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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 105, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/105\hfil Sierpinski gasket]
{Optimization problems on the Sierpinski gasket}

\author[M. Galewski \hfil EJDE-2016/105\hfilneg]
{Marek Galewski}

\address{Marek Galewski \newline
Institute of Mathematics,
Lodz University of Technology,
W\'o{\l}czanska 215, 90--924 {\L}\'odz, Poland}
\email{marek.galewski@p.lodz.pl}

\thanks{Submitted January 18, 2016. Published April 22, 2016.}
\subjclass[2010]{35J20, 28A80, 49J20}
\keywords{Control problem; Sierpinski gasket;  direct variational method;
\hfill\break\indent continuous dependence on parameters}

\begin{abstract}
 We investigate the existence of an optimal process for such an optimal
 control problem in which the dynamics is given by the Dirichlet problem
 driven by weak Laplacian on the Sierpinski gasket. To accomplish this task
 using a direct variational approach with no global growth conditions on the
 nonlinear term, we consider the existence of solutions, their uniqueness and
 their dependence on a functional parameter for mentioned Dirichlet problem.
 This allows us to prove that the optimal control problem admits at least
 one solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Let $V$ stand for the Sierpi\'{n}ski gasket, $V_0$  its intrinsic
boundary. Let $\Delta $ denote the weak Laplacian on $V$ and let measure
$\mu $ denote the restriction to $V$ of normalized
${\log N}/{\log 2}$-dimensional Hausdorff measure, so that $\mu (V)=1$.
Let $M$ be a compact interval of ${\mathbb{R}}$ and denote
$L_{M}=\{ u\in L^2(V,\mu ):u(y) \in M\text{ for a.e. }y\in V\}$.
The aim of this article is to consider an optimal control problem of
minimizing the action functional
\[
J_0=\int_{V}f_0(y,x(y) ,u(y) ) d\mu
\]
where the admissible pairs satisfy
\begin{equation}
\begin{gathered}
\Delta x(y)+a(y)x(y)=f(y,x(y),u(y) )+g(y) \quad
\text{for a.e. }y\in V\setminus V_0, \\
x|_{V_0}=0,
\end{gathered}  \label{rownanie}
\end{equation}
and where we assume, apart from the growth conditions, that
$g\in L^1(V,\mu )$, $g\neq 0$ a.e. on $V$,
$f:V\times \mathbb{R}\times M\to {\mathbb{R}}$ is a Caratheodory function
and $a\in L^1(V,\mu )$ and $a\leq 0$ almost everywhere in $V$. Solutions to
 \eqref{rownanie} are understood in the weak sense which we will describe in a
more detail later. Define $F:V\times {\mathbb{R}}\times M\to {\mathbb{R}}$
by $F(y,\xi ,u)=\int_0^{\xi }f(y,x,u)dx$, for a.e. $y\in V$
and every $u\in M$. Concerning the nonlinear term, we will employ the
following conditions
\begin{itemize}
\item[(H1a)]  for any fixed $u\in M$  and a.e.
$y\in V$ the function $x\to F(y,u,v)$
is convex on $\mathbb{R}$;

\item[(H1b)] there exists constants $0<A<\frac{1}{2(2N+3)}$,
$B\in{\mathbb{R}}$ such that
\[
F(y,x,u) \geq -A| x| ^2+B
\]
for all $x\in {\mathbb{R}}$, $u\in M$ and a. e.
$y\in V$;

\item[(H2)] for each $r>0$  there exist functions $f_{r}$,
$g_{r}\in L^1(V,\mu ) $  such that for all
$(x,u) \in H_0^1(V)\times L_{M}$  satisfying
$\| x\| _{H_0^1(V)}\leq r$  and for a.e.
$y\in V$ it holds
\[
| F(y,x(y) ,u(y) ) |
\leq f_{r}(y) , | f(y,x(y) ,u(y) ) | \leq g_{r}(y);
\]

\item[(H3)]  $f_0:V\times {\mathbb{R}}\times M\to {\mathbb{R}}$
 is measurable with respect to the first variable and continuous
with respect to the two last variables and convex in $u$.
Moreover, for any $d>0$ there exists a function $\psi _{d}\in L^1(
V,\mu ) $ such that  $| f_0(y,x,u) | \leq \psi _{d}(y) $ a.e. on $V$
for all $x\in [-d,d] $  and $u\in M$.
\end{itemize}
In case of weakly convergent sequence of parameters, we would require some
structure condition on a nonlinear term, i.e.
$f(y,x,u) =f_1(y,x) +f_2(y) u$. We define
\[
F_1(y,x) =\int_0^{x}f_1(y,s) ds\quad \text{for a.e. } y\in V.
\]
Now we replace (H2) with the assumption
\begin{itemize}
\item[(H4)] $f_1:V\times {\mathbb{R}}\to {\mathbb{R}}$  is
a Caratheodory function, $f_2\in L^2(V,\mu ) $;
for each $r>0$ there exist functions $f_{r}$,
$g_{r}\in L^1(V,\mu ) $ such that for all $x\in H_0^1(V)$
 satisfying $\| x\| _{H_0^1(V)}\leq r$
 and for a.e. $y\in V$ it holds
\[
| F_1(y,x(y) ) | \leq f_{r}(y) , \quad | f_1(y,x(y) ) | \leq g_{r}(y) .
\]
\end{itemize}

The main idea to tackle this optimization problem is first to examine the
continuous dependence on a functional parameter of problem \eqref{rownanie}
in case of strongly and weakly convergent sequence of parameters. Having
obtained these results we can construct the set on which the optimization
problem can be minimized, i.e. the set containing all admissible pairs and
then minimize $J_0$ on this set. In order to get the solution to the
optimization problem considered, we require only weak convergence of the
sequence of parameters. However, we believe that the continuous dependence
on parameters results are of independent interest on fractal domains since
these to the best knowledge of the author have not been investigated yet.
Similar problems for a system described by second order PDE's of the
elliptic type considered on domains in ${\mathbb{R}}^n$ with Dirichlet
boundary data were investigated in \cite{ledzSW1} and \cite{ledzSW2} using
direct variational methods and for second order ODE in \cite
{LedzewiczWalczak}. We base on the approach used in the sources mentioned
with necessary modifications due to the setting of Sierpinski gasket.
However we modify the ideas from these works by putting emphasis on working
mainly with weak solutions and using a kind of a iterative technique. To the
authors' knowledge such problem was not studied in the setting of Sierpinski
gasket before. Necessary conditions of optimality for second-order systems
of ordinary differential equations with Dirichlet boundary conditions were
given by Goebel and Raitums in \cite{goebelraitums} and by Idczak in \cite
{idczakopt} (see also the references therein).

The Sierpinski gasket has the origin in a paper by Sierpinski \cite{Sie}.
This fractal domain can be described as a subset of the plane obtained from
an equilateral triangle by removing the open middle inscribed equilateral
triangle of $1/4$ of the area, removing the corresponding open triangle from
each of the three constituent triangles and continuing in this way.

The study of the Laplacian on fractals started in physical sciences in \cite
{A} and \cite{Ra,Ra2}. The Laplacian on the Sierpi\'{n}ski gasket was first
constructed in \cite{Ku} and\cite{gold}. Among the contributions to the
theory of nonlinear elliptic equations on fractals we mention \cite
{BRV,Fa99,FaHu} and \cite{K}, \cite{stribook}. Concerning some recent
results by variational methods and critical point theory pertaining to the
existence and the multiplicity of solutions by the recently developed
variational tools we must mention the following sources \cite{bonanno1},
\cite{bonanno2}, \cite{molica1}, \cite{BRaduV}.

\section{Remarks on the abstract setting}

Concerning the Sierpinski gasket we employ the following definition and
remarks, these follow remarks collected in \cite{bonanno1}. Let $N\geq 2$ be
a natural number and let $p_1,\dots ,p_{N}\in {\mathbb{R}}^{N-1}$ be so
that $|p_i-p_j|=1$ for $i\neq j$. Define, for every $i\in \{1,\dots ,N\}$,
the map $S_i:{\mathbb{R}}^{N-1}\to {\mathbb{R}}^{N-1}$ by
\[
S_i(x)=\frac{1}{2}\,x+\frac{1}{2}\,p_i\,.
\]
Let ${\mathcal{S}}:=\{S_1,\dots ,S_{N}\}$ and denote by
$G\colon {\mathcal{P}}({\mathbb{R}}^{N-1})\to {\mathcal{P}}({\mathbb{R}}^{N-1})$
the map assigning to a subset $A$ of ${\mathbb{R}}^{N-1}$ the set
\[
G(A)=\cup_{i=1}^{N}S_i(A).
\]
It is known that there is a unique non-empty compact subset $V$ of
${\mathbb{R}}^{N-1}$, called the attractor of the family ${\mathcal{S}}$,
such that $G(V)=V$; see,  Falconer \cite[Theorem 9.1]{Fa}.

The set $V$ is called the \emph{Sierpi\'nski gasket} in ${\mathbb{R}}^{N-1} $.
 It can be constructed inductively as follows:

Put $V_0:=\{p_1,\dots ,p_{N}\}$ which is called the
\emph{intrinsic boundary} of $V$ and define $V_m:=G(V_{m-1})$,
for $m\geq 1$, and put $V_{\ast }:=\cup_{m\geq 0}V_m$.
Since $p_i=S_i(p_i)$ for $i\in
\{1,\dots ,N\}$, we have $V_0\subseteq V_1$, hence $G(V_{\ast })=V_{\ast }$.
Taking into account that the maps $S_i$, $i\in \{1,\dots ,N\}$, are
homeomorphisms, we conclude that $\overline{V_{\ast }}$ is a fixed point of
$G$. On the other hand, denoting by $C$ the convex hull of the set
$\{p_1,\dots ,p_{N}\}$, we observe that $S_i(C)\subseteq C$ for
$i={1,\dots ,N}$. Thus $V_m\subseteq C$ for every $m\in {\mathbb{N}}$, so
$\overline{V_{\ast }}\subseteq C$. It follows that $\overline{V_{\ast }}$ is
non-empty and compact, hence $V=\overline{V_{\ast }}$.

The set $V$ is  endowed with the relative topology induced from the
Euclidean topology on ${\mathbb{R}}^{N-1}$.

Denote by $C(V)$ the space of real-valued continuous functions on $V$ and by
\[
C_0(V):=\{u\in C(V): d u|_{V_0}=0\}.
\]
The spaces $C(V)$ and $C_0(V)$ are endowed with the usual supremum norm
$ \| \cdot \| _{\infty }$. The space $L^2(V,\mu ) $
equipped with the product
\[
\langle v,h \rangle =\int_{V}v(y) h(y) d\mu
\]
is obviously a Hilbert space.

For a function $u:V\to {\mathbb{R}}$ and for $m\in {\mathbb{N}}$ let
\begin{equation}
W_m(u)=\big(\frac{N+2}{N}\big) ^{m}\sum_{|x-y|=2^{-m},\, x,y\in V_m}
(u(x)-u(y))^2.  \label{defWm}
\end{equation}
Since $W_m(u)\leq W_{m+1}(u)$ for every natural $m$, we can put
\begin{equation}
W(u)=\lim_{m\to \infty }W_m(u).  \label{defW}
\end{equation}
Define now
\[
H_0^1(V):=\{u\in C_0(V)\mid W(u)<\infty \}.
\]
$H_0^1(V)$ is a dense linear subset of $L^2(V,\mu )$ equipped with the
$\| \cdot \| _2$ norm. We now endow $H_0^1(V)$ with the norm
\[
\| u\| =\sqrt{W(u)}.
\]
There is an inner product defining this norm:
for $u,v\in H_0^1(V)$ and $m\in {\mathbb{N}}$ let
\[
{\mathcal{W}}_m(u,v)=(\frac{N+2}{N}) ^{m}
\sum_{|x-y|=2^{-m},\, x,y\in V_m} (u(x)-u(y))(v(x)-v(y)).
\]
Put
\[
{\mathcal{W}}(u,v)=\lim_{m\to \infty }{\mathcal{W}}_m(u,v).
\]

Note that ${\mathcal{W}}(u,v)\in {\mathbb{R}}$ and the space $H_0^1(V)$, equipped
with the inner product ${\mathcal{W}}$, which induces the norm $\| \cdot\| $,
becomes real Hilbert spaces. Moreover,
\begin{equation}
\| u\| _{\infty }\leq (2N+3)\| u\| ,\quad \text{for every }
u\in H_0^1(V),  \label{embeddingconstant}
\end{equation}
and the embedding
\begin{equation}
(H_0^1(V),\| \cdot \| )\hookrightarrow (C_0(V),\| \cdot \|
_{\infty })  \label{embedding}
\end{equation}
is compact, see \cite{FuSc} for further details.

Note that $(H_0^1(V),\| \cdot \| )$ is a Hilbert space which is
dense in $L^2(V,\mu )$, that ${\mathcal{W}}$ is a Dirichlet form on
$L^2(V,\mu )$. Let $Z$ be a linear subset of $H_0^1(V)$ which is dense
in $L^2(V,\mu )$. Then, in \cite{FaHu} it is defined a linear self-adjoint
operator $\Delta \colon Z\to L^2(V,\mu )$, the \emph{(weak)
Laplacian} on $V$, such that
\[
-{\mathcal{W}}(u,v)=\int_{V}\Delta u\cdot vd\mu ,\quad \text{for every }(u,v)\in
Z\times H_0^1(V).
\]

Let $H^{-1}(V)$ be the closure of $L^2(V,\mu )$ with respect to the
pre-norm
\[
\| u\| _{-1}=\sup_{\| h\| =1,\,h\in H_0^1(V)}
|\langle u,h\rangle|,
\]
$v\in L^2(V,\mu )$ and $h\in H_0^1(V)$. Then $H^{-1}(V)$ is a Hilbert
space. Then the relation
\[
-{\mathcal{W}}(u,v)=\langle \Delta u,v\rangle ,\quad \forall v\in H_0^1(V),
\]
uniquely defines a function $\Delta u\in H^{-1}(V)$ for every $u\in H_0^1(V)$.

\section{Existence and uniqueness}

A function $x\in H_0^1(V)$ is called a \emph{weak solution} of
\eqref{rownanie} if
\begin{equation}
\begin{aligned}
&{\mathcal{W}}(x,v)-\int_{V}a(y)x(y)v(y)d\mu \\
&+\int_{V}f(y,x(y),u(y))v(x)d\mu +\int_{V}g(y) x(y) d\mu =0,
\end{aligned} \label{weak-sol}
\end{equation}
for every $v\in H_0^1(V)$. Further on whenever we write that we obtain a
solution to \eqref{rownanie} we mean the weak one. The functional $
J:H_0^1(V)\to {\mathbb{R}}$ given by
\begin{equation}
J(x)=\frac{1}{2}\| x\| ^2-\frac{1}{2}\int_{V}a(y)x^2(y)d\mu
+\int_{V}F(y,x(y),u(y) )d\mu +\int_{V}g(y) x(
y) d\mu , \label{energyfunctional}
\end{equation}
for all $x\in H_0^1(V)$,
is the Euler action functional attached to problem \eqref{rownanie}.
Employing calculations as in the classical case on the domain in
 ${\mathbb{R}}^n$ and the reasoning from \cite[Proposition 2.19]{FaHu}
and further in \cite{bonanno1} we obtain the following result.

\begin{lemma} \label{criticalpoint}
Assume either {\rm (H2)} or {\rm (H4)}. Then, the
functional $J\colon H_0^1(V)\to {\mathbb{R}}$ defined by
relation \textrm{\eqref{energyfunctional}} is a
$C^1(H_0^1(V),{\mathbb{R}})$ functional. Moreover,
\[
J'(x)(w)={\mathcal{W}}(u,w)-\int_{V}a(y)x(y)w(x)d\mu
+\int_{V}f(y,x(y),u(y))d\mu +\int_{V}g(y) w(y) d\mu,
\]
for all $w\in H_0^1(V)$
for each point $x\in H_0^1(V)$. In particular, $x\in H_0^1(V)$ is a
weak solution of problem \eqref{rownanie} if and only if $x$ is a critical
point of $J$.
\end{lemma}

\begin{lemma}\label{ExistUniqLemma}
Assume that either {\rm (H1a), (H2)} or
{\rm (H1b), (H2)} hold. Let $u\in L^2(V,\mu ) $ be
fixed. Then $J$ is continuously G\^{a}teaux differentiable, weakly
lower semicontinuous  and
coercive and its critical points correspond to the weak solutions of
\eqref{rownanie}.
\end{lemma}

\begin{proof}
Let us take any weakly convergent sequence
$\{ x_k\}_{k=1}^{\infty }\subset H_0^1(V)$.
Then by \eqref{embedding} a sequence
$\{ x_k\} _{k=1}^{\infty }$ has a subsequence
$\{x_{k_n}\} _{n=1}^{\infty }$ which is strongly convergent in
$L^2(V,\mu ) $ and also convergent in $C(V) $.
Denote by $\overline{x}\in H_0^1(V)$ the weak limit of
$\{x_{k_n}\} _{n=1}^{\infty }$. Since
$\{ x_{k_n}\}_{n=1}^{\infty }$ is bounded\ there exist a constant
$r>0$ such that $\| v_{k_n}\| _{H_0^1(V)}\leq r$ for all $n\in N$.
Thus from (H3) there exists a function
$g_{r}\in L^1(V,\mu) $ such that
$| F(y,x_{k_n}(y) ,u(y) ) | \leq g_{r}(y) $ for a.e.
 $y\in V$. Than by the Lebesgue Dominated Convergence Theorem we obtain
\[
\int_{V}F(y,x_{k_n}(y) ,u(y) ) d\mu \to \int_{V}F(y,\overline{x}(y) ,u(y)) d\mu .
\]
Therefore, $J$ is weakly l.s.c. on $H_0^1(V)$ since all other terms of
$J $ are weakly l.s.c. on $H_0^1(V)$

Consider first case (H1a). Since $f$ is nondecreasing, it follows
that $F$ is convex in the second variable. We see that for all $v\in {
\mathbb{R}}$, $u\in M$ and a.e. $y\in V$ it follows
\begin{equation}
F(y,v,u) \geq f(y,0,u) y+F(y,0,u)
\label{nier_convexity}
\end{equation}
By (H2) there exist functions $f_0$, $g_0\in L^1(V,\mu) $ such that
\[
| F(y,0,u(y) ) | \leq f_0(y),\quad  | f(y,0,u(y) )| \leq g_0(y)
\]
We see by \eqref{embeddingconstant} that for every $y\in V$,
\begin{equation}
| x(y) | \leq \| x\|_{\infty }\leq (2N+3)\| x\| _{H_0^1(V)}.
\label{nier_C_H01}
\end{equation}
Then we see that
\begin{align*}
\int_{V}| f(y,0,u(y) ) | | x(y) | d\mu
&\leq \| x\| _{\infty }\int_{V}| f(y,0,u(y) ) | d\mu \\
&\leq ((2N+3)\int_{V}| g_0(y) | d\mu ) \| x\| _{H_0^1(V)}
\end{align*}
for any $x\in H_0^1(V)$. Thus by \eqref{nier_convexity},
\begin{equation}
\int_{V}F(y,x(y) ,u(y) ) d\mu 
\geq -((2N+3)\int_{V}| g_0(y) | d\mu) \| x\| _{H_0^1(V)}
-\int_{V}g_0(y) d\mu  \label{coercivity}
\end{equation}
for any $x\in H_0^1(V)$. Next we see that
\[
\int_{V}g(y) x(y) d\mu \leq \| x\|
_{\infty }\int_{V}| g(y) | d\mu \leq (
(2N+3)\int_{V}| g(y) | d\mu )
\| x\| _{H_0^1(V)}
\]
Since $a$ is non-positive, it now follows that $J$ is coercive in $x$ for
any fixed $u$. Indeed
\[
J(x) \geq \frac{1}{2}\| x\| ^2
-\Big( (2N+3)\int_{V}| g_0(y) | d\mu
+\int_{V}| g(y) | d\mu \Big) \|
x\| _{H_0^1(V)} -\int_{V}g_0(y) d\mu
\]
so $J(x) \to \infty $ as $\| x\| \to \infty $.

To prove coercivity in case (H1b) we see using the first inequality
\eqref{nier_C_H01} and the fact that $\mu (V) =1$
\[
\| x\| _{L^2(V,\mu ) }\leq \|x\| _{\infty }\leq (2N+3)\| x\| _{H_0^1(V)}
\]
for any $x\in H_0^1(V)$. Thus
\[
\int_{V}F(y,x(y) ,u(y) ) d\mu \geq -A(2N+3)\| x\| _{H_0^1(V)}^2-B
\]
Now by the assumptions on $A$, we see that $J$ is coercive.
\end{proof}

Replacing (H2) with (H4) we obtain the following result.

\begin{corollary}
Assume that either {\rm (H1a), (H4)} or {\rm (H1b), (H4)}
hold. Let $u\in L^2(V,\mu ) $ be fixed. Then $J$ is
continuously G\^{a}teaux differentiable, weakly l.s.c and coercive and its
critical points correspond to the weak solutions of \eqref{rownanie}.
\end{corollary}

The above assertion follows from Lemma \ref{criticalpoint}.
 we obtain the last assertion.

\begin{theorem} \label{thm3.4}
Let $u\in L^2(V,\mu ) $ be fixed. Then Problem \eqref{rownanie}
 has exactly one solution $\overline{x}_u\in H_0^1(V)$ in case
{\rm (H1a), (H2)} and at least one solution in case {\rm (H1b),
(H2)}. Note that all solutions are non-trivial.
\end{theorem}

\begin{proof}
By Lemma \ref{ExistUniqLemma} $J$ is G\^{a}teaux differentiable, weakly
l.s.c. and coercive on $H_0^1(V)$. Therefore there exists $\overline{x}
_u\in H_0^1(V)$ such that $J(\overline{x}_u) =\inf_{v\in
H_0^1(V)}J(v) $ and thus $\overline{x}_u$ satisfies \eqref{rownanie}.
Since in case (H1a), (H2), (H3) functional $J$ is strictly convex,
the critical point is unique. Assuming that $0$ is a
weak solution, we arrive at contradiction since \emph{then we obtain }
$g(y) =0$ for a.e. $y\in V$, which contradicts the
assumption on $g$.
\end{proof}

\begin{corollary}
Let $u\in L^2(V,\mu ) $ be fixed.  Problem \eqref{rownanie}
 has exactly one solution $\overline{x}_u\in H_0^1(V)$ in case
{\rm (H1a), (H4)} and at least one solution in case {\rm (H1b),
(H4)}. Note that all solutions are non-trivial.
\end{corollary}

\section{Continuous dependence on parameters}

Having shown the existence and in one case the uniqueness of a solution, we
investigate the dependence on a sequence of parameters. These results will
be indispensable in proving the existence of solutions to the optimal
control problem. We note that it is not necessary to use the uniqueness of
solutions in demonstrating the continuous dependence on parameters.
Therefore we would not distinguish between the unique and the non-unique
case as far as the methods in the proof are used. We mainly use the
iterative technique together with the definition of the weak solution
together with coercivity which appears to be uniform in $u$.

\subsection{Case of strongly convergent sequence of parameters}

\begin{theorem}\label{dep_direct_strong}
Assume that either {\rm (H1a), (H2)} or
{\rm (H1b), (H2)} hold. Assume that $\{ u_n\}_{n=1}^{\infty }$ satisfies
that $u_k\to u_0\in L^2(V,\mu ) $. Then, for any sequence
$\{ x_k\}_{k=1}^{\infty }$ of solutions to \eqref{rownanie} corresponding
to $u_k$, there exists a subsequence
$\{ x_{n_k}\} _{k=1}^{\infty}\subset H_0^1(V)$ and an element
$x_0\in H_0^1(V)$ such that $x_{n_k}\to x_0$ (strongly) in $H_0^1(V)$
and that $x_0$ is a weak solution to \eqref{rownanie} corresponding to $u_0$.
\end{theorem}

\begin{proof}
We define a sequence $\{ x_n\} _{n=1}^{\infty }$, where $x_n$
is a solution to \eqref{rownanie} with $u=u_n$. Thus the following holds
\begin{equation}
\begin{gathered}
-\Delta x_n(y)+a(y)x_n(y)=f(y,x_n(y),u_n(y) )\quad \text{for a.e. }
y\in V\setminus V_0, \\
x_n|_{V_0}=0.
\end{gathered} \label{iterative_scheme}
\end{equation}
We shall investigate the convergence of $\{ x_n\}_{n=1}^{\infty }$.
By definition of $J$ and by \eqref{coercivity} there
exists a constant $r>0$ such that $\| x_n\|_{H_0^1(V)}\leq r$ for $n\in N$.
Indeed, each $x_n$ is the argument of a minimum to $J$, so we see that
\begin{align*}
0=J(0) &\geq \frac{1}{2}\| x_n\| ^2-\frac{1}{2}
\int_{V}a(y)x_n^2(y)d\mu -\int_{V}F(y,x_n(y),u_n(y) )d\mu \\
&\quad -\int_{V}g(y) x_n(y) d\mu\\
& \geq \frac{1}{2}\| x_n\| ^2-\Big((2N+3)\int_{V}| g_0(
y) | d\mu +\int_{V}| g(y) |
d\mu \Big) \| x_n\| _{H_0^1(V)} \\
&\quad -\int_{V}f_0(y) d\mu .
\end{align*}
Hence $\{ x_n\} _{n=1}^{\infty }$ is weakly convergent in
$H_0^1(V)$\ to some $x_0$, possibly up to a subsequence which we assume
to be chosen. We shall observe that $x_0$ is a solution to \eqref{rownanie}
 corresponding to $u_0.$ Observe that by by \eqref{embedding}
 $\{x_n\} _{n=1}^{\infty }$ is also convergent in $C(V) $
and therefore in $L^2(V,\mu ) $. \ Note that since
$\{x_n\} _{n=1}^{\infty }$ is bounded by some $r$ say, we obtain by
\eqref{embeddingconstant} for any $v\in H_0^1(V)$ with $\|v\| \leq r$
\[
| f(y,x_m(y) ,u_m(y))v(y)| d\mu
\leq | f(y,x_m(y) ,u_m(y))| (2N+3)r
\leq r(2N+3)g_{r}(y)
\]
Since $\{ x_n\} _{n=1}^{\infty }$, $\{ u_n\}_{n=1}^{\infty }$ are convergent
in $L^2(V,\mu ) $ it follows by the Lebesgue Dominated Theorem that
\begin{equation}
\int_{V}f(y,x_m(y) ,u_m(y))v(y)d\mu \to
\int_{V}f(y,x_0(y) ,u_0(y))v(y)d\mu
\label{krasnosel-converg}
\end{equation}
We see by the definition of the weak solution that
\[
{\mathcal{W}}(x_m,v)-\int_{V}a(y)x_m(y)v(y)d\mu +\int_{V}f(y,x_m(
y) ,u_m(y))v(y)d\mu =0.
\]
Since ${\mathcal{W}}(x_m,v)\to {\mathcal{W}}(x_0,v)$ and
$\int_{V}a(y)x_m(y)v(y)d\mu \to \int_{V}a(y)x_0(y)v(y)d\mu $ as
$m\to \infty $, we see by \eqref{krasnosel-converg}
\[
{\mathcal{W}}(x_0,v)-\int_{V}a(y)x_0(y)v(y)d\mu +\int_{V}f(y,x_0(
y) ,u_0(y))v(y)d\mu =0
\]
for any $v\in H_0^1(V)$, so $x_0$ is a weak solution to \eqref{rownanie}
 corresponding to $u_0$.

Now we further examine the convergence of $\{ x_n\}_{n=1}^{\infty }$.
Namely, we shall show that it is in fact strong. Since
each $x_n$ for $n\in\mathbb{N}$ is a critical point we see that for
any $m\geq k$ we have
\begin{equation}
0=\langle J'(x_k) ,x_k\rangle -\langle J'(x_m) ,x_m\rangle . \label{Jk=jm}
\end{equation}
Writing \eqref{Jk=jm} explicitly we obtain
\begin{align*}
0&={\mathcal{W}}(x_k,x_k)-{\mathcal{W}}(x_m,x_m) \\
&\quad -\int_{V}a(y)x_k^2(y)d\mu +\int_{V}f(y,x_k(y)
,u_k(y))x_k(x)d\mu \\
&\quad +\int_{V}a(y)x_m^2(y)d\mu -\int_{V}f(y,x_m(y)
,u_m(y))x_m(x)d\mu
\end{align*}
As already mentioned since $\{ x_n\} _{n=1}^{\infty }$ is
weakly convergent in $H_0^1(V)$\ by \eqref{embedding} it is strongly
convergent in $L^2(V,\mu ) $. Thus for some fixed $\varepsilon
>0$ there exists $N_{\varepsilon }^1$ that for all
$m\geq k\geq N_{\varepsilon }^1$
\[
-\frac{\varepsilon }{2}
<\int_{V}a(y)x_m^2(y)d\mu -\int_{V}a(y)x_k^2(y)d\mu <\frac{\varepsilon }{2}.
\]
By \eqref{krasnosel-converg} for all $m\geq k\geq N_{\varepsilon }^2$,
where $N_{\varepsilon }^2$ is some number
\[
-\frac{\varepsilon }{2}
<\int_{V}f(y,x_m(y) ,u_m(y))x_m(y)d\mu -\int_{V}f(y,x_k(y),u_k(y))x_k(y)d\mu
<\frac{\varepsilon }{2}
\]
Therefore for all $m\geq k\geq N_{\varepsilon }:=\max \{ N_{\varepsilon
}^1,N_{\varepsilon }^2\} $
\[
-\frac{\varepsilon }{2}<{\mathcal{W}}(x_k,x_k)-{\mathcal{W}}
(x_m,x_m)<\frac{\varepsilon }{2}
\]
This means that $\{ {\mathcal{W}}(x_n,x_n)\} _{n=1}^{\infty }$
is a Cauchy sequence, and since $H_0^1(V)$ is complete we see that
\[
{\mathcal{W}}(x_n,x_n)\to {\mathcal{W}}(x_0,x_0)\quad \text{as }n\to \infty .
\]
Since also $\{ x_n\} _{n=1}^{\infty }$ is weakly convergent to $x_0$,
it converges strongly by the properties of the scalar product.
\end{proof}

\subsection{Case of weakly convergent sequence of parameters}

In Theorem \ref{dep_direct_strong} the convergence of a sequence of
parameters was a strong one. We are now interested in the case when this
convergence is weak.

\begin{theorem} \label{dep_direct_weak}
Assume that either {\rm (H1a), (H4)} or
{\rm (H1b), (H4)} hold. Let $\{ u_k\} _{k=1}^{\infty }$
satisfy that $u_k\rightharpoonup u_0$ (weakly)
$L^2(V,\mu ) $. Then, for any sequence $\{ x_k\}_{k=1}^{\infty }$ of solutions
 to \eqref{rownanie} corresponding to $u_k$,
there exists a subsequence $\{ x_{k_n}\} _{n=1}^{\infty}\subset H_0^1(V)$
and an element $x_0\in H_0^1(V)$ such that $x_{k_n}\rightharpoonup x_0$ (weakly)
in $H_0^1(V)$ and that $x_0$ is a classical solution to \eqref{rownanie}
corresponding to $u_0$.
\end{theorem}

\begin{proof}
Following the proof of Theorem \ref{dep_direct_strong}  we obtain the
weak convergence of a subsequence $\{ x_{k_n}\} _{n=1}^{\infty} $ of
solutions corresponding to a subsequence of parameters. The only
change is that now we apply the Lebesgue Dominated Theorem to function
$f_1 $ and we observe that for any $v\in H_0^1(V)$,
\[
\int_{V}f_2(y) u_{n_k}(y) v(y) d\mu
\to \int_{V}f_2(y) \overline{u}(y) v(y) d\mu
\]
by the weak convergence of $\{ u_k\} _{k=1}^{\infty }$. Then
we obtain that $x_0$ is a weak solution corresponding to $u_0$.
\end{proof}

\section{Solvability of the optimal control problem}

We construct a set $A\subset H_0^1(V)\times L^2(V,\mu ) $
consisting of pairs $(x_u,u)$ chosen as follows: we fix a
function $u\in L_{M}$ and next we take $x_u$ as all solutions to
\eqref{rownanie} corresponding to $u$ with assumptions
{\rm (H1a), (H4), (H3)} or {(H1b), (H4), (H3)} We recall that to some
$u$ there may exist more than one solution whether we employ convexity or
not.

\begin{remark} \label{prop_A} \rm
Since the functions from $L_{M}$ are pointwisely equibounded
we obtain $\lim_{k\to \infty }u_k=\overline{u}$\textbf{\ }weakly in
$L^2(V,\mu ) $, up to a subsequence by \eqref{nier_C_H01}, for
any sequence $\{ u_k\} _{k=1}^{\infty }\subset L_{M}$.
Moreover, any sequence $\{ x_k\} _{k=1}^{\infty }$ of solutions
to \eqref{rownanie} corresponding to such $\{ u_k\}_{k=1}^{\infty }$
is necessarily bounded in $H_0^1(V) $ as follows from the proof of
Theorem \ref{dep_direct_weak}. Thus by relation
\eqref{embeddingconstant} there exists a $d>0$ such that
$x_k(y) \in [-d,d] $ for all $k=1,2,\dots $ and for all $y\in V$.
Note that the last relation holds for all $y\in V$ since $x_k$ is
continuous for all $k=1,2,\dots $.
\end{remark}

\begin{theorem} \label{FirstExTh}
Assume that either {\rm (H1a), (H4), (H3)}
or {\rm (H1b), (H4), (H3)} hold. There exists a pair
$(\overline{x},\overline{u}) \in A$ such that
$J(\overline{x},\overline{u}) =\inf_{(x_u,u) \in A}J(x_u,u) $.
\end{theorem}

\begin{proof}
From Remark \ref{prop_A} it follows that any sequence in $A$ is bounded. Any
bounded sequence in $H_0^1(V) $ has a uniformly convergent
subsequence and by convexity of $f_0$\ with respect to $u$ we see that
$J_0$ is weakly l.s.c. on $H_0^2(V) \times L^2(V,\mu ) $. Assumption (H3)
and Remark \ref{prop_A} provide that the functional $J_0$ is bounded
from below on $A$. Thus we may choose
a minimizing sequence $\{ x_u^{k},u^{k}\} _{k=1}^{\infty }$ for
a functional $J$ such that $\{ u^{k}\} _{k=1}^{\infty }$ is
weakly convergent in $L^2(V,\mu ) $ to a certain $\overline{u}\in L_{M}$.
Theorem \ref{dep_direct_weak} asserts that $\{x_u^{k}\} _{k=1}^{\infty }$
converges, possibly up to a subsequence, strongly in $C(V) $, weakly
in $H_0^1(V) $ to a certain $\overline{x}$ solving \eqref{rownanie}
for $\overline{u}$ in the weak sense. Thus
\[
J_0(\overline{x},\overline{u}) =\lim \inf_{k\to \infty
}J(x_u^{k},u^{k}) \geq J(\overline{x},\overline{u})
\geq \inf_{(x,u) \in A}J(x,u) .
\]
Therefore $(\overline{x},\overline{u}) $ solves our optimization problem.
\end{proof}

\section{Examples}

We conclude the paper with some examples on nonlinear terms satisfying our
assumptions related to both the continuous dependence on parameters results
and the optimization problem.

\begin{example} \rm
Let $g\in L^1(V,\mu ) $, $g\neq 0$ a.e. on $V$,
$h\in L^2(V,\mu ) $ and let $f:{\mathbb{R}}\to {\mathbb{R}}$
be a continuous nondecreasing function. Consider
\begin{equation}
\begin{gathered}
\Delta x(y)+a(y)x(y)+f(x(y) ) +h(y)
u(y) =g(y) , \\
x|_{V_0}=0.
\end{gathered}
\label{exampleNON}
\end{equation}
Then we see that problem \eqref{exampleNON} satisfies the assumptions of
Theorem \ref{dep_direct_weak} in the convex case.
\end{example}

\begin{example} \rm
Let $g\in L^1(V,\mu ) $, $g\neq 0$ a.e. on $V$, $h\in L^2(V,\mu ) $.
Let $m$ be an odd number arbitrarily fixed.
Consider
\begin{equation}
\begin{gathered}
-\Delta x(y)+a(y)x(y)+x^{m}(y) e^{-u^2(y)
}+h(y) u(y) =g(y) ,  \\
x|_{V_0}=0.
\end{gathered} \label{example2}
\end{equation}
Again we see that  \eqref{example2} satisfies the assumptions of
Theorem \ref{dep_direct_strong} again in the convex case.
\end{example}

We conclude with an example of the integrand $f_0$.

\begin{example} \rm
Let $f_0(y,x,u) =h(y) e^{-x^2(y)}g(u) $, where $h\in L^1(V,\mu ) $
is positive a.e. on $V$ and $g:{\mathbb{R}}\to {\mathbb{R}}$ be a convex
continuous function. Function $\psi _{d}$ reads
$\psi _{d}(y) =h(y) e^{-d^2}\max_{u\in M}g(u) $.
\end{example}

\subsection*{Acknowledgements}
This research was supported by grant no. 2014/15/B/ST8/ 02854 
``Multiscale, fractal, chemo-hygro-thermo-mechanical models for
 analysis and prediction the durability of cement based composites''


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\end{document}