\ref{risolv}),
we have proved that 
\beq \label{convris10}
\raen f \to \rao f \quad \textrm{in } \ldue \qquad \forall f\in \ldue.
\eeq
Let us show now that 
\beq \label{convptsg}
\etaen f \to \etao f \quad \textrm{in } \ldue \qquad \forall f\in
\ldue, \ \forall t>0.
\eeq
We recall that, by the usual representation of analytic semigroups as
Dunford integrals, we have
$$\etaen f- \etao f = \frac{1}{2 \pi i} \int_{\gamma} e^{\lambda t}
\apq R(\lambda,\aen)-R(\lambda,\ao)\chq f \ d\lambda \qquad \forall f\in
\ldue,$$
where $\gamma$ is a path from $+\infty e^{-i\vartheta}$ to $+\infty
e^{i\vartheta}$, with $0<\vartheta<\vartheta_0$\ , contained in
$S_{\vartheta_0}$ (see Proposition \ref{Agisga}) and leaving $0$ on its left
side. Using this representation and a simple change of variable, as the
integrand is a holomorphic function of $\lambda$, we have for all $f\in
\ldue$: 
\beq \begin{array}{l}
\|\etaen f- \etao f\|^2_{\ldue} = \vspace{2mm}\\
\qquad = \left\| \frac{1}{2 \pi i} \int_{\gamma} e^{\sigma} \apq
R(\frac{\sigma}{t},\aen) - R(\frac{\sigma}{t},\ao)\chq f\ \frac{d\sigma}{t}
\right\|_{\ldue} \le \vspace{2mm} \\
\qquad \le C t^{-1} \int_{\gamma} e^{Re\ \sigma} \|[R(\frac{\sigma}{t},\aen)
- R(\frac{\sigma}{t},\ao)] f\|_{\ldue} \ |d\sigma|. \end{array} 
\eeq
Now fix $\delta \in ]1/2,1[$: by (\ref{stispe}) and (\ref{strAe}) we can
write 
$$\begin{array}{l}
\|\etaen f- \etao f\|_{\ldue} \le \vspace{2mm} \\
\qquad \le C(\delta) \|f\|_{\ldue}^{\delta} t^{\delta -1} \int_{\gamma}
e^{Re\ \sigma} \sigma^{-\delta} \|[R(\frac{\sigma}{t},\aen) -
R(\frac{\sigma}{t},\ao)]f\|^{1-\delta}_{\ldue} \ |d\sigma|.\end{array}
$$ 
We observe that
$$\frac{e^{Re\ \sigma}}{\sigma^{\delta}} \left\|\left[R \apt
\frac{\sigma}{t},\aen \cht - R \apt \frac{\sigma}{t},\ao
\cht\right]f\right\|^{1-\dl}_{\ldue} \le C\ \frac{e^{Re \sigma}}{\sigma}
T^{1-\dl} \|f\|_{\ldue}\ ,$$ 
and that the right-hand side has finite integral over $\gamma$; thus,
recalling (\ref{convris10}), by Lebesgue Theorem we get (\ref{convptsg}). \\ 
Moreover, since
\beq \begin{array}{l}
\|\etaen f- \etao f\|_{\ldue} \le C(\delta) t^{\delta-1} \
\|f\|^{\delta}_{\ldue} \apq \int_{\gamma} e^{Re\
\sigma}\frac{|d\sigma|}{|\sigma|}\chq^{1/2} T^{1-\delta} \end{array}
\eeq
and $\delta > 1/2$, applying again the Lebesgue Theorem we conclude that
(\ref{convsg}) holds true. This completes the proof of Lemma \ref{lemgch}.
\vspace{2mm} \quad \qed \\
We prove now that hypothesis {\bf (ii)} in Theorem \ref{condsuffglim} is
fulfilled too.
\begin{LEMMA} \label{lemcb}
Assume {\em (H0), (H1), (H2), (H3)}, let $B_{\enp}$ and $B$ be defined by
(\ref{defopcosto}) and let $Y$ and ${\cal U}$ be given by (\ref{evspazi}).
If $u_{\enp} \to u$ in ${\cal U}$, then $B_{\enp}(u) \to B(u)$ in $Y$.
\end{LEMMA}
{\bf Proof.} Set $\ven := B_{\enp}(u_{\enp})$ and write $u:=(u_0,u_1)$.
By Theorem \ref{fdrapprox}, $\ven$ is the solution of 
\beq \label{convb0}
\left\{ \begin{array}{rcll} \frac{\de \ven}{\de t}-A(\p,D)\ven & = & 0 &
\quad \textrm{in } [0,T] \times \cho \vspace{1mm} \\ 
\teen \frac{\de \ven}{\de \nu_A} + (1-\teen) \ven & = & u_{\enp} & \quad
\textrm{in } [0,T] \times \de \Omega \vspace{1mm}\\
\ven(0,\p) & = & 0 & \quad \textrm{in } \cho. \end{array} \right.
\eeq
Then, by Theorem \ref{eupbapprox} and Remark \ref{stimepbapprox}, $\ven$
satisfies
\beq \label{convb1}
\begin{array}{l} {\dis \sup_{s \in [0,T]}} \|\ven(s,\cdot)\|^2_{\ldue}
+ \int_0^T \|D \ven \|^2_{\ldue} \ dt+ \\
\qquad + \int_0^T \int_{\gao} \frac{\teen}{1-\teen} \apm
\frac{\de \ven}{\de \nu_A} \chm^2 d\sigma \ dt + \int_0^T \int_{\gai}
\frac{1-\teen}{\teen} |\ven|^2 \ d\sigma \ dt \le \vspace{2mm}\\
\qquad \le C \int_0^T \apt \|u_0\|_{\humoogo}^2
+ \|u_1\|_{\hmumgi}^2 \cht \ dt. \end{array} 
\eeq
By compactness, passing possibly to a subsequence still denoted by
$\{\ven\}$, there exists a function $v \in L^2(0,T;H^1(\Omega))$
such that
\beq  \label{convb2}
\ven \rightharpoonup v \qquad \textrm{in } L^2(0,T;H^1(\Omega)).
\eeq
Since this implies $\ven \rightharpoonup v$ in
$L^2(0,T;L^2(\partial\Omega))$, we also have
\beq \label{convb2a}
\teen \frac{\de \ven}{\de \nu_A} = u_{\enp}-(1-\teen)\ven \rightharpoonup
u_0-v \qquad \textrm{in }  L^2(0,T;L^2(\gao)).
\eeq 
Now, using again compactness, (\ref{convb1}) implies that there exists a
function $w \in L^2(0,T;L^2(\gao))$ such that
$$
\teen \frac{\de \ven}{\de \nu_A} = \sqrt{(1-\teen)\teen}\ 
\sqrt{\frac{\teen}{1-\teen}}\ \frac{\de \ven}{\de \nu_A} \rightharpoonup 0
\cdot  w =0 \qquad \textrm{in } L^2(0,T;L^2(\gao)).
$$
Then, by uniqueness, (\ref{convb2a}) implies
\beq \label{convb3}
 v=u_0 \qquad \textrm{on } \gao.
\eeq
Moreover, since the operator $\frac{\de}{\de \nu_A}$ is closed from
$H^1(\Omega)$  into $\hmunmeb$, by (\ref{convb2}) we have
\beq \label{convb33}
\frac{\de \ven}{\de \nu_A} \rightharpoonup \frac{\de v}{\de \nu_A} \qquad
\textrm{in } L^2(0,T;\hmunmeb).
\eeq
Then, recalling that $\teen=1-\en$ on $\gai$ we obtain
$$\teen \frac{\de \ven}{\de \nu_A} \rightharpoonup \frac{\de v}{\de \nu_A}
\qquad \textrm{in }  L^2(0,T;[\humoogi]^*),$$
so that, by (\ref{convb33}), we get
\beq \label{convb4}
\frac{\de v}{\de \nu_A}=u_1 \qquad \textrm{on } \Gamma_1.
\eeq
Finally, letting  $n \to \infty$ in problem (\ref{convb0}),
by (\ref{convb2}), (\ref{convb3}) and (\ref{convb4}) we conclude that
$v$ solves the equation 
\beq 
\left\{ \begin{array}{rcll} \frac{\de v}{\de t}-A(\p,D)v & = & 0 &
\quad \textrm{in } [0,T] \times \cho \vspace{1mm} \\
v & = & u_0 & \quad \textrm{in } [0,T] \times \Gamma_0 \vspace{1mm}\\ 
\frac{\de v}{\de \nu_A} & = & u_1 & \quad \textrm{in } [0,T] \times \Gamma_1
\vspace{1mm}\\
v(0,\p) & = & 0 & \quad \textrm{in } \cho. \end{array} \right.
\eeq
Therefore, by uniqueness and by Theorem \ref{formrappr}, we deduce $v=B(u)$;
hence we have proved that $B_{\enp}(u_{\enp}) \rightharpoonup B(u)$ in
$L^2(0,T;\huno)$. By Rellich Theorem we also get $B_{\enp}(u_{\enp}) \to
B(u)$ in $Y$. \vspace{2mm} \quad \qed \\
\
We are finally ready to prove our main result.
\begin{THEO}
Assume {\em (H0), (H1), (H2), (H3)}. In the spaces $Y$, ${\cal U}$ introduced
in (\ref{evspazi}), let $M_{\enp}$, $M$, $B_{\enp}$ and
$B$ be defined by (\ref{defopstato}) and (\ref{defopcosto}) 
respectively, and let $J$ be given by (\ref{costi}). Then the sequence
of optimal control problems 
$$\min \apg J(y,u)+ \chi_{\{M_{\enp}(y)=B_{\enp}(u)\}}(y,u) : (y,u) \in
Y \times {\cal U} \chg \leqno({\cal P}_n)$$
$\Gamma$-converges to the optimal control problem
$$\min \apg J(y,u)+ \chi_{\{M(y)=B(u)\}}(y,u) : (y,u) \in Y \times {\cal U}
\chg. \leqno({\cal P})$$
\end{THEO}
{\bf Proof.} We apply Theorem \ref{condsuffglim}. In Lemma \ref{lemgch}
and in Lemma \ref{lemcb} we proved that, under our present assumptions,
hypotheses {\bf (i)} and {\bf (ii)} in Theorem \ref{condsuffglim} hold true.
On the other hand it is easy to see that the cost functional verifies {\bf
(iii)}. Thus by Theorem \ref{condsuffglim} and Theorem \ref{minglim} we
conclude that the sequence of problems $({\cal P}_n)$ $\Gamma$-converges to
the optimal control problem $({\cal P})$. \vspace{2mm} \quad \qed \\
The above theorem holds for an arbitrary sequence $\varepsilon_n \to 0^+$;
this allows us to say that problems $({\cal P}_{\varepsilon})$
$\Gamma$-converge to problem $({\cal P})$ as $\varepsilon \to 0^+$. 
\begin{REM}
{\em For the sake of simplicity we considered a cost functional with a very
simple form. One can also deal with more general cost functionals such as
$$\int_0^T \apt \|y(s,\cdot) \|_H^2 + \|u(s,\cdot)\|_V^2 \cht ds +
\langle P_T y(T),y(T) \rangle_{\ldue}$$
with $P_T\in {\cal L}(\ldue)$, provided the operator $P_T$ is regular enough
in order to have existence of the optimal controls and to satisfy
 condition {\bf (iii)} of Theorem \ref{condsuffglim}
(see \cite{AFT91} and \cite{AT96}). The cost functional might also depend
explicitly on $\varepsilon$: in that case one has in addition to determine
the $\Gamma$-limit in (\ref{glimcosto}).} 
\end{REM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}