\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 135--147.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{135}

\begin{document}

\title[\hfilneg EJDE/Conf/14 \hfil Optimal controls of nonlinear evolution 
systems]{Optimal controls for a class of nonlinear evolution systems}

\author[A. Benbrik, M. Berrajaa, S. Lahrech \hfil EJDE/Conf/14 \hfilneg]
{Abdelhaq Benbrik, Mohammed Berrajaa, Samir Lahrech}  % in alphabetical order

\address{Abdelhaq Benbrik \newline
D\'epartement de Math\'ematiques et Informatique \\
Facult\'e des Sciences \\
Universit\'e Mohammed 1er, Oujda, Maroc}
\email{benbrik@sciences.univ-oujda.ac.ma}

\address{Mohammed Berrajaa \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences\\
Universit\'e Mohammed 1er, Oujda, Maroc}
\email{berrajaa@sciences.univ-oujda.ac.ma}

\address{Samir Lahrech \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences\\
Universit\'e Mohammed 1er, Oujda, Maroc}
\email{lahrech@sciences.univ-oujda.ac.ma}

\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{49J20, 49K20} 
\keywords{Optimal control; monotone operator; compact embedding;
\hfill\break\indent   $p$-Laplacian; bilinear system}

\begin{abstract}
 We consider the abstract nonlinear evolution equation
 $\dot{z}+ Az =uBz +f$. Viewing $u$ as control, we seek to minimize
 $J(u)=\int_{0}^{T}L(z(t),u(t))\,dt$.
 Under suitable hypotheses, it is shown that there exists an
 optimal control $\overline{u}$ and that it satisfies the
 appropriate optimality system. An example involving the $p$-Laplacian
 operator demonstrates the applicability of our results.
\end{abstract}

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

\section{Introduction}

In this paper, we investigate the optimal control problem governed by
the abstract non linear evolution equation
\begin{equation}
    \dot{z}+Az =uBz+f  \label{eq1}
\end{equation}
These systems  with linear operators $A$ and $B$
are called bilinear systems (see. \cite{b2,b3,m3}).
They appear in many mathematical
models from physical processes, for example, models
involving the $p$ Laplacian operator (see \cite{c1}).
 Our aim is to investigate the case where $A$ is not linear.

We organize this work as follows: After  formulating the
problem, we  address the question of existence and uniqueness
of solutions to these systems.
In section 3, we prove the existence theorem of optimal control and we
give the necessary conditions of optimality.
Finally, we present an example involving the $p$-Laplacian operator
which illustrates the applications of the abstract framework and the
results of the theory developed in the previous sections.

\section{Setting of the problem}

Throughout the paper, $H$ denotes a separable Hilbert space and $V$ a
subspace of $H$ having the structure of a reflexive Banach space which
is continuously and densely embedded in $H$.

Identifying $H$ with its dual $H'$, we have the Gelfand triplet
$V\hookrightarrow H \hookrightarrow V'$ where $V'$ is the dual of
$V$. We  suppose that these embeddings are compact. Let
$\langle .,.\rangle$ be the duality pairing between $V$ and $V'$
as well as the inner product on $H$. Let $\|\cdot \|$, $|\cdot |$ and
$\|\cdot \|_{V'}$ denote the norms on $V$, $H$ and $V'$ respectively.
Given a fixed real number $T>0$ and $2< p <+\infty$ we introduce the
 spaces:
$$
L^{p}(V)=L^{p}(0,T;V),\quad
L^{p}(H)=L^{p}(0,T;H),\quad
L^{p'}(V')=L^{p'}(0,T;V'),
$$
where $(\frac{1}{p}+ \frac{1}{p'}=1)$ and
$W=\big\{w\in L^{p}(V): \dot{w}\in L^{p'}(V') \big\}$. Here the derivative is
understood in the sense of vector valued distributions.

It is well known  that every $w\in W$ is after eventual
modification on a set of measure zero, continuous from $[0,T]$ in
$H$ and the embedding $W\hookrightarrow \mathcal{C}([0,T];H)$ is
continuous \cite{l1,l2}. Furthermore, if $V\hookrightarrow H$ compactly, then
also $W \hookrightarrow L^p(H)$ compactly.

We study the  control problem
\begin{equation} \label{P}
\inf_{u} J(u)
\end{equation}
subject to the state equation
\begin{gather*}
    \dot{z}+ Az(t)=  u(t) Bz(t)+f(t)  \\
    z(0)=  z_{0},
\end{gather*}
where the cost functional is
$$
J(u)=\int_{0}^{T}L(z(t),u(t))\,dt.
$$
Our aim is to provide conditions under which the optimal solutions
\eqref{P} exist. By an optimal solution we mean a control
$\overline{u}$ on which the infimum is attained.

For  problem \eqref{P} we need the following
hypotheses:
\begin{itemize}
\item[(H1)]  $A\colon V\to V'$ is such that:
 \begin{itemize}
    \item[(i)] $\Vert A\varphi \Vert_{V'}\leq \alpha_{1} \Vert
    \varphi\Vert^{p-1}$ with $\alpha_{1}>0$.
    \item[(ii)] $\langle A\varphi,\varphi \rangle \geq \alpha_{2} \Vert
    \varphi\Vert^{p}$ with $\alpha_{2}>0$.
    \item[(iii)] $\langle A\varphi_1 -A\varphi_2,\varphi_1 -\varphi_2
    \rangle \geq \alpha_{3} \Vert     \varphi\Vert^{2}$ with $\beta >0$.
    \item[(iv)] $\varphi \to A(\varphi)$ is continuously Frechet
    differentiable.
    \end{itemize}
\item[(H2)]  $B\colon H\to H$ is linear and continuous with
    $| B\varphi | \leq b|\varphi|$, $b>0$.

\item[(H3)]  $u\in L^{r}(0,T)$ with $r=p/(p-2)$.

\item[(H4)]  $f\in L^{p'}(V')$.

\item[(H5)]  $z_{0}\in H$.

\item[(H6)]  $L\colon H\times \mathbb{R}\to \mathbb{R}$ is a integrand convex such that:
  \begin{itemize}
   \item[(i)] $L$ is coercive: $\lim_{\Vert u\Vert_{ L^{r}(0,T)} \to \infty}
    \int_{0}^{T} L(z(t),u(t))\,dt=+\infty$
 \item[(ii)] $(x,u) \to L(x,u)$  is continuously Frechet
    differentiable.
 \item[(iii)] for every $x \in\mathcal{C}([0,T],H)$ and every
   $u \in L^{r}(0,T)$, $J(u)$ is finite.
\end{itemize}
\end{itemize}

\begin{remark} \label{rmk1} \rm
    $A(\varphi)\in L^{p'}(V')$ if $\varphi\in L^{p}(V)$ and then
    $$
    \Vert A\varphi\Vert_{L^{p'}(V')} \leq \alpha_{1} \Vert \varphi
    \Vert^{p-1}_{L^{p}(V)}.
    $$
  For $u\in L^{r}(0,T)$ and $\varphi\in L^{p}(V)$ we have
    $uB\varphi\in L^{p'}(V')$ and
    $$
    \Vert uB\varphi\Vert_{L^{p'}(V')} \leq \beta_{1} \Vert u
    \Vert_{L^{r}(0,T)} \Vert \varphi \Vert_{L^{p}(V)},
    $$
    where $\beta_{1} >0$.
Therefore, the choice of control space is compatible with the equation.
\end{remark}

\section{Results on the evolution problem}

We consider the evolution problem
\begin{equation}
\begin{gathered} \label{ezuf}
    \dot{z}(t)+ Az(t)=  u(t) Bz(t) +f  \\
    z(0)=  z_{0}
\end{gathered}
\end{equation}
We recall that by a solution to the above problem, we mean a
function $z\in W$ that satisfies \eqref{ezuf}.

\begin{theorem} \label{thm2.1}
    Under hypothesis (H1)(i), (H1)(ii), (H1)(iii), (H2), (H3), (H4) and
    (H5),  equation \eqref{ezuf} admits a unique
    solution $z$ such that $ z\in L^{\infty}(H)$ and $z\in W$.
\end{theorem}

\begin{proof}
    \textbf{Uniqueness:}
    if $z_{1}$ and $z_{2}$ are solutions of \eqref{ezuf}, then
    $z=z_{1}-z_{2}$ satisfies
\begin{equation}\label{eE0}
\begin{gathered}
    \dot{z}+Az_{1} -Az_{2} =uBz \\
    z(0)= 0
\end{gathered}
\end{equation}
and for $t\in [0,T]$,
$$
    \frac{1}{2}| z(t) |^{2} \leq b\int_{0}^{t}
   | u(\tau)| |  z(\tau)|^{2}\,d\tau\,.
$$
    Using the Gronwall lemma, we obtain $z_{1}=z_{2}$.

The existence follows from a standard application of the
Galerkin method  \cite{l1} and the a priori estimates given in
Lemma \ref{lem2.1}. We  remark that by Theorem \ref{thm2.1},
$z\in  \mathcal{C}([0,T];H)$.
\end{proof}

\begin{lemma} \label{lem2.1}
    Under the hypothesis of Theorem \ref{thm2.1}, if $z$ is a solution of
    \eqref{ezuf} then
    \begin{gather}
 \Vert z\Vert_{L^{p}(V)} \leq K_{1} \big[| z_{0}|^{2}
 + \Vert  u\Vert^{r}_{L^{r}(0,T)} + \Vert f\Vert^{p'}_{L^{p'}(V')}
 \big]^{1/p} \label{eE1}
 \\
 \Vert z\Vert_{L^{\infty}(H)} \leq K_{2} \big[| z_{0}|^{2}
 + \Vert         u\Vert^{r}_{L^{r}(0,T)} + \Vert f\Vert^{p'}_{L^{p'}(V')}
  \big]^{1/2} \label{eE2}
 \\
 \Vert \dot{z}\Vert_{L^{p'}(V')} \leq K_{3} \big[\Vert
        z\Vert^{p-1}_{L^{p}(V)}
 + \Vert  u\Vert^{p-1/p-2}_{L^{r}(0,T)} + \Vert f\Vert_{L^{p'}(V')}
        \big] \label{eE3}
 \end{gather}
\end{lemma}

\begin{proof}
 Let $z$ be a solution of \eqref{ezuf}, then
     $$
    \int_{0}^{T}\langle \dot{z}(t), z(t) \rangle \, dt +
    \int_{0}^{T}\langle A {z}(t), z(t) \rangle \, dt =
    \int_{0}^{T}\langle u(t)B{z}(t), z(t) \rangle \, dt +
    \int_{0}^{T}\langle f(t), z(t) \rangle \, dt.
    $$
    Using  (H1)(ii), (H2) and the continuity of the
    embedding $V\hookrightarrow H$, we have
    \begin{align*}
    &\frac{1}{2}| z(T) |^{2} - \frac{1}{2}| z_{0} |^{2} +
    \alpha_{2}\int_{0}^{T} \Vert z(\tau)\Vert^{p}_{V}\,d\tau \\
    &\leq  K'_{1}\int_{0}^{T}| u(\tau)| \Vert z(\tau)\Vert^{2}\,d\tau +
    \int_{0}^{T}\Vert f(\tau)\Vert_{V'} \Vert z(\tau)\Vert\,d\tau\,.
    \end{align*}
    By the Young inequality \cite{m2}, for $\frac{1}{r}+\frac{2}{p}=1$,
 we have
 \begin{gather*}
   K'_{1}\int_{0}^{T}| u(\tau)| \Vert
    z(\tau)\Vert^{2}\,d\tau \leq \frac{\alpha_{2}}{4} \Vert
    z\Vert^{p}_{L^{p}(V)} + K_{2}'\Vert u \Vert^{r}_{L^{r}(0,T)},
\\
\int_{0}^{T}\Vert f(\tau)\Vert_{V} \Vert z(\tau)\Vert\,d\tau
    \leq \frac{\alpha_{2}}{4} \Vert
    z\Vert^{p}_{L^{p}(V)} + K_{3}'\Vert f \Vert^{p'}_{L^{p'}(V')}.
\end{gather*}
    Hence
    $$
    \frac{\alpha_{2}}{2} \Vert z\Vert^{p}_{L^{p}(V)} \leq
    \frac{1}{2}| z_{0}|^{2} +  K_{2}'\Vert u \Vert^{r}_{L^{r}(0,T)} +
    K_{3}'\Vert f \Vert^{p'}_{L^{p'}(V')},
    $$
 from which, we deduce then \eqref{eE1}.

 Multiplying \eqref{ezuf} by $z$ and
    integrating on $[0,t]$ we obtain
    $$
    \frac{1}{2}| z(t)|^{2} -\frac{1}{2}| z_{0}|^{2}
    +\frac{\alpha_{2}}{2} \int_{0}^{t}\Vert z(\tau)\Vert^{p}\,d\tau \leq
     K_{2}''\Vert u \Vert^{r}_{L^{r}(0,T)} + K''\Vert f \Vert^{p'}_{L^{p'}(V')}
    $$
and then
    $$
    | z(t) |_{H} \leq K_{2} \big[ | z_{0}|^{2} + \Vert u
    \Vert^{r}_{L^{r}(0,T)} + \Vert f \Vert^{p'}_{L^{p'}(V' )}  \big]^{1/2}
    $$
which implies \ref{eE2}.

 Multiplying \ref{eE0} by  $\xi\in L^{p}(V)$, we have
\begin{align*}
& |\int_{0}^{T}\langle \dot{z}(t), \xi(t)\rangle\,dt |\\
&\leq  |\int_{0}^{T}\langle A{z}(t), \xi(t)\rangle\,dt | +
      |\int_{0}^{T}\langle u(t)B{z}(t), \xi(t)\rangle |
     + |\int_{0}^{T}\langle f(t), \xi(t)\rangle |.
 \end{align*}
     Hence
    $$
    \big|\int_{0}^{T}\langle \dot{z}(t), \xi(t)\rangle\,dt \big| \leq
    \big[\alpha_{1} \Vert z \Vert^{p-1}_{L^{p}(V)} + \beta_{1} \Vert
    u\Vert_{L^{r}(0,T)} \Vert z \Vert_{L^{p}(V)} + \Vert f\Vert_{L^{p'}(V')}
    \big] \Vert \xi\Vert_{L^{p}(V)},
    $$
which by Young inequality implies \eqref{eE3}.
\end{proof}


\section{Optimal controls}

The aim of this section is to prove the existence of optimal controls
for problem \eqref{P}.
The differentiability of the mapping $u\mapsto z$ permits the
characterization of the optimal control $\overline{u}$ by necessary
conditions corresponding to $J'(\overline{u})=0$.

\subsection*{Existence theorem for the control problem}

\begin{theorem} \label{thm3.1}
    If  (H1), (H2), (H3), (H4),
    (H5) and (H6) hold, then  \eqref{P} admits
    an optimal solution.
\end{theorem}

\begin{proof}
    Let $(u_{n})_{n}$ be a minimizing sequence for \eqref{P}, i.e.
    the pairs $(z_{n},u_{n})$ are admissible for \eqref{P} and
    $$
    \lim_{n} J(u_{n})= \overline{J}.
    $$
    From (H6) we have $\Vert u_{n}\Vert_{L^{r}(0,T)}\leq M$.

    And from Lemma \ref{lem2.1}, we know that $(z_{n})_{n}$ belongs to a bounded
    subset of ${L^{\infty}(H)}\cap W$.
    By passing to a subsequence if necessary, we may assume that
    \begin{gather*}
        u_{n}\rightharpoonup \overline{u} \quad w-L^{r}(0,T)  \\
        z_{n} \rightharpoonup \overline{z}\quad w*-L^{\infty}(H)  \\
        z_{n}\rightharpoonup \overline{z} \quad w-L^{p}(V)  \\
        Az_{n}\rightharpoonup \chi \quad w-L^{p'}(V')  \\
        u_{n}Bz_{n} \rightharpoonup \Psi \quad w-L^{p'}(V')  \\
        \dot{z}_{n}\rightharpoonup \Lambda \quad w-L^{p'}(V')
 \end{gather*}
\noindent \textbf{1.} Using the convergence
    $\sigma(\mathcal{D}(0,T;V);\mathcal{D}'(0,T;V'))$ we obtain
    $\Lambda=\dot{\overline{z}}$.\\
\textbf{2.} $V\hookrightarrow H$ compactly implies that
\begin{gather*}
    z_{n} \to \overline{z} \quad s-L^{p}(H) \\
    z_{n}(t) \to \overline{z}(t)\quad s-H \quad\text{ for all } t\in
    [0,T].
\end{gather*}
    For $\varphi  \in L^{p}(V)$, we have
\begin{align*}
&\int_{0}^{T} \langle u_{n}(t)Bz_{n}(t), \varphi(t)  \rangle\,dt \\
&=  \int_{0}^{T} \langle u_{n}(t) B(z_{n}(t)-\overline{z}(t);
        \varphi(t)\rangle\,dt + \int_{0}^{T} \langle u_{n}(t)
        B\overline{z}(t), \varphi(t) \rangle\,dt.
\end{align*}
Note that
$$
 \int_{0}^{T} \langle u_{n}(t) B(z_{n}(t)-\overline{z}(t);
        \varphi(t)\rangle\,dt
\leq K_{1} \Vert u_{n}\Vert_{L^{r}(0,T)} \Vert
    z_{n} -\overline{z} \Vert_{L^{p}(H)} \Vert \varphi\Vert_{L^{p}(H)}
$$
and
$$
 \int_{0}^{T} u_{n}(t)  \langle B\overline{z}(t), \varphi(t)
    \rangle\,dt \to \int_{0}^{T}  \overline{u}(t)  \langle
    B\overline{z}(t), \varphi(t) \rangle\,dt
$$
because $\langle B\overline{z},\varphi\rangle \in L^{r'}(0,T)$.
    We deduce that $\Psi=\overline{u}B\overline{z}$.

\noindent\textbf{3.}  For $y\in L^{p}(V)$, we set
    $$
    X_{m}=\int_{0}^{T} \langle Az_{m}(t)-Ay(t); z_{m}(t)-y(t)
    \rangle\,dt\,.
    $$
    We have
    $$
    X_{m}=\int_{0}^{T} \langle Az_{m}(t); z_{m}(t)
    \rangle\,dt - \int_{0}^{T} \langle Az_{m}(t);y(t) \rangle\,dt -
    \int_{0}^{T}\langle Ay(t);z_{m}(t)-y(t)
    \rangle\,dt
    $$
and
    \begin{align*}
&\int_{0}^{T}\langle Az_{m}(t),z_{m}(t)\rangle\,dt \\
&=\frac{1}{2} | z_{m,0}|^{2} - \frac{1}{2} | z_{m}(T)|^{2} +
    \int_{0}^{T} \langle u_{m}Bz_{m}(t), z_{m}(t) \rangle\,dt
    + \int_{0}^{T} \langle f(t), z_{m}(t) \rangle\,dt.
\end{align*}
    But
\begin{align*}
&\int_{0}^{T}\big(\langle u_{m}(t)Bz_{m}(t),z_{m}(t)\rangle -
        \langle \overline{u}(t)B\overline{z}(t),\overline{z}(t)\rangle
        \big)\,dt\\
&=  \int_{0}^{T}\langle u_{m}(t)Bz_{m}(t),z_{m}(t)-\overline{z}(t)
        \rangle\,dt + \int_{0}^{T}\langle u_{m}(t)Bz_{m}(t)-
        \overline{u}(t)B\overline{z}(t) ,\overline{z}(t)
        \rangle\,dt
\end{align*}
The first integral in the right-hand side approaches zero
because $z_{m}\to \overline{z}$ ($s-L^{p}(H)$).
The second integral approaches zero because
$u_{m}Bz_{m}\rightharpoonup     \overline{u}B\overline{z}$ ($w-L^{p'}(V')$).
 We deduce that
    \begin{align*}
&\limsup_{m}\int_{0}^{T}\langle Az_{m}(t), z_{m}(t) \rangle\,dt \\
&\leq \frac{1}{2} | z_{0}|^{2} - \frac{1}{2} |
    \overline{z}(T)|^{2} + \int_{0}^{T} \langle \overline{u}(t)
    B\overline{z}(t), \overline{z}(t) \rangle\,dt
    +\int_{0}^{T}\langle f(t), \overline{z}(t) \rangle\,dt\,.
\end{align*}
Since $\overline{z}$ satisfies
 \begin{gather*}
        \dot{\overline{z}}+ \chi=  \overline{u}B\overline{z}+f  \\
        \overline{z}(0)= z_{0}
    \end{gather*}
it follows that
    \begin{equation*}
    \frac{1}{2} | z_{0}|^{2} - \frac{1}{2} |
    \overline{z}(T)|^{2} + \int_{0}^{T} \langle \overline{u}(t)
    B\overline{z}(t), \overline{z}(t) \rangle\,dt
    +\int_{0}^{T}\langle f(t), \overline{z}(t) \rangle\,dt =
    \int_{0}^{T} \langle \chi(t),\overline{z}(t) \rangle\,dt.
    \end{equation*}
    Hence
    $$
    0\leq \limsup_{m} X_{m} \leq \int_{0}^{T}\langle
    \chi(t)-Ay(t),\overline{z}(t)-y(t)\rangle\,dt \text{ for all }y\in
    L^{p}(V)
    $$
    Using the continuity of the operator $A$ we
    obtain     $\chi=A\overline{z}$.
    We deduce that $(\overline{z},\overline{u})$ is admissible for
    \eqref{P}. From (H6) we have
    $$
    \int_{0}^{T}L(\overline{z}(t),\overline{u}(t))\,dt \leq
    \liminf_{m}
    \int_{0}^{T} L(z_{m}(t),u_{m}(t))\,dt=\overline{J}
    $$
    Hence $\overline{u}$ is an optimal control.
\end{proof}

\subsection*{Optimality conditions}

Before proceeding with investigation of the mapping $\Theta\colon u\mapsto
z$, where $z$ is defined by \eqref{ezuf}, we introduce a technical
lemma generalizing the Gronwall inequality.

\begin{lemma} \label{lem3.1}
    Let $T>0$ and $c\geq 0$. Assume that $\lambda$ and $m$ are
    integrable in $[0,T]$ with positive values.
    Let $\varphi\colon [0,T] \to\mathbb{R}^{+}$ be such that:
\begin{itemize}
\item[(a)] $\lambda\varphi$ and $\lambda\varphi^{2}$ are integrable on
    $[0,T]$.
\item[(b)] $\frac{1}{2}\varphi^{2}(t) \leq \frac{1}{2}c^{2}+
    \int_{0}^{t}\lambda(s)\varphi(s)\,ds +
    \int_{0}^{t}m(s)\varphi^{2}(s)\,ds$ for $t\geq 0$.
\end{itemize}
Then
    $$
    \varphi(t)\leq \Big[c+ \int_{0}^{t}\lambda(s)\,ds \Big]
    \exp\Big(\int_{0}^{t}m(s)\,ds \Big).
    $$
\end{lemma}

\begin{proof}
    Set
    $$
    \Psi(t) = \Big[c^{2}+ 2
    \int_{0}^{t}\lambda(s)\varphi(s)\,ds + 2
    \int_{0}^{t}m(s)\varphi^{2}(s)\,ds\Big]^{1/2}.
    $$
    We have that $\varphi(t)\leq \Psi(t)$ and $\dot{\Psi}\leq
    \lambda(t) + m(t)\Psi(t)$. Then
    $$
    \frac{d}{dt}\Big[\Psi(t)\exp\Big(-\int_{0}^{t}m(s)\,ds \Big)
    -\int_{0}^{t}\lambda(s)\exp\Big(-\int_{0}^{s}m(\tau)\,d\tau \Big)
    \Big] \leq 0.
    $$
    Hence
    $$
    \Psi(t)\leq \exp\Big(\int_{0}^{t}m(\tau)\,d\tau \Big)
    \Big[c+\int_{0}^{t}\lambda(\tau)\exp\Big(-\int_{0}^{\tau}m(s)\,ds \Big)
    d\tau\Big],
    $$
which completes the proof.
\end{proof}

\begin{lemma} \label{lem3.2}
    Suppose the hypothesis (H1), (H2), (H3), (H4) and
    (H5) hold,
    then the mapping $\Theta\colon L^{r}(0,T) \to L^{\infty}(H)\cap
    L^{2}(V)$, $u\mapsto z$ is locally Lipschitz.
\end{lemma}

\begin{proof}
    Let $\overline{u}$ and $h$ be in $L^{r}(0,T)$ with
$\Vert h\Vert_{L^{r}(0,T)}\leq 1$. Set
    $\overline{z}=\Theta(\overline{u})$, $z_{h}=\Theta(\overline{u}+h)$
    and $z=z_{h}-\overline{z}$. Then $z$ satisfies
    \begin{gather*}
    \dot{z}+Az_{h}-A\overline{z} =  \overline{u}Bz +hBz_{h}\\
    z(0)=  0
    \end{gather*}
    Multiplying by $z$ and integrating on $[0,t]$ we have
    $$
    \frac{1}{2} | z(t)|^{2} + \beta\int_{0}^{t} \Vert
    z(\tau)\Vert^{2}_{V}\,d\tau \leq b\int_{0}^{t} |
    \overline{u}(\tau)| | z(\tau)|^{2}\,d\tau +
    b\int_{0}^{t} |
    h(\tau)| | z_{h}(\tau)| | z(\tau)|\,d\tau\,.
    $$
    Invoking the  Lemma \ref{lem3.1}, we have
    $$
    | z(t)|_{H} \leq \exp \Big(b \int_{0}^{t}|
    \overline{u}(\tau)| \,d\tau\Big)\Big[  b\int_{0}^{t} |
    h(\tau)| | z_{h}(\tau)| \,d\tau\Big]
    $$
    but
    $$
    \int_{0}^{t} |
    h(\tau)| | z_{h}(\tau)|\,d\tau \leq K \Vert
    h\Vert_{L^{r}(0,T)} \Vert z_{h}\Vert_{L^{\infty}(H)}
    $$
    and
    $$
    \Vert z_{h}\Vert_{L^{\infty}(H)}\Vert \leq K_{1}\left[|
    z_{0}|^{2} + \Vert \overline{u}+h\Vert^{r}_{L^{r}(0,T)} + \Vert
    f\Vert^{p'}_{L^{p'(V')}} \right]^{1/2} \leq
    K'
    $$
    where $K'$ is a positive constant depending on $z_{0}$,
    $\overline{u}$ and $f$ (because $\Vert h\Vert \leq 1$).
    We obtain
\begin{gather*}
    \Vert z\Vert_{L^{\infty}(H)} \leq K'_{1}\Vert h\Vert_{L^{r}(0,T)},
\\
    \Vert z\Vert_{L^{2}(V)} \leq K'_{2}\Vert h\Vert_{L^{r}(0,T)}
\end{gather*}
\end{proof}

\begin{theorem} \label{thm3.2}
    Suppose that:
\begin{itemize}
\item[(i)] The hypothesis of Lemma \ref{lem3.2} are satisfied with $f=0$.
\item[(ii)] For $\varphi$
    and $\Psi$ in $\mathcal{C}([0,T);H)$ with $\Vert
    \Psi\Vert_{\mathcal{C}([0,T];H)}\leq 1$ we have
    $$
    \Vert A'(\varphi(t)+\Psi(t)) - A'(\varphi(t))\Vert_{\mathcal{L}(H)}
    \leq \gamma(t) | \Psi(t)|_{H}
    $$
    where $\gamma\in L^{1}(0,T)$.
\end{itemize}
    Then the mapping $\Theta\colon L^{r}(0,T)\to L^{\infty}(H) \cap
    L^{2}(V)$ is Fr\'echet differentiable and the derivative
    $\Theta'_{\overline{u}}.h$ is a solution of
\begin{equation} \label{eS}
    \begin{gathered}
        \dot{y}(t) + A'_{\overline{z}(t)}y(t)=  \overline{u}(t)By(t) +
        h(t)B\overline{z}(t)  \\
        y(0)=  0
    \end{gathered}
\end{equation}
    where $\overline{z}=\Theta(\overline{u})$.
\end{theorem}

\begin{proof}
    \textbf{1.} Since $A$ is strongly monotone. For $\lambda>0$,
    $\varphi$ and $\Psi$ in $V$, we have
    $$
    \big\langle \frac{1}{\lambda}(A(\varphi+\lambda\Psi)-A(\varphi)), \Psi
    \big\rangle
    \geq \beta\Vert \Psi\Vert^{2}_{V}.
    $$
    Hence
    $ \langle A'_{\varphi}\Psi, \Psi \rangle  \geq
\beta\Vert \Psi\Vert^{2}_{V} $

\noindent \textbf{2.} For $\overline{u}\in L^{r}(0,T)$, the mapping $h\mapsto y$
    defined by \eqref{eS} is linear.
Multiplying \eqref{eS} by $y$ and integrating on $[0,t]$ we obtain
    $$
    \frac{1}{2}| y(t)|^{2} + \beta\int_{0}^{t}\Vert
    y(\tau)\Vert^{2}_{V}\,d\tau \leq b\int_{0}^{t} |
    \overline{u}(\tau)| | y(\tau)|^{2}\,d\tau + |
    h(\tau)| | \overline{z}(\tau)|| y(\tau)|\,d\tau
    $$
    By  Lemma \ref{lem3.2},
    $$
    | y(t)| \leq b\int_{0}^{t} | h(\tau)| |
    \overline{z}(\tau)|\,d\tau \exp\Big[b\int_{0}^{t}|
    \overline{u}(\tau)| \Big],
    $$
    but
    $$
    \Vert \overline{z}\Vert_{L^{\infty}(H)}\leq K_{1} \big[|
    z_{0}|^{2} + \Vert \overline{u}\Vert^{r}_{L^{r}(0,T)} \big]^{1/2}.
    $$
    Then
\begin{gather*}
    \Vert y\Vert_{L^{\infty}(H)} \leq K'_{1} \Vert h\Vert_{L^{r}(0,T)},
 \\
    \Vert y\Vert_{L^{2}(V)} \leq K'_{2} \Vert h\Vert_{L^{r}(0,T)},
\end{gather*}
    where $K'_{i}$ are positive constants depending on $z_{0}$ and
    $\overline{u}$. Hence the mapping $h\mapsto y$ is continuous.

\noindent \textbf{3.} Set $\overline{z}=\Theta(\overline{u})$,
    $z_{h}=\Theta(\overline{u}+h)$, $z=z_{h}-\overline{z}$ and $w=z-y$
    where $y$ is a solution of \eqref{eS}. We have
\begin{equation} \label{eS'}
    \begin{gathered}
        \dot{w}(t) + A'_{\overline{z}(t)}w(t)=  \overline{u}(t) Bw(t) +
        h(t) Bz(t) + g(t)  \\
        w(0)=  0
    \end{gathered}
\end{equation}
    where
\begin{align*}
        g(t)  &= A'_{\overline{z}(t)}z(t) - \big(Az_{h}(t)
        -A\overline{z}(t) \big)\\
          &= \int_{0}^{1}[A'\overline{z}(t)
- A'(\overline{z}(t)+sz(t))]z(t)\,ds.
    \end{align*}
Then
    $$
    | g(t) |_{H} \leq \int_{0}^{1} \gamma(t) | sz(t)|
    | z(t)|\,ds =\frac{\gamma(t)}{2} | z(t)|^{2}
    $$
On the other hand, multiplying \eqref{eS'} by $w$ and integrating on
    $[0,t]$ we obtain
\begin{align*}
  &\frac{1}{2} | w(t) |^{2} + \beta\int_{0}^{t} \Vert w(\tau)
    \Vert^{2}_{V}\,d\tau \\
& \leq b\int_{0}^{t} |
    \overline{u}(\tau)| | w(\tau)|^{2}\,d\tau + b\int_{0}^{t} |
    h(\tau)| | z(\tau)| | w(\tau)|\,d\tau
    +\frac{1}{2}\int_{0}^{t}\gamma(\tau) | z(\tau) |^{2} |
    w(\tau) |\,d\tau
\end{align*}
    Then  Lemma \ref{lem3.2} gives
    $$
    | w(t)| \leq \exp \Big(b\int_{0}^{t} |\overline{u}(\tau)
    |\,d\tau  \Big)\Big[b\int_{0}^{t} | h(\tau) | |
    z(\tau)|\,d\tau + \frac{1}{2}\int_{0}^{t}\gamma(\tau) |
    z(\tau)|^{2}\,d\tau \Big],
    $$
    but
    $\Vert z\Vert_{L^{\infty}(H)} \leq K \Vert h\Vert_{L^{r}(0,T)}$
    then
\begin{gather*}
    \Vert w\Vert_{L^{\infty}(H)} \leq K_{1} \Vert h\Vert^{2}_{L^{r}(0,T)},
\\
    \Vert w\Vert_{L^{2}(V)} \leq K_{2} \Vert h\Vert^{2}_{L^{r}(0,T)}.
\end{gather*}
    It follows that $\Theta$ is fr\'echet differentiable from
    $L^{r}(0,T)$ on $L^{\infty}(H) \cap L^{2}(V)$ and
    $\Theta'_{\overline{u}}.h$ is a solution of \eqref{eS}.
\end{proof}

\begin{theorem} \label{thm3.3}
    Assume the  hypotheses of Theorem \ref{thm3.2} and (H6) hold. Then an optimal
    control $\overline{u}$, its corresponding state $\overline{z}$, and
    its adjoint state $p$ are necessarily tied by the
    optimality system:
    \begin{itemize}
\item[(1)] $ \dot{\overline{z}}+A\overline{z}=\overline{u}B\overline{z}$
$\overline{z}(0)=z_{0}$
\item[(2)] $ -\dot{p}+{A'}^{*}_{\overline{z}}p= \overline{u}B^{*}p+
        \partial_{1}L(\overline{z}(t),\overline{u}(t))$
    $p(T)=0$
\item[(3)] $\langle B\overline{z}(t),p(t)\rangle  +\partial_{2}
        L(\overline{z}(t),\overline{u}(t))=0$ a.e. in $[0,T]$
\end{itemize}
\end{theorem}

\begin{proof}
    Since $L$ is Fr\'echet differentiable, we deduce that the functional
    $$
    J(u)=\int_{0}^{T} L(z(t),u(t))\,dt
    $$
    is  Fr\'echet differentiable on $L^{r}(0,T)$.
    Since $\overline{u}$ is a minimum point for $J$,
    $$
    J'_{\overline{u}}.h=0,\quad \forall h\in L^{r}(0,T)
    $$
    but
    $$
    J'_{\overline{u}}.h= \int_{0}^{T} \langle \partial_{1}
    L(\overline{z}(t),\overline{u}(t),y(t)\rangle\,dt + \int_{0}^{T} \langle
    \partial_{2}
    L(\overline{z}(t),\overline{u}(t),h(t)\rangle\,dt
    $$
    where $y=\Theta'_{\overline{u}}.h$.
    We define $p$ by (2), then
    \begin{align*}
        J'(u).h &= \int_{0}^{T} \langle -\dot{p}(t) +
        {A'}^{*}_{\overline{z}(t)}p(t) -\overline{u}(t)B^{*}p(t),
        y(t)\rangle\,dt + \int_{0}^{T} h(t)
        \partial_{2}L(\overline{z}(t),\overline{u}(t))\,dt   \\
         & =\int_{0}^{T}\langle p(t),\dot{y}(t)+A'_{\overline{z}(t)}y(t) -
         \overline{u}(t)By(t)\rangle\,dt +  \int_{0}^{T} h(t)
        \partial_{2}L(\overline{z}(t),\overline{u}(t))\,dt  \\
         & =\int_{0}^{T}\big[\langle p(t),B\overline{z}(t)\rangle +
         \partial_{2}L(\overline{z}(t),\overline{u}(t)) \big]h(t)\,dt
    \end{align*}
 Hence part (3) of the theorem is consequence of the above equality.
\end{proof}

\section{Example}

In this section, we present an example which illustrates the
application of the results of the theory developed in the previous
sections.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ with smooth
boundary $\Gamma=\partial \Omega$. We consider the
control problem \eqref{P}
with
$$
 J(u)=\int_{Q} | z(x,t)|^{4}\,dx\,dt + \int_{0}^{T}
| u(t)|^{2}_{\mathbb{R}^{N}}\,dt
$$
Where $z$ satisfies  the
nonlinear evolution equation
\begin{equation} \label{eE}
\begin{gathered}
    \frac{\partial z}{\partial t}-\mathop{\rm div} (| \nabla z|^{2} \nabla
    z) =\sum_{i=1}^{N}u_{i}(t)\frac{\partial z}{\partial x_{i}}  \quad
    \text{in } Q=\Omega\times ]0,T[  \\
    z  =0 \quad \text{in } \Sigma=\Gamma\times ]0,T[  \\
    z(x,0)  = z_{0}(x)
\end{gathered}
\end{equation}
 Setting $V=W^{1,4}_{0}(\Omega)$, $H=L^{2}(\Omega)$ and
$V'=W^{-{1},4/3}(\Omega)$ we have $V\hookrightarrow
H\hookrightarrow V'$ continuously and densely.
Furthermore $V \hookrightarrow H$ compactly.

The equation \eqref{eE} can be written in the form
\begin{gather*}
    \dot{z}(t)+Az(t)  = u(t) Bz(t)  \\
    z(0)  =z_{0}
\end{gather*}
where
\begin{enumerate}
\item $A\colon V\to V'$, $\varphi\mapsto -\mathop{\rm div}
(| \nabla \varphi|^{2}\nabla \varphi)$ which satisfies
(H1) (see \cite{l1}).

\item  $B=(B_{1},\ldots,B_{N})$
with $B_{i}\colon V\to H$, $\varphi\mapsto B_{i}\varphi=\varphi_{x_i}$.
Hence $\Vert B_{i}\varphi\Vert_{H}\leq b_{i} \Vert\varphi\Vert_{V}$,
$b_{i}>0$ and $\Vert B\varphi\Vert_{H^{N}} \leq b \Vert\varphi\Vert_{V}$.

\item $u=(u_{1},\ldots,u_{N})\in \mathcal{U}=L^{2}(0,T;\mathbb{R}^{N})$ .
Here
$$
u(t)Bz(t)= \sum_{i=1}^{N} u_{i}(t) B_{i}z(t)= \sum_{i=1}^{N}
u_{i}(t) \frac{\partial z(t)}{\partial x_{i}}.
$$

\end{enumerate}
The cost function becomes
$$
J(u)= \Vert u\Vert^{2}_{L^{2}(0,T;\mathbb{R}^{N})} + \Vert
z\Vert^{4}_{L^{4}(0,T;Q)}
$$
Since $\int_{\Omega} u(t)Bz(x,t)z(x,t)\,dx=0$, the a priori estimates
given by Lemma \ref{lem2.1} become
\begin{gather*}
        \Vert z\Vert_{L^{4}(V)} \leq K_{1} | z_{0}|^{1/2},  \\ %\label{E1}
        \Vert z\Vert_{L^{\infty}(H)} \leq K_{2} | z_{0}| ,  \\ %\label{E2}
        \Vert \dot{z}\Vert_{L^{4/3}(V')} \leq K_{3} \big[\Vert
        z\Vert^{3/2}_{L^{4}(V)} + \Vert
        u\Vert^{3/2}_{L^{2}(0,T;\mathbb{R}^{N})} \big] \\ %\label{E3}
\end{gather*}

\begin{corollary} \label{coro4.1}
    For $z_{0}$ in $L^{2}(\Omega)$ and $u$ in $L^{2}(0,T;\mathbb{R}^{N})$,
the  equation \eqref{ezuf} with $f=0$ admits a unique solution which
    satisfies
\begin{gather*}
    z\in L^{\infty}(0,T;L^{2}(\Omega)) \cap L^{4}(0,T;W^{1,4}_{0}(\Omega)),
\\
    \dot{z}\in L^{4/3}(0,T;W^{-1,4/3}(\Omega))
\end{gather*}
\end{corollary}

\begin{proposition} \label{prop4.1}
    The mapping $\Theta\colon \mathcal{U}\to \mathcal{C}([0,T];H)$,
    $u\mapsto z$, with $z$ the solution of \eqref{ezuf} with $f=0$.
  is     differentiable in the sense of Fr\'echet,
    and  $\Theta'_{\overline{u}}.h$ satisfies
\begin{equation}
    \begin{gathered} \label{eS2}
    \dot{y}+ A'_{\overline{z}(t)}y(t) = \overline{u}(t)By(t)
    +  h(t)B\overline{z}(t)  \\
     y(0) =0,
    \end{gathered}
\end{equation}
    where  $z=(\Theta(u) )(t)$ and
$$
A'_{\varphi}.h= -\sum_{i=1}^{N}\big[ | \nabla
\varphi|^{2}h_{x_{i}} + 2 \langle \nabla \varphi,\nabla
h\rangle_{1}\varphi_{x_{i}} \big]_{x_{i}}
$$
with
$\langle \nabla \varphi,\nabla h\rangle_{1}= \sum_{i=1}^{N} \varphi_{x_{i}}
h_{x_{i}}$ ($\varphi$ and $h\in V$).
\end{proposition}

\begin{proof}
\textbf{1.} For $\varphi\in V$, the mapping,  $A'_{\varphi}\colon V\to V'$,
  $$
    h\mapsto A'_{\varphi}h=-\sum_{i=1}^{N}\left[| \nabla
\varphi|^{2}h_{x_{i}} + 2 \langle \nabla \varphi,\nabla
h\rangle_{1} \varphi_{x_{i}}\right]_{x_{i}}
    $$
    is linear and for $v\in V$ we have
    $$
    \langle A'_{\varphi}h,v\rangle_{V',V}
= \sum_{i=1}^{N}\int_{\Omega} f_{i}v_{x_{i}}\,dx
    $$
    with $f_{i}=| \nabla \varphi|^{2}h_{x_{i}} + 2 \langle \nabla
    \varphi,\nabla h\rangle_{1} \varphi_{x_{i}}$. Furthermore,
\begin{align*}
    \Vert f_{i}\Vert^{4/3}_{L^{4/3}(\Omega)}
&\leq  K_{1}\Big[\int_{\Omega}| \nabla
\varphi|^{8/3}| h_{x_{i}}|^{4/3}\,dx +
\int_{\Omega}| \langle \nabla \varphi,\nabla
h\rangle_{1}|^{4/3} | \varphi_{x_{i}}|^{4/3}\,dx\Big]\\
&\leq 2K_{1} \Big[ \Big(\int_{\Omega}| \nabla
\varphi|^{4} \Big)^{1/4}\Big(\int_{\Omega}| \nabla
h|^{4} \Big)^{1/3}\Big].
\end{align*}
    Then
    $$
    \Vert f_{i}\Vert_{L^{4/3}(\Omega)} \leq K \Vert
    \varphi\Vert^{2}_{W^{1,4}_{0}(\Omega)} \Vert
    h\Vert_{W^{1,4}_{0}(\Omega)}.
    $$
    Using the norm in $V'$ it follows that
$A'_{\varphi}\in \mathcal{L}(V,V')$.

 For $\varphi$ and $h$ in $V$, we have
    $$
    A(\varphi+h)-A(\varphi)-A'_{\varphi}(h)= F(\varphi,h)
    $$
    where
    $$
    F(\varphi,h)= -\sum_{i=1}^{N}[| \nabla
    h|^{2}h_{x_{i}} + | \nabla
    h|^{2}\varphi_{x_{i}} + 2 \langle \nabla \varphi,\nabla h\rangle_{1}
    h_{x_{i}}]_{x_{i}}\,.
    $$
For $v\in V$,
    $$
    \langle F,v\rangle_{V',V} = \sum_{i=1}^{N}\int_{\Omega}
    f_{i}v_{x_{i}}\,dx,
    $$
    where
    $$
    f_{i} = | \nabla h|^{2}\varphi_{x_{i}} + | \nabla
    h|^{2}h_{x_{i}} + 2 \langle \nabla \varphi,\nabla h\rangle_{1}
    h_{x_{i}}.
    $$
Then
    \begin{align*}
    \Vert f_{i}\Vert^{4/3}_{L^{4/3}(\Omega)}
& \leq K' \Big[ \int_{\Omega}| \nabla \varphi|^{4/3} | \nabla
    h|^{8/3}\,dx + \int_{\Omega}| \nabla \varphi|^{4/3} | \nabla
    h|^{8/3}\,dx + \int_{\Omega} | \nabla
    h|^{4}\,dx \Big]\\
& \leq K'' \Big[\Vert
    \varphi\Vert^{4/3}_{W^{1,4}_{0}(\Omega)} \Vert
    h\Vert^{8/3}_{W^{1,4}_{0}(\Omega)} + \Vert
    h\Vert^{4}_{W^{1,4}_{0}(\Omega)}\Big].
\end{align*}
    We deduce that
    $$
    \Vert A(\varphi+h)-A(\varphi)-A'_{\varphi}(h)\Vert_{V'}
\leq K [\Vert  \varphi\Vert_{V} \Vert
    h\Vert^{2}_{V} + \Vert  h\Vert^{3}_{V} \Vert]
    $$
Hence $A$ is differentiable in the sense of Frechet.

 \noindent \textbf{2.} The equation \eqref{eS2} admits  a unique solution
satisfying
    $$
    y\in L^{2}(V)\cap L^{\infty}(H), \quad
    \dot{y}\in L^{2}(V').
    $$
 The existence follows from a standard application of the
    Galerkin method  and the a priori estimates
    obtained for \eqref{eS}.
    We remark that $y\in \mathcal{C}([0,T];H)$.

\noindent \textbf{3.} The mapping
$A'\colon V\to \mathcal{L}(V,V')$, $\varphi\mapsto A'_{\varphi}$ is locally
Lipschitz.
Let $\varphi$ and $\psi$ be in $V$ with $\psi$ in neighbourhood
    of 0.
    For $h$ in $V$, we have
\[
        A'_{\varphi+\psi}h=-\sum_{i=1}^{N}\big[ |
        \nabla(\varphi+\psi)|^{2}h_{x_{i}} +2 \langle
        \nabla(\varphi+\psi),\nabla h\rangle_{1}
        ((\varphi+\psi))_{x_{i}}\big]_{x_{i}}
\]
    and $(A'_{\varphi+\psi}-A'_{\varphi})h=F$, where
    \begin{align*}
 F&=-\sum_{i=1}^{N}\Big[ |
        \nabla \psi|^{2}h_{x_{i}} +2 \langle
        \nabla\varphi,\nabla \psi\rangle_{1}h_{x_{i}}
        + 2\langle  \nabla\varphi,\nabla h\rangle_{1}\psi_{x_{i}}\\
&\quad  + 2\langle  \nabla\psi,\nabla h\rangle_{1}\varphi_{x_{i}}
        + 2\langle  \nabla\psi,\nabla h\rangle_{1}\psi_{x_{i}}\Big]_{x_{i}}.
 \end{align*}
    Then for $v\in V$,
    $$
    \langle F,v\rangle_{V',V}=\sum_{i=1}^{N}\int_{\Omega}
    f_{i}v_{x_{i}}\,dx,
    $$
    where
$$
f_{i}= | \nabla \psi|^{2}h_{x_{i}} + 2 \langle
        \nabla\varphi,\nabla \psi\rangle_{1}h_{x_{i}}
        + 2\langle
        \nabla\psi,\nabla h\rangle_{1}\varphi_{x_{i}}
 + 2\langle  \nabla\psi,\nabla h\rangle_{1}\psi_{x_{i}}+ 2\langle
        \nabla\varphi,\nabla h\rangle_{1}\psi_{x_{i}}.
$$
Hence
 $$
        \Vert f_{i}\Vert_{L^{p'}(\Omega)} \leq K\bigg[\Vert \psi\Vert^{2}_{V}
+ \Vert \psi\Vert_{V} \Vert \varphi\Vert_{V} \bigg]\Vert h\Vert_{V}.
 $$
        Since $\psi$ is in neighbourhood of 0,
 $$
        \Vert (A'_{\varphi+\psi}-A'_{\varphi})h\Vert_{V'} \leq K'\Vert \varphi\Vert_{V} \Vert
        \psi\Vert_{V} \Vert h\Vert_{V}.
 $$
        Hence
$$
        \Vert A'_{\varphi+\psi}-A'_{\varphi}\Vert_{\mathcal{L}(V,V')} \leq K''
        \Vert \psi\Vert_{V}.
$$
        It follows by theorem \ref{thm3.2}. that $\Theta$ is
        Frechet differentiable and its derivative $\Theta'_{u}.h$
        satisfies \eqref{eS2}
\end{proof}

Now the functional $J$ can be written as
$J(u)=\int_{0}^{T}L(z(t),u(t))\,dt$
 with $L$ satisfying (H6).

 The differentiability of $\Theta$ and
the norm ensures the differentiability of $J$ and the expression
of derivative is
 $$
    dJ(u).h = 4\int_{Q} | z(x,t)|^{2}
    z(x,t)y(x,t)\,dx\,dt + 2\int_{0}^{T}\langle
    u(t),h(t)\rangle_{\mathbb{R}^{N}}\,dt
$$
where $y=\Theta'_{u}h$.

 From Theorems \ref{thm2.1}, \ref{thm3.2} and \ref{thm3.3},
we get the following result.

\begin{corollary} \label{coro4.2}
    An optimal control $\overline{u}$, its corresponding state
    $\overline{z}$, and its adjoint state $p$ are necessarily tied by the
 optimality system:
For $1\leq i \leq N$ and $t\in [0,T]$,
\begin{gather*}
        \overline{u}_{i}(t)  = -2\int_{\Omega}p(x,t)\frac{\partial
        z}{\partial x_{i}} (x,t)\,dx  \\
\frac{\partial \overline{z}}{\partial
     t} - \mathop{\rm div} (|\nabla\overline{z}|^{2}\nabla \overline{z}
 =\sum_{i=1}^{N}\overline{u}_{i}(t)\frac{\partial\overline{z}}{\partial
     x_{i}} \quad  \text{ in }Q\\
 \overline{z}(x,t)  =0 \quad   \text{in }\Sigma  \\
 \overline{z}(x,0)  =z_{0}(x) \quad   \text{in }\Omega  \\
 -\frac{\partial \overline{p}}{\partial  t}
+ A'_{\overline{z}(t)} \overline{p} =
 -\sum_{i=1}^{N}\overline{u}_{i}(t)\frac{\partial\overline{p}}{\partial
     x_{i}} +| \overline{z}(x,t)|^{2}\overline{z}(x,t) \quad \text{ in }Q\\
    \overline{p}(x,t)  =0  \quad  \text{in }\Sigma  \\
    \overline{p}(x,T)  =0  \quad   \text{in }\Omega
\end{gather*}
\end{corollary}

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