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\begin{Titul}
{\large \bf \uppercase{Optimal control for a controlled \\ ill-posed wave equation\\[4pt] without requiring the Slater hypothesis}}\footnote{This work was supported by the Directorate-General for Scientific Research and Technological
Development (DGRSDT).}\\[3ex]
{{\bf Abdelhak Hafdallah} \\[2ex] {\small
Laboratory of Mathematics, Informatics and Systems,
University of Larbi T\'{e}bessi,\\ Rue
de Constantine 12002, T\'{e}bessa, Algeria \\ \hrefemail{abdelhak.hafdallah@univ-tebessa.dz} }\\[5ex]}
\end{Titul}
\begin{Annot}
{\bf Abstract:}  In this paper, we investigate the problem of optimal control for an ill-posed wave equation without
 using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a
 controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data
initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control
distributed systems with incomplete data. After approximating the no-regret control by a low-regret control
sequence, we characterize the optimal control by a singular optimality system.

{\bf Keywords:}  Ill-posed wave equation, No-regret control, Incomplete data, Carleman estimates, Null-controllability.
\end{Annot}

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\section{Introduction}
The first systematic study of optimal control of ill-posed problems was by
J.L.~Lions in his book ``Control of distributed singular systems''~\cite{11},
exactly when he focused on an ill-posed heat equation (backward heat
equation). In his study, he required the set of admissible controls $U_{ad}$
to have a non-empty interior. This condition is the so-called Slater
hypothesis. Regrettably, a difficulty starts when we need to use some sets
like the positive cone $\left( L^{2}\right) ^{+}$, which has an empty
interior, as a set of admissible controls, where the hypothesis of Slater
doesn't hold. To avoid such kind of obstacle, we propose to take a different
approach to the regularization approach proposed in \cite{3} and \cite{5},
to get an optimality system characterizing the optimal control without
requiring Slater extra-hypothesis, it's the controllability approach.

The aim of our work is to generalize existing results \cite{3, 5} where we
seek to get an optimality system characterizing the optimal control for an
ill-posed wave equation \cite{4,5}, to reach our goal, we start by assuming
that when taking the control in some dense space of $L^{2}\left( Q\right) $,
the problem becomes well-posed. Then, by null-controllability of the
well-posed wave equation, we seek to retrieve\ the second order time
condition in the ill-posed equation. Hence, we get an optimal control
problem for a controlled wave equation with incomplete data where we apply
the no-regret control method introduced by Lions \cite{12} (the original
idea was introduced by Savage in \cite{15}) for optimal control problems
with incomplete data.

On the contrary of \cite{5}, this work leads us to characterize the optimal
control by an optimality system which has a simpler form than the one given
in \cite{5}, this will be very beneficial in a numerical analysis viewpoint.

A few studies have been published in the context of optimal control of PDEs
with missing data after that by Lions himself \cite{13}.

Later, many papers are published such as \cite{14} and \cite{9} where
authors studied an age-structured population dynamics of incomplete data. In
\cite{1}, authors applied the notion of no-regret control on a fractional
wave equation with incomplete data. Afterward, a control coupled systems and
a wave equation both with incomplete data were treated in \cite{7} and \cite{6} respectively, extended recently to more general and abstract systems
in \cite{8} .

\ Actually, the method of no-regret control consists of taking only controls
$v$ such that%
\begin{equation*}
J\left( v,g\right) \leq J\left( 0,g\right),
\end{equation*}
($J$ is the cost function) for every missing data $g$, where we guarantee
the belonging of the optimal control to this set of controls. To avoid the
difficulty of characterizing the no-regret control, we should relax the
definition by making a quadratic perturbation in no-regret control
definition, i.e. \
\begin{equation*}
J\left( v,g\right) \leq J\left( 0,g\right) +\gamma \left\Vert g\right\Vert
^{2},\quad\gamma >0.
\end{equation*}

In this way, we define a sequence of low-regret controls expected to be
converging to the no-regret control.

The recent paper is organized as follows: in the next section we present
some preliminaries for the main problem, in the third section we prove
existence and uniqueness for the controllability problem, in the fourth we
introduce the optimal control problem with missing data, in the fifth, we
give an optimality system for the optimal control problem, and we finish
with a conclusion.

\section{Preliminaries}
\setcounter{equation}{0}
Let $\Omega \subset
\mathbb{R}^{N}$ be an open bounded domain with smooth boundary $\Gamma $, $\Gamma _{0}$
is a non-empty subset of $\Gamma $, denote $Q=\Omega \times $ $\left(
0,T\right) $, $\Sigma =\Gamma \times $ $\left( 0,T\right) $, $\Sigma
_{0}=\Gamma _{0}\times \left( 0,T\right) $ and $T>0$. Consider the following
wave equation given by:
\begin{equation}\label{Haf1}%\tag{1}
\left\{
\begin{array}{l}
y^{\prime \prime }-\triangle y=v \\
y\left( x,0\right) =0,\quad y\left( x,T\right) =0 \\
y\left( x,t\right) =0%
\end{array}%
\right.
\begin{array}{c}
\text{in }Q\text{,} \\
\text{in }\Omega \text{,} \\
\text{on }\Sigma \text{,}%
\end{array}
\end{equation}
where $v$ is a distributed control in
\begin{equation*}
U_{ad}^{\rho }=\left\{ v\in L_{\rho }^{2}\left( Q\right) :v\geq 0\text{
almost everywhere in }Q\right\} \text{,}
\end{equation*}%
it's the closed convex cone $\left( L_{\rho }^{2}\left( Q\right) \right)
^{+} $, where
\begin{equation*}
L_{\rho }^{2}\left( Q\right) =\left\{ w\in L^{2}\left( Q\right) \text{ such
that }\rho w\in L^{2}\left( Q\right) \right\}
\end{equation*}%
and $\rho $ is a positive function defined on $Q$ such that ${1}/{\rho }$
is bounded in $Q$. It's well known that~\eqref{Haf1} is ill-posed \cite{5}.

On the other hand, let's consider a null controllability problem for the
following wave equation:
\begin{equation} \label{Haf2}
\left\{
\begin{array}{l}
y^{\prime \prime }-\triangle y=v \\
y\left( x,0\right) =0, \quad y^{\prime }\left( x,0\right) =g \\
y\left( x,t\right) =\left\{
\begin{array}{l}
\theta \\
0%
\end{array}%
\right.%
\end{array}%
\right.
\begin{array}{l}
\text{in }Q\text{,} \\
\text{in }\Omega \text{,} \\
\text{on }\Sigma _{0}\text{,} \\
\text{on }\Sigma \diagdown \Sigma _{0}\text{,}%
%\end{array}%
\end{array}
%\tag{2}
\end{equation}
where $g\in $ $L^{2}\left( \Omega \right) $ is a missing initial
condition.

The following geometric and time conditions hold:
\begin{gather} \label{Haf3}
\exists x_{0}\notin \overline{\Omega }\text{ such that }\left\{ x\in
\partial \Omega :\left( x-x_{0}\right) .\nu \left( x\right) \geq 0\right\}
\subset \Gamma _{0}\text{,} \\ %\tag{3}
\label{Haf4}
T>2\,\sup\limits_{x\in \Omega } \vert x-x_{0}\vert,
%\tag{4}
\end{gather}
and $\nu \left( x\right)$ denotes the external unit normal vector at $x$.

Note that for every $\left( v,g;\theta \right) \in U_{ad}^{\rho }\times
L^{2}\left( \Omega \right) \times L^{2}\left( \Sigma _{0}\right) $, the
system \eqref{Haf2}--\eqref{Haf4} has a unique solution $y\left(
v,g;\theta \right) =y\left( v,g;\theta \right) \left( x,t\right) $ in some
sense (see \cite[Ch.~4, p.~325]{10}).

Actually, we want to find a function $\theta \in L^{2}\left( \Gamma
_{0}\times \left( 0,T\right) \right) $ such that for every $v\in
U_{ad}^{\rho }$ and every missing initial condition $g\in $ $L^{2}\left(
\Omega \right) $ the solution of \eqref{Haf2} verifies the following
null controllability property
\begin{equation} \label{Haf5}
y( x,T) =y^{\prime }( x,T) =0.  %\tag{5}
\end{equation}

In this way, by the controllability of~\eqref{Haf2} we retrieve the
initial condition $y^{\prime }\left( x,0\right) $ in~\eqref{Haf1} but
with an unknown value and~\eqref{Haf1} becomes a well-posed equation
with missing data.

After this, we want to find a control function $v$ in $U_{ad}^{\rho }$
solution to the following optimal control problem%
\begin{equation}\label{Haf6}
\underset{v\in U_{ad}^{\rho }}{\inf }J_{\rho }\left( v,g\right) \quad\text{such
that}\quad
J_{\rho }\left( v,g\right) =\left\Vert y\left( v,g\right)
-y_{d}\right\Vert _{\rho }^{2}+N\left\Vert v\right\Vert _{\rho }^{2}\text{,
\ \ }  %\tag{6}
\end{equation}
where $y_{d}$ is a target function in $L^{2}\left( Q\right) $, $N>0$
all are given.

Now, let's prove the existence of a solution for the null-controllability
problem \eqref{Haf2}--\eqref{Haf5}.

\section{Existence for the null-controllability problem (\ref{Haf2})--(\ref{Haf5})}
\setcounter{equation}{0}
Before treating the controllability problem~\eqref{Haf2}--\eqref{Haf5} we announce the following theorem giving a so-called Carleman
inequality type, which will be the main tool to solve the controllability problem.
\begin{theoen}\label{Th1}
\ Denote the operator $L={\partial ^{2}}/{\partial t^{2}}-\Delta $ in
distribution sense, under the geometric and time conditions \eqref{Haf3}--\eqref{Haf4} there exists a $C^{2}$ weighted positive function $\rho $
defined on $Q$ such that ${1}/{\rho }$ is bounded in $Q$ and $C=C\left(
\Omega ,T,\Gamma _{0},\rho \right) >0$ such that:%
\begin{equation}\label{Haf7}
\int_{0}^{T}\int_{\Omega }\frac{1}{\rho ^{2}}\left\vert q\right\vert
^{2}dxdt\leq C\bigg[ \int_{0}^{T}\int_{\Omega }\frac{1}{\rho ^{2}}%
\left\vert Lq\right\vert ^{2}dxdt+\int_{0}^{T}\int_{\Gamma _{0}}\frac{1}{%
\rho ^{2}}\Big\vert \frac{\partial q}{\partial \nu }\Big\vert ^{2}d\Gamma
dt\bigg]   %\tag{7}
\end{equation}%
for every
$$q\in \mathcal{V=}\Big\{ \varphi \in L^{2}\left(
0,T;H_{0}^{1}\left( \Omega \right) \right) :L\varphi \in L^{2}\left(
Q\right) , \dfrac{\partial \varphi }{\partial \nu }\Big\vert _{\Sigma
_{0}}\in L^{2}\left( \Sigma _{0}\right) \Big\},$$
where
\begin{equation*}
L^{2}\left( 0,T;H_{0}^{1}\left( \Omega \right) \right) =\Big\{
\varphi :\left[ 0,T\right] \rightarrow H_{0}^{1}\left( \Omega \right) \text{
measurable such that }
\displaystyle\int_{0}^{T}\text{ }\left\Vert \varphi \left( t\right) \right\Vert
_{H_{0}^{1}\left( \Omega \right) }^{2}dt<\infty
\Big\}.
\end{equation*}
\end{theoen}
\proofen    It leads from a Carleman inequality, it can be found in \cite[ Theorem~1.1]{2}.

The inequality~\eqref{Haf7}  allows us to introduce the following real
inner product:%
\begin{equation}\label{Haf8}
a\left( r,s\right) =\int_{0}^{T}\int_{\Omega }\frac{1}{\rho ^{2}}%
Lr\cdot Ls\,dxdt+\int_{0}^{T}\int_{\Gamma _{0}}\frac{1}{\rho ^{2}}\frac{\partial r%
}{\partial \nu }\frac{\partial s}{\partial \nu }d\Gamma dt  %\tag{8}
\end{equation}
on $V$ the Hilbert space completion of $\mathcal{V}$, with its
associated norm $\|\cdot\|_{a}=\sqrt{a\left(\cdot,\cdot\right) }$.\hfill$\square$\\[1ex]

 \medskip\emph{Remark~1.}
 We can characterize the structure of $\ \mathcal{V}$ as a subspace of a
weighted Sobolev space. Indeed, let $H_{\rho }(Q)$ be the weighted Hilbert
space defined by%
\begin{equation*}
H_{\rho }(Q)=\Big\{ v\in L^{2}\left( Q\right)\ \text{such that:} \
\int_{0}^{T}\int_{\Omega }\frac{1}{\rho ^{2}}\left\vert v\right\vert
^{2}dxdt<\infty \Big\},
\end{equation*}
endowed with the natural norm $$\|\cdot\|_{H_{\rho
}(Q)}=\left( \int_{0}^{T}\int_{\Omega }\frac{1}{\rho ^{2}}|
\cdot|^{2}dxdt\right)^{{1}/{2}}.$$ This shows that $V$ is
embedded continuously in $H_{\rho }(Q)$ as:
\begin{equation*}
\exists C>0:\left\Vert v\right\Vert _{H_{\rho }(Q)}\leq C\left\Vert
v\right\Vert _{a}\quad\text{for every}\quad
v\in V.
\end{equation*}
By the boundedness of ${1}/{\rho ^{2}}$ on $Q$, we also see that $
L^{2}\left( Q\right) $ is continuously embedded in $H_{\rho }(Q)$.
\begin{prpen}
Fix $\left( v,g\right) \in \emph{U}_{ad}^{\rho }\times L^{2}\left( \Omega
\right) $. Define on $V$ the linear form%
\begin{equation*}
l_{\left( v,g\right) }\left( s\right) =\int_{0}^{T}\int_{\Omega
}vsdxdt+\int_{\Omega }gs\left( 0\right) dx,
\end{equation*}
then there exists a unique solution $\widetilde{p}\left( v,g\right) \in V$
to the following variational equation:%
\begin{equation}\label{Haf9}
a\left( r,s\right) =l_{\left( v,g\right) }\left( s\right),\quad\forall s\in V. %\tag{9}
\end{equation}
Also, we have%
\begin{equation}\label{Haf10}
\exists C>0:\left\Vert \widetilde{p}\left( v,g\right) \right\Vert _{a}\leq
C\left( \left\Vert v\rho \right\Vert _{L^{2}\left( Q\right) }+\left\Vert
g\right\Vert _{L^{2}\left( \Omega \right) }\right). %\tag{10}
\end{equation}
Moreover, if we choose%
\begin{equation}\label{Haf11}
y\left( v,g\right) =\frac{1}{\rho ^{2}}L\widetilde{p}\left( v,g\right), \quad\theta \left( v,g\right) =\frac{1}{\rho ^{2}}\frac{%
\partial \widetilde{p}\left( v,g\right) }{\partial \nu }\Big\vert _{\Sigma
_{0}},  %\tag{11}
\end{equation}
the pair $\left\{ y\left( v,g\right) ,\theta \left( v,g\right) \right\} $ is
a solution of the null controllability problem~\eqref{Haf2}--\eqref{Haf5}.
\end{prpen}
\proofen
The result is obtained by application of the Lax--Milgram theorem with using
Carleman inequality~\eqref{Haf7} to prove that the inner product~\eqref{Haf8} is coercive. Using~\eqref{Haf9} and integration by parts we get
the null controllability property~\eqref{Haf5}. \hfill$\square$

\section{Optimal control of the controlled wave equation with incomplete
data}
\setcounter{equation}{0}
In this principal section, we focus on the following controlled wave
equation missing initial condition
\begin{equation}\label{Haf12}
\left\{
\begin{array}{l}
\text{\ }y^{\prime \prime }-\triangle y=v\\ %\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
y\left( x,0\right) =0\text{ },\quad y^{\prime }\left( x,0\right) =g \\
\text{\ }y=\left\{
\begin{array}{l}
\theta \left( v,g\right) \\
0%
\end{array}%
\right. %\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }%
\end{array}%
\right.
\begin{array}{l}
\text{in }Q\text{,} \\
\text{in }\Omega , \\
\text{on }\Sigma _{0}, \\
\text{on }\Sigma \diagdown \Sigma _{0},
\end{array}
%\tag{12}
\end{equation}%
with%
\begin{equation}\label{Haf13}
y\left( x,T\right) =0,\quad y^{\prime }\left( x,T\right) =0,
%\tag{13}
\end{equation}
where $\theta \left( v,g\right) $ is given by~\eqref{Haf11}.

We solve the optimal control problem~\eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}  regardless of the values of the missing initial condition $%
g$, where $L_{\rho }^{2}\left( Q\right) $ endowed with the inner product $%
\left(\cdot,\cdot\right) _{\rho }=\left( \rho \cdot,\rho \cdot\right) _{L^{2}\left(
Q\right) }$ and the associated norm $\| \cdot\|_{\rho }=\sqrt{(\cdot,\cdot)_{\rho }}$.

In order to ensure the existence of the optimal control for~\eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6} we need an extra hypothesis of
Slater (see \cite[Ch.~4, Remark 1.4]{11}) which requires that
\begin{equation}\label{Haf14}
U_{ad}^{\rho }\quad\text{has a non-empty interior.}  %\tag{14}
\end{equation}

\bigskip Unfortunately, the extra hypothesis~\eqref{Haf14} is not
fulfilled by $U_{ad}^{\rho }$ because it's well known that~$\left(
L^{2}\left( Q\right) \right) ^{+}$ has an empty interior.

However, we propose an approach where there is no need to~\eqref{Haf14}, it's the method of \textit{no-regret control} which was introduced by J.L.~Lions in~\cite{12}, to solve optimal control problems with some
incomplete data.

First of all, let's give a definition of the no-regret control for the
controlled system with missing data~\eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}.
 \begin{dfnen}[\cite{12}] We say that $u\in U_{ad}^{\rho }$ is a no-regret control for~\eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6} if $u$ is the solution
of:%
\begin{equation}\label{Haf15}
\underset{v\in U_{ad}^{\rho }}{\inf }\big( \underset{g\in L^{2}\left(
\Omega \right) }{\sup }\left( J_{_{\rho }}\left( v,g\right) -J_{_{\rho
}}\left( 0,g\right) \right) \big).  %\tag{15}
\end{equation}
 \end{dfnen}
 In the following lemma, we try to rewrite the main quantity in the last definition to isolate the missing data in some way.
  \begin{lemen}
Let $M$ be an operator defined from $L^{2}\left( \Omega \right) $ to $%
L^{2}\left( \Sigma _{0}\right) $ by $Mg=\dfrac{\partial \widetilde{p}}{%
\partial \nu }\left( 0,g\right) $, where $\widetilde{p}\left( v,g\right) $
is the unique solution to \eqref{Haf9}. Then, $M$ is a linear bounded
operator on $L^{2}\left( \Omega \right) $, and we have%
\begin{equation}\label{Haf16}
J_{\rho }\left( v,g\right) -J_{\rho }\left( 0,g\right) =J_{\rho }\left(
v,0\right) -J_{\rho }\left( 0,0\right) +2\left( S\left( v\right) ,g\right)
_{L^{2}\left( \Omega \right) },  %\tag{16}
\end{equation}
where $S$ is also a linear bounded operator from $U_{ad}^{\rho }$ to $%
L^{2}\left( \Omega \right) $ given by%
\begin{equation*}\label{Haf17}
S\left( v\right) =\widetilde{p}\left( v,0\right) \left( 0\right) -M^{\ast
}\left( \theta \left( v,0\right) \right).  %\tag{17}
\end{equation*}
  \end{lemen}
\proofen
 It's clear that $M$ is linear, also $M$ is bounded. In fact, we know that $\widetilde{p}\left( v,0\right) $ solves~\eqref{Haf9} for every $s\in V$%
, we choose an $s$ such that%
\begin{equation*}
\left\{
\begin{array}{ll}
Ls=0 &\text{    in }Q,\\
s\left( x,0\right) =g,\quad s^{\prime }\left( x,0\right) =0 &\text{    in }\Omega,
\\
\dfrac{\partial s}{\partial \nu }=\left\{
\begin{array}{l}
\dfrac{\partial \widetilde{p}\left( 0,g\right) }{\partial \nu } \\
0%
\end{array}%
\right.
&\begin{array}{l}
\text{on }\Sigma _{0}, \\[1ex]
\text{on }\Sigma \diagdown \Sigma _{0}%
\end{array}%
\end{array}%
\right.
%\begin{array}{l}
%\text{in }Q, \\
%\text{in }\Omega, \\
%\text{on }\Sigma _{0}, \\
%\text{on }\Sigma \diagdown \Sigma _{0}%
%\end{array}%
\end{equation*}
to get%
\begin{equation*}
\inf\limits_{\Sigma _{0}}\frac{1}{\rho ^{2}}\int_{0}^{T}\int_{\Gamma
_{0}}\Big\vert \frac{\partial \widetilde{p}\left( 0,g\right) }{\partial \nu
}\Big\vert ^{2}d\Gamma dt\leq \int_{0}^{T}\int_{\Gamma _{0}}\frac{1}{%
\rho ^{2}}\Big\vert \frac{\partial \widetilde{p}\left( 0,g\right) }{%
\partial \nu }\Big\vert ^{2}d\Gamma dt=\left\Vert g\right\Vert
_{L^{2}\left( \Omega \right) }^{2}\text{.}
\end{equation*}

From linearity in~\eqref{Haf9}, we get $y\left( v,g\right) =y\left(
v,0\right) +y\left( 0,g\right) +y\left( 0,0\right) $, and by a simple
calculation we get%
\begin{equation*}
J_{\rho }\left( v,g\right) -J_{\rho }\left( 0,g\right) =J_{\rho }\left(
v,0\right) -J_{\rho }\left( 0,0\right) +2\left( y\left( v,0\right) ,y\left(
0,g\right) \right) _{\rho }.
\end{equation*}%
Use Green formula to prove%
\begin{gather*}
\left( y\left( v,0\right) ,y\left( 0,g\right) \right) _{\rho } =\left( L%
\widetilde{p}\left( v,0\right) ,y\left( 0,g\right) \right) _{L^{2}\left(
Q\right) } \\
=\left( \widetilde{p}\left( v,0\right) ,Ly\left( 0,g\right) \right)
_{L^{2}\left( Q\right) }-\Big( \frac{\partial \widetilde{p}\left(
v,0\right) }{\partial \nu },\theta \left( 0,g\right) \Big) _{L^{2}\left(
\Sigma _{0}\right) }
+\left( g,\widetilde{p}\left( v,0\right) \left( 0\right) \right)
_{L^{2}\left( \Omega \right) } \\
=\left( \widetilde{p}\left( v,0\right) \left( 0\right) ,g\right)
_{L^{2}\left( \Omega \right) }-\Big( \frac{\partial \widetilde{p}\left(
v,0\right) }{\partial \nu },\frac{1}{\rho ^{2}}\frac{\partial \widetilde{p}}{%
\partial \nu }\left( 0,g\right) \Big) _{L^{2}\left( \Sigma _{0}\right) } \\
=\left( \widetilde{p}\left( v,0\right) \left( 0\right) ,g\right)
_{L^{2}\left( \Omega \right) }-\left( M^{\ast }\left( \theta \left(
v,0\right) \right) ,g\right) _{L^{2}\left( \Omega \right) }.
\end{gather*}

We know that $\theta \left( v,0\right) :U_{ad}^{\rho }$ $\rightarrow $ $%
L^{2}\left( \Sigma _{0}\right) $ solves~\eqref{Haf10}, choose $s=%
\widetilde{p}\left( v,0\right) $ and use~\eqref{Haf13} to find
\begin{gather*}
\inf\limits_{\Sigma _{0}} \frac{1}{\rho ^{2}}\int_{0}^{T}\int_{\Gamma
_{0}}\Big\vert \frac{\partial \widetilde{p}\left( v,0\right) }{\partial \nu
}\Big\vert ^{2}d\Gamma dt \leq \int_{0}^{T}\int_{\Gamma _{0}}\frac{1}{%
\rho ^{2}}\Big\vert \frac{\partial \widetilde{p}\left( v,0\right) }{%
\partial \nu }\Big\vert ^{2}d\Gamma dt\leq \int_{0}^{T}\int_{\Omega }v%
\widetilde{p}\left( v,0\right) dxdt \\
\leq \left\Vert v\right\Vert _{\rho }\left\Vert \widetilde{p}\left(
v,0\right) \right\Vert _{H_{\rho }(Q)}
\leq C\left\Vert v\right\Vert _{\rho }^{2},
\end{gather*}%
which proves that $\theta \left( v,0\right) $ is bounded. Moreover, the map $%
\widetilde{p}\left( v,0\right) \left( 0\right) :U_{ad}^{\rho }$ $\rightarrow
$ $L^{2}\left( \Omega \right) $ is continuous. In fact, by a Carleman
estimate given in  \cite[Corollary~2.8]{2}, under the same condition of
Theorem~\ref{Th1} there exists a $C^{2}$ weighted positive function $\rho $ on $Q$
such that ${1}/{\rho }$ is bounded in $Q$ and ${C=C\left( \Omega
,T,\Gamma _{0},\rho \right) >0}$ such that:
\begin{equation*}
\int_{\Omega }\frac{1}{\rho \left( 0\right) ^{2}}\left\vert q\left(
0\right) \right\vert ^{2}dxdt\leq C\bigg[ \int_{0}^{T}\int_{\Omega }\frac{1%
}{\rho ^{2}}\left\vert Lq\right\vert ^{2}dxdt+\int_{0}^{T}\int_{\Gamma
_{0}}\frac{1}{\rho ^{2}}\left\vert \frac{\partial q}{\partial \nu }%
\right\vert ^{2}d\Gamma dt\bigg]
\end{equation*}
for every $q\in \mathcal{V}$. Choose $q=\widetilde{p}\left( v,0\right) $ to
find%
\begin{equation*}
\int_{\Omega }\frac{1}{\rho \left( 0\right) ^{2}}\left\vert \widetilde{p}%
\left( v,0\right) \left( 0\right) \right\vert ^{2}dxdt \leq C\bigg[
\int_{0}^{T}\int_{\Omega }\frac{1}{\rho ^{2}}\left\vert L\widetilde{p}%
\left( v,0\right) \right\vert ^{2}dxdt+\left\Vert \theta \left( v,0\right)
\right\Vert _{L^{2}\left( \Sigma _{0}\right) }^{2}\bigg]
\leq C\left\Vert v\right\Vert _{\rho }^{2}.
\end{equation*}

Finally, $S$ is also a linear bounded operator. \hfill$\square$\\[1ex]
%\end{proof}

Unfortunately, we encounter a big difficulty when characterizing the
no-regret control where we need to know the structure of the set
\begin{equation*}
\big\{ v\in U_{ad}^{\rho }:\left( S\left( v\right) ,g\right) _{L^{2}\left(
\Omega \right) }=0\text{ for every }g\text{ in }L^{2}\left( \Omega \right)
\big\} ,
\end{equation*}%
which is difficult to do, this requires on us to relax no-regret control definition by making some quadratic perturbation, then, we announce:
 \begin{dfnen}[\cite{12}] \bigskip We say that $u_{\gamma }\in U_{ad}^{\rho }$ is a
low-regret control for~\eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}
if $u_{\gamma }$ is the solution of  the problem\textup:%
\begin{equation*}\label{Haf18}
\inf\limits_{v\in U_{ad}^{\rho }}\Big({\sup\limits_g\in L^{2}(
\Omega) } \big( J_{\rho }( v,g) -J_{\rho }(
0,g) -\gamma \Vert g\Vert _{L^{2}(\Omega )}^{2}\big)
\Big), \quad\gamma >0. %\tag{18}
\end{equation*}
 \end{dfnen}
 From \eqref{Haf16}, we get for all $v\in U_{ad}^{\rho }$
\begin{gather*}
\underset{g\in L^{2}(\Omega )}{\sup }\left( J_{\rho }\left( v,g\right)
-J_{\rho }\left( 0,g\right) -\gamma \left\Vert g\right\Vert _{L^{2}(\Omega
)}^{2}\right)\\ =J_{\rho }\left( v,0\right) -J_{\rho }\left( 0,0\right)
+\underset{g\in L^{2}(\Omega )}{\sup }\left( 2\left( S\left( v\right)
,g\right) _{L^{2}(\Omega )}-\gamma \left\Vert g\right\Vert _{L^{2}(\Omega
)}^{2}\right) \text{ \ \ } \\
=J_{\rho }\left( v,0\right) -J_{\rho }\left( 0,0\right) +\frac{1}{\gamma }%
\left\Vert S\left( v\right) \right\Vert _{L^{2}(\Omega )}^{2}.
\end{gather*}

Thus, our optimal control problem is transformed into a standard optimal
control problem (i.e.~a problem with complete data) given by
\begin{equation}\label{Haf19}
\underset{v\in U_{ad}^{\rho }}{\inf }\mathcal{J}_{\rho }^{\gamma }\left(
v\right) \text{,}  %\tag{19}
\end{equation}
where%
\begin{equation}\label{Haf20}
\mathcal{J}_{\rho }^{\gamma }\left( v\right) =J_{\rho }\left( v,0\right)
-J_{\rho }\left( 0,0\right) +\frac{1}{\gamma }\left\Vert S\left( v\right)
\right\Vert _{L^{2}(\Omega )}^{2}\text{.}  %\tag{20}
\end{equation}
 \begin{lemen}
 The problem \eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}, \eqref{Haf19}, \eqref{Haf20} has a unique solution $u_{\gamma }\in
U_{ad}^{\rho }$.
 \end{lemen}
 \proofen
 We have for every $v\in \mathcal{U}_{ad}^{\rho }$ : $\mathcal{J}_{\rho
}^{\gamma }\left( v\right) \geq -J_{\rho }\left( 0,0\right) =-\left\Vert
y_{d}\right\Vert _{\rho }^{2}$ then $d_{\gamma }=\underset{v\in \mathcal{U}%
_{ad}^{\rho }}{\inf }\mathcal{J}_{\rho }^{\gamma }\left( v\right) $ exists.
Let $\left( v_{n}^{\gamma }\right) $ be a minimizing sequence such that $%
d_{\gamma }=\underset{n\rightarrow \infty }{\lim }$ $\mathcal{J}_{\rho
}^{\gamma }\left( v_{n}^{\gamma }\right) $. We know that%
\begin{equation*}
\mathcal{J}_{\rho }^{\gamma }\left( v_{n}^{\gamma }\right) =J_{\rho }\left(
v_{n}^{\gamma },0\right) -J_{\rho }\left( 0,0\right) +\frac{1}{\gamma }%
\left\Vert S\left( v_{n}^{\gamma }\right) \right\Vert _{L^{2}\left( \Omega
\right) }^{2}\leq d_{\gamma }+1\text{.}
\end{equation*}
This implies the following bounds%
\begin{equation*}
\left\Vert v_{n}^{\gamma }\right\Vert _{\rho }\leq C_{\gamma },\quad
\left\Vert y\left( v_{n}^{\gamma },0\right) \right\Vert _{\rho }\leq
C_{\gamma },\quad\frac{1}{\sqrt{\gamma }}\left\Vert S\left( v_{n}^{\gamma
}\right) \right\Vert _{L^{2}\left( \Omega \right) }\leq C_{\gamma }\text{,}
\end{equation*}
where $C_{\gamma }$ is a positive constant independent of $n$. Then, there
exists $u_{\gamma }$ such that $v_{n}^{\gamma }\rightharpoonup $ $u_{\gamma
} $ weakly in $U_{ad}^{\rho }$ (closed), also $y\left( v_{n}^{\gamma
},0\right) \rightharpoonup y\left( u_{\gamma },0\right) $ weakly in $L_{\rho
}^{2}\left( Q\right) $\ because of continuity w.r.t. the data.

Since $S$ is bounded, then
\begin{equation*}
S\left( v_{n}^{\gamma }\right) \rightharpoonup S\left( u_{\gamma }\right)
\text{ weakly in }L^{2}\left( \Omega \right) \text{,}
\end{equation*}
with%
\begin{equation*}
\mathcal{J}_{\rho }^{\gamma }\left( u_{\gamma }\right) \leq \underset{%
n\rightarrow \infty }{\lim \inf }\mathcal{J}_{\rho }^{\gamma }\left(
v_{n}^{\gamma }\right)
\end{equation*}
and we conclude that%
\begin{equation*}
\mathcal{J}_{\rho }^{\gamma }\left( u_{\gamma }\right) =\underset{v\in
U_{ad}^{\rho }}{\inf }\mathcal{J}_{\rho }^{\gamma }\left( v\right) \text{.}
\end{equation*}
Since $\mathcal{J}_{\rho }^{\gamma }\left( v\right) $ is strictly convex, $%
u_{\gamma }$ is unique. \hfill$\square$\\[1ex]

It still remains to obtain an optimality system giving a characterization
for low-regret control $u_{\gamma }$ as follows
  \begin{theoen}
  The low-regret control $u_{\gamma }\in $ $U_{ad}^{\rho }$ which is a solution to
\eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}, \eqref{Haf19}, \eqref{Haf20} is characterized by the following optimality system
\begin{equation}\label{Haf21}
\left\{
\begin{array}{ll}
Ly_{\gamma }=u_{\gamma }; &Lp_{\gamma }=y_{\gamma }-y_{d}\\
y_{\gamma }\left( x,0\right) =0,\quad y_{\gamma }^{\prime }\left(
x,0\right) =0, & p_{\gamma }\left( x,0\right) =0,\quad p_{\gamma }^{\prime }\left( x,0\right) =0,  \\
y_{\gamma }\left( x,T\right) =0,\quad y_{\gamma }^{\prime }\left(
x,T\right) =0;& p_{\gamma }\left( x,T\right) =0,\quad p_{\gamma }^{\prime }\left( x,T\right) =0  \\
y_{\gamma }=\left\{
\begin{array}{l}
\theta \left( u_{\gamma },0\right) \\
0%
\end{array}%
\right. ; & p_{\gamma }=\left\{
\begin{array}{l}
\lambda _{\gamma }, \\
0,%
\end{array}%
\right.
\end{array}%
\right.
\begin{array}{l}
\text{in }Q,\\
\\
\text{in }\Omega, \\
\text{on }\Sigma _{0}, \\
\text{on }\Sigma \diagdown \Sigma _{0},%
\end{array}
%\tag{21}
\end{equation}
where $\gamma >0$, $y_{\gamma }=y\left( u_{\gamma },0\right) $ and $%
p_{\gamma }=p\left( u_{\gamma }\right) $, with the following variational
inequality%
\begin{equation}\label{Haf22}
\Big( T^{\ast }\left( Lp_{\gamma }\right) +Nu_{\gamma }+\frac{1}{\gamma }%
S^{\ast }S\left( u_{\gamma }\right) ,v-u_{\gamma }\Big) _{\rho }\geq 0\ \
\quad\forall \ v\in U_{ad}^{\rho }\text{,}  %\tag{22}
\end{equation}
where $T:v\rightarrow y\left( v,0\right) $ from $U_{ad}^{\rho }$\ to $%
L_{\rho }^{2}\left( Q\right) $ is a linear bounded operator.
  \end{theoen}
\proofen
A first order necessary condition of Euler--Lagrange~\cite{10} for  \eqref{Haf19}, \eqref{Haf20} gives for every~$v\in U_{ad}^{\rho }$
\begin{equation}\label{Haf23}
\left( y\left( u_{\gamma },0\right) -y_{d},y\left( v-u_{\gamma },0\right)
\right) _{\rho }+N\left( u_{\gamma },v-u_{\gamma }\right) _{\rho }+\frac{1}{%
\gamma }\left( S\left( u_{\gamma }\right) ,S\left( v-u_{\gamma }\right)
\right) _{L^{2}(\Omega )}\geq 0.  %\tag{23}
\end{equation}

Denote  $y_{\gamma }=y\left( u_{\gamma },0\right) $ and let $\sigma
_{\gamma }=\sigma \left( u_{\gamma }\right) $ be the unique solution of the
following variational equation
\begin{equation}\label{Haf24}
a\left( \sigma _{\gamma },q\right) =\int_{Q}\left( y_{\gamma }-y_{d}\right)
qdxdt\quad\forall q\in V\text{.}  %\tag{24}
\end{equation}

Consider the pair $\left( p_{\gamma },\lambda _{\gamma }\right) $ given by
\begin{equation*}
p_{\gamma }=\frac{1}{\rho ^{2}}L\sigma _{\gamma },\quad\lambda _{\gamma
}=\left. \frac{1}{\rho ^{2}}\frac{\partial \sigma _{\gamma }}{\partial \nu }%
\right\vert _{\Sigma _{0}}\text{,}
\end{equation*}
then $\left( p_{\gamma },\lambda _{\gamma }\right) $ is the solution of the
following backward wave equation%
\begin{equation}\label{Haf25}
\left\{
\begin{array}{l}
Lp_{\gamma }=y_{\gamma }-y_{d} \\
p_{\gamma }\left( x,T\right) =0,\quad p_{\gamma }^{\prime }\left(
x,T\right) =0 \\
p_{\gamma }=\left\{
\begin{array}{c}
\lambda _{\gamma } \\
0%
\end{array}%
\right.%
\end{array}%
\right.
\begin{array}{l}
\text{in }Q\text{,} \\
\text{in }\Omega \text{,} \\
\text{on }\Sigma _{0}\text{,} \\
\text{on }\Sigma \diagdown \Sigma _{0}\text{,}%
\end{array}
%\tag{25}
\end{equation}
with the null controllability propriety
\begin{equation*}
p_{\gamma }\left( x,0\right) =p_{\gamma }^{\prime }\left( x,0\right) =0\text{%
.}
\end{equation*}
Rewrite the optimality condition  \eqref{Haf23} to be in the following
form%
\begin{equation*}
\big( Lp_{\gamma },y\left( v-u_{\gamma },0\right) \big) _{\rho }+\Big(
Nu_{\gamma }+\frac{1}{\gamma }S^{\ast }S\left( u_{\gamma }\right)
,v-u_{\gamma }\Big)_{\rho }\geq 0\quad\forall \text{\ }v\in
U_{ad}^{\rho }\text{,}
\end{equation*}
where
\begin{equation*}
\big( Lp_{\gamma },y\left( v-u_{\gamma },0\right) \big)_{\rho }=\left(
T^{\ast }\left( Lp_{\gamma }\right) ,v-u_{\gamma }\right) _{\rho }\quad\forall
\text{\ }v\in U_{ad}^{\rho }\text{,}
\end{equation*}
which gives optimality condition  \eqref{Haf22}.

The boundedness of $T$ follows from the continuity of the solution to
 \eqref{Haf2},  \eqref{Haf5} w.r.t. data. \hfill$\square$

 \section{No-regret control optimality system (Optimal control for the
ill-posed wave equation)}
\setcounter{equation}{0}
In this section, we will give an optimality system characterizing
the optimal control (or the no-regret control) solution to  \eqref{Haf12},
 \eqref{Haf13},  \eqref{Haf6},  \eqref{Haf15} by taking the limits of $%
u_{\gamma }$, $y_{\gamma }$, $p_{\gamma },\theta \left( u_{\gamma },0\right)
$ and $\lambda _{\gamma }$ when $\gamma \rightarrow 0$.

\begin{theoen}
There exists a positive constant $C$ independent of $\gamma $ such that
\begin{gather} \label{Haf26a} \tag{5.1a}
\left\Vert u_{\gamma }\right\Vert _{\rho }\leq C,\quad\left\Vert
y_{\gamma }-y_{d}\right\Vert _{\rho }\leq C,\quad\left\Vert y_{\gamma
}\right\Vert _{\rho }\leq C,\quad\left\Vert S\left( u_{\gamma }\right)
\right\Vert _{L^{2}(\Omega )}\leq C\sqrt{\gamma }, \\ %\tag{26.a}
\label{Haf26b} \tag{5.1b}
\left\Vert \sigma _{\gamma }\right\Vert _{\rho }\leq C,\quad\left\Vert \theta
\left( u_{\gamma },0\right) \right\Vert _{L^{2}\left( \Sigma _{0}\right)
}\leq C,\quad\left\Vert \lambda _{\gamma }\right\Vert _{L^{2}\left( \Sigma
_{0}\right) }\leq C, \\ %\tag{26.b}
\notag \left\Vert p_{\gamma }\right\Vert _{L^{\infty }\left( 0,T;H_{0}^{1}(\Omega)\right) }\leq C,\quad
\left\Vert p_{\gamma }^{\prime }\right\Vert_{L^{\infty
}\left( 0,T;L^{2}(\Omega )\right) }\leq C.   %\tag
\end{gather}
 \end{theoen}
 \proofen
 Let $u_{\gamma }$ is the unique solution for \eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}, \eqref{Haf19}, \eqref{Haf20}, then
\begin{equation*}
\mathcal{J}_{\rho }^{\gamma }\left( u_{\gamma }\right) \leq \mathcal{J}%
_{\rho }^{\gamma }\left( 0\right) =0\text{ i.e. }J_{\rho }\left( u_{\gamma
},0\right) +\frac{1}{\gamma }\left\Vert S\left( u_{\gamma }\right)
\right\Vert _{L^{2}(\Omega )}^{2}\leq J_{\rho }\left( 0,0\right)
\end{equation*}

which give \eqref{Haf26a}.

Choose $q=\sigma _{\gamma }$ in \eqref{Haf24} with Lax--Milgram theorem
stability estimates to prove that%
\begin{equation*}
\exists C>0:\left\Vert \sigma _{\gamma }\right\Vert _{\rho }\leq C\text{,}
\end{equation*}
where $C$ is independent of $\gamma $. From continuity and \eqref{Haf26a}, we deduce the boundedness of $\theta \left( u_{\gamma
},0\right) $ and $\lambda _{\gamma }$ in $L^{2}\left( \Sigma _{0}\right) $.

Multiply \eqref{Haf25} by $p_{\gamma }^{\prime }$, integrate by parts,
and use \eqref{Haf26a} with \eqref{Haf26b} to find%
\begin{equation*}
\left\Vert p_{\gamma }^{\prime }\right\Vert _{L^{\infty }\left(
0,T;L^{2}(\Omega )\right) }^{2}+\left\Vert p_{\gamma }\right\Vert
_{L^{\infty }\left( 0,T;H_{0}^{1}(\Omega )\right) }^{2}\leq C\left(
\left\Vert y_{\gamma }-y_{d}\right\Vert _{L^{2}\left( Q\right)
}^{2}+\left\Vert \lambda _{\gamma }\right\Vert _{L^{2}\left( \Sigma
_{0}\right) }^{2}\right) \leq C\text{.}
\end{equation*}
\hfill$\square$

  \begin{lemen}\label{Lem3}
 The low-regret control $u_{\gamma }$ solution to \eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}, \eqref{Haf19}, \eqref{Haf20} converges in $%
U_{ad}^{\rho }$ to the no-regret control $u$ solution to \eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}, \eqref{Haf15}.
  \end{lemen}
\proofen
By \eqref{Haf26a}, we have%
\begin{equation*}
\left\Vert Ly_{\gamma }\right\Vert _{\rho }\leq C
\end{equation*}
and%
\begin{equation*}
u_{\gamma }\rightharpoonup u\quad\text{weakly in }U_{ad}^{\rho }\text{,}
\end{equation*}%
\begin{equation*}
y_{\gamma }\rightharpoonup y\quad\text{weakly in }L_{\rho }^{2}\left( Q\right)
\text{.}
\end{equation*}
And by \eqref{Haf26b} we have
\begin{equation*}
\theta \left( u_{\gamma },0\right) \rightharpoonup \theta \left( u,0\right)\quad
\text{weakly in }L^{2}\left( \Sigma _{0}\right) \text{.}
\end{equation*}
We conclude that $y$ solves%
\begin{equation*}
\left\{
\begin{array}{l}
\text{\ }Ly=u \\
y\left( 0\right) =0,\quad  y^{\prime }\left( 0\right) =0 \\
y\left( T\right) =0,\quad y^{\prime }\left( T\right) =0 \\
y=\left\{
\begin{array}{l}
\theta \left( u,0\right) \\
0%
\end{array}%
\right. \text{\ \ \ \ \ \ \ \ \ \ \ }%
\end{array}%
\right.
\begin{array}{l}
\text{in }Q, \\
\text{in }\Omega , \\
\text{in }\Omega , \\
\text{on }\Sigma _{0}\text{,} \\
\text{on }\Sigma \diagdown \Sigma _{0}.%
\end{array}%
\end{equation*}

Again, from \eqref{Haf26a}
\begin{equation*}
S\left( u_{\gamma }\right) \rightarrow 0\quad\text{strongly in}\quad L^{2}\left(
\Omega \right) \text{,}
\end{equation*}
then $\left( S\left( u\right) ,g\right) _{L^{2}\left( \Omega \right) }=0$
for every $g$ in $L^{2}\left( \Omega \right) $, which means that $u$ is a
no-regret control solution to \eqref{Haf12}, \eqref{Haf13}, \eqref{Haf6}, \eqref{Haf15}. \hfill$\square$\\[1ex]

Finally, we can announce the following our main theorem characterizing the
optimal control for the ill-posed wave equation \eqref{Haf13}.
 \begin{theoen}
 The no-regret control $u\in $ $U_{ad}^{\rho }$ solution to \eqref{Haf12},
\eqref{Haf13}, \eqref{Haf6}, \eqref{Haf16} is characterized by the
following optimality system%
\begin{equation}\label{Haf27}\tag{5.2}
\left\{
\begin{array}{ll}
\text{\ }Ly=u; &Lp=y-y_{d}\\
y\left( x,0\right) =0,\quad y^{\prime }\left( x,0\right) =0, & p\left( x,0\right) =0,\quad p^{\prime }\left( x,0\right) =0,\\
y\left( x,T\right) =0,\quad y^{\prime }\left( x,T\right) =0;
&p\left( x,T\right) =0,\quad
p^{\prime }\left( x,T\right) =0 \\
y=\left\{
\begin{array}{l}
\theta \left( u,0\right) \\
0%
\end{array}%
\right.; &p=\left\{
\begin{array}{l}
\lambda \\
0%
\end{array}%
\right.%
\end{array}%
\right.
\begin{array}{c}
\text{in }Q, \\
\\
\text{in }\Omega , \\
\text{on }\Sigma _{0}\text{,} \\
\text{on }\Sigma \diagdown \Sigma _{0}\text{,}%
\end{array}
%\tag{27}
\end{equation}

where $\lambda =\underset{\gamma \rightarrow 0}{\lim }\lambda _{\gamma },$ $%
y=y\left( u,0\right) $ and $p=p\left( u\right) $, with the following
variational inequality%
\begin{equation*}
\left( T^{\ast }\left( Lp\right) +N\rho ^{2}u+S^{\ast }S\left( u\right)
,v-u\right) _{L^{2}(Q)}\geq 0\quad \forall \ v\in U_{ad}^{\rho }\text{.}
\end{equation*}
 \end{theoen}
 \proofen
 We have already proved the convergence of $y_{\gamma }$ to $y$, and $%
u_{\gamma }$ to $u$ in the proof of Lemma~\ref{Lem3}. For the rest, use \eqref{Haf26a}
to get
\begin{equation*}
\left\Vert Lp_{\gamma }\right\Vert _{\rho }\leq C,
\end{equation*}
and \eqref{Haf26b}, to find
\begin{equation*}
\lambda _{\gamma }\rightharpoonup \lambda \text{ weakly in }L^{2}\left(
\Sigma _{0}\right),
\end{equation*}
Passing to the limit when $\gamma \rightarrow 0$ in \eqref{Haf21} we obtain
the optimality system \eqref{Haf27}.
\hfill$\square$





\section{Conclusion}
To sum up, our work leads to solving the optimal control problem for an
ill-posed wave equation without requiring the extra hypothesis of Slater.
The main idea was to make a null controllability approach to deal with  a
well-posed equation with a missing initial condition. Then, we have applied
the no-regret control method to solve the optimal control with incomplete
data. The optimality system describing the optimal control is built by an
overdetermined optimal state and adjoint state.

\section*{Acknowledgements}
%This work was supported by the Directorate-General for Scientific Research
%and Technological Development (DGRSDT).
The author wish to acknowledge the editor and anonymous reviewers for their
insightful comments, which have improved the quality of this publication.

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