\documentclass[JEP,XML,SOM,Unicode,NoEqCountersInSection]{cedram}
\datereceived{2016-06-18}
\dateaccepted{2017-01-26}
\dateepreuves{2017-02-08}

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\begin{document}
\frontmatter
\title{Local exact controllability of the~$2$D-Schrödinger-Poisson system}

\author[\initial{K.} \lastname{Beauchard}]{\firstname{Karine} \lastname{Beauchard}}
\address{IRMAR, École Normale Supérieure de Rennes, UBL\\
Avenue Robert Schumann, 35170 Bruz, France}
\email{Karine.Beauchard@ens-rennes.fr}
\urladdr{http://w3.bretagne.ens-cachan.fr/math/people/karine.beauchard/}

\author[\initial{C.} \lastname{Laurent}]{\firstname{Camille} \lastname{Laurent}}
\address{Sorbonne Universités, CNRS UMR 7598 and UPMC Univ.\ Paris 06, Laboratoire Jacques-Louis Lions\\
F-75005, Paris, France}
\email{camille.laurent@upmc.fr}
\urladdr{https://www.ljll.math.upmc.fr/~laurent/}

\thanks{The authors were partially supported by the ``Agence Nationale de la Recherche'' (ANR), Projet Blanc EMAQS number ANR-2011-BS01-017-01.}

\begin{abstract}
In this article, we investigate the exact controllability of the $2D$-Schrödinger-Poisson system, which couples a Schrödinger equation on a bounded domain of $\mathbb{R}^2$ with a Poisson equation for the electrical potential. The control acts on the system through a Neumann boundary condition on the potential, locally distributed on the boundary of the space domain. We prove several results, with or without nonlinearity and with different boundary conditions on the wave function, of Dirichlet type or of Neumann type.
\end{abstract}

\subjclass{35Q40, 35Q41, 93C10, 93C20}

\keywords{Control of partial differential equations, Schrödinger-Poisson
system, bilinear control}

\altkeywords{Contrôle d'équations aux dérivées partielles, système de
Schrödinger-Poisson, contrôle bilinéaire}

\alttitle{Contrôlabilité locale exacte du système de Schrödinger-Poisson
2D}

\begin{altabstract}
Dans cet article, nous étudions la contrôlabilité exacte du système de Schrödinger-Poisson 2D, qui couple une équation de Schrödinger sur un ouvert borné 2D, avec une équation de Poisson pour le potentiel électrique. Le contrôle agit sur le système via une condition de Neumann sur le potentiel, localement distribuée sur le bord du domaine spatial. Nous démontrons plusieurs résultats, avec ou sans non-linéarité, avec différents types de conditions de bord sur la fonction d'onde, de type Dirichlet ou de type Neumann.
\end{altabstract}

\maketitle
\vspace*{\baselineskip}\enlargethispage{-\baselineskip}
\tableofcontents
\mainmatter

\section{Introduction}

Microelectronics industry has driven transistor sizes to the nanometer
scale. This has led to the possibility of building nanostructures like
single electron transistors or single electron memories, which involve the
transport of only a few electrons. In general, such devices consist in an
active region, on which the electrical potential can be tuned by an
electrode (the gate). In many applications, the performance of the device
will depend on the possibility of controlling the electrons by acting on
the gate voltage.

At the nanometer scale, quantum effects become important and a quantum
transport model is necessary. In this paper, we analyze the
controllability of a simplified mathematical model, of the quantum
transport of electrons trapped in a two-dimensional device. The model
consists in a single Schrödinger equation, on a 2D bounded domain,
coupled to the Poisson equation for the electrical potential. The control
acts on the system through a Neumann boundary condition on the potential,
on a part of the boundary, modeling the gate. Such a model has already
been studied in \cite{Mehats_P_S} for controllability purposes.

It would be physically relevant to impose Dirichlet boundary conditions on
the wave function and to take into account the self-consistent potential
modeling interactions between electrons. However, the mathematical
analysis of this configuration is quite complicated, thus we investigate
two simpler configurations:
\begin{enumerate}
\item a first configuration, in which Dirichlet boundary conditions are
imposed on the wave function, but we neglect the self-consistent
potential,
\item a second configuration, in which we take into account the
self-consistent potential, but Neumann boundary conditions are imposed on
the wave function.
\end{enumerate}
This work is a first step towards more realistic models.

\subsection{Linear PDE with Dirichlet boundary conditions}

First, we consider the system
\begin{equation} \label{Schro_Poisson}
\begin{cases}
(i\partial_t + \Delta) \psi(t,x) = v(t,x) \psi(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\psi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\psi(0,x)=\psi_0(x) & x \in \Omega, \\
(-\Delta+1)v(t,x)=0, & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} v(t,x)=g(t,x) 1_{\Gamma_c}(x), & (t,x) \in (0,T)\times\partial \Omega,
\end{cases}
\end{equation}
where $\Omega$ is a bounded open subset of $\mathbb{R}^2$,
$\Gamma_c$ is an open subset of $\partial \Omega$
and $1_{\Gamma_c}$ is its characteristic function.
The control is the real valued function $g$ and we want to control the wave function $\psi$.

\subsubsection{Definitions and notation}

Let $\Omega$ be either a smooth bounded open subset of~$\R^2$ or the rectangle $(0,\pi) \times (0,L)$ with $L>0$.

We denote
by $\langle.\,,.\rangle$ the complex valued scalar product on $L^2(\Omega,\mathbb{C})$,
$$\langle f, g \rangle := \int_{\Omega} f(x) \overline{g(x)} \,dx,$$
by $\mathcal{S}$ the $L^2(\Omega,\mathbb{C})$-sphere
$$\mathcal{S}:=\{ \xi \in L^2(\Omega,\mathbb{C}) \sep \|\xi\|_{L^2(\Omega)}=1 \},$$
by $(-\Delta_D)$ the Laplace operator associated with Dirichlet boundary conditions
$$D(-\Delta_D):=H^2 \cap H^1_0(\Omega,\mathbb{C}), \quad -\Delta_D \xi:=-\Delta \xi,$$
by $(\lambda_k)_{k \in \mathbb{N}^*}$ the nondecreasing sequence of its eigenvalues,
by $(\varphi_k)_{k \in \mathbb{N}^*}$ one associated orthonormal basis of eigenfunctions,
$$\begin{cases}
- \Delta \varphi_k (x) = \lambda_k \varphi_k(x), \quad & x \in \Omega, \\
\varphi_k(x)=0, & x \in\partial \Omega, \\
\|\varphi_k \|_{L^2(\Omega)}=1,
\end{cases}$$
by $H^3_{(0)}(\Omega)$ the Sobolev space
$$H^3_{(0)}(\Omega):= D\big((-\Delta_D)^{3/2} \big) = \big\{ \xi \in H^3 \cap H^1_0(\Omega,\mathbb{C}) \sep \Delta \xi \in H^1_0(\Omega,\mathbb{C}) \big\},$$
and by $\mathbb{P}_K$, for $K \in \mathbb{N}^*$, the projection
\begin{align*}
\mathbb{P}_K: H^{-1}(\Omega,\mathbb{C}) & \to \text{Adh}_{H^{-1}(\Omega)} \left(\text{Span}(\varphi_k\sep k\geqslant K) \right) \\
& \xi \mto \xi-\sum_{k=1}^{K-1} \langle \xi, \varphi_k \rangle_{H^{-1},H^1_0}\, \varphi_k,
\end{align*}
where $\langle.\,,.\rangle_{H^{-1},H^1_0}$ is the duality product between $H^{-1}(\Omega)$ and $H^1_0(\Omega)$.
We also use the weak observability of the Schrödinger equation, defined below;
several configurations $(\Omega,\Gamma_c)$ for which it has been proved are recalled in Section \ref{subsec:H2}.

\begin{Def}[Observability and weak observability of Schrödinger equation] \label{Def:Weak_Obs_of_Schro}
Let $\Gamma$ be an open subset of $\partial \Omega$ and $0 \leqslant \tau_1<\tau_2 < \infty$.
The Schrödinger equation on $\Omega$ is observable (\resp weakly observable) on $(\tau_1,\tau_2)\times\Gamma$ if
there exists $\mathcal{C}_0>0$ such that
\begin{equation}
\label{Obs_Schro}\begin{split}
\|\phi_T\|_{H^1_0(\Omega)} &\leqslant \mathcal{C}_0 \|\partial_{\nu} \phi \|_{L^2((\tau_1,\tau_2)\times\Gamma)},
\quad \forall \phi_T \in H^1_0(\Omega,\mathbb{C})\\
(\resp
\|\phi_T\|_{H^1_0(\Omega)} &\leqslant \mathcal{C}_0 \big(\|\partial_{\nu} \phi \|_{L^2((\tau_1,\tau_2)\times\Gamma)}
+ \|\phi_T\|_{H^{-1}(\Omega)} \big), \quad \forall \phi_T \in H^1_0(\Omega,\mathbb{C})),
\end{split}
\end{equation}
where $\phi(t):=e^{i\Delta_D(t-T)} \phi_T$, \ie $\phi$ is the solution of
\begin{equation} \label{Adjoint}
\begin{cases}	
\big(i\partial_t +\Delta\big) \phi(t,x)=0, \quad & (t,x) \in (0,T)\times\Omega, \\
\phi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\phi(T,x)=\phi_T(x), & x \in \Omega.
\end{cases}
\end{equation}
The Schrödinger equation on $\Omega$ is observable (\resp weakly observable) on $\Gamma$ if
it is observable (\resp weakly observable) on $(\tau_1,\tau_2)\times \Gamma$ for some $0\leqslant \tau_1 < \tau_2 < \infty$.
\end{Def}

In the inequality \eqref{Obs_Schro}, the last term $\|\phi_T\|_{H^{-1}(\Omega)}$ may be equivalently
replaced by any term of the form $\|\phi_T\|_{H^{s}(\Omega)}$ with $-1<s<1$, by an interpolation argument.

\subsubsection{Local exact controllability of high frequencies}

Our first result is the local exact controllability of high frequencies of the system \eqref{Schro_Poisson}, around any trajectory of the free system.

\begin{thm} \label{Thm:controle_HF}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^2$,
$\Gamma_c$ be a non-empty open subset of $\partial \Omega$,
$T>0$, $\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$ and $\psi_{\rref}(t):=e^{i \Delta_D t} \psi_0$.
We assume that
\begin{itemize}
\item[\HH1] either $\Omega$ is of class $C^{\infty}$ and locally on one side of $\partial \Omega$;
or $\Omega = (0,\pi)\times(0,L)$ for some $L>0$ and $\overline\Gamma_c$ does not contain any vertex of $\Omega$,
\item[\HH2] the Schrödinger equation on $\Omega$ is weakly observable on $(0,\widetilde{T})\times\Gamma_c$
for some $\widetilde{T} \in (0,T)$,
\item[\HH3] $|\partial_{\nu} \psi_{\rref}(t,x)| \geqslant m >0$, $\forall (t,x) \in (T',T'')\times\Gamma_c$
for some $T', T'' \in [0,T]$ such that $T'' - T' > \widetilde{T}$.
\end{itemize}
Then, there exists $K, \delta>0$ and a $C^1$-map
$$\Upsilon:\mathcal{V} \to L^2((0,T)\times\Gamma_c,\mathbb{R}),$$
where $\mathcal{V}:=\{\psi_f \in \mathbb{P}_K[H^3_{(0)}(\Omega,\mathbb{C})] \sep \| \psi_f - \mathbb{P}_K[\psi_{\rref}(T)] \|_{H^3_{(0)}} < \delta \}$,
such that
\begin{itemize}
\item $\Upsilon(\mathbb{P}_K[\psi_{\rref}(T)])=0$,
\item for every $\psi_f \in \mathcal{V}$, the solution of \eqref{Schro_Poisson} with control $g=\Upsilon(\psi_f)$ satisfies
\[
\mathbb{P}_K[\psi(T)]=\psi_f.
\]
\end{itemize}
As a consequence, there exists $K' \geqslant K$ and $g \in L^2((0,T)\times\Gamma_c,\mathbb{R})$ such that the solution of \eqref{Schro_Poisson}
satisfies $\mathbb{P}_{K'}[\psi(T)]=0$; in particular, $x \mto \psi(T,x)$ is a smooth function.
\end{thm}

Such a result may be used to prove global exact controllability of the Schrödinger-Poisson system in $H^3_{(0)}(\Omega,\mathbb{C})$
with controls $g \in L^2((0,T),\mathbb{R})$, by following the strategy of \cite{KB_CL_HF}. This will be at the core of future works by the authors.

In particular, Theorem \ref{Thm:controle_HF} applies, for arbitrary $T>0$ and $\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$,
when (see Propositions \ref{Prop:[H2]_exemples} and \ref{Prop:hyp_HF})
\begin{itemize}
\item $\Omega = (0,\pi)\times(0,L)$ for some $L>0$ and $\Gamma_c$ contains both a horizontal and a vertical segment,
\item $\Omega$ is a disk and $\Gamma_c$ is arbitrary.
\end{itemize}
When the Geometric Control Condition is fulfilled for $(\Omega,\Gamma_c)$, then Theorem \ref{Thm:controle_HF} applies for any $T>0$ if
$\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$ satisfies \HH3
(see Proposition \ref{Prop:[H2]_exemples}).

Assumption \HH1 is important for the system \eqref{Schro_Poisson} to be well-posed in $H^3_{(0)}(\Omega)$
with control $g \in L^2((0,T)\times\Gamma_c,\mathbb{R})$ (see Section \ref{subsec:WP}).
When $\Omega$ is a rectangle, it is important to assume that its vertices do not belong to $\overline\Gamma_c$,
in order to take advantage of the usual elliptic regularity on the potential $v$.

\subsubsection{Local exact controllability around an eigenstate}

Our second result is the local exact controllability of the system \eqref{Schro_Poisson} around an eigenstate.

\begin{thm} \label{thm:control_eig}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^2$,
$\Gamma_c$ be an open subset of $\partial \Omega$,
$T>0$, $R \in \mathbb{N}^*$ and $\psi_{\rref}(t):=\varphi_R(x)e^{-i\lambda_R t}$.
We assume \HH1, \HH2,
\begin{itemize}
\item[\HH{3'}] $|\partial_{\nu} \varphi_R(x)| \geqslant m >0, \forall x \in \Gamma_c$,
\item[\HH4] $\lambda_R$ is a simple eigenvalue of $(-\Delta_D)$
and for any eigenvector $\Phi$ of $(-\Delta_D)$, the solution $w$ of
$$\begin{cases}
(-\Delta+1)w(x)=\varphi_R(x) \Phi(x), \quad & x \in \Omega, \\
\partial_{\nu} w(x)=0, & x \in\partial \Omega,
\end{cases}$$
does not identically vanish on $\Gamma_c$: $w_{\left|\Gamma_c\right.} \not \equiv 0$.
\end{itemize}

Then, there exists $ \delta>0$ and a $C^1$-map
$$\Upsilon:\mathcal{V} \to L^2((0,T)\times\Gamma_c,\mathbb{R}),$$
where
\[
\mathcal{V}:=\big\{(\psi_0,\psi_f) \in [ H^3_{(0)}(\Omega,\mathbb{C}) \cap \mathcal{S} ]^2 \sep
\| \psi_0 - \psi_{\rref}(0) \|_{H^3_{(0)}} + \| \psi_f - \psi_{\rref}(T) \|_{H^3_{(0)}} < \delta \big\},
\]
such that $\Upsilon[ \psi_{\rref}(0), \psi_{\rref}(T)]=0$ and,
for every $(\psi_0,\psi_f) \in \mathcal{V}$, the solution of \eqref{Schro_Poisson} with control $g=\Upsilon(\psi_0,\psi_f)$ satisfies
$\psi(T)=\psi_f$.
\end{thm}

Note that, under Assumption \HH1, then Assumption \HH{3'} systematically holds with $R=1$.
The assumptions of Theorem \ref{thm:control_eig} could look quite technical, but roughly speaking the main assumptions are:
\begin{itemize}
\item the weak observability (Assumption \HH2), that ensures the controllability of high frequencies, as in Theorem \ref{Thm:controle_HF},
\item the unique continuation property (Assumption \HH4), that ensures the controllability of low frequencies, see Proposition \ref{Prop_LF}).
\end{itemize}
If \HH2 holds, but \HH4 is not satisfied, then the linearized system around $\psi_{\rref}$ is not controllable:
it misses a finite number of directions corresponding to the eigenfunctions~$\Phi$ for which \HH4 is not satisfied, see Remark \ref{rkmisseddirection}.
It would be interesting to recover these directions by power series expansions \cite[Chap.\,8]{Coronlivre}.

Theorem \ref{thm:control_eig} applies, in particular, when $\Omega = (0,\pi)\times(0,L)$ for some $L>0$,
$\Gamma_c$ is an open subset of $\partial \Omega$ that contains both a horizontal and a vertical segment
and $\lambda_R$ is a simple eigenvalue of $(-\Delta_D)$ (see Proposition \ref{Prop:[H4]_rect}).

The validity of Theorem \ref{thm:control_eig} on the disk $\Omega=\{(x,y) \in \mathbb{R}^2 \sep x^2 + y^2 < 1 \}$ is an open problem:
checking Assumption \HH4 would require some study on the product of Bessel functions, which might require lengthy computations.

Following arguments by Méhats, Privat and Sigalotti \cite{Mehats_P_S}
(relying on methods of Privat-Sigallotti \cite{PrivatSigal}),
it should be possible to prove that \HH4 holds generically with respect to the domain $\Omega$.
This becomes clearer with some equivalent ways of expressing Assumption \HH4 described in Proposition \ref{propequivUCP}.

Assumption \HH4 is quite unusual in control theory.
It looks like a classical unique continuation property for eigenfunctions but it does not seem that we can refer to some already
known results to prove it in great generality. We have indeed chosen to discuss about it more precisely in Subsection \ref{subsec:H4}.

Exact controllability as in Theorem \ref{thm:control_eig} probably holds close to
any reference trajectory $\psi_{\rref}$ as done in Theorem \ref{Thm:controle_HF}. Yet, Assumption \HH4 should be replaced by a unique continuation assumption complicated to state and depending on time. We have chosen to state Theorem \ref{thm:control_eig} only close to an eigenfunction because the unique continuation assumption takes the nice form of \HH4.

\Subsection{Controllability of Schrödinger equation with real valued control}

For the Schrödinger-Poisson system, the control $g$ corresponds to a potential and therefore needs to be real valued to have a physical meaning.
Therefore, as an inter\-mediary result, we will also get the exact controllability of Schrödinger equation with
\emph{real valued} controls (instead of complex valued ones in the existing literature),
when the equation is weakly observable. The result that we need for the control of the Schrödinger-Poisson system will actually be more complicated.
Yet, we believed that it could be useful for other contexts and we give a simpler proof in the simpler context
of the free Schrödinger equation (see for instance Araruna-Cerpa-Mercado-Santos \cite{ArarunaAlSchrodKdV} where related questions are raised).

\begin{thm} \label{thm:anecdotic}
Let $\Omega$ be an open subset of $\mathbb{R}^2$, $\Gamma$ be an open subset of $\partial \Omega$ and \hbox{$0 < \widetilde{T} < T < \infty$}.
If the Schrödinger equation on $\Omega$ is weakly observable on $(0,\widetilde{T})\times\Gamma$
then, for every $\psi_f \in H^{-1}(\Omega,\mathbb{C})$, there exists a \emph{real valued} control
$u \in L^2((0,T)\times\nobreak\partial\Omega,\mathbb{R})$ such that the solution of
$$\begin{cases}
\left(i\partial_t + \Delta \right) \psi=0, \quad & (t,x) \in (0,T)\times\Omega, \\
\psi(t,x)=u(t,x)1_{\Gamma}(x), & (t,x) \in (0,T)\times\partial\Omega, \\
\psi(0,x)=0, & x \in \Omega,
\end{cases}$$
satisfies $\psi(T)=\psi_f$.
\end{thm}
The proof will be given in Section \ref{sectrealcontrol}.

\Subsection{Nonlinear PDE, on a rectangle, with Neumann boundary conditions}

\label{subsectionintrononlin}
Now, we consider the nonlinear Schrödinger-Poisson system on a rectangle
\begin{equation} \label{NL_syst_toy_init}
\begin{cases}
i\partial_t \psi(t,x) = - \Delta \psi(t,x) + \widetilde{v}(t,x) \psi(t,x), & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu \psi(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\psi(0,x)=\psi_0(x), & x \in \Omega, \\
(-\Delta+1) \widetilde{v}(t,x)=\epsilon |\psi(t,x)|^2, & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu \widetilde{v}(t,x)=g(t,x) 1_{\Gamma_c}(x), & (t,x) \in (0,T)\times\partial \Omega,
\end{cases}
\end{equation}
where
$\epsilon \in \mathbb{R}$,
$x=(x_1,x_2) \in \Omega:=(0,\pi) \times (0,L)$, $L>0$\, and
$\Gamma_c$ is an open subset of $\partial \Omega$.
The control is the real valued function $g$ and we want to control the wave function~$\psi$.

For the nonlinear system \eqref{NL_syst_toy_init}, our main result is the local exact controllability around the reference trajectory, constant in space,
\begin{equation} \label{Traj_Ref}
\Big(\psi_{\rref}(t,x)=\frac{e^{-i\sfrac{\epsilon t }{\sqrt{\pi L}} }}{\sqrt{\pi L}}, \widetilde{v}_{\rref}(t,x)=0, g_{\rref}(t,x)=0 \Big).
\end{equation}
We denote by $\Delta_N$ the Laplace operator associated with Neumann boundary conditions
\begin{align*}
D(\Delta_N)&=H^2_{N}(\Omega,\mathbb{C}):=\{ \varphi \in H^2 (\Omega,\mathbb{C}) \sep\partial_\nu \varphi = 0 \text{ on }\partial \Omega \}
\\
\Delta_N \varphi :&= \Delta \varphi.
\end{align*}

\begin{thm} \label{Main_Thm_loc}
Let $L>0$, $\Omega:=(0,\pi)\times(0,L)$,
$\Gamma_c$ be an open subset of $\partial \Omega$ such that~$\overline\Gamma_c$ does not contain any vertex of $\Omega$,
$T>0$, $\epsilon \in \mathbb{R}$ be such that
\begin{equation} \label{hyp:epsilon}
\epsilon > - \frac{\pi L}{2} m(m+1)^2, \quad \text{ where } \quad m:=\min\big\{ 1 \sep \left(\sfrac{\pi}{L}\right)^2 \big\}.
\end{equation}
and $\psi_{\rref}$ be defined by \eqref{Traj_Ref}.
There exists $\delta>0$ and a $C^1$-map
$$\Upsilon:\mathcal{V} \to L^2((0,T)\times\partial \Omega,\mathbb{R}),$$
where
$$\mathcal{V} := \left\{ (\psi_0,\psi_f) \in [ H^2_{N}(\Omega,\mathbb{C}) \cap \mathcal{S} ]^2 \sep
\|\psi_0-\psi_{\rref}(0)\|_{H^2} + \|\psi_f-\psi_{\rref}(T)\|_{H^2} < \delta \right\},$$
such that $\Upsilon(\psi_{\rref}(0),\psi_{\rref}(T))=0$ and for every $(\psi_0,\psi_f) \in \mathcal{V}$, the solution of
\eqref{NL_syst_toy_init} with control $g=\Upsilon(\psi_0,\psi_f)$ satisfies $\psi(T)=\psi_f$.
\end{thm}

This part with the nonlinear Schrödinger equation only deals with $\Omega$ a rectangle, where the boundary is flat.
Indeed, with the Neumann boundary control, the smoothing effect, required in our proof, is not well-understood for a general open set $\Omega$.
The nonhomogeneous boundary Cauchy problem is then quite complicated.
This was for instance investigated for the wave equation by Tataru \cite{Tataruwavetrace} and
earlier papers by Lasiecka and Triggiani \cite{LasiTrigrregwave}. The curvature of the boundary has important consequences in this case.

\Subsection{Bibliography}

\subsubsection{Schrödinger equation with bilinear control}

The Schrödinger equation with bilinear control has been widely studied in the literature,
and is related to the system under study in this article. This model reads
\begin{equation} \label{Schro_BL}
\begin{cases}
(i\partial_t + \Delta - V) \psi(t,x)=u(t) \mu(x) \psi(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\psi(t,x)=0, & (t,x) \in (0,T) \times\partial \Omega,
\end{cases}
\end{equation}
where $V, \mu:\Omega \to \mathbb{R}$ are given functions,
the state $\psi$ lives in $\mathcal{S}$
and the control is the real valued function $u:(0,T)\to \mathbb{R}$.
When $V$ and $\mu$ solve an appropriate Poisson equation,
then the system \eqref{Schro_BL} is a particular case of the Schrödinger-Poisson system~\eqref{Schro_Poisson}
(take $v(t,x)=V(x)+u(t)\mu(x)$).

\subsubsection*{A negative result}

A negative control result was proved by Turinici in \cite{Defranceschi-LeBris},
as a consequence of a general result by Ball, Marsden and Slemrod in \cite{ball-marsden-slemrod}.
It states that,
for $V=0$,
for a given function $\mu \in C^2(\Omega,\mathbb{R})$,
for a given initial condition $\psi_0 \in (H^2 \cap H^1_0)(\Omega,\mathbb{C}) \cap \mathcal{S}$,
and by using controls $u \in L^{r}_\loc((0,\infty),\mathbb{R})$ with $r>1$,
one may only reach a subset of $ (H^2 \cap H^1_0)(\Omega) \cap \mathcal{S}$ that has an empty interior in \hbox{$(H^2 \cap H^1_0)(\Omega,\mathbb{C}) \cap \mathcal{S}$}.
Recently, Boussaid, Caponigro and Chambrion extended this negative result to the case of controls in $L^1_\loc((0,\infty),\mathbb{R})$, see \cite{Boussaid_al_hal}.
However, this negative results are actually due to an inappropriate choice of functional setting, as emphasized in the next paragraph.

Note that this type of negative results is specific to systems with a bilinear control, which is a function of the time variable $t$
(and not a function of both $t$ and $x$). Thus, it does not apply to the Schrödinger-Poisson system \eqref{Schro_Poisson}
studied in this article. See also Section \ref{sectioncommentsut} for some comments about this type of control in our context.

\subsubsection*{Local exact results in 1D}

Beauchard proved in \cite{KB-JMPA} the exact controllability of Equation \eqref{Schro_BL},
locally around the ground state in $H^7$,
with controls $u \in H^1((0,T),\mathbb{R})$
in large time $T$, in the case $N=1$, $\Omega=(-1/2,1/2)$, $\mu(x)=x$ and $V=0$.
The proof of~\cite{KB-JMPA} relies on Coron's return method and Nash-Moser's theorem.

The reference \cite{KB-CL}, by the authors of this paper, improves this result and establishes the exact controllability of Equation \eqref{Schro_BL},
locally around the ground state in $H^3$,
with controls $u \in L^2((0,T),\mathbb{R})$,
in arbitrary time $T>0$,
and with generic functions $\mu$ when $N=1$, $\Omega=(0,1)$.
This result, proved with $V=0$ in \cite{KB-CL}, can be extended to an arbitrary potential $V$, as explained in \cite{Morancey_Nersesyan_simult}.
The proof relies on a smoothing effect, that allows one to conclude with the inverse mapping theorem (instead of Nash-Moser's one).

Then, Morancey and Nersesyan developed this strategy
to control a Schrödinger equation with a polarizability term \cite{Morancey_Nersesyan_polar}
and a finite number of Schrödinger equations with one control \cite{Morancey_simult, Morancey_Nersesyan_simult}.

The goal of this article is to extend this type of strategy (previously applied in~1D)
to the 2D Schrödinger-Poisson system \eqref{Schro_Poisson}. New difficulties appear at the two levels of the proof:
\begin{itemize}
\item in controlling the linearized system, because a new observability inequality is required,
\item in proving the appropriate smoothing effect or the 2D-system \eqref{Schro_Poisson}.
\end{itemize}

\subsubsection*{Global approximate results}

Three strategies have been developed to study approximate controllability for Equation \eqref{Schro_BL}

The first strategy is a variational argument introduced by Nersesyan in \cite{Nersesyan_CMP}.
It proves the global controllability to the ground state,
approximately in $H^3$,
with smooth controls $u \in C^\infty_c((0,T),\mathbb{R})$,
in large time $T$,
for generic functions $(\mu,V)$,
in arbitrary dimension $N$.

Note that this global approximate control result may be coupled to the previous local exact controllability results to provide
global exact controllability, see
\cite{Nersesyan_IHP} for Equation \eqref{Schro_BL},
\cite{Morancey_Nersesyan_polar} for a Schrödinger equation with a polarizability term,
\cite{Morancey_Nersesyan_simult} for finite number of Schrödinger equations with the same control.

A second strategy consists in deducing approximate controllability in regular spaces (containing $H^3$) from
exact controllability results in infinite time by Nersesyan and Nersisyan \cite{Nersesyan_Nersisyan_JMPA}

A third strategy, due to Chambrion, Mason, Sigalotti, and Boscain \cite{Chambrion-et-al},
relies on geometric techniques for the controllability of the Galerkin approximations.
It proves (under appropriate assumptions on $V$ and $\mu$) the approximate controllability of \eqref{Schro_BL} in $L^2$,
with piece-wise constant controls.
The hypotheses of this result were refined by Boscain, Caponigro, Chambrion, and Sigalotti in \cite{Boscain_al_CMP}.
The approximate controllability is proved in higher Sobolev norms in \cite{Boussaid_al_hal} for one equation,
and in \cite{Boscain_al_JDE} for a finite number of equations with one control.
For more details and more references about the geometric techniques, we refer the reader to the recent survey \cite{Boscain_al_survey}.

\subsubsection{Schrödinger-Poisson system}

In \cite{Mehats_P_S}, Méhats, Privat and Sigalotti prove the approximate controllability in $L^2$
for a Schrödinger-Poisson system, with mixed boundary conditions
(of Dirichlet or Neumann type, depending on the place where we are on the boundary).
This result holds for generic domains $\Omega$ and generic control supports $\Gamma_c$ on the boundary.
The proof relies on the general result of \cite{Chambrion-et-al} and analyticity arguments to obtain genericity.

\subsection{Structure of the article}

This article is organized in 7 Sections.

Section 2 is a discussion about our assumptions \HH2, \HH3 and \HH4,
in particular when $\Omega$ is a rectangle domain or a disk.

Section 3 is dedicated to the well-posedness of the system \eqref{Schro_Poisson}, in appropriate spaces for our control problem.
The starting point is a smoothing effect proved by Puel in~\cite{Puelsmoothing}, that needs to be recast in our context.

In Section 4, we prove local exact controllability of high frequencies, around any trajectory of the free system, \ie Theorem \ref{Thm:controle_HF}.

Section 5 is devoted to the proof of the local exact controllability, around any eigenstate, \ie Theorem \ref{thm:control_eig}.

The goal of Section 6 it to explain how the strategy of Section 5
can be used to get controllability for the Schrödinger equation with \emph{real-valued} boundary controls
(instead of complex valued ones in the literature), \ie Theorem \ref{thm:anecdotic}.

Finally, in Section 7, we prove the local exact controllability of the nonlinear Schrödinger-Poisson system \eqref{NL_syst_toy_init}
on a rectangle, \ie Theorem \ref{Main_Thm_loc}.

\subsection{Notation}

Implicitly functions take values in $\mathbb{C}$, otherwise we specify, for instance $L^2((0,T)\times\partial \Omega, \mathbb{R})$.
The volume element on $\Omega$ is denoted $dx$ and the surface element on $\partial \Omega$ is denoted $d\sigma(x)$.
When $\varphi \in \mathcal{S}$ then $T_{\mathcal{S}} \varphi$ denotes the tangent space to the sphere $\mathcal{S}$ at point $\varphi$
$$T_{\mathcal{S}} \varphi := \left\{ \xi \in L^2(\Omega,\mathbb{C}) \sep \Re \left(\int_{\Omega} \varphi(x) \overline{\xi(x)} \,dx \right) = 0 \right\}.$$

\section{Comments on our assumptions}
\label{sec:Discussion on our assumptions}

The goal of this section is to prove that our assumptions \HH2, \HH3 and \HH4
hold for appropriate configurations $(\Omega,\Gamma_c)$ and to explain why \HH4 is related to the control of low frequencies.

\subsection{Weak observability of Schrödinger equation \HH2}
\label{subsec:H2}

The goal of this section is to recall known results about Assumption \HH2.

\begin{Prop} \label{Prop:[H2]_exemples}
Property \HH2 holds in the following cases.
\begin{enumerate}
\item $\Omega=(0,\pi) \times (0,L)$ for some $L>0$, $\widetilde{T}>0$ and $\Gamma_c$ is an open subset of $\partial \Omega$
that contains both a horizontal and a vertical segment with non-zero length
(necessary and sufficient condition).
\item $\Omega$ is smooth, does not have a contact of infinite order with its tangents and $(\Omega,\Gamma_c)$ satisfies the Geometric Control Condition.
\item $\Omega=\left\{(x,y)\sep x^2+y^2<1\right\}$ and $\Gamma_c$ is an arbitrary open set of the boundary.
\end{enumerate}
\end{Prop}

Statement 1 is proved by Tenenbaum and Tucsnak in \cite[Th.\,1.4]{Tenenbaum}.
Precisely, they prove that the Schrödinger equation is observable on $(0,\widetilde{T}) \times \Gamma_c$ iff
$\Gamma_c$ contains both a horizontal and a vertical segment with non-zero length.
Thus weak observability also holds in this configuration.
Moreover, the same counter-example as in \cite[p.\,967]{Tenenbaum} proves that this condition is necessary:
with $\Gamma_c:=(a,b)\times\{0\}$, $0 \leqslant a<b \leqslant \pi$,
$m \in \mathbb{N}^*$ and $\phi_T(x,y):=\sin(mx)\sin(\pi y/L)$, we get
$$C m \leqslant \|\phi_T\|_{H^1_0(\Omega)} \quad \text{ and } \quad
\|\partial_{\nu} \phi \|_{L^2((T',T'')\times\Gamma_c)} + \|\phi_T\|_{H^{-1}(\Omega)} \leqslant C',$$
for some constants $C,C'>0$ independent of $m$.

Statement 2 is proved by Lebeau in \cite{control-lin1} when $\Omega$ is analytic.
The result is actually also true if $\Omega$ is smooth (see for instance Burq-Zworski \cite{Burq}).
The assumptions of contact of finite order is made in order to ensure the uniqueness of the broken geodesic flow.
It is certainly also true if $\Omega$ is only $C^3$ using the result by Burq \cite{Burqcontrolnonreg}
for the wave equation and transmutation or resolvent methods (see Miller \cite{MillerViolent,Miller} for instance).

Statement 3 is proved by Anatharaman, Léautaud and Macià in \cite{AnanthaLeauMaciDisk}.

\subsection{Simultaneous validity of \HH2 and \HH3}
\label{subsec:H23}

The goal of this section is to prove the claim written after Theorem \ref{Thm:controle_HF}.

\begin{Prop} \label{Prop:hyp_HF}
We assume that
\begin{itemize}
\item either $\Omega$ is a disk and $\Gamma_c$ is a non-empty open subset of $\partial \Omega$,
\item or $\Omega = (0,\pi)\times(0,L)$ for some $L>0$ and $\Gamma_c$ contains both a horizontal and a vertical segment.
\end{itemize}
Let $T>0$, $\psi_0 \in H^3_{(0)}(\Omega,\mathbb{C}) \cap \mathcal{S}$ and $\psi_{\rref}(t):=e^{i\Delta_D t}\psi_0$.
Then Assumptions \HH2 and~\HH3 are satisfied when $\Gamma_c$ is replaced by an appropriate non-empty open subset~$\Gamma_c'$ of~$\Gamma_c$.
As a consequence, the conclusion of Theorem \ref{Thm:controle_HF} holds.
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop:hyp_HF}]
Let us assume that $\Omega$ is a disk and $\Gamma_c$ is a non-empty open subset of $\partial \Omega$.
By unique continuation for the Schrödinger equation, we know that the continuous function $\partial_\nu \psi_{\rref}$ does not identically vanish on $(0,T) \times \Gamma_c$. This is a consequence of the Holmgren theorem (see Theorem 8.6.5 of \cite{HormanderI}) since $\partial \Omega$ is analytic here and any hypersurface $\left\{\Phi=0\right\}$ with $\nabla_x \Phi\neq 0$ is non-characteristic for the Schrödinger operator. We can also get this directly as a consequence of the observability estimate of \cite{AnanthaLeauMaciDisk}.
Thus, there exists $0 \leqslant T' < T'' \leqslant T$ and an open subset $\Gamma_c'$ of $\Gamma_c$ such that
$|\partial_\nu \psi_{\rref}(t,x)| \geqslant m >0$ for every $(t,x) \in (T',T'') \times \Gamma_c'$.
Let $\widetilde{T} \in (0,T''-T')$. By the previous proposition, the Schrödinger equation on $\Omega$ is
weakly observable on $(0,\widetilde{T})\times\Gamma_c'$.
Therefore, both \HH2 and \HH3 hold with $\Gamma_c$ replaced by $\Gamma_c'$.
Then, Theorem \ref{Thm:controle_HF} provides controls supported in $(0,T)\times\Gamma_c'$ that are, a fortiori, also supported in $(0,T)\times \Gamma_c$.

Now, let us assume that $\Omega = (0,\pi)\times(0,L)$ for some $L>0$ and $\Gamma_c$ contains both a horizontal segment $\Gamma_H$ and a vertical segment $\Gamma_V$.
By unique continuation for the Schrödinger equation (by application of the Holmgren theorem for instance), we know that $\partial_\nu \psi_{\rref}$ does not identically vanish on $(0,T) \times \Gamma_H$.
Thus, there exists $0 \leqslant T_H' < T_{H}'' \leqslant T$ and an open subset $\Gamma_{H}'$ of $\Gamma_H$ such that
$|\partial_\nu \psi_{\rref}(t,x)| \geqslant m_H >0$ for every $(t,x) \in (T_H',T_H'') \times \Gamma_H'$.
By unique continuation for the Schrödinger equation, we know that $\partial_\nu \psi_{\rref}$ does not identically vanish on $(T_H',T_H'') \times \Gamma_V$.
Thus, there exists $T_H' \leqslant T' < T'' \leqslant T_H''$ and an open subset $\Gamma_{V}'$ of $\Gamma_V$ such that
$|\partial_\nu \psi_{\rref}(t,x)| \geqslant m_V >0$ for every $(t,x) \in (T',T'') \times \Gamma_V'$.
Then $|\partial_\nu \psi_{\rref}(t,x)| \geqslant m := \min \{m_H\sep m_V\} >0$ for every $(t,x) \in (T',T'') \times \Gamma_c'$
with $\Gamma_c' := \Gamma_H' \cup \Gamma_V'$. The conclusion comes as in the previous case.
\end{proof}

\subsection{Unique continuation assumption \HH4}
\label{subsec:H4}

The goal of this section is to prove that \HH4 holds on rectangular domains
and is related to the controllability of low frequencies.

\Subsubsection{The case of a rectangle}

\begin{Prop} \label{Prop:[H4]_rect}
Let $L>0$,
$\Omega:=(0,\pi) \times (0,L)$,
$\Gamma_c$ be a non-empty open subset of~$\partial\Omega$,
$R_1, R_2 \in \mathbb{N}^*$ be such that $R_1^2 + \left(\sfrac{R_2 \pi}{L} \right)^2$ is a simple eigenvalue of $(-\Delta_D)$
and $\varphi_R(x,y) := \frac{2}{\sqrt{\pi L}} \sin(R_1 x) \sin\left(\sfrac{R_2 \pi y}{L} \right)$.
Then \HH4 is satisfied.
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop:[H4]_rect}]
Without loss of generality, one may assume that $\Gamma_c$ contains $(a,b) \times \{0\}$ for some $0 \leqslant a<b \leqslant \pi$.
Let $\lambda \in \text{Sp}(-\Delta_D)$, set
\[
J:=\{(p,n)\in (\mathbb{N}^*)^2 \sep p^2+\left(\sfrac{n\pi}{L} \right)^2 = \lambda \},
\]
which is a finite set,
and let
\[
\Phi=\sum_{(p,n) \in J} c_{p,n} \sin(px) \sin\left(\sfrac{n\pi y}{L} \right)
\]
be an eigenfunction of $(-\Delta_D)$ associated to the eigenvalue $\lambda$.
From the relation $2\sin(a)\sin(b)=\cos(a-b)-\cos(a+b)$ we deduce that
$$\varphi_R \Phi = \sum_{(p,n) \in J} \sum_{\substack{\epsilon_1=\pm 1,\\ \epsilon_2=\pm 1}}
\frac{c_{p,n} \epsilon_1 \epsilon_2 }{2 \sqrt{\pi L}}\, \cos[(p+\epsilon_1 R_1)x] \cos\left[ \sfrac{(n+\epsilon_2 R_2)\pi y}{L} \right]$$
thus
$$w(x,y)=\sum_{(p,n) \in J}\ \sum_{\substack{\epsilon_1=\pm 1,\\ \epsilon_2=\pm 1}}
\frac{c_{p,n} \epsilon_1 \epsilon_2 \cos[(p+\epsilon_1 R_1)x] \cos\left[ \sfrac{(n+\epsilon_2 R_2)\pi y}{L} \right]}{
2\sqrt{\pi L} \left[ (p+\epsilon_1 R_1)^2 + \left(\sfrac{(n+\epsilon_2 R_2)\pi}{L}\right)^2 +1 \right]
}$$
and in particular
$$w(x,0)=\sum_{(p,n) \in J}\ \sum_{\substack{{c}\epsilon_1=\pm 1,\\ \epsilon_2=\pm 1}}
\frac{c_{p,n} \epsilon_1 \epsilon_2 \cos[(p+\epsilon_1 R_1)x]}{2\sqrt{\pi L}\left[ (p+\epsilon_1 R_1)^2 + \left(\sfrac{(n+\epsilon_2 R_2)\pi}{L}\right)^2 +1 \right]}.$$
Let
\[
P:=\max\{p \in \mathbb{N}^* \sep \exists n \in \mathbb{N}^* \text{ such that } (p,n) \in J \text{ and } c_{p,n} \neq 0\}.
\]
Let $N \in \mathbb{N}^*$ be the unique integer such that $(P,N) \in J$. Then $x \mto w(x,0)$ is a finite sum of cosine functions
$$w(x,0)=\sum_{k=0}^{P+R_1} \alpha_k \cos(kx)$$
and its fastest oscillating term has coefficient
\begin{multline*}
\alpha_{P+R_1}=\frac{c_{P,N}}{2\sqrt{\pi L}} \Bigg(\frac{1}{\left[ (P+R_1)^2 + \left(\sfrac{(N+R_2)\pi}{L}\right)^2 +1 \right]}\\[-5pt]
- \frac{1}{\left[ (P+R_1)^2 + \left(\sfrac{(N-R_2)\pi}{L}\right)^2 +1 \right]} \Bigg).
\end{multline*}
If $w(x,0)=0$ for every $x \in (a,b)$, then $\alpha_k=0$ for $k=0,\dots,P+R_1$
and in particular $\alpha_{P+R_1}=0$. This is impossible because
$c_{P,N} \neq 0$ and $(N+R_2)^2\neq (N-R_2)^2$ (note that $N, R_2 \in \mathbb{N}^*$).
\end{proof}

\subsubsection{General comments}

First, we prove the equivalence between Assumption \HH4 and the existence of some non-vanishing integral quantities.
To this end, we introduce the following notation: for $\chi\in L^2(\partial\Omega)$, $V_{\chi}$ denotes the unique solution in $H^{\sfrac{3}{2}}(\Omega)$ of
\begin{equation} \label{def:V_chi}
\begin{cases}
(-\Delta+1) V_{\chi}(x) = 0, \quad & x \in \Omega, \\
\partial_{\nu}V_{\chi}(x) = \chi(x), & x \in\partial\Omega.
\end{cases}
\end{equation}

\begin{Prop} \label{propequivUCP}
Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^2$ and
$(\varphi_k)_{k\in \N}$ be an orthogonal basis of eigenfunctions of $(-\Delta_D)$, $R \in \mathbb{N}^*$.
If the spectrum of $(-\Delta_D)$ is simple, then the following statements are equivalent.
\begin{enumerate}
\item \label{equiv1}
For any eigenvector $\Phi$ of $(-\Delta_D)$, the solution $w$ of
\begin{equation}\label{eqnwprop}\begin{cases}
(-\Delta+1)w(x)=\varphi_R(x) \Phi(x), \quad & x \in \Omega, \\
\partial_{\nu} w(x)=0, & x \in\partial \Omega,
\end{cases}
\end{equation}
does not identically vanish on $\Gamma_c$: $w_{\left|\Gamma_c\right.} \not \equiv 0$.
\item \label{equiv2}There is an open dense set in $L^2(\Gamma_c)$ of functions $\chi$ such that $\int_{\Omega}V_{\chi}\varphi_R\varphi_{k} \neq 0$ for any $k\in \N^*$
\item \label{equiv3}There is one $\chi\in L^2(\Gamma_c)$ such that $\int_{\Omega}V_{\chi}\varphi_R \varphi_{k} \neq 0$ for any $k\in \N^*$.
\end{enumerate}
If the spectrum of $(-\Delta_D)$ is not simple, we still have $ \eqref{equiv1} \Rightarrow \eqref{equiv2}\Rightarrow \eqref{equiv3}$.
\end{Prop}

\begin{proof}
We have obviously $\eqref{equiv2} \Rightarrow \eqref{equiv3}$. The main tool for the other implications will be the following formula using Equation \eqref{eqnwprop} (the same as in \HH4) satisfied by $w$
\begin{align*}
\int_{\Omega}V_{\chi} \varphi_R \Phi
& = \int_{\Omega} V_{\chi} (-\Delta+1)w \\
& = \int_{\Omega} (-\Delta+1)V_{\chi}w + \int_{\partial \Omega} [\partial_{\nu} V_{\chi} w - V_{\chi}\partial_{\nu}w ] \\
& = \int_{\partial \Omega}\chi w.
\end{align*}
In the case of simple eigenvalues, we easily get $\eqref{equiv3}\Rightarrow \eqref{equiv1}$ by selecting $\chi\in L^2(\Gamma_c)$
such that $\int_{\Omega} V_{\chi} \varphi_R \varphi_{k} = \int_{\partial \Omega} \chi w_{k}\neq 0$.
So, the main task is to prove $ \eqref{equiv1}\Rightarrow \eqref{equiv2}$.
For $k \in \mathbb{N}^*$, we introduce the set
$$\mathcal{G}_k:=\left\{ \chi \in L^2(\Gamma_c,\mathbb{R}) \sep \int_{\Omega}V_{\chi}\varphi_R(x) \varphi_k(x) \,dx \neq 0 \right\}.$$
By Assumption \eqref{equiv1}, for any $k$, $w_k$ is not identically zero on $\Gamma_c$, so there exists $\chi$ such that
$\int_{\partial \Omega}\chi w_k = \int_{\Omega}V_{\chi}\varphi_R \varphi_k \neq 0$. In particular, $\mathcal{G}_k$ is not empty for every $k \in \N^*$.
Moreover, if $\chi_0 \in L^2(\Gamma_c,\R) \setminus \mathcal{G}_k$ and $\chi \in
\mathcal{G}_k$ then $\chi_0+\epsilon\chi$ belongs to $\mathcal{G}_k$ for every $\epsilon \in
\R^*$ and converges to $\chi_0$ when $\epsilon \to 0$. Thus $\mathcal{G}_k$
is a dense open subset of $L^2(\Gamma_c,\R)$.

By Baire's Lemma $\bigcap_{k \in \mathbb{N}^*} \mathcal{G}_k$ is a dense subset of $L^2(\Gamma_c,\mathbb{R})$. This proves \eqref{equiv2}.
\end{proof}

With the previous proposition in hands, it should be possible to prove that \HH4 is generic with respect to perturbations of the domain,
as done in \cite{Mehats_P_S}. Indeed, the previous proposition proves that \HH4 is equivalent to the existence of one $\chi$ so that
some integral quantities are not zero. It is then often a good starting point to prove genericity with respect to $\chi$ and deformation of $\Omega$.
This would then directly imply that \HH4 is generic with respect to perturbations of the domain.

Note also that with the same proof, $\chi$ can be chosen in $C^{k}_0(\Gamma_c)$ for some $k\in \N$.

\subsubsection{Control of low frequencies}
\label{subsectlow}

Actually, the condition \HH4 is a condition of controllability of the low frequencies for a control only depending on time. More precisely, we have the following result.

\begin{Prop} \label{Prop_LF}
Assume all the eigenvalues of $\Delta_D$ are simple and \HH4 is fulfilled for one $R\in \N^*$.
Let $\psi_{\rref}(t):=\varphi_R(x)e^{-i\lambda_R t}$.
Then there is one $\chi\in C^4(\Gamma_c)$ such that $\int_{\Omega}V_{\chi}\varphi_R\Phi_{k} \neq 0$ for any $k\in \N^*$.
This implies the following local controllability property for low frequencies.

For any $N\geq 0$, there exists $ \delta>0$ such that
for every
\[
\psi_f\in \mathcal{V}:=\{\psi_f \in H^3_{(0)}(\Omega) \cap \mathcal{S} \sep
\| \psi_f - \psi_{\rref}(T) \|_{H^3_{(0)}} < \delta \},
\]
there exists $u \in L^2((0,T),\R)$ such that the solution of
\begin{equation} \label{Schro_PoissonthmLF}
\begin{cases}
(i\partial_t + \Delta) \psi(t,x) = u(t)V_{\chi}(x) \psi(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\psi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\psi(0,x)=\varphi_R(x), & x \in \Omega,
\end{cases}
\end{equation}
satisfies $\mathbb{P}_N^{\perp}\psi(T)=\mathbb{P}_N^{\perp}\psi_f$, where $\mathbb{P}_N^{\perp}$ is the orthogonal projection on the $N$ first eigenfunctions.
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop_LF}]
The map
\begin{align*}
\Theta:  L^2((0,T),\mathbb{R}) & \to  \mathbb{P}_N^{\perp}[\mathcal{S}] \\
 u & \mto  \mathbb{P}_N^{\perp} [\psi(T)]
\end{align*}
is of class $C^1$ and
\begin{align*}
d\Theta(0):  L^2((0,T),\mathbb{R}) & \to  \mathbb{P}_N^{\perp}[T_{\psi_{\rref}(T)}\mathcal{S}] \\
 u & \mto  \mathbb{P}_N^{\perp} [\Psi(T)],
\end{align*}
where
\begin{align*}
\Psi(T) & = -i \int_0^T e^{i(T-t)\Delta_D} \big(u(t) V_{\chi} \psi_{\rref}(t) \big) \,dt \\
& = -i \sum_{k=1}^\infty \left\langle V_{\chi}\varphi_R,\varphi_k\right\rangle
\bigg(\int_0^T u(t) e^{i(\lambda_k-\lambda_R)t} \,dt \bigg) e^{-i\lambda_k T} \varphi_k.
\end{align*}
The surjectivity of $d\Theta(0)$ can be formulated in terms of a finite trigonometric moment problem on $u$.
Since $\left\langle V_{\chi}\varphi_R,\varphi_k\right\rangle\neq 0$ for every $k$ and the frequencies $(\lambda_k-\lambda_R)$ are all different,
this moment problem has a solution $u \in L^2((0,T),\mathbb{R})$.
We can therefore apply the inverse function theorem to $\Theta$.
We skip the details since it is a simpler version of arguments used several times in the paper.
\end{proof}

\section{Well-posedness of the Schrödinger-Poisson system and linearization}

The goal of this section is to prove the well-posedness of the system \eqref{Schro_Poisson} in functional spaces that are appropriate for
the controllability problem. This proof requires several preliminary results.
The first one concerns Sobolev embeddings, trace theorems and elliptic estimates; they are recalled in Section \ref{subsec:Prel}.
The second preliminary result concerns smoothing effects of the Schrödinger equation on a bounded domain:
a~smoothing effect concerning the normal derivative is recalled in Section~\ref{subsec:smoothing_derivee_normale},
a smoothing effect concerning the source term is justified in Section~\ref{subsec:smoothing}.
Finally, in Section \ref{subsec:WP}, we prove the well-posedness of the system \eqref{Schro_Poisson} in $H^3_{(0)}(\Omega)$
when the control $g$ lives in $L^2((0,T)\times\Gamma_c)$; the smoothing effects are crucial at this point.
In Section \ref{subsec:Lin}, we prove the $C^1$-regularity of the end-point map.

\Subsection{Preliminary}
\label{subsec:Prel}

\subsubsection{Sobolev embeddings}

In the whole section $\Omega$ is an open subset of $\mathbb{R}^n$ such that \HH1 is satisfied.
For $s\geq 0$ and $p\geq 1$, we use the definitions
\begin{align*}
W^{s,p}(\R^2) &:= \big\{ \widetilde{f} \in \mathcal{S}'(\mathbb{R}^2) \sep \mathcal{F}^{-1}[(1+|\cdot|^s)\mathcal{F}\widetilde{f}] \in L^p(\mathbb{R}^2) \big\},\\
\|\widetilde{f}\|_{W^{s,p}(\R^2)} &:= \big\|\mathcal{F}^{-1}[(1+|\cdot|^s)\mathcal{F}\widetilde{f}] \big\|_{L^p(\mathbb{R}^2)},\\
W^{s,p}(\Omega)&:=\big\{ \widetilde{f}|_{\Omega} \sep \widetilde{f} \in W^{s,p}(\R^2) \big\},\\
\nor{f}{W^{s,p}(\Omega)}&:=\inf\big\{\norb{\widetilde{f}}{W^{s,p}(\R^2)}\sep \widetilde{f}\in W^{s,p}(\R^2), f=\widetilde{f} \textnormal{ on }\Omega \big\}.
\end{align*}
If $p=2$, we use the notation $H^s$ instead of $W^{s,2}$.

Remark that if $s=1$, we also have
$$W^{1,p}(\Omega):=\{f\in L^p(\Omega) \sep \nabla f\in L^p(\Omega)^n \}\,$$
with the obvious norm equivalent to the one previously introduced.

The standard Sobolev embeddings
\begin{equation} \label{Sobolev_emb_3}
\begin{split}
H^{1}(\Omega) &\subset L^q(\Omega), \quad \forall q \in [2,\infty),\\
W^{1,p}(\Omega) &\subset L^\infty(\Omega), \quad \forall p > 2
\end{split}
\end{equation}
(see \cite[Cor.\,IX.14]{Brezis_An_Fonc}) and
$$W^{s,p}(\R^2)\subset W^{s_1,p_1}(\R^2) \quad\text{ for } s-\frac{2}{p}=s_1-\frac{2}{p_1},\quad 1<p\leq p_1<+\infty,$$
(see \cite[Th.\,6.5.1]{Bergh}) yield to
\begin{gather} \label{Sobolev_emb_1}
H^{\sfrac{3}{2}}(\Omega) \subset L^\infty(\Omega),\\
\label{Sobolev_emb_2}
H^{\sfrac{1}{2}}(\Omega) \subset L^4(\Omega),\\
\label{Sobolev_emb_4}
\forall \epsilon \in(0,\sfrac{1}{2}),\ \exists p \in (1,2) \text{ such that } W^{1,p}(\Omega) \subset H^{\sfrac{1}{2} + \epsilon}(\Omega).
\end{gather}
If $s\geq 0$, we denote
\bna
H^{-s}(\Omega) := \left(H^s_0(\Omega)\right)',
\ena
which is a space of distributions.

\Subsubsection{Regularity for elliptic boundary value problems on a smooth domain}

In this section, we recall classical results used in the article
(see \cite[Th.\,5.4 p.\,176, Th.\,6.5, Th.\,6.7 p.\,192]{LionsMagenes1}).

\begin{Prop} \label{Prop:trace+DN+elliptiq_Cinfty}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^2$, of class $C^\infty$ and locally on one side of $\partial \Omega$.
\begin{enumerate}
\item The mapping
\begin{align*}
C^\infty(\overline{\Omega}) & \to  C^\infty(\partial\Omega) \\
f & \mto  f|_{\partial \Omega}
\end{align*}
has a unique continuous extension from
$D(\Delta,L^2(\Omega)):=\{ f \in L^2(\Omega) \sep \Delta f \in L^2(\Omega) \}$
into $H^{-\sfrac{1}{2}}(\partial\Omega)$.
\item The mapping
\begin{align*}
H^2(\Omega) & \to  L^2(\partial \Omega) \\
f & \mto\partial_{\nu} f
\end{align*}
has a unique continuous extension from
$\{f \in H^{\sfrac{3}{2}}(\Omega) \sep \Delta f \in L^2(\Omega) \}$
into $L^2(\partial\Omega)$.
\item For every $f \in L^2(\Omega)$ and $g \in L^2(\partial \Omega)$, there exists a unique $v \in H^{\sfrac{3}{2}}(\Omega)$ such that
$$\begin{cases}
(-\Delta+1)v=f \text{ in } \Omega, \\
\partial_{\nu} v=g \text{ on }\partial \Omega.
\end{cases}$$
Moreover, there exists $C(\Omega)>0$ such that
$$\| v \|_{H^{\sfrac{3}{2}}(\Omega)} \leqslant C \big[ \|f\|_{L^2(\Omega)} + \|g\|_{L^2(\partial \Omega)} \big].$$
\end{enumerate}
\end{Prop}

\Subsubsection{Regularity and Green formula for elliptic boundary value problems on a rectangle}

The goal of this section is to state the following results.

\begin{Prop} \label{Prop:Trace_rect}
Let $\Omega=(0,\pi) \times (0,L)$ for some $L>0$,
$(\Gamma_j)_{1\leqslant j \leqslant 4}$ be its edges,
$(\nu_j)_{1 \leqslant j \leqslant 4}$ be its unitary exterior normal vectors,
$(S_j)_{1\leqslant j \leqslant 4}$ be its vertices
and $p \geqslant 1$.
\begin{enumerate}

\item The mapping $u \mto (u|_{\Gamma_j},\partial_{\nu_j} u)$,
which is well-defined for $u \in W^{2,p}(\Omega)$ has a unique extension as an operator from
$$D(\Delta,L^p(\Omega)):=\{u \in L^p(\Omega) \sep \Delta u \in L^p(\Omega) \}$$
into
\begin{itemize}
\item $W^{-\sfrac{1}{p},p}(\Gamma_j) \times W^{-1-\sfrac{1}{p},p}(\Gamma_j)$ when $p \neq 2$,
\item $H^{-\sfrac{1}{2}-\epsilon}(\Gamma_j) \times H^{-\sfrac{3}{2}-\epsilon}(\Gamma_j)$, for every $\epsilon>0$, when $p=2$.
\end{itemize}

\item Moreover,
$$ \int_{\Omega} \big((\Delta u) v - u(\Delta v) \big) \,dx =
\sum_{j=1}^4 \int_{\Gamma_j} \big((\partial_{\nu_j} u) v - u (\partial_{\nu_j} v) \big) d\sigma_j(x)$$
for every $u \in D(\Delta,L^p(\Omega))$ and $v \in W^{2,p'}(\Omega)$ such that
\begin{itemize}
\item $v(S_j)=0$ for $j=1,\dots,4$ when $p>2$,
\item $v(S_j)=0$ and $\nabla v(S_j)=0$ for $j=1,\dots,4$ when $p<2$,
\item $v \equiv 0$ on a neighborhood of $S_j$ for $j=1,\dots,4$ when $p=2$.
\end{itemize}

\item For every $f \in L^2(\Omega)$ and $g \in L^2(\partial \Omega)$, there exists a unique $v \in H^{\sfrac{3}{2}}(\Omega)$ such that
\bnan
\label{eqnvrect}
\begin{cases}
(-\Delta+1)v=f \text{ in } \Omega, \\
\partial_{\nu} v=g \text{ on }\partial \Omega.
\end{cases}
\enan
Moreover, there exists $C(\Omega)>0$ such that
$$\| v \|_{H^{\sfrac{3}{2}}(\Omega)} \leqslant C \big[ \|f\|_{L^2(\Omega)} + \|g\|_{L^2(\partial \Omega)} \big].$$
\end{enumerate}
\end{Prop}

For the first two statements, see \cite[Th.\,1.5.3.4 \& Th.\,1.5.3.6]{Grisvard}.
The third one is a consequence of the same one on $(\R/2\pi\Z)\times (0,L)$ after symmetrisation and periodization.
More precisely, by linearity, one may assume that $\Supp(g) \subset [0,\pi] \times \{0\}$.
Then we extend $g$ as a function $\widetilde{g}: (\R/\pi\Z) \times \{0\} \to \mathbb{R}$ such that
$\widetilde{g}(-x_1,0)=g(x_1,0)$ for every $x_1 \in (0,\pi)$ and $\widetilde{g}(x_1,0)=\widetilde{g}(x_1+2\pi,0)$. Perform similar symmetrisation and periodization for $f$ to produce $\widetilde{g}$ well-defined on $(\R/2\pi\Z)\times (0,L)$.

Then, since $(\R/2\pi\Z)\times (0,L)$ is a smooth compact manifold with boundary, there exists a unique $\widetilde{v} \in H^{\sfrac{3}{2}}((\R/2\pi\Z)\times (0,L))$ such that
$$\begin{cases}
(-\Delta+1)\widetilde{v}=\widetilde{f} \text{ in } \Omega, \\
\partial_{\nu} \widetilde{v}=\widetilde{g} \text{ on } (\R/\pi\Z) \times \{0,1\}.
\end{cases}$$

We easily check that the symmetry gives that if $f\in C^{\infty}_0(\Omega)$ and $g\in C^{\infty}_0(0,\pi)$, then, $\partial_{x_1}\widetilde{v}(x_1,x_2)=0$ for $x_1\in \{0,\pi\}$ and $x_2\in (0,L)$. Therefore, $v=\widetilde{v}_{\left|\Omega\right.}$ satisfies~\eqref{eqnvrect} in the weak sense. The uniqueness can be obtained by following the same process.

\subsection{Smoothing effect on the normal derivative}
\label{subsec:smoothing_derivee_normale}

The goal of this section is to state the following results.

\begin{Prop} \label{Prop:smoothin_effect_DN}
Let $T>0$ and $\Omega$ be a bounded open subset of $\mathbb{R}^2$ which is either~$C^\infty$ or a rectangle.
There exists $C=C(T,\Omega)>0$ such that,
for every $\psi_0 \in H^1_0(\Omega)$ and $h \in L^1((0,T),H^1_0(\Omega))$,
the solution $\psi$ of
\begin{equation} \label{Schro_h}
\begin{cases}
(i\partial_t + \Delta)\psi (t,x)=h(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\psi(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\psi(0,x)=\psi_0(x), & x \in \Omega,
\end{cases}
\end{equation}
satisfies
\begin{equation} \label{majo_DN}
\int_0^T \int_{\partial \Omega} |\partial_\nu \psi(t,x)|^2 |\partial_\nu \varphi_1 (x)| d\sigma(x) \,dt
\leqslant C \big(\|\psi_0\|_{H^1_0(\Omega)} + \|h\|_{L^1((0,T),H^1_0(\Omega))} \big).
\end{equation}
In particular, if $\Gamma_c$ is an open subset of $\partial \Omega$ such that \HH1 holds,
then the following linear mapping is continuous
$$\begin{array}{rclcl}
H^1_0(\Omega) & \times & L^1((0,T),H^1_0(\Omega)) & \dpl\to & L^2((0,T)\times\Gamma_c) \\[3pt]
(\psi_0 &, & \varphi) & \dpl\mto &\partial_\nu \psi.
\end{array}$$
\end{Prop}

When $\Omega$ is smooth, Puel proves these results in \cite[Lem.\,3.1]{Puelsmoothing}.
Precisely, he first proves the inequality \eqref{majo_DN} for smooth data $(\psi_0,h) \in C^\infty_c(\Omega) \times C^\infty_c((0,T)\times\Omega)$,
by applying the multiplier $\nabla \varphi_1$ to Equation \eqref{Schro_h}: under these assumptions $\psi$ is regular,
$\sfrac{\partial \psi}{\partial \nu}$ makes perfect sense and integrations by parts are legitimate.
Then, the inequality \eqref{majo_DN} holds for $(\psi_0,h) \in H^1_0(\Omega)\times L^1((0,T),H^1_0(\Omega))$ by a density argument.
Finally, Puel gets the final result because $|\partial_\nu \varphi_1 (x)| \geqslant \beta >0, \forall x \in\partial \Omega$

When $\Omega$ is a rectangle, Puel's proof of the inequality \eqref{majo_DN} is still valid.
Then, the conclusion of Proposition \ref{Prop:smoothin_effect_DN} follows because
$|\partial_\nu \varphi_1 (x)| \geqslant \beta >0, \forall x \in \Gamma_c$;
this is one of the reasons why we need to assume that $\Gamma_c$ does not touch the vertices of $\Omega$.

\subsection{Smoothing effect on the source term}
\label{subsec:smoothing}

The goal of this section is to justify the following result.

\begin{thm} \label{thm:Puel}
Let $T>0$, $\Omega$ be an open subset of $\mathbb{R}^2$ and $\Gamma_c$ be an open subset of $\partial \Omega$ such that \HH1 holds.
For every $\psi_0 \in H^3_{(0)}(\Omega)$,
$\mu_1 \in L^2((0,T),H^2 \cap H^1_0(\Omega))$
and $\mu_2 \in L^1((0,T),H^3_{(0)}(\Omega))$
such that
\begin{equation} \label{HYP:mu1}
\Delta^2 \mu_1 \equiv 0, \quad
\Supp \left(\Delta \mu_1 \big{|}_{\partial \Omega} \right) \subset [0,T]\times\Gamma_c, \quad
\Delta \mu_1 \big{|}_{\Gamma_c} \in L^2((0,T)\times \Gamma_c),
\end{equation}
the solution of
$$\begin{cases}
(i\partial_t + \Delta) \psi(t,x) =(\mu_1 +\mu_2) (t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\psi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\psi(0,x)=\psi_0(x), & x \in \Omega,
\end{cases}$$
satisfies $\psi \in C^0([0,T],H^3_{(0)}(\Omega))$.
Moreover, there exists a constant $C>0$ (independent of $\psi_0, \mu_1, \mu_2$) such that
\begin{multline} \label{Puel_estimee}
\|\psi\|_{L^\infty([0,T],H^3_{(0)}(\Omega))}\\ \leqslant C \big(\|\psi_0\|_{H^3_{(0)}(\Omega)}
+ \norb{\Delta \mu_1\big{|}_{\Gamma_c}}{L^2((0,T)\times \Gamma_c)} + \|\mu_2\|_{L^1((0,T),H^3_{(0)}(\Omega))} \big).
\end{multline}
\end{thm}

\begin{rk}
When $\Omega$ is smooth, the trace $\Delta \mu_1 |_{\partial \Omega}$ is well-defined in the space
$L^2((0,T),H^{-\sfrac{1}{2}}(\partial \Omega))$
(see Proposition \ref{Prop:trace+DN+elliptiq_Cinfty}), which gives a sense to the last two requirements in \eqref{HYP:mu1}.

When $\Omega$ is a rectangle,
then the traces along the four sides $\Delta \mu_1 |_{\Gamma_j}$,
$1 \leqslant j \leqslant 4$,
are well-defined in $L^2((0,T),H^{-\sfrac{1}{2}-\epsilon}(\Gamma_j))$ (see Proposition \ref{Prop:Trace_rect}).
The last two requirements in \eqref{HYP:mu1} have to be interpreted in the following sense:
\begin{gather*}
\Gamma_j \cap \Supp \left(\Delta \mu_1 \big{|}_{\partial \Omega} \right) \subset \Gamma_j \cap \Gamma_c, \quad \forall j \in \{1,\dots,4\},\\
\Delta \mu_1 \big{|}_{\Gamma_c \cap \Gamma_j} \in L^2((0,T)\times \Gamma_c \cap \Gamma_j), \quad \forall j \in \{1,\dots,4\}.
\end{gather*}
\end{rk}

Theorem \ref{thm:Puel} emphasizes a regularizing effect, concerning the source term,
because~$\mu_1$ is not assumed to belong to $L^1((0,T),H^3_{(0)}(\Omega))$.
When $\Omega$ is a regular domain and $\Gamma_c=\partial \Omega$, Puel proves this result in \cite[Th.\,2.1]{Puelsmoothing}.
His proof, by transposition, relies on the smoothing effect of Proposition \ref{Prop:smoothin_effect_DN}.

Puel does not treat the case of a rectangle domain, but his proof would probably lead to Theorem \ref{thm:Puel} in this case too.
For the sake of completeness, we propose below an alternative argument.

\begin{proof}[Proof of Theorem \ref{thm:Puel} on a rectangle]
Let $L>0$, $\Omega:=(0,\pi)\times(0,L)$,
$\Gamma_c$ be an open subset of $\partial \Omega$,
$\psi_0 \in H^3_{(0)}(\Omega)$,
$\mu_1 \in L^2((0,T),H^2 \cap H^1_0(\Omega))$
and $\mu_2 \in L^1((0,T),H^3_{(0)}(\Omega))$ be such that \eqref{HYP:mu1} holds.
By linearity, we may assume that $\Gamma_c \subset (a,b) \times \{0\}$ with $0<a<b<\pi$.
Let $m \in \mathbb{N}^*$ be such that $(m-1)\sfrac{2L^2}{\pi} < T \leqslant T_0 := \sfrac{2 m L^2}{\pi}$ and $t \in (0,T)$.

By the Duhamel formula, we have
$$\psi(t)= e^{i\Delta_D t} \psi_0 -i \int_0^t e^{i\Delta_D(t-\tau)} \big(\mu_1(\tau) + \mu_2(\tau) \big) d\tau.$$
The first and third terms in the right-hand side belong to $C^0([0,T],H^3_{(0)}(\Omega))$ because $H^3_{(0)}(\Omega)$ is stable by $e^{i\Delta_D t}$.
Thus, we focus on the second one. We will make the proof in two steps: the first one when $\mu_1$ is regular enough to perform integration by parts and the second one to get to the first case with a suitable regularization.

\subsubsection*{Step~1: $\mu_1$ regular enough}

We assume moreover that $\Delta \mu_1\in L^2((0,T),L^p(\Omega))$, with $p>2$, in this step.

Using the equality $-\Delta \varphi_k = \lambda_k \varphi_k$ and integrations by part,
that are licit because $\mu_1 \in L^2((0,T),H^2 \cap H^1_0(\Omega))$, we get
\begin{align*}
S=\left\| \int_0^t e^{i\Delta(t-\tau)} \mu_1(\tau) d\tau \right\|_{H^3_{(0)}(\Omega)}^2
& = \sum_{k=1}^\infty \left| \lambda_k^{3/2} \int_0^t \int_{\Omega} \mu_1(\tau,x) \varphi_k(x) \,dx e^{i \lambda_k \tau} d\tau \right|^2
\\ & = \sum_{k=1}^\infty \left| \frac{1}{\sqrt{\lambda_k}} \int_0^t \int_{\Omega} \Delta \mu_1(\tau,x) \Delta \varphi_k(x) \,dx e^{i \lambda_k \tau} d\tau \right|^2.
\end{align*}

We can apply Green's formula (see Proposition \ref{Prop:Trace_rect}) because
$\Delta \mu_1(\tau,.) \in D(\Delta,L^p(\Omega))$ for almost every $\tau \in (0,T)$ and
$\varphi_k \in H^2(\Omega)$ vanishes on the vertices of~$\Omega$.
Using $\Delta^2 \mu_1 \equiv 0$, this leads to
\begin{align*}
S & = \sum_{k=1}^\infty \left| \frac{1}{\sqrt{\lambda_k}}
\int_0^t \int_{\partial \Omega} \Delta \mu_1(\tau,x)\partial_{\nu} \varphi_k(x) d\sigma(x) e^{i \lambda_k \tau} d\tau \right|^2
\\ & = \sum_{k=1}^\infty \left| \frac{1}{\sqrt{\lambda_k}}
\int_0^t \int_{\Gamma_c} \Delta \mu_1(\tau,x)\partial_{y} \varphi_k(x) d\sigma(x) e^{i \lambda_k \tau} d\tau \right|^2
\\ & \leqslant C \sum_{p,n \in \mathbb{N}^*} \bigg|
\frac{\sfrac{n\pi}{L}}{\sqrt{p^2+\left(\sfrac{n\pi}{L}\right)^2}} \int_0^t \int_0^\pi
\Delta \mu_1(\tau,x_1,0) \sin(px_1) e^{-i\left(p^2+\left(\sfrac{n\pi}{L}\right)^2 \right) \tau} \,dx_1 d\tau \bigg|^2
\\ & \leqslant
C \sum_{p \in \mathbb{N}^*} \sum_{n \in \mathbb{N}^*} \bigg| \int_0^{T_0}
1_{[0,t]}(\tau) \int_0^\pi \Delta \mu_1(\tau,x_1,0) \sin(px_1) \,dx_1 e^{-i\left(p^2+\left(\sfrac{n\pi}{L}\right)^2 \right) \tau} d\tau \bigg|^2
\\ & \leqslant
C \sum_{p \in \mathbb{N}^*} \int_0^{T_0} \left|
1_{[0,t]}(\tau) \int_0^\pi \Delta \mu_1(\tau,x_1,0) \sin(px_1) \,dx_1 e^{-i p^2 \tau} \right|^2 d\tau
\\ & \leqslant
C \sum_{p \in \mathbb{N}^*} \int_0^t \left| \int_0^\pi \Delta \mu_1(\tau,x_1,0) \sin(px_1)\,dx_1 \right|^2 d\tau
\\ & \leqslant
C \int_0^t \| \Delta \mu_1(\tau,.,0) \|_{L^2(0,\pi)}^2 d\tau.
\end{align*}
To go from the first line to the second one, we have used:
$\Supp(\Delta \mu_1 |_{\partial \Omega}) \subset [0,T]\times\Gamma_c$, $\varphi_k \equiv 0$ on $\Gamma_c$.
To go from the fourth line to the fifth line, we have used Bessel-Parseval's inequality.

Note that $\Delta \mu_1$ can be computed explicitly in terms of $\Delta \mu_1\big{|}_{\Gamma_c}$,
thanks to the rectangular form of the domain. This explicit expression shows the existence of a constant $C=C(\Omega)>0$ such that
$$\|\Delta \mu_1 \|_{L^2((0,t)\times\Omega))} \leqslant C \norb{\Delta \mu_1\big{|}_{\Gamma_c}}{L^2((0,t)\times\Gamma_c)}^2, \quad \forall t \in [0,T].$$
Thus, we get
$$\left\| \int_0^t e^{i\Delta(t-\tau)} \mu_1(\tau) d\tau \right\|_{H^3_{(0)}(\Omega)}^2
\leqslant C \norb{\Delta \mu_1\big{|}_{\Gamma_c}}{L^2((0,t)\times \Gamma_c)}^2.$$
This proves that the map $t \in [0,T] \mto \int_0^t e^{i\Delta(t-\tau)} \mu_1(\tau) d\tau$
takes values in $H^3_{(0)}(\Omega)$ and is continuous at $t=0$.
The same argument proves the continuity at any $t \in [0,T]$.

\subsubsection*{Step~2: regularization}

We have to be a bit careful with the regularization to keep the required conditions. Let $f_{\eps}\in C^{\infty}_0((0,T)\times \Gamma_c)$ converging strongly to $\Delta \mu_1\big{|}_{\Gamma_c}$ in $L^2((0,T)\times \Gamma_c)$. Denote
$$\begin{cases}
\Delta g_{\eps}(t,x) =0, \quad & (t,x) \in (0,T)\times\Omega, \\
g_{\eps}(t,x)=f_{\eps}(t,x), & (t,x) \in (0,T)\times\partial \Omega, \\
\end{cases}$$
By elliptic regularity (similar to Proposition \ref{Prop:Trace_rect}), $g_{\eps}$ converges strongly to $\Delta \mu_1$ in $L^2((0,T),H^{1/2}(\Omega))$.

Now, denote $\mu_{1,\eps}$ the solution of
$$\begin{cases}
\Delta\mu_{1,\eps}(t,x) =g_{\eps}(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\mu_{1,\eps}(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega.
\end{cases}$$
By elliptic regularity, $\mu_{1,\eps}$ converges to $\mu_{1}$ in $L^2((0,T),H^2 \cap H^1_0(\Omega))$. The estimate~\eqref{Puel_estimee} (that is proven for $\mu_{1,\eps}$ regular enough thanks to Step~1) allows us to prove that the related $\psi_{\eps}$ make a Cauchy sequence in $L^\infty([0,T],H^3_{(0)}(\Omega))$. The limit can only be the expected $\psi$ by uniqueness of the solution and convergence of $\mu_{1,\eps}$ to $\mu_1$. The estimate~\eqref{Puel_estimee} follows in this case.
\end{proof}

\subsection{Well-posedness of the linear Schrödinger-Poisson system}
\label{subsec:WP}

The goal of this section is the proof of the following result, thanks to Theorem \ref{thm:Puel}.

\begin{thm} \label{thm:WP_H3}
Let $T>0$, $\Omega$ be an open subset of $\mathbb{R}^2$ and $\Gamma_c$ be an open subset of~$\partial \Omega$ such that \HH1 holds.
For every $\psi_0 \in H^3_{(0)}(\Omega)$ and $g \in L^2((0,T)\times\partial \Omega,\mathbb{R})$,
there exists a unique solution $\psi \in C^0([0,T],H^3_{(0)}(\Omega))$ of the system \eqref{Schro_Poisson}.
\end{thm}

To this end, we will need the following preliminary result.

\begin{Prop} \label{Prop:WP_prel}
Let $T>0$ and $\psi \in C^0([0,T],H^3_{(0)}(\Omega))$.
For every function $g \in L^2((0,T)\times\partial\Omega,\mathbb{R})$, the solution of
\begin{equation} \label{eq_potentiel}
\begin{cases}
(-\Delta+1)v(t,x)=0, \quad & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu v(t,x)=g(t,x) 1_{\Gamma_c}(x), \quad & (t,x) \in (0,T)\times\partial\Omega,
\end{cases}
\end{equation}
satisfies $v \in L^2((0,T),H^{\sfrac{3}{2}}(\Omega))$, $v \psi \in L^2((0,T),H^2 \cap H^1_0(\Omega))$ and $\Delta^2[ v \psi]$ belongs to $L^2((0,T),H^{-1}(\Omega))$.
Moreover, the operator
\begin{align*}
L^2((0,T)\times\partial\Omega,\mathbb{R}) & \to L^2((0,T),H^{-1}(\Omega)) \\
g & \mto \Delta^2(v\psi).
\end{align*}
is continuous.
\end{Prop}

\skpt
\begin{proof}[Proof of Proposition \ref{Prop:WP_prel}]

\subsubsection*{Step~1}
We prove that $v \psi \in L^2((0,T),H^2 \cap H^1_0(\Omega))$.
By Proposition \ref{Prop:trace+DN+elliptiq_Cinfty} and \ref{Prop:Trace_rect}, we know that $v \in L^2((0,T),H^{\sfrac{3}{2}}(\Omega))$.
Thus $v \psi \in L^2((0,T),H^{\sfrac{3}{2}}(\Omega) \cap H^1_0(\Omega))$ because $H^{\sfrac{3}{2}}(\Omega)\subset L^{\infty}$ is an algebra.
To end Step~1, it is sufficient to prove that $\Delta(v\psi) \in L^2((0,T),L^2(\Omega))$.
Using $\Delta v = v$ in $(0,T) \times \Omega$, we get
$$\Delta(v\psi)=v(\psi+\Delta \psi) + 2 \nabla v \cdot \nabla \psi.$$
\begin{itemize}
\item The Sobolev embedding \eqref{Sobolev_emb_1}
justifies that
\[
H^{\sfrac{3}{2}}(\Omega) * H^1_0(\Omega) \subset L^\infty(\Omega) * L^2(\Omega) = L^2(\Omega)
\]
(where $*$ stands for the multiplication of scalar valued functions).
Thus $v(\psi+\Delta \psi) \in L^2((0,T),L^2(\Omega))$.
\item The Sobolev embedding \eqref{Sobolev_emb_1} justifies that
\[
H^{\sfrac{1}{2}}(\Omega)*H^2(\Omega) \subset L^2(\Omega)*L^\infty(\Omega) = L^2(\Omega).
\]
Thus $\nabla v \cdot \nabla \psi \in L^2((0,T),L^2(\Omega))$.
\end{itemize}

\subsubsection*{Step~2}
We prove that $\Delta^2[v \psi] \in L^2((0,T),H^{-1}(\Omega))$.
Using $\Delta v = v$, we get
\begin{equation} \label{dvpt:Delta2(vpsi)}
\Delta^2[ v \psi] = v(\psi+2\Delta \psi+\Delta^2 \psi) + 4 \nabla v \cdot \nabla(\psi+\Delta \psi) + 4 \text{Tr}[D^2 v\cdot D^2 \psi].
\end{equation}

\begin{itemize}
\item The Sobolev embeddings \eqref{Sobolev_emb_1} and \eqref{Sobolev_emb_2}
justify that $H^{\sfrac{3}{2}}(\Omega) * H^1_0(\Omega) \subset H^1_0(\Omega)$.
Thus, by duality, $H^{\sfrac{3}{2}}(\Omega) * H^{-1}(\Omega) \subset H^{-1}(\Omega)$.
This proves that $v(\psi+2\Delta \psi+\Delta^2 \psi) \in L^2((0,T),H^{-1}(\Omega))$.

\item The Sobolev embedding \eqref{Sobolev_emb_2} justifies that
\[
H^{\sfrac{1}{2}}(\Omega) * H^1_0(\Omega) \subset L^4(\Omega)* L^4(\Omega) \subset L^2(\Omega).
\]
Thus, by duality, $H^{\sfrac{1}{2}}(\Omega) * L^2(\Omega) \subset H^{-1}(\Omega)$.
This proves that $\nabla v \cdot \nabla(\psi+\Delta \psi) \in L^2((0,T),H^{-1}(\Omega))$.

\item The Sobolev embedding \eqref {Sobolev_emb_3} and then \eqref{Sobolev_emb_4}
justify that
\[
H^1(\Omega)*H^1_0(\Omega) \subset H^{\sfrac{1}{2} + \epsilon}(\Omega).
\]
Thus, by duality, $H^1(\Omega)*H^{-\sfrac{1}{2}-\epsilon}(\Omega) \subset H^{-1}(\Omega)$.
This proves that $\text{Tr}[D^2 v\cdot D^2 \psi] \in L^2((0,T),H^{-1}(\Omega))$.\qedhere
\end{itemize}
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:WP_H3}]
Let $T>0$, $\psi_0 \in H^3_{(0)}(\Omega)$, $g \in L^2((0,T)\times\partial \Omega,\mathbb{R})$ and $v$ be the solution of \eqref{eq_potentiel}
We apply the fixed point theorem to the map
\begin{align*}
F: C^0([0,T],H^3_{(0)}(\Omega)) & \to C^0([0,T],H^3_{(0)}(\Omega)) \\
\psi & \mto \xi,
\end{align*}
where $\xi=F(\psi)$ is the solution of
$$\begin{cases}
(i\partial_t + \Delta) \xi(t,x)=v(t,x) \psi(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\xi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\xi(0,x)=\psi_0(x), & x \in \Omega.
\end{cases}$$

\subsubsection*{Step~1}
We prove that $F$ takes values in $C^0([0,T],H^3_{(0)}(\Omega))$.

Let $\psi \in C^0([0,T],H^3_{(0)}(\Omega))$. We introduce the solution $\mu_2$ of
\begin{equation} \label{def:equation_mu2}
\begin{cases}
\Delta^2 \mu_2(t,x)=\Delta^2[ v \psi](t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\mu_2 (t,x)=\Delta \mu_2(t,x)=0, & (t,x) \in (0,T) \times\partial \Omega,
\end{cases}
\end{equation}
and the function $\mu_1 := v \psi - \mu_2$. To prove that $\xi=F(\psi)$ belongs to $C^0([0,T],H^3_{(0)}(\Omega))$,
it suffices to prove that $\mu_1$ and $\mu_2$ satisfy the assumptions of Theorem \ref{thm:Puel}.

By Proposition \ref{Prop:WP_prel}, $\Delta^2[v\psi]$ belongs to $L^2((0,T),H^{-1}(\Omega))$ thus,
by elliptic regularity, $\mu_2 \in L^2((0,T),H^3_{(0)}(\Omega))$.
Then, $\mu_1$ belongs to $L^2((0,T),H^2 \cap H^1_0(\Omega))$, by Proposition~\ref{Prop:WP_prel}.
Moreover, $\Delta^2 \mu_1 \equiv 0$ by \eqref{def:equation_mu2} and
$\Delta \mu_1 |_{\partial \Omega} = 2 g\,\partial_{\nu} \psi\, 1_{\Gamma_c}$.

\subsubsection*{Step~2}
We prove that $F$ is a contraction when $g$ is small enough.
Let $\psi, \widetilde{\psi} \in C^0([0,T],H^3_{(0)}(\Omega))$.
From \eqref{Puel_estimee} and elliptic regularity, we get
\begin{align*}
\| F(\psi) - F(\widetilde{\psi})\|&_{C^0([0,T],H^3_{(0)}(\Omega))}\\
&\leqslant C \big(\|\Delta(\mu_1-\widetilde{\mu}_1)\|_{L^2((0,T) \times \Gamma_c)} + \|\mu_2-\widetilde{\mu}_2\|_{L^1((0,T),H^3_{(0)}(\Omega))} \big) \\
&\leqslant C' \|g\|_{L^2((0,T)\times\Gamma_c)} \|\psi-\widetilde{\psi}\|_{C^0([0,T],H^3_{(0)}(\Omega))}.
\end{align*}
for some constant $C'$ independent of $\psi$ and $\widetilde{\psi}$.
Thus $F$ is a contraction when $C' \|g\|_{L^2((0,T)\times\Gamma_c)} <1$.
Otherwise, one may subdivide the interval $(0,T)$ into a finite number of intervals on which this assumption is satisfied
and iterate the previous result.
\end{proof}

\Subsection{$C^1$-regularity of the end-point map}
\label{subsec:Lin}

\begin{Prop} \label{Prop:C1}
Let $T>0$, $\Omega$ be an open subset of $\mathbb{R}^2$ and $\Gamma_c$ be an open subset of $\partial \Omega$ such that \HH1 holds.
Consider the end-point map
$$\begin{array}{crclcl}
\Theta: & [H^3_{(0)}(\Omega) \cap \mathcal{S} ] & \times & L^2((0,T)\times\Gamma_c,\mathbb{R}) & \dpl\to & H^3_{(0)}(\Omega) \cap \mathcal{S} \\[3pt]
& (\psi_0 &, & g) & \dpl\mto & \psi(T),
\end{array}$$
where $\psi$ is the solution of \eqref{Schro_Poisson}.
Then, $\Theta$ is $C^1$ and for every $\psi_0 \in [H^3_{(0)}(\Omega) \cap \mathcal{S} ]$,
$$\begin{array}{crclcl}
d\Theta(\psi_0,0): & [H^3_{(0)}(\Omega) \cap T_{\mathcal{S}} \psi_0 ] & \times & L^2((0,T)\times\Gamma_c,\mathbb{R}) & \dpl\to & H^3_{(0)}(\Omega) \cap T_{\mathcal{S}} \psi_{\rref}(T) \\[3pt]
& (\Psi_0 &, & G) & \dpl\mto & \Psi(T),
\end{array}$$
where $\psi_{\rref}(t):=e^{i \Delta_D t} \psi_0$ and $\Psi$ is the solution of the linearized system
\begin{equation} \label{Schro_Poisson_linearise}
\begin{cases}
(i\partial_t + \Delta) \Psi(t,x) = V(t,x) \psi_{\rref}(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\Psi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\Psi(0,x)=\Psi_0(x), & x \in \Omega, \\
(-\Delta+1)V(t,x)=0, & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} V(t,x)=G(t,x) 1_{\Gamma_c}(x), & (t,x) \in (0,T)\times\partial \Omega. \\
\end{cases}
\end{equation}
\end{Prop}

This result is a consequence of estimate \eqref{Puel_estimee}.
Its proof is classical and follows the same steps as in \cite[Proof of Prop.\,3, pp.\,531--532]{KB-CL}.

\section{Local exact control of high frequencies}

The goal of this section is the proof of Theorem \ref{Thm:controle_HF}, via a perturbation argument.
In Section \ref{subsec:control_HF_lin}, we prove the controllability of the linearized system at high frequency,
thanks to the weak observability inequality of the adjoint system and the Hilbert Uniqueness Method.
In Section \ref{subsec:control_HF_nl}, we prove Theorem \ref{Thm:controle_HF} by applying the inverse mapping theorem.

\subsection{Control of high frequencies for the linearized system}
\label{subsec:control_HF_lin}

The goal of this section is to prove the following result.

\begin{Prop} \label{Prop:control_lin_H3}
Let $T>0$, $\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$ and $\psi_{\rref}(t):=e^{i \Delta_D t} \psi_0$.
We assume that \HH1, \HH2 and \HH3 hold.
Then, there exists $K \in \mathbb{N}^*$ and a continuous linear map
$$L: \mathbb{P}_K[ H^3_{(0)}(\Omega)] \to L^2((0,T)\times\Gamma_c,\mathbb{R}) $$
such that, for every $\Psi_f \in \mathbb{P}_K[ H^3_{(0)}(\Omega)]$, the solution of
\eqref{Schro_Poisson_linearise} with $\Psi_0=0$ and control $G=L(\Psi_f)$ satisfies $\mathbb{P}_K[\Psi(T)]=\Psi_f$.
\end{Prop}

The following proposition will allow us to work on $r:=\Delta^2 \Psi$ instead of $\Psi$.

\begin{Prop} \label{Prop:sol_transp}
Let $T>0$, $\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$, $\psi_{\rref}(t):=e^{i \Delta_D t} \psi_0$,
$G \in L^2((0,T)\times\Gamma_c,\mathbb{R})$ and $\Psi \in C^0([0,T],H^{3}_{(0)}(\Omega))$ be the solution of the system \eqref{Schro_Poisson_linearise} with $\Psi_0=0$.
Then, the function $r:=\Delta^2(\Psi)$ belongs to $C^0([0,T],H^{-1}(\Omega))$ and is the solution by transposition of
\begin{equation} \label{Eq:r}
\begin{cases}
(i\partial_t + \Delta) r(t,x) = \mathcal{K}(G)(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
r(t,x)= 2 G\partial_{\nu} \psi_{\rref} 1_{\Gamma_c}, & (t,x) \in (0,T)\times\partial \Omega, \\
r(0,x)=0, & x \in \Omega,
\end{cases}
\end{equation}
where
\begin{equation} \label{def:K}
\begin{split}
\mathcal{K}: L^2((0,T)\times\Gamma_c,\mathbb{R}) & \to L^2((0,T),H^{-1}(\Omega)) \\
G & \mto \Delta^2(V \psi_{\rref}), \text{ where }
\left\lbrace \begin{array}{l}
(-\Delta+1)V=0 \text{ in } (0,T)\times\Omega, \\
\partial_{\nu} V = G 1_{\Gamma_c} \text{ on } (0,T)\times\partial \Omega.
\end{array}\right.
\end{split}
\end{equation}
This means that, for every $\phi_T \in H^1_0(\Omega)$ and $\varphi \in L^1((0,T),H^1_0(\Omega))$ the following equality holds
\begin{multline} \label{def:sol_transp}
\int_0^T \langle r(t), \overline{\varphi(t)} \rangle_{{H^{-1},H^1_0}} \,dt =
\int_0^T \langle \mathcal{K}(G)(t), \overline{\phi(t)} \rangle_{H^{-1},H^1_0} \,dt - i \langle r(T), \overline{\phi_T} \rangle_{{H^{-1},H^1_0}}
\\
+ 2 \int_0^T \int_{\Gamma_c} G(t,x)\partial_{\nu} \psi_{\rref}(t,x) \overline{\partial_{\nu} \phi(t,x)} d\sigma(x) \,dt,
\end{multline}
where $\phi \in C^0([0,T],H^1_0(\Omega))$ is the solution of
$$\begin{cases}
(i\partial_t + \Delta) \phi(t,x) = \varphi(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\phi(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega\, \\
\phi(T,x)=\phi_T(x), & x \in \Omega.
\end{cases}$$
\end{Prop}

Note that there is no ambiguity in this definition of a solution by transposition of \eqref{Eq:r}. Indeed, for every $\phi_T \in H^1_0(\Omega)$ and
$\varphi \in L^1((0,T),H^1_0(\Omega))$, then $\partial_\nu \phi$ belongs to
$L^2((0,T)\times\Gamma_c)$ by Proposition \ref{Prop:smoothin_effect_DN}, thus, the last integral in \eqref{def:sol_transp} is well-defined. Moreover, $\Delta^2(V \psi_{\rref})\in L^2((0,T),H^{-1}(\Omega))$ thanks to Proposition \ref{Prop:WP_prel}.

\begin{proof}[Proof of Proposition \ref{Prop:sol_transp}]
When $G \in C^\infty_c((0,T)\times\Gamma_c)$, $\varphi \in C^\infty_c((0,T)\times\Omega)$ and $\phi_T \in C^\infty_c(\Omega)$,
then, formula \eqref{def:sol_transp} can be proved with integrations by part. The conclusion follows thanks to the continuity of the maps
$\mathcal{K}$ (see Proposition \ref{Prop:WP_prel}) and the continuity of the following linear mappings
\begin{gather*}
\begin{array}{ccccc}
L^2((0,T)\times \Gamma_c) & \dpl\to & C^0([0,T],H^3_{(0)}(\Omega)) & \dpl\to & C^0([0,T],H^{-1}(\Omega)) \\
G & \dpl\mto & \Psi & \dpl\mto & r=\Delta^2(\Psi)
\end{array}\\[3pt]
\begin{split}
L^1((0,T),H^1_0(\Omega)) \times H^1_0(\Omega) & \to L^2((0,T) \times \Gamma_c) \\
(\varphi,\phi_T) & \mto\partial_\nu \phi
\end{split}
\end{gather*}
stated in Theorem \ref{thm:Puel} and Proposition \ref{Prop:smoothin_effect_DN}.
\end{proof}

From now on, we denote by $\mathcal{K}^*$ the adjoint operator of $\mathcal{K}$,
$$\mathcal{K}^*: L^2((0,T),H^1_0(\Omega)) \to L^2((0,T)\times\Gamma_c,\mathbb{R}),$$
\ie
\begin{equation} \label{relation_K_adjoint}
\Re \bigg(\int_0^T \langle \mathcal{K}(G), \overline{\xi} \rangle_{H^{-1},H^1_0} \,dt \bigg)
= \int_0^T \int_{\Gamma_c} G \mathcal{K}^* (\xi) d\sigma(x) \,dt
\end{equation}
for every $\xi \in L^2((0,T),H^1_0(\Omega))$ and $G \in L^2((0,T)\times\Gamma_c,\mathbb{R})$.

Thus Proposition \ref{Prop:control_lin_H3} is a consequence of the following result.

\begin{Prop} \label{Prop:control_lin_H1}
Let $T>0$, $\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$ and $\psi_{\rref}(t):=e^{i \Delta_D t} \psi_0$.
We assume that \HH1, \HH2 and \HH3 hold.
Then, there exists $K \in \mathbb{N}^*$ and a continuous linear map
$$\widetilde{L}: \mathbb{P}_K[ H^{-1}(\Omega)] \to L^2((0,T)\times\Gamma_c,\mathbb{R})$$
such that, for every $r_f \in \mathbb{P}_K[ H^{-1}(\Omega)]$, the solution of
\eqref{Eq:r} with control $G=\widetilde{L}(r_f)$ satisfies $\mathbb{P}_K[r(T)]=r_f$.
\end{Prop}

The controllability result of Proposition \ref{Prop:control_lin_H1} is a consequence of the following weak observability result.

\begin{Prop} \label{Prop:Obs_faible}
Let $T>0$, $\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$ and $\psi_{\rref}(t):=e^{i \Delta_D t} \psi_0$.
We assume that \HH1, \HH2 and \HH3 hold.
There exists $\mathcal{C}_1>0$ such that, for every $\phi_T \in H^1_0(\Omega,\mathbb{C})$ the solution of
\begin{equation} \label{adjoint_syst}
\begin{cases}
(i\partial_t + \Delta) \phi(t,x) = 0, \quad & (t,x) \in (0,T)\times\Omega, \\
\phi(t,x)= 0, & (t,x) \in (0,T)\times\partial \Omega, \\
\phi(T,x)=\phi_T, & x \in \Omega, \\
\end{cases}
\end{equation}
satisfies
\begin{equation} \label{IO_faible}
\|\phi_T\|_{H^1_0(\Omega)} \leqslant \mathcal{C}_1 \left(
\| 2\Im(\partial_{\nu} \psi_{\rref} \overline{\partial_{\nu} \phi}) + \mathcal{K}^*(i\phi) \|_{L^2((0,T)\times\Gamma_c)}
+ \|\phi_T\|_{H^{-1}(\Omega)} \right).
\end{equation}
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop:control_lin_H1}, assuming Proposition \ref{Prop:Obs_faible}]
By \cite[Th.\,II.10 \& II.19]{Brezis_An_Fonc}, the exis\-tence of a continuous right inverse to the continuous operator
\begin{align*}
F_K: L^2((0,T)\times\Gamma_c,\mathbb{R}) & \to \mathbb{P}_K[H^{-1}(\Omega)] \\
G & \mto \mathbb{P}_K[r(T)]
\end{align*}
is equivalent to the existence of $\mathcal{C}_2>0$ such that
\begin{equation} \label{IO_HF}
\|\phi_T\|_{H^1_0(\Omega)} \leqslant \mathcal{C}_2 \| F_K^*(\phi_T) \|_{L^2((0,T)\times\Gamma_c)},
\quad \forall \phi_T \in \mathbb{P}_K[H^1_0(\Omega)],
\end{equation}
where $F_K^*:\mathbb{P}_K[H^1_0(\Omega)] \to L^2((0,T)\times\Gamma_c,\mathbb{R})$ is the adjoint operator associated to $F_K$.

For $G \in L^2((0,T)\times\Gamma_c,\mathbb{R})$ and $\phi_T \in \mathbb{P}_K[H^1_0(\Omega)]$, we have
(note that $\mathcal{K}$ is $\mathbb{R}$-linear)
\begin{align*}
\Re \big(\langle F_K(G),\overline{\phi_T} \rangle_{H^{-1},H^1_0} \big)
&= \Re \big(\langle r(T), \overline{\phi_T} \rangle_{H^{-1},H^1_0} \big) \quad\text{ because }\phi_T \in \mathbb{P}_K[H^1_0(\Omega)] \\
&= \Re \bigg(- i \int_0^T \int_{\Gamma_c} 2 G\partial_{\nu} \psi_{\rref} \overline{\partial_{\nu} \phi} d\sigma(x) \,dt\\
&\hspace*{2cm}- i \int_0^T \int_{\Omega} \langle \mathcal{K}(G), \overline{\phi} \rangle_{H^{-1},H^1_0} \,dx \,dt \bigg)
\quad \text{ by \eqref{def:sol_transp}} \\
&= \int_0^T \int_{\Gamma_c} G \big(2 \Im (\partial_{\nu} \psi_{\rref} \overline{\partial_\nu \phi}) + \mathcal{K}^* (i\phi)\big) d\sigma(x) \,dt
\quad \text{ by } \eqref{relation_K_adjoint}.
\end{align*}
Thus $F_K^*$ has the following explicit expression
\begin{align*}
F_K^*: \mathbb{P}_K[H^1_0(\Omega)] & \to L^2((0,T)\times\Gamma_c,\mathbb{R}) \\
\phi_T & \mto 2\Im(\partial_{\nu} \psi_{\rref} \overline{\partial_{\nu} \phi}) + \mathcal{K}^*(i\phi).
\end{align*}
Let $\mathcal{C}_1$ be as in Proposition \ref{Prop:Obs_faible}. There exists $K \in \mathbb{N}^*$ such that
$$\mathcal{C}_1 \|\phi_T\|_{H^{-1}(\Omega)} \leqslant \frac{1}{2} \|\phi_T\|_{H^1_0(\Omega)}, \quad \forall \phi_T \in \mathbb{P}_K[H^1_0(\Omega)]$$
and then \eqref{IO_HF} follows from \eqref{IO_faible} with $\mathcal{C}_2:=2\mathcal{C}_1$.
\end{proof}

\begin{proof}[Proof of Proposition \ref{Prop:Obs_faible}]
Let $\widetilde{T}$ be as in \HH2 and $T', T''$ be as in \HH3.
Let $\epsilon:=(T''-T'-\widetilde{T})/2$ and $\rho \in C^\infty_c(\mathbb{R},\mathbb{R})$ be such that $\rho \equiv 1$ on $(T'+\epsilon,T'+\epsilon+\widetilde{T})$
and $\Supp(\rho) \subset (T',T'')$.

\subsubsection*{Step~1}
We prove the existence of $C_1>0$ such that, for every $\phi_T \in H^1_0(\Omega)$, the solution of \eqref{adjoint_syst} satisfies
\begin{equation} \label{step1}
\|\phi_T\|_{H^1_0(\Omega)} \leqslant C_1 \left(
\| 2\Im(\rho\partial_{\nu} \psi_{\rref} \overline{\partial_{\nu} \phi}) \|_{L^2(\mathbb{R}\times\Gamma_c)}
+ \|\phi_T\|_{H^{-1}(\Omega)} \right).
\end{equation}
Let $\phi_T \in H^1_0(\Omega)$. By \HH2, we have
\begin{align*}
\|\phi_T\|_{H^1_0(\Omega)}^2
& \leqslant 2 \mathcal{C}_0^2 \bigg[ \int_{T'+\epsilon}^{T'+\epsilon+\widetilde{T}} \int_{\Gamma_c} |\partial_{\nu} \phi(t,x)|^2 d\sigma(x) \,dt
+ \|\phi_T\|_{H^{-1}}^2 \bigg] \\
& \leqslant 2 \mathcal{C}_0^2 \bigg[ \int_{\mathbb{R}} \int_{\Gamma_c} \big| \rho(t) \overline{\partial_{\nu} \phi(t,x)} \big|^2 d\sigma(x) \,dt
+ \|\phi_T\|_{H^{-1}}^2 \bigg]
\end{align*}
because $\rho \equiv 1$ on $(T'+\epsilon,T'+\epsilon+\widetilde{T})$.
Then, taking into account that $\Supp(\rho) \subset (T',T'')$ and
$|\partial_{\nu} \psi_{\rref}(t,x)| \geqslant m >0$ for every $(t,x) \in (T',T'') \times \Gamma_c$, i.e \HH3, we obtain
\begin{multline*}
\|\phi_T\|_{H^1_0(\Omega)}^2
\leqslant \frac{3 \mathcal{C}_0^2}{m^2} \int_{\mathbb{R}} \int_{\Gamma_c}
\big| \rho(t)\,\partial_{\nu} \psi_{\rref}(t,x)\, \overline{\partial_{\nu} \phi(t,x)} \big |^2 d\sigma(x) \,dt \\
- \mathcal{C}_0^2 \int_{\mathbb{R}} \int_{\Gamma_c} \left| \rho(t)\,\partial_{\nu} \phi(t,x) \right|^2 d\sigma(x) \,dt + 2 \mathcal{C}_0^2 \|\phi_T\|_{H^{-1}}^2.
\end{multline*}
Moreover,
\begin{multline*}
2 \int_{\mathbb{R}} \int_{\Gamma_c} \left| \rho\,\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi} \right|^2 d\sigma \,dt
= \int_{\mathbb{R}} \int_{\Gamma_c} \Big(\left| \rho\,\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi} \right|^2
+ \left| \rho\,\partial_{\nu} \overline{\psi_{\rref}}\,\partial_{\nu} \phi \right|^2 \Big) d\sigma \,dt \\
= \int_{\mathbb{R}} \int_{\Gamma_c} \Big(
\left| 2 \Im (\rho\,\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi}) \right|^2
+ 2 \Re ((\rho\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi})^2) \Big) d\sigma \,dt.
\end{multline*}
Note here that in the end of the proof, we will prove that the second term is compact. It will be crucial for that to notice that it is $\Re [ (\rho\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi})^2 ]$ and not $\left|\Re [ \rho\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi}] \right|^2 $.

Thus
\begin{multline*}
\|\phi_T\|_{H^1_0(\Omega)}^2 \leqslant
\frac{3 \mathcal{C}_0^2}{2 m^2} \int_{\mathbb{R}} \int_{\Gamma_c} \Big(\left| 2 \Im (\rho\,\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi}) \right|^2
+ 2 \Re [ (\rho\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi})^2 ] \Big) d\sigma \,dt \\
- \mathcal{C}_0^2 \int_{\mathbb{R}} \int_{\Gamma_c} \left| \rho\,\partial_{\nu} \phi \right|^2 d\sigma \,dt
+ 2 \mathcal{C}_0^2 \|\phi_T\|_{H^{-1}}^2.
\end{multline*}
In order to get \eqref{step1}, it suffices to prove the existence of $C>0$ such that
\begin{equation} \label{interm1}
\int_{\mathbb{R}} \int_{\Gamma_c} \Re [ (\rho\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi})^2 ] d\sigma \,dt
\leqslant \frac{m^2}{3} \int_{\mathbb{R}} \int_{\Gamma_c} \left| \rho\,\partial_{\nu} \phi \right|^2 d\sigma \,dt
+ C \|\phi_T\|_{H^{-1}(\Omega)}^2.
\end{equation}
Let $R \in \mathbb{N}^*$ that will be chosen later on,
\begin{equation} \label{def:psirefH}
\psi_{\rref}^H(t):=\mathbb{P}_R[\psi_{\rref}(t)] \quad \text{ and } \quad \psi_{\rref}^L:= \psi_{\rref} - \psi_{\rref}^H.
\end{equation}
Then
\begin{multline*}
\int_{\mathbb{R}} \int_{\Gamma_c} \Re [ (\rho\,\partial_{\nu} \psi_{\rref}\, \overline{\partial_{\nu} \phi})^2 ] d\sigma \,dt \\
= \int_{\mathbb{R}} \int_{\Gamma_c} \Re \Big((\rho\,\partial_{\nu} \psi_{\rref}^L \, \overline{\partial_{\nu} \phi})^2
+ (\rho\, \overline{\partial_{\nu} \phi})^2\,\partial_{\nu} \psi_{\rref}^H (2\,\partial_{\nu} \psi_{\rref}^L +\partial_{\nu} \psi_{\rref}^H) \Big) d\sigma \,dt.
\end{multline*}
In order to get \eqref{interm1}, it is sufficient to prove that, for $R$ large enough
\begin{align} \label{interm2}
\int_{\mathbb{R}} \int_{\Gamma_c} \Re [ (\rho\,\partial_{\nu} \psi_{\rref}^L \, \overline{\partial_{\nu} \phi})^2 ] d\sigma \,dt
&\leqslant C \|\phi_T\|_{H^{-1}},
\\ \label{interm3}
\int_{\mathbb{R}} \int_{\Gamma_c} \big| (\rho\, \overline{\partial_{\nu} \phi})^2\,\partial_{\nu} \psi_{\rref}^H (2\,\partial_{\nu} \psi_{\rref}^L +\partial_{\nu} \psi_{\rref}^H) \big| d\sigma \,dt
&\leqslant \frac{m^2}{2} \int_{\mathbb{R}} \int_{\Gamma_c} \left| \rho\,\partial_{\nu} \phi \right|^2 d\sigma \,dt.
\end{align}
Note that $\psi_{\rref}^H \to 0$ in $C^0([0,T],H^3_{(0)}(\Omega))$ when $R \to + \infty$ because $\psi_0 \in H^3_{(0)}(\Omega)$.
Thus, $\partial_\nu \psi_{\rref}^H \big{|}_{\Gamma_c} \to 0$ in $L^\infty((0,T)\times \Gamma_c)$ when $R \to + \infty$,
because of the Sobolev embeddings $\partial_\nu [ H^3_{(0)}(\Omega) ] \subset H^{3/2}(\Gamma_c) \subset L^\infty(\Gamma_c)$.
As a consequence, there exists $R_0$ such that for any $R\geq R_0$, we have
$$\|\partial_{\nu} \psi_{\rref}^H (2\partial_{\nu} \psi_{\rref}^L +\partial_{\nu} \psi_{\rref}^H) \|_{L^\infty((0,T)\times\Gamma_c)}
\leqslant \frac{m^2}{2}.$$
This implies \eqref{interm3}. Until the end of Step~2, $R$ is fixed.
The function $\rho$ is $C^\infty_c(\mathbb{R})$ thus, for every $N \in \mathbb{N}^*$, there exists $C_N>0$ such that
$$ \bigg| \int_{\mathbb{R}} \rho(t)^2 e^{i[\lambda_j+\lambda_J-\lambda_k-\lambda_K]t} \,dt \bigg| \leqslant \frac{C_N}{(\lambda_j+\lambda_J)^N},\quad
\forall j,J \in \mathbb{N}^*,\; k,K \in \{1,\dots,R-1\}.$$
We have
\begin{align*}
\bigg | \int_{\mathbb{R}} \int_{\Gamma_c} \Re [ (\rho\partial_{\nu} &\psi_{\rref}^L \, \overline{\partial_{\nu} \phi})^2 ] d\sigma \,dt \bigg |\\
&= \bigg| \sum_{k,K=1}^{R-1} \sum_{j,J=1}^\infty \psi^0_k \psi^0_{K} \overline{\phi^0_j \phi^0_J}
\left(\int_{\mathbb{R}} \rho(t)^2 e^{i[\lambda_j+\lambda_J-\lambda_k-\lambda_K]t} \,dt \right)\\[-5pt]
&\hspace*{5cm}\cdot\left(\int_{\Gamma_c}\partial_{\nu} \varphi_k\,\partial_{\nu} \varphi_K\,\partial_{\nu} \varphi_j\,\partial_{\nu} \varphi_J d\sigma \right) \bigg | \\
&\leqslant \sum_{k,K=1}^{R-1} \sum_{j,J=1}^\infty |\psi^0_k \psi^0_{K} \phi^0_j \phi^0_J|
\frac{C_N}{(\lambda_j+\lambda_J)^N} \lambda_k^{\alpha} \lambda_K^{\alpha} \lambda_j^{\alpha} \lambda_J^{\alpha}\quad \text{ for some } \alpha>0\\
&\leqslant C(\psi_0,R,N) \bigg(\sum_{j}^\infty \lambda_j^{\alpha-N} |\phi^0_j| \bigg)^2 \\
&\leqslant C'(\psi_0,R,N) \| \phi_T\|_{H^{-1}(\Omega)}^2 \text{ for } N \text{ large enough.}
\end{align*}
Here, we have used Weyl law to get that for another $N$ large enough, $\sum_j^{\infty}\lambda_j^{-N}<+\infty$. Inequality \eqref{interm2} is proved, which ends Step~1.

\subsubsection*{Step~2}
We prove the existence of $C_2>0$ such that, for every $\phi_T \in H^1_0(\Omega)$, the solution of \eqref{adjoint_syst} satisfies
\begin{equation} \label{step2}
\|\phi_T\|_{H^1_0(\Omega)} \leqslant C_2 \left(
\| 2\Im(\partial_{\nu} \psi_{\rref} \overline{\partial_{\nu} \phi}) + \mathcal{K}^*(i\phi) \|_{L^2((0,T)\times\Gamma_c)}
+ \|\phi_T\|_{H^{-1}(\Omega)} \right).
\end{equation}
From \eqref{step1}, we deduce that
\begin{multline*}
\|\phi_T\|_{H^1_0(\Omega)}\\ \leqslant C_1 \left(
\| 2\Im(\rho\partial_{\nu} \psi_{\rref} \overline{\partial_{\nu} \phi}) + \mathcal{K}^*(i\phi) \|_{L^2(\mathbb{R}\times\Gamma_c)}
+ \| \mathcal{K}^*(i\phi) \|_{L^2(\mathbb{R}\times\Gamma_c)}
+ \|\phi_T\|_{H^{-1}(\Omega)} \right).
\end{multline*}
In order to prove \eqref{step2}, it suffices to prove the existence of $C>0$ such that,
\begin{equation} \label{interm4}
\| \mathcal{K}^*(i\phi) \|_{L^2((0,T)\times\Gamma_c)}
\leqslant \frac{1}{2C_1} \|\phi_T\|_{H^1_0(\Omega)} + C \|\phi_T\|_{H^{-1}}, \quad \forall \phi_T \in H^1_0(\Omega).
\end{equation}
Let $\widetilde{R} \in \mathbb{N}^*$ that will be chosen later on and $\psi_{\rref}^H$ be defined as in \eqref{def:psirefH} but with $R$ replaced by $\widetilde{R}$.
From \eqref{def:K} and the expansion \eqref{dvpt:Delta2(vpsi)}, we see that $\mathcal{K}=\mathcal{K}_1 + \mathcal{K}_2$, where
\begin{align*}
\mathcal{K}_1: L^2((0,T)\times\Gamma_c,\mathbb{R}) & \to L^2((0,T),H^{-1}(\Omega)) \\
G & \mto \Delta^2(V \psi_{\rref}^H)
\end{align*}
and
\begin{align*}
\mathcal{K}_2: L^2((0,T)\times\Gamma_c,\mathbb{R}) & \to L^2((0,T),H^{-1/2-\eps}(\Omega)) \\
G & \mto \Delta^2(V \psi_{\rref}^L)
\end{align*}
is continuous. We denote by $\|\mathcal{K}_1\|$ and $\|\mathcal{K}_2\|$ their operator norm in this functional frame.

Note that $\psi_{\rref}^H \to 0$ in $C^0([0,T],H^3_{(0)}(\Omega))$ when $\widetilde{R} \to + \infty$ because $\psi_0 \in H^3_{(0)}(\Omega)$.
Thus, there exists $\widetilde{R}\in \mathbb{N}^*$ such that
$$\| \mathcal{K}_1 \| \leqslant \frac{1}{2C_1\sqrt{T}}.$$
From now on $\widetilde{R}$ is fixed. Then, for every $\phi_T \in H^1_0(\Omega)$,
\begin{align*}
\| \mathcal{K}^*(i\phi) \|_{L^2((0,T)\times\Gamma_c)}
& \leqslant \| \mathcal{K}_1^*(i\phi) \|_{L^2((0,T)\times\Gamma_c)} + \| \mathcal{K}_2^*(i\phi) \|_{L^2((0,T)\times\Gamma_c)} \\
& \leqslant \frac{1}{2C_1\sqrt{T}} \|\phi\|_{L^2((0,T),H^1_0(\Omega))} + \| \mathcal{K}_2 \| \| \phi \|_{L^2((0,T),H^{1/2+\eps}(\Omega))} \\
& \leqslant \frac{1}{2C_1} \|\phi_T\|_{H^1_0(\Omega)} + \sqrt{T}\, \| \mathcal{K}_2 \| \|\phi_T\|_{H^{1/2+\eps}(\Omega)}.
\end{align*}
This implies \eqref{interm4} after interpolation, which ends Step~2.
\end{proof}

\subsection{Control of high frequencies for the nonlinear system}
\label{subsec:control_HF_nl}

The goal of this section is the proof of Theorem \ref{Thm:controle_HF}.
Thus, in the whole section, $T>0$, $\psi_0 \in H^3_{(0)}(\Omega) \cap \mathcal{S}$, $\psi_{\rref}(t):=e^{i \Delta_D t} \psi_0$ are fixed
and \HH1, \HH2, \HH3 are assumed to hold. We consider the end-point map
\begin{align*}
\widetilde{\Theta}_K: L^2((0,T)\times\Gamma_c,\mathbb{R}) & \to \mathbb{P}_K [H^3_{(0)}(\Omega)] \\
g & \mto \mathbb{P}_K [\psi(T)],
\end{align*}
where $\psi$ solves \eqref{Schro_Poisson}. By Proposition \ref{Prop:C1}, $\widetilde{\Theta}_K$ is of class $C^1$ and
\begin{align*}
d\widetilde{\Theta}_K(0): L^2((0,T)\times\Gamma_c,\mathbb{R}) & \to \mathbb{P}_K [H^3_{(0)}(\Omega)] \\
G & \mto \mathbb{P}_K [\Psi(T)],
\end{align*}
where $\Psi$ solves \eqref{Schro_Poisson_linearise} with $\Psi_0=0$.
By Proposition \ref{Prop:control_lin_H3}, there exists $K \in \mathbb{N}^*$ such that $d\widetilde{\Theta}_K(0)$ has a continuous right inverse.
By the inverse mapping theorem, $\widetilde{\Theta}_K$ is a local $C^1$-diffeomorphism on a neighborhood of $0$.
The first statement of Theorem \ref{Thm:controle_HF} holds with $\Upsilon:= \widetilde{\Theta}_{K}^{-1}$ which is locally well-defined.

There exists $K'\geqslant K$ such that
$$\|\psi_{\rref}(T)-\mathbb{P}_{K'}[\psi_{\rref}(T)]\|_{H^3_{(0)}} < \delta.$$
Then, $\psi_f:=(\mathbb{P}_K -\mathbb{P}_{K'})[\psi_{\rref}(T)] ]$ belongs to $\mathcal{V}$
thus $g:=\Upsilon(\psi_f)$ is well-defined in $L^2((0,T)\times\Gamma_c,\mathbb{R})$
and the associated solution of \eqref{Schro_Poisson} satisfies
$\mathbb{P}_K[\psi(T)]=\psi_f$ thus $\mathbb{P}_{K'}[\psi(T)]=0$.
As a consequence $\psi(T)$ is a finite sum of eigenfunctions of $(-\Delta_D)$
so it is a smooth function.\qed

\section{Local exact control around eigenfunctions}
The goal of this section is to prove Theorem \ref{thm:control_eig},
by following essentially the same strategy as in the previous section.
In Section \ref{subsec:Obs_eig}, we prove the observability of the adjoint of the linearized system.
In Section \ref{subsec:thm:control_eig}, we prove Theorem \ref{thm:control_eig}.

\subsection{Observability results}
\label{subsec:Obs_eig}

The goal of this section is to prove the following observability inequality.

\begin{Prop} \label{Prop:Obs_eig}
Let $T>0$, $R \in \mathbb{N}^*$. We assume that \HH1, \HH2, \HH{3'} and \HH4 hold.
There exists $\mathcal{C}_2>0$ such that, for every $\phi_T \in H^1_0(\Omega) \cap T_{\mathcal{S}} \varphi_R$ the solution~of
\begin{equation} \label{adjoint_syst_eig}
\begin{cases}
\big(i\partial_t+\Delta+\lambda_R\big) \phi(t,x) = 0, \quad & (t,x) \in (0,T)\times\Omega, \\
\phi(t,x)= 0, & (t,x) \in (0,T)\times\partial \Omega, \\
\phi(T,x)=\phi_T, & x \in \Omega, \\
\end{cases}
\end{equation}
satisfies
\begin{equation} \label{IO_eig}
\|\phi_T\|_{H^1_0(\Omega)} \leqslant \mathcal{C}_2
\| 2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi}) + \widetilde{\mathcal{K}}^*(i\phi) \|_{L^2((0,T)\times\Gamma_c)},
\end{equation}
where
\begin{equation} \label{def:K_tilde}
\begin{split}
\widetilde{\mathcal{K}}: L^2(\Gamma_c,\mathbb{R}) & \to L^2(\Omega,\mathbb{C}) \\[-10pt]
G & \mto \Delta^2(V \varphi_R), \quad \text{ where }
\left\lbrace \begin{array}{l}
(-\Delta+1)V=0, \text{ in } \Omega, \\
\partial_{\nu} V = G 1_{\Gamma_c} \text{ on }\partial \Omega. \\
\end{array}\right.
\end{split}
\end{equation}
\end{Prop}

The proof of Proposition \ref{Prop:Obs_eig} relies on two key ingredients:
the weak observability of the Schrödinger equation \HH2
and a unique continuation result given by the following statement.

\begin{Prop} \label{Prop:UC}
Let $T>0$, $R \in \mathbb{N}^*$. We assume that \HH1, \HH2, \HH{3'} and~\HH4 hold.
Then for every non-zero $\phi_T \in H^1_0(\Omega) \cap T_{\mathcal{S}}\varphi_R$, the function
$2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi}) + \widetilde{\mathcal{K}}^*(i\phi)$ is not identically zero on $(0,T)\times\Gamma_c$,
where $\phi(t):=e^{i\Delta_D(t-T)} \phi_T$.
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop:UC}]
Let $T' \in (\widetilde{T},T)$, where $\widetilde{T}$ is as in \HH2.
By Proposition \ref{Prop:Obs_faible} (with $T$ replaced by $T'$)
and the change of phase $\phi(t,x) \leftarrow \phi(t,x) e^{i\lambda_R t}$,
we know that there exists $\mathcal{C}_1>0$ such that,
for every $\phi_T \in H^1_0(\Omega)$, the solution of \eqref{adjoint_syst_eig} satisfies
\begin{equation} \label{IO_faible_eig}
\|\phi_T\|_{H^1_0(\Omega)} \leqslant \mathcal{C}_1 \big(
\| 2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi}) + \widetilde{\mathcal{K}}^*(i\phi) \|_{L^2((0,T')\times\Gamma_c)}
+ \|\phi_T\|_{H^{-1}(\Omega)} \big).
\end{equation}
We introduce
$$N_T:=\left\{ \phi_0 \in H^1_0(\Omega)\cap T_{\mathcal{S}}\varphi_R \sep 2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi}) + \widetilde{\mathcal{K}}^*(i\phi) =0
\text{ in } L^2((0,T)\times\Gamma_c) \right\},$$
where $\phi(t) := e^{i(\Delta_D+\lambda_R)t} \phi_0 :=S(t)\phi_0$.
$N_T$ is a $\mathbb{R}$-vector subspace of $H^1_0(\Omega)$. We want to prove that $N_T=\{0\}$.

\subsubsection*{Step~1}
We prove that $N_T$ contains eigenfunctions of $-\Delta_D$.
Let $\phi_0 \in N_T$.
Since $\phi_0 \in H^1_0(\Omega)$, then $\phi_\epsilon :=\psfrac{S(\epsilon)\phi_0-\phi_0}{\epsilon}$ is bounded in $H^{-1}(\Omega)$ uniformly when $\epsilon \to 0$.
Moreover, for $\epsilon<T-T'$, $\phi_\epsilon$ belongs to $N_{T'}$.
Applying \eqref{IO_faible_eig} to $\phi_\epsilon$, with $\epsilon < T-T'$, we get $\| \phi_\epsilon \|_{H^1_0(\Omega)} \leqslant \mathcal{C}_1 \|\phi_\epsilon\|_{H^{-1}}$.
Thus $\phi_\epsilon$ is bounded in $H^1_0(\Omega)$ when $\epsilon \to 0$.
Therefore $\phi_0 \in H^3_{(0)}(\Omega)$. Indeed, by Fatou's lemma,
\begin{align*}
\|\phi_0\|_{H^3_{(0)}}^2  = \sum_{k=1}^\infty \big| \lambda_k^{3/2} \langle \phi_0, \varphi_k \rangle \big|^2
& = \sum_{k=1}^\infty \liminf_{\epsilon \to 0}
\Big| \sqrt{\lambda_k} \frac{e^{i \lambda_k \epsilon} - 1}{\epsilon} \langle \phi_0, \varphi_k \rangle \Big|^2 \\
& \leqslant \liminf_{\epsilon \to 0} \sum_{k=1}^\infty
\Big| \sqrt{\lambda_k} \frac{e^{i \lambda_k \epsilon} - 1}{\epsilon} \langle \phi_0, \varphi_k \rangle \Big|^2 \\
& \leqslant \liminf_{\epsilon \to 0} \|\phi_\epsilon\|_{H^1_0} < \infty.
\end{align*}
For $T''<T$, the map
\begin{align*}
\mathcal{J}_{T''}: H^1_0(\Omega) & \to L^2((0,T'')\times\Gamma_c) \\
\phi_0 & \mto 2 \Im[\partial_\nu \varphi_R \overline{\partial_\nu \phi}]+\mathcal{K}^*(i\phi)
\end{align*}
is continuous (see Propositions \ref{Prop:smoothin_effect_DN} and \ref{Prop:WP_prel}, which imply the continuity of
$\mathcal{K}:L^2(\Gamma_c,\mathbb{R}) \to L^2(\Omega)$ and thus of $\mathcal{K}^*:L^2(\Omega) \mto L^2(\Gamma_c,\mathbb{R})$).
Moreover, $\mathcal{J}_{T''}(\phi_\epsilon)=0$ for every \hbox{$\epsilon<T-T''$},
and $\phi_\epsilon \to i(\Delta_D+\lambda_R)\phi_0$ in $H^1_0(\Omega)$ thus
$\mathcal{J}_{T''}[i(\Delta_D+\lambda_R)\phi_0]=0$.
This holds for every $T'' \in (0,T)$, thus $i(\Delta_D+\lambda_R)\phi_0 \in N_T$.

We have proved that $N_T$ is stable by $i(\Delta_D+\lambda_R)$ and only contains smooth functions in $D[(-\Delta_D)^s]$, $\forall s\in \mathbb{N}^*$.
Applying again the estimate \eqref{IO_faible_eig} to $i(\Delta_D+\lambda_R)\phi_0$, we get
$$\| (\Delta_D+\lambda_R)\phi_0 \|_{H^1_0(\Omega)} \leqslant \mathcal{C}_1 \| (\Delta_D+\lambda_R)\phi_0 \|_{H^{-1}(\Omega)}.$$
This shows that the unit ball of $N_T$ is compact (for the $H^1_0(\Omega)$-topology), thus $N_T$ has finite dimension (Rellich's theorem).

Then the finite dimensional space $N_T$ is stable by the real symmetric operator $-(\Delta_D+\lambda_R)^2 = [i(\Delta_D+\lambda_R)]^2$,
thus $-(\Delta_D+\lambda_R)^2$ has an eigenfunction in $N_T$.
This candidate is also an eigenfunction of $(-\Delta_D)$.

\subsubsection*{Step~2}
We prove that
\begin{equation} \label{Step2_UC}
\widetilde{\mathcal{K}}^*(z \varphi_k) = \Re(z) \big(\lambda_k^2 w_k - 2\,\partial_{\nu} \varphi_R\,\partial_{\nu} \varphi_k \big)_{\left|\Gamma_c\right.}, \quad \forall z \in \mathbb{C}, k \in \mathbb{N}^*,
\end{equation}
where $w_k$ is defined in \HH4.
Let $z \in \mathbb{C}$ and $k \in \mathbb{N}^*$. For every $G \in L^2 (\Gamma_c,\mathbb{R})$, we have
$$\int_{\Gamma_c} G \widetilde{\mathcal{K}}^*(z \varphi_k) d\sigma(x)
= \Re\left(\int_{\Omega} \widetilde{\mathcal{K}}(G) \overline{z \varphi_k} \,dx \right)
= \Re(z) \int_{\Omega} \widetilde{\mathcal{K}}(G) \varphi_k \,dx, $$
thus $\widetilde{\mathcal{K}}^*(z \varphi_k)=\Re(z)\widetilde{\mathcal{K}}^*(\varphi_k)$.

If $G \in C^\infty_c((0,T)\times\Gamma_c,\mathbb{R})$, then the potential $V$ defined in \eqref{def:K_tilde}
belongs to $C^\infty((0,T)\times\Omega,\mathbb{R})$ thus the following integrations by part are legitimate:
\begin{align*}
\int_{\Omega} \widetilde{\mathcal{K}}(G) \varphi_k \,dx &= \int_{\Omega} \Delta^2(V\varphi_R) \varphi_k \,dx \quad\text{ with } V \text{ as in \eqref{def:K_tilde} }\\
&= \int_{\partial \Omega}\hspace*{-1mm} \big(\partial_{\nu}[\Delta(V\varphi_R)] \varphi_k - \Delta(V\varphi_R)\,\partial_{\nu} \varphi_k \big) d\sigma(x)
+ \int_{\Omega}\hspace*{-1mm} \Delta(V\varphi_R) \Delta \varphi_k \,dx \\
&= -2 \int_{\Gamma_c} G\partial_{\nu} \varphi_R\,\partial_{\nu} \varphi_k\, d\sigma(x) - \lambda_k \int_{\Omega} \Delta(V\varphi_R) \varphi_k \,dx,\\
\int_{\Omega} \Delta(V\varphi_R) \varphi_k \,dx & = - \lambda_k \int_{\Omega} V\varphi_R \varphi_k \,dx \\
& = - \lambda_k \int_{\Omega} V (-\Delta+1) w_k \,dx \\
& = - \lambda_k \int_{\Gamma_c} G w_k d\sigma(x).
\end{align*}
This proves that
$$\int_\Omega \widetilde{\mathcal{K}}(G)\varphi_k \,dx = \int_{\Gamma_c} G \left(\lambda_k^2 w_k - 2\partial_{\nu} \varphi_R\,\partial_{\nu} \varphi_k \right) d\sigma$$
for every $G \in C^\infty_c((0,T) \times \Gamma_c,\mathbb{R})$.
The same equality holds by density for every \hbox{$G \in L^2((0,T)\times\Gamma_c,\mathbb{R})$}.
Therefore, $\widetilde{\mathcal{K}}^*(\varphi_k) = \lambda_k^2 w_k - 2\partial_{\nu} \varphi_R\,\partial_{\nu} \varphi_k$.

\subsubsection*{Step~3}
We prove that $N_T = \{0\}$.
Working by contradiction, we assume that $N_T \neq \{0\}$.
By Step~1, there exists $z \in \mathbb{C} \setminus \{0\}$ and $k \in \mathbb{N}^*$ such that $z \varphi_k \in N_T$.
Moreover, by Step~2,
$$2\Im \big(\partial_{\nu} \varphi_R \overline{\partial_{\nu} [z \varphi_k e^{i(\lambda_R-\lambda_k)t}]} \big) + \widetilde{\mathcal{K}}^*(i z \varphi_k e^{i(\lambda_R-\lambda_k)t})
= \Re(i z e^{i(\lambda_R-\lambda_k)t}) \lambda_k^2 w_k.$$
Thus $\Re(i z e^{i(\lambda_R-\lambda_k)t}) w_k$ vanishes on $(0,T) \times \Gamma_c$.
By Assumption \HH4, we deduce that
\begin{equation} \label{Formule_UC_1}
\Re(i z e^{i(\lambda_R-\lambda_k)t}) \equiv 0 \quad \text{ on } (0,T).
\end{equation}

\noindent \emph{First case: $k=R$.} Then $z \in i \mathbb{R}$ because $z \varphi_k \in T_{\mathcal{S}}\varphi_R$.
From \eqref{Formule_UC_1}, we deduce that $0=\Re(i z)=iz$, which is impossible because $z \neq 0$.

\noindent \emph{Second case: $k \neq R$.} Then $\lambda_k \neq \lambda_R$ because $\lambda_R$ is a simple eigenvalue (see \HH4).
From \eqref{Formule_UC_1}, we deduce that $z=0$, which is a contradiction.
\end{proof}

\begin{rk} \label{rkmisseddirection}
Note that Assumption \HH4 is necessary for the controllability of the linearized system \ref{Schro_Poisson_linearise} (with $\Psi_0=0$) around
$(\psi_{\rref}(t)=\varphi_R e^{-i\lambda_R t}, g_{\rref} \equiv 0)$.
Indeed, for any eigenfunction $\Phi$ of $(-\Delta_D)$ associated with the eigenvalue $\lambda_\Phi$, we have
\begin{align*}
\langle \Psi(T), \Phi \rangle
& = \Big\langle i \int_0^T e^{-i\Delta_D(T-t)} \big(V(t) \psi_{\rref}(t) \big) \,dt, \Phi \Big\rangle \\
& = i e^{-i\lambda_\Phi T} \int_0^T e^{-i(\lambda_R-\lambda_\Phi)t} \big\langle V(t), \varphi_R \Phi \big\rangle \,dt \\
& = i e^{-i\lambda_\Phi T} \int_0^T e^{-i(\lambda_R-\lambda_\Phi)t} \langle G(t), w \rangle_{L^2(\Gamma_c)} \,dt,
\end{align*}
where $w$ is as in \HH4 (apply the Green formula, as in the proof of Proposition 2). In particular, if \HH4 is not fulfilled, we are in one of the following situation
\begin{itemize}
\item if $\lambda_R$ is a multiple eigenvalue, then,
for any eigenfunction $\Phi$ associated to the eigenvalue $\lambda_\Phi=\lambda_R$ and linearly independent of $\varphi_R$,
one cannot control the complex component $i \langle \Psi(T), \Phi \rangle e^{i\lambda_\Phi T}$ because it only takes real values
$$i \langle \Psi(T), \Phi \rangle e^{i\lambda_\Phi T} = \int_0^T \langle G(t), w \rangle \,dt \in \mathbb{R},$$
\item if there exists an eigenfunction $\Phi$ for which $w \equiv 0$ on $\Gamma_c$, then, one cannot control the component $\langle \Psi(T), \Phi \rangle$. Indeed, whatever the control $G$ is, the final state satisfies
$$\langle \Psi(T), \Phi \rangle = i e^{-i\lambda_\Phi T} \int_0^T e^{-i(\lambda_R-\lambda_\Phi)t} \langle G(t), w \rangle_{L^2(\Gamma_c)} \,dt = 0.$$
\end{itemize}
\end{rk}

\begin{proof}[Proof of Proposition \ref{Prop:Obs_eig}]
Working by contradiction, we assume that, for every $n \in \nobreak\mathbb{N}^*$, there exists $\phi_T^n \in H^1_0(\Omega)$ such that
\begin{equation} \label{Obs_eig_absurd}
1=\|\phi_T^n\|_{H^1_0(\Omega)} > n \| 2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi^n}) + \widetilde{\mathcal{K}}^*(i\phi^n) \|_{L^2((0,T)\times\Gamma_c)}.
\end{equation}
Up to a subsequence, one may assume that $\phi_T^n \rightharpoonup\phi_T^\infty$ weakly in $H^1_0(\Omega)$.
The strong continuity of the operator
\begin{align*}
H^1_0(\Omega) & \to L^2((0,T)\times\Gamma_c)\\
\phi_T & \mto 2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi}) +\widetilde{\mathcal{K}}^*(i\phi)
\end{align*}
implies its continuity for the weak topology \cite[Th.\,III.9]{Brezis_An_Fonc}.
Thus
$$2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi^n}) + \widetilde{\mathcal{K}}^*(i\phi^n)
\underset{n \to \infty}{\rightharpoonup}
2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi^\infty}) + \widetilde{\mathcal{K}}^*(i\phi^\infty)
\text{ in } L^2((0,T)\times\Gamma_c).$$
From \eqref{Obs_eig_absurd}, we deduce that
$$2\Im(\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi^\infty}) + \widetilde{\mathcal{K}}^*(i\phi^\infty) =0 \text{ on } (0,T)\times\Gamma_c.$$
By Proposition \ref{Prop:UC}, $\phi^\infty_T=0$. Thus, up to a subsequence, $\|\phi_T^n\|_{H^{-1}(\Omega)} \to 0$ when $n \to \infty$.
We deduce from \eqref{IO_faible_eig} that
$$1=\|\phi_T^n\|_{H^1_0(\Omega)} \leqslant \mathcal{C}_1 \Big(
\frac{1}{n} + \|\phi_T^n\|_{H^{-1}(\Omega)} \Big) \underset{n \to \infty}{\to} 0,$$
which is a contradiction.
\end{proof}

\subsection{Proof of Theorem \ref{thm:control_eig}}
\label{subsec:thm:control_eig}

Let $T>0$, $R \in \mathbb{N}^*$ and $\psi_{\rref}(t):=\varphi_R(x)e^{-i\lambda_R t}$.
We assume that \HH1, \HH2, \HH{3'} and \HH4 hold.
By Proposition \ref{Prop:C1}, the end-point map
$$\begin{array}{crclcl}
\widetilde{\Theta}: & [H^3_{(0)}(\Omega) \cap \mathcal{S} ] & \times & L^2((0,T)\times\Gamma_c,\mathbb{R}) & \dpl\to & [H^3_{(0)}(\Omega) \cap \mathcal{S} ]^2 \\[3pt]
& (\psi_0 &, & g) & \dpl\mto & (\psi_0,\psi(T) e^{i \lambda_R T})
\end{array}$$
is of class $C^1$ and
$$\begin{array}{crclcl}
d\widetilde{\Theta}(\varphi_R,0): & [H^3_{(0)}(\Omega) \cap T_{\mathcal{S}} \varphi_R ] & \times & L^2((0,T)\times\Gamma_c,\mathbb{R}) & \dpl\to &
[H^3_{(0)}(\Omega) \cap T_{\mathcal{S}} \varphi_R]^2 \\[3pt]
& (\Psi_0 &, & G) & \dpl\mto & (\Psi_0,\Psi(T)),
\end{array}$$
where
\begin{equation} \label{Schro_Poisson_linearise_eig}
\begin{cases}
\big(i\partial_t+\Delta+\lambda_R\big) \Psi(t,x) = V(t,x) \varphi_R, \quad & (t,x) \in (0,T)\times\Omega, \\
\Psi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\Psi(0,x)=\Psi_0(x), & x \in \Omega, \\
(-\Delta+1)V(t,x)=0, & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} V(t,x)=G(t,x) 1_{\Gamma_c}(x), & (t,x) \in (0,T)\times\partial \Omega. \\
\end{cases}
\end{equation}
The same arguments as in the previous section prove that $d\widetilde{\Theta}(\varphi_R,0)$ has a continuous right inverse
thanks to Proposition \ref{Prop:Obs_eig}. The inverse mapping theorem proves that $\widetilde{\Theta}$ is a local $C^1$-diffeomorphism. \qed

\subsection{Some comments about control only depending on time}
\label{sectioncommentsut}
In this subsection, we comment on the information given by our previous study for a control that would be of the form\vspace*{-5pt}
\begin{equation} \label{Schro_Poissondept}
\quad\begin{cases}
(i\partial_t + \Delta) \psi(t,x) = v(t,x) \psi(t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\psi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\psi(0,x)=\psi_0(x) & x \in \Omega, \\
(-\Delta+1)v(t,x)=0, & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} v(t,x)=g(t)\mu(x) 1_{\Gamma_c}(x) & (t,x) \in (0,T)\times\partial \Omega,
\end{cases}
\end{equation}
where the control $g\in L^2(0,T)$ only depends on time and $\mu\in L^{\infty}(\partial \Omega)$ is fixed.

This is very close to the configuration studied in several papers described in the introduction (except that we impose that the potential $v=g(t)V_{\mu}$ is harmonic).

If we followed the framework described in the previous section, we would obtain that the observability estimate necessary to obtain the controllability of the linearized system, would be of the form\vspace*{-5pt}\enlargethispage{.2\baselineskip}%
\begin{equation} \label{IO_eigdept}\quad
\|\phi_T\|_{H^1_0(\Omega)}^2 \leqslant \mathcal{C}_2
\int_0^T \left|\int_{\partial \Omega }2\Im(\mu(x)\partial_{\nu} \varphi_R \overline{\partial_{\nu} \phi}) + \mathcal{K}^*(i\phi)d\sigma(x) \right|^2 \,dt,
\end{equation}
instead of the classical observability estimate \eqref{IO_eig}, where $\mathcal{K}$ would be another (but similar) compact operator.

Then, it becomes quite clear that \eqref{IO_eigdept} is really not likely to be true in general. Indeed, to contradict \eqref{IO_eigdept}, it suffices to find a sequence of solutions with $H^1_0$ norm equals to $1$ but whose Neumann trace on the boundary weakly converges to zero. This can be easily done for instance
\begin{itemize}
\item on the square $(0,\pi)^2$: with the normalized initial data
$\phi_n(x,y)\!=\!c_n\sin(nx)\mkern-1mu\sin(ny)$
\item on the disk: with some initial data $\phi_n(r,\theta)= e^{in \theta} g_n(r)$, where $g_n$ are appropriate normalized Bessel functions.
\end{itemize}
Also, this analysis can also help to identify some class of initial data that could be observable. It is necessary that there is no oscillation of the trace on the boundary. This could be true for instance
\begin{itemize}
\item on the square: with some initial data that would depend only on $x$ or $y$.
\item on the disk: with some radial initial data.
\end{itemize}
Also, from a microlocal point of view, we can see that \eqref{IO_eigdept} can only be true for sequences of solutions whose trace do not weakly converge to zero. That means some data whose Neumann trace on the boundary is concentrated in the frequency $\xi'=0$. Yet, the theorems of propagation of microlocal defect measure (described for example for the wave equation in \cite{Burqbourbakimesures}) could suggest that it implies that the data are concentrated close to some rays that intersect the boundary orthogonally.

\section{Schrödinger equation with real-valued boundary controls}
\label{sectrealcontrol}
The goal of this section is to prove Theorem \ref{thm:anecdotic}.
The arguments have already been developed in the Step~1 of the proof of Proposition \ref{Prop:Obs_faible}
but here, we give a simpler proof in this simpler case.

\begin{Prop} \label{Prop:Weak_Obs_Im}
Let $\Omega$ be an open subset of $\mathbb{R}^2$, $\Gamma$ be an open subset of $\partial \Omega$ and $0 < \widetilde{T} < T < \infty$.
If the Schrödinger equation on $\Omega$ is weakly observable on $(0,\widetilde{T})\times\Gamma$
then, there exists $\mathcal{C}_1>0$ such that
\begin{equation}
\|\phi_0\|_{H^1_0(\Omega)} \leqslant \mathcal{C}_1 \big(\| \Im(\partial_{\nu} \phi) \|_{L^2((0,T)\times\Gamma)}
+ \|\phi_0\|_{H^{-1}(\Omega)} \big), \quad \forall \phi_0 \in H^1_0(\Omega,\mathbb{C}),
\end{equation}
where $\phi(t):=e^{it\Delta_D} \phi_0$.
\end{Prop}
\bnp

Let $T',T'' \in (0,T)$ and $\rho \in C^\infty_c(\mathbb{R},\mathbb{R}^+)$
be such that $T''-T'>\widetilde{T}$, $\rho \equiv 1$ on $(T',T'')$ and $\Supp(\rho) \subset (0,T)$.
The assumption and conservation of the norm give
\begin{equation}
\|\phi_0\|_{H^1_0(\Omega)}^2 \leqslant \mathcal{C}_0 \big(\| \rho\partial_{\nu} \phi \|_{L^2(\R_t\times\Gamma)}^2
+ \|\phi_0\|_{H^{-1}(\Omega)}^2 \big), \quad \forall \phi_0 \in H^1_0(\Omega,\mathbb{C})
\end{equation}
Since $\left|\Im(\partial_{\nu} \phi)\right|^2=\sfrac{|\partial_{\nu} \phi|^2}{2}+\sfrac{\Re\left((\partial_{\nu} \phi)^2\right)}{2}$, it is enough to prove
\bna
\left|\int_{\R_t} \int_{\Gamma }\rho^2(\partial_{\nu} \phi)^2\right|\leq C\|\phi_0\|_{H^{-1}(\Omega)}^2.
\ena
This inequality is obviously false if $(\partial_{\nu} \phi)^2$ is replaced by $|\partial_{\nu} \phi|^2$.
It is actually true because $(\partial_{\nu} \phi)^2$ is the product of two terms that oscillate in the same direction, which is false for $|\partial_{\nu} \phi|^2$.

More precisely, write $\phi_0=\sum_k \varphi_k\phi_{0,k}$, where $(\varphi_k)_{k\in \N^*}$ is an orthonormal basis of eigenfunctions
for $\Delta_D$ with eigenvalues $-\lambda_k$. We have $\phi= \sum_k e^{-i\lambda_k}\varphi_k\phi_{0,k}$
\bna
\int_{\R_t} \int_{\Gamma }\rho^2(\partial_{\nu} \phi)^2=\sum_{k,l}\phi_{0,l}\phi_{0,k}\left(\int_{\R_t}
e^{-i(\lambda_k+\lambda_l)t}\rho^2\right)\left(\int_{\Gamma }(\partial_{\nu}\varphi_k)(\partial_{\nu}\varphi_l)\right).
\ena
The boundary term can be bounded for instance by trace estimates (actually, finer estimates would give a better exponent $\lambda_k^{1/2}$)
\begin{align*}
\int_{\Gamma }\left|(\partial_{\nu}\varphi_k)(\partial_{\nu}\varphi_l)\right|
& \leq \nor{\partial_{\nu}\varphi_k}{L^2(\Gamma)}\nor{\partial_{\nu}\varphi_l}{L^2(\Gamma)}
\leq C\nor{\varphi_k}{H^{2}(\Omega)}\nor{\varphi_l}{H^{2}(\Omega)}
\\
& \leq C\lambda_k \lambda_l
\leq C(\lambda_k+\lambda_l)^2.
\end{align*}
The function $\rho$ is $C^\infty_c(\mathbb{R})$ thus, for every $N \in \mathbb{N}^*$, there exists $C_N>0$ (whose value may change) such that
$$ \left| \int_{\mathbb{R}} \rho(t)^2 e^{-i[\lambda_k+\lambda_l]t} \,dt \right| \leqslant \frac{C_N}{(\lambda_k+\lambda_l)^N}\leqslant \frac{C_N}{(\lambda_k\lambda_l)^{N/2}},\quad
\forall k,l \in \mathbb{N}^*$$
So, choosing $N$ large enough (and replacing $N/2-2$ by $N$), we obtain the bound
\bna
\left|\int_{\R_t} \int_{\Gamma }\rho^2(\partial_{\nu} \phi)^2\right|\leq \sum_{k,l}|\phi_{0,l}\phi_{0,k}|\frac{C_N}{(\lambda_k\lambda_l)^N}\leq C_N\bigg(\sum_{k}\frac{|\phi_{0,k}|}{\lambda_k^N}\bigg)^2\leq C\|\phi_0\|_{H^{-1}(\Omega)}^2
\ena
when $N$ is chosen large enough so that $\lambda_k^{1/2-N}$ is summable, which is always possible (for instance using Weyl law).
\enp
The following result can be proved thanks to the arguments developed in the proof of Proposition \ref{Prop:UC}
(in a simpler way, because $\mathcal{K}^*$ does not appear anymore),
together with Proposition \ref{Prop:Weak_Obs_Im} and the time-oscillation of $\partial_\nu \phi$.

\begin{Prop} \label{Prop:UC_Schro_classic}
Let $\Omega$ be an open subset of $\mathbb{R}^2$, $\Gamma$ be an open subset of $\partial \Omega$ and $0 < \widetilde{T} < T < \infty$.
If the Schrödinger equation on $\Omega$ is weakly observable on $(0,\widetilde{T})\times\Gamma$
then, for every non-zero $\phi_0 \in H^1_0(\Omega) \cap T_{\mathcal{S}}\varphi_R$,\vspace*{-2pt}\enlargethispage{2pt}%
$$\Im(\partial_{\nu} \phi) \not \equiv 0 \quad \text{ on } (0,T)\times\Gamma_c,$$
where $\phi(t):=e^{it\Delta_D} \phi_0$.
\end{Prop}

\bnp
Since the method is classical and was already performed in a more complicated case, we only detail the difference with Proposition \ref{Prop:UC}.
We define similarly\vspace*{-2pt}
$$N_T:=\left\{ \phi_0 \in H^1_0(\Omega)\sep \Im(\partial_{\nu} \phi) =0
\text{ in } L^2((0,T)\times\Gamma_c) \right\},$$
where $\phi(t) := e^{i\Delta_Dt} \phi_0$.

Following Step~1, we obtain that if $N_T\neq \{0\}$, $N_T$ only contains one eigenfunction.
Assume $\varphi\in N_T\setminus \{0\}$ is an eigenfunction of $\Delta_D$. Then, $e^{it\Delta_D} \varphi=e^{-it\lambda} \varphi$. Note that the Dirichlet boundary condition implies $\lambda\neq 0$ and therefore $\varphi\in N_T$ implies $\partial_{\nu} \varphi=0$ on $\Gamma_c$. By unique continuation for eigenfunctions, we get $\varphi=0$, a contradiction.
\enp

\begin{rk}
This result would be false if we considered Neumann boundary conditions.
The real constants solutions $\Phi(t)=c$ with $c\in \R$ create a zero observation.
The system is therefore not controllable with a real control.
Yet, the space of non-controllable data is only one dimensional.
\end{rk}
Then, arguing by contradiction as in the proof of Proposition \ref{Prop:Obs_eig},
we obtain the following observability Proposition which directly implies Theorem \ref{thm:anecdotic} by the HUM method (a variant with $\R$-linear vector space).

\begin{Prop} \label{Prop:Obs_Im}
Let $\Omega$ be an open subset of $\mathbb{R}^2$, $\Gamma$ be an open subset of $\partial \Omega$ and $0 < \widetilde{T} < T < \infty$.
If the Schrödinger equation on $\Omega$ is weakly observable on $(0,\widetilde{T})\times\Gamma$
then, there exists $\mathcal{C}_1>0$ such that\vspace*{-3pt}
\begin{equation}
\|\phi_0\|_{H^1_0(\Omega)} \leqslant \mathcal{C}_1 \| \Im(\partial_{\nu} \phi) \|_{L^2((0,T)\times\Gamma)}, \quad \forall \phi_0 \in H^1_0(\Omega,\mathbb{C}),
\end{equation}
where $\phi(t):=e^{it\Delta_D} \phi_0$.
\end{Prop}

\section{Nonlinear equation on a rectangle}

The goal of this section is to prove Theorem \ref{Main_Thm_loc}.
Note that the system \eqref{NL_syst_toy_init} may be written
\begin{equation} \label{NL_syst_toy}
\begin{cases}
i\partial_t \psi = - \Delta \psi + (-\Delta_N+1)^{-1} (\epsilon|\psi|^2) \psi + v \psi, & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu \psi(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\psi(0,x)=\psi_0(x), & x \in \Omega, \\
(-\Delta+1) v=0, & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu v(t,x)=g(t,x) 1_{\Gamma_c}(x), & (t,x) \in (0,T)\times\partial \Omega.
\end{cases}
\end{equation}
In Section \ref{Subsec:smoothing_NL}, we prove a smoothing effect.
In Section \ref{WP_NL}, we prove the well-posedness of the system \eqref{NL_syst_toy} in functional spaces appropriate for the controllability problem.
In Section \ref{Subsec:C1_NL}, we prove the $C^1$-regularity of the end-point map.
In Section \ref{Subsec:Linearized_NL}, we prove the controllability of the linearized system.
In Section \ref{Subsec:TIL_NL}, we prove Theorem~\ref{Main_Thm_loc} by applying the inverse mapping theorem.

\subsection{Smoothing effect on the source term}
\label{Subsec:smoothing_NL}

We introduce the operator $\mathcal{A}$ defined~by
$$D(\mathcal{A}):=H^2_N(\Omega,\mathbb{C}), \quad \quad \mathcal{A}\phi := - \Delta \phi + \frac{2\epsilon}{\pi L} (-\Delta_N+1)^{-1} (\Re(\phi)).$$
On $H^2_N(\Omega,\mathbb{C})$, we use the norm
$$\|\varphi\|_{H^2_N}:=\| (-\Delta+1)\varphi \|_{L^2(\Omega)}
= \bigg(\sum_{p,n \in \mathbb{N}} \big|\big(p^2+\left(\sfrac{n\pi}{L}\right)^2 +1\big) \langle \varphi, \xi_{p,n} \rangle \big|^2 \bigg)^{1/2},$$
where $\langle.\,,.\rangle$ is the usual $L^2(\Omega)$-scalar product and
\begin{gather} \label{def:xi_pn}
\xi_{p,n}(x_1,x_2)=\zeta_p(x_1)\phi_n(x_2)\\
\notag
\zeta_p(x_1)=\begin{cases}
\sfrac{1}{\sqrt{\pi}} \text{ if } p=0, \\
\sqrt{\sfrac{2}{\pi}} \cos(px_1) \text{ if } p \in \mathbb{N}^*,
\end{cases}
\quad
\phi_n(x_2)=\begin{cases}
\sfrac{1}{\sqrt{L}} \text{ if } n=0, \\
\sqrt{\sfrac{2}{L}} \cos\left(\sfrac{n\pi x_2}{L} \right) \text{ if } n \neq 0.
\end{cases}
\end{gather}
The goal of this section is the proof of the following result.

\begin{thm} \label{thm:Puel_Neumann}
Let $T>0$, $L>0$, $\Omega=(0,\pi) \times (0,L)$ and
$\Gamma_c$ be an open subset of~$\partial \Omega$ such that $\overline\Gamma_c$ does not contain any vertex of $\Omega$.
For every $\psi_0 \in H^2_N(\Omega)$,
$\mu_1 \in\nobreak L^2((0,T),H^{3/2}(\Omega))$
and $\mu_2 \in L^1((0,T),H^2_N(\Omega))$
such that
\begin{equation} \label{HYP:mu1_NL}
(-\Delta+1) \mu_1 \equiv 0, \quad
\partial_\nu \mu_1 \in L^2((0,T)\times\partial \Omega), \quad
\Supp (\partial_\nu \mu_1) \subset \Gamma_c,
\end{equation}
the solution of
$$\begin{cases}
(i\partial_t-\mathcal{A}) \psi(t,x) = \Big(\mu_1 +\mu_2 \Big) (t,x), \quad & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu \psi(t,x)=0, & (t,x) \in (0,T)\times\partial \Omega, \\
\psi(0,x)=\psi_0(x), & x \in \Omega,
\end{cases}$$
satisfies $\psi \in C^0([0,T],H^2_N(\Omega))$.
Moreover, there exists a constant $C>0$ (independent of $\psi_0, \mu_1, \mu_2$) such that
\begin{equation} \label{Puel_estimee_NL}
\|\psi\|_{L^0([0,T],H^2_{N}(\Omega))} \leqslant C \big(\|\psi_0\|_{H^2_{N}(\Omega)}
+ \|\partial_\nu \mu_1\|_{L^2((0,T)\times \Gamma_c)} + \|\mu_2\|_{L^1((0,T),H^2_{N}(\Omega))} \big).
\end{equation}
\end{thm}

\begin{rk}
By Proposition \ref{Prop:Trace_rect}, $\partial_\nu \mu_1 |_{\Gamma_j}$ is well-defined in $L^2((0,T),H^{-\sfrac{3}{2}-\epsilon}(\Gamma_j))$, for $j=1,\dots,4$.
This gives a sense to the second assumption in \eqref{HYP:mu1_NL}
\ie $\partial_\nu \mu_1 |_{\Gamma_j} \in L^2((0,T)\times \Gamma_j)$ for $j=1,\dots,4$.
\end{rk}

To prove Theorem \ref{thm:Puel_Neumann}, we need the following preliminary result.

\begin{Prop} \label{Prop_mu1}
Under the assumptions of the previous statement, the map
$$\mathcal{H}:t \mto \int_0^t e^{-i\Delta_N s} \mu_1(s)\,ds$$
belongs to $C^0([0,T],H^2_N(\Omega))$ and
\bnan
\label{inegneuman}
\| \mathcal{H} \|_{L^\infty((0,T),H^2_N(\Omega))} \leqslant C \|\partial_\nu \mu_1 \|_{L^2((0,T)\times\Gamma_c)},
\enan
for some constant $C=C(T)>0$ independent of $\mu_1$.
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop_mu1}]
The proof is quite close to that of Theorem \ref{thm:Puel} on the rectangle. We only perform Step~1, since the step~2 of regularization is very similar. Actually, following similar arguments as Step~2, we can first assume ${\partial_{\nu}\mu_1}_{\left|\partial \Omega \right.}\in\nobreak C^{\infty}_0((0,T)\times\nobreak\Gamma_c)$ which gives by elliptic regularity $\mu_1 \in C^{\infty}((0,T)\times \Omega)$. The inequality \eqref{inegneuman} that we obtain for smooth functions will then allow us to get the same result under the above regularity assumptions.

So, up to now, we assume that $\mu_1$ is regular enough. By linearity, we may assume that $\Gamma_c \subset (a,b) \times \{0\}$ with $0<a<b<\pi$.
We have
\begin{align*}
S=\|\mathcal{H}(t)\|_{H^2_N(\Omega)}^2
& = \sum_{p,n \in \mathbb{N}} \left|\big(p^2+\left(\sfrac{n\pi}{L}\right)^2 +1\big)
\int_0^t \langle \mu_1(s), \xi_{p,n} \rangle e^{-i \left(p^2 + \left(\sfrac{n\pi}{L} \right)^2 \right) s}\,ds \right|^2 \\
& = \sum_{p,n \in \mathbb{N}} \left|
\int_0^t \langle \mu_1(s), (-\Delta+1) \xi_{p,n} \rangle e^{-i \left(p^2 + \left(\sfrac{n\pi}{L} \right)^2 \right) s}\,ds \right|^2.
\end{align*}
We can apply the Green formula because
$\mu_1 $ and
$\xi_{p,n}$ are smooth.
This leads to
\begin{align*}
S & = \sum_{p,n=0}^\infty \left| \int_0^t e^{-i \left(p^2 + \left(\sfrac{n\pi}{L} \right)^2 \right) s}
\int_{\Gamma_c}\partial_{\nu} \mu_1(s,x) \xi_{p,n}(x) d\sigma(x)\,ds \right|^2
\\ & \leqslant C \sum_{p=0}^\infty \sum_{n=0}^\infty
\left| \int_0^{T_0} 1_{(0,t)}(s) e^{-i p^2 s}
\int_0^\pi\partial_{x_2} \mu_1(s,x_1,0) \cos(px_1) \,dx_1\, e^{-i \left(\sfrac{n\pi}{L}\right)^2 s}\,ds \right|^2
\\ & \leqslant C \sum_{p=0}^\infty \int_0^{T_0}
\left| 1_{(0,t)}(s) e^{-i p^2 s} \int_0^\pi\partial_{x_2} \mu_1(s,x_1,0) \cos(px_1) \,dx_1 \right|^2\,ds
\end{align*}
thanks to the Bessel-Parseval inequality in $L^2(0,T_0)$ with $T_0 \geqslant T$ of the form $\sfrac{2 m L^2}{\pi}$ with $m \in \mathbb{N}^*$.
Thus, we have
\begin{align*}
S & \leqslant C \sum_{p=0}^\infty \int_0^{t} \left| \int_0^\pi\partial_{x_2} \mu_1(s,x_1,0) \cos(px_1) \,dx_1 \right|^2\,ds \\
& \leqslant C \int_0^{t} \sum_{p=0}^\infty \left| \int_0^\pi\partial_{x_2} \mu_1(s,x_1,0) \cos(px_1) \,dx_1 \right|^2\,ds \\
& \leqslant C \int_0^{t} \|\partial_{x_2} \mu_1(s,.,0) \|_{L^2(0,\pi)}^2\,ds
\end{align*}
thanks to the orthogonality of the functions $(\cos(px_1))_{p \in \mathbb{N}}$ in $L^2(0,\pi)$.

Note that $\mu_1$ can be computed explicitly in terms of $\partial_\nu \mu_1\big{|}_{\Gamma_c}$ thanks to the rectangular form of the domain $\Omega$.
This explicit expression shows the existence of a constant $C=C(\Omega)>0$ such that
$$\|\mu_1\|_{L^2((0,t),H^1(\Omega))}^2 \leqslant C \|\partial_\nu \mu_1 \|_{ L^2((0,t) \times\partial \Gamma_c)}, \quad \forall t \in (0,T).$$
Therefore,
$$\|\mathcal{H}(t)\|_{H^2_N(\Omega)}^2
\leqslant C(T) \|\partial_\nu \mu_1 \|_{ L^2((0,t) \times\partial \Gamma_c)}, \quad \forall t \in (0,T).$$
This inequality proves that the map $\mathcal{H}$ takes values in $H^2_N(\Omega)$ on $[0,T]$ and that
$\mathcal{H}:[0,T] \to H^2_N(\Omega)$ is continuous at $t=0$.
The same proof shows that $\mathcal{H}$ is continuous at any $t \in (0,T]$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:Puel_Neumann}]
By the Duhamel formula, we have
\begin{equation} \label{duhamel_N}
\psi(t)= e^{i \Delta_N t} \psi_0 - i \int_0^t e^{i \Delta_N (t-s)} \big(\widetilde{\mu}_2(s) + \mu_1(s) \big)\,ds,
\end{equation}
where
$$\widetilde{\mu}_2(t):=\mu_2(t) - \frac{2 \epsilon}{\pi L} (-\Delta_N+1)^{-1}(\Re \psi).$$
In particular, $\widetilde{\mu}_2 \in L^1((0,T),H^2_N(\Omega))$ and
\begin{equation} \label{borne_mu2tilde}
\|\widetilde{\mu}_2\|_{L^1((0,T),H^2_N(\Omega))} \leqslant \| \mu_2 \|_{L^1((0,T),H^2_N(\Omega))} + \frac{2 \epsilon T}{\pi L} \|\psi\|_{L^\infty((0,T),L^2(\Omega))}.
\end{equation}
For every $\tau \in \mathbb{R}$, the operator $e^{i \Delta_N \tau}$ is an isometry of $H^2_N(\Omega)$ thus,
the first two terms in the right-hand side of \eqref{duhamel_N} belong to $C^0([0,T],H^2_N(\Omega))$.
The third term also does, by Proposition \ref{Prop_mu1}, thus $\psi \in C^0([0,T],H^2_N(\Omega))$.

\subsubsection*{Step~1}
Proof of \eqref{Puel_estimee_NL} when $T$ is small enough so that
\begin{equation} \label{T_pt_NL}
\frac{2 \epsilon T}{\pi L}\leqslant \frac{1}{2}.
\end{equation}
We deduce from \eqref{duhamel_N} that, for every $t \in (0,T)$
$$\|\psi(t)\|_{H^2_N(\Omega)} \leqslant
\|\psi_0\|_{H^2_N(\Omega)} + \int_0^t \|\widetilde{\mu}_2(s)\|_{H^2_N(\Omega)}\,ds + \left\| \int_0^t e^{- i \Delta_N s} \mu_1(s)\,ds \right\|_{H^2_N(\Omega)}.$$
Using \eqref{borne_mu2tilde}, Proposition \ref{Prop_mu1} and \eqref{T_pt_NL} we obtain
$$\|\psi\|_{L^\infty((0,T),H^2_N(\Omega))} \leqslant 2 \big(
\|\psi_0\|_{H^2_N(\Omega)} + \| \mu_2 \|_{L^1((0,T),H^2_N(\Omega))} + C \|\partial_\nu \mu_1\|_{L^2((0,T)\times\Gamma_c)} \big).$$

\subsubsection*{Step~2}
Proof of \eqref{Puel_estimee_NL} when $T$ is arbitrary.
Let $0=T_1<T_2<\cdots<T_m=T$ be a subdivision of $[0,T]$ such that $\sfrac{2 \epsilon (T_{k+1}-T_k)}{\pi L}\leqslant \sfrac{1}{2}$ for $k=1,\dots,m-1$.
Applying Step~1 on each subinterval $(T_k,T_{k+1})$ we obtain \eqref{Puel_estimee_NL}.
\end{proof}

\subsection{Well-posedness of the nonlinear Schrödinger-Poisson system}
\label{WP_NL}

The goal of this section is the proof of the following result.

\begin{Prop} \label{Prop:WP}
Let $T>0$. There exists $\rho, \delta>0$ such that,
for every $\psi_0 \in H^2_N(\Omega)$, $g \in L^2((0,T) \times\partial \Omega)$ with
$\|\psi_0-\psi_{\rref}(0)\|_{H^2_N} < \rho$ and $\|g\|_{L^2((0,T)\times\partial \Omega)}<\delta$,
there exists a unique solution $\psi \in C^0([0,T],H^2_N(\Omega))$ of \eqref{NL_syst_toy}.
Moreover $\|\psi(t)\|_{L^2(\Omega)}=\|\psi_0\|_{L^2(\Omega)}$ for every $t \in [0,T]$.
\end{Prop}

We search $\psi$ in the form
$$\psi(t,x,y)=\frac{e^{-\sfrac{i \epsilon t}{\pi L}}}{\sqrt{\pi L}} \left(1+\zeta(t,x,y) \right),$$
where $\zeta$ is the solution of
\begin{equation} \label{zeta_syst_toy}
\begin{cases}
(i\partial_t-\mathcal{A}) \zeta = \Phi(\zeta) + v (1+\zeta), & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu \zeta(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\zeta(0,x)=\zeta_0(x), & x \in \Omega, \\
(-\Delta+1) v(t,x)=0, & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu v(t,x)=g(t,x) 1_{\Gamma}(x), & (t,x) \in (0,T)\times\partial\Omega,
\end{cases}
\end{equation}
and
\begin{equation} \label{def:Phi(zeta)}
\Phi(\zeta) :=
\frac{2 \epsilon}{\pi L} (-\Delta_N+1)^{-1} \left(\Re(\zeta) \right)\zeta
+ \frac{\epsilon}{\pi L} (-\Delta_N+1)^{-1} \left(|\zeta|^2 \right) (1+\zeta).
\end{equation}

Proposition \ref{Prop:WP} is a consequence of the following result.

\begin{Prop} \label{Prop:WP_zeta}
Let $T>0$. There exists $\rho, \delta>0$ such that,
for every $\zeta_0 \in H^2_N(\Omega)$, $g \in L^2((0,T)\times\partial\Omega,\mathbb{R})$ with
$\|\zeta_0\|_{H^2_N} < \rho$ and $\|g\|_{L^2((0,T)\times\partial\Omega)}<\delta$,
there exists a unique weak solution $\zeta \in C^0([0,T],H^2_N(\Omega))$ of \eqref{zeta_syst_toy}.
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop:WP_zeta}]
Let $T>0$.

\subsubsection*{Step~1}
We prove the existence of $C_0>0$ such that,
for every $\zeta \in H^2_N(\Omega)$ and $v \in H^{\sfrac{3}{2}}(\Omega)$ satisfying $(-\Delta+1)v=0$ in $\Omega$,
then $(-\Delta+v)(v\zeta) \in L^2(\Omega)$ and
$$\| (-\Delta+v)(v\zeta) \|_{L^2(\Omega)} \leqslant C_0 \|v\|_{H^{\sfrac{3}{2}}(\Omega)} \|\zeta\|_{H^2_N(\Omega)}.$$
We have
$$(-\Delta+1)[v \zeta] =-2 \nabla v \cdot\nabla \zeta - v \Delta \zeta.$$
\begin{itemize}
\item We have $\Delta \zeta \in L^2(\Omega)$ and $v \in L^\infty(\Omega)$, by the Sobolev embedding \eqref{Sobolev_emb_1}, thus $v \Delta \zeta \in L^2(\Omega)$.
\item We have $\nabla v \in H^{1/2}(\Omega)$ and $\nabla \zeta \in H^1(\Omega)$.
From the Sobolev embedding \eqref{Sobolev_emb_2}, we deduce that $\nabla v$ and $\nabla \zeta$ belong to $L^4(\Omega)$.
Thus $\nabla v \cdot\nabla \zeta \in L^2(\Omega)$.
\end{itemize}

\subsubsection*{Step~2}
Choice of $R, \delta, \rho$.
Let $C>0$ be as in Proposition \ref{Prop_mu1} and $C'>0$ be the constant of the third statement of Proposition \ref{Prop:Trace_rect}.
Let $C_1, C_2, C_3>0$ be such that
\begin{align*}
\|\Phi(\zeta)\|_{H^2_N(\Omega)} &\leqslant C_1 \|\zeta\|_{H^2_N(\Omega)}^2,
\quad \forall \zeta \in H^2_N(\Omega) \text{ such that } \|\zeta \|_{H^2_N(\Omega)} \leqslant 1,\\
\|\zeta\|_{L^\infty(\overline{\Omega})} &\leqslant C_2 \|\zeta\|_{H^2_N(\Omega)}, \quad \forall \zeta \in H^2_N(\Omega),\\
\|\Phi(\zeta)-\Phi(\widetilde{\zeta})\|_{H^2_N(\Omega)}
&\leqslant C_3 \|\zeta-\widetilde{\zeta}\|_{H^2_N(\Omega)} \max\big\{ \|\zeta\|_{H^2_N(\Omega)} \sep \|\widetilde{\zeta}\|_{H^2_N(\Omega)} \big\},
\\
&\hspace*{2cm}\forall \zeta, \widetilde{\zeta} \in H^2_N(\Omega) \text{ such that } \|\zeta \|_{H^2_N(\Omega)}, \|\widetilde{\zeta}\|_{H^2_N(\Omega)} \leqslant 1,
\end{align*}
where $\Phi(\zeta)$ is defined in \eqref{def:Phi(zeta)}. Let
\begin{equation} \label{def:R_rho_delta}
\begin{split}
R&:=\min\left\{ 1 \sep \frac{1}{4C_1 T} \sep \frac{1}{4 C_3 T} \right\}, \quad \quad \rho:=\frac{R}{2}
\\
\delta&:=\min\left\{ \frac{R}{4[C(1+C_2)+C_0 C' \sqrt{T}]} \sep \frac{1}{4(C C_2 + C_0 C' T)} \right\},
\end{split}
\end{equation}
and $\zeta_0 \in H^2_N(\Omega)$ be such that $\|\zeta_0\|_{H^2_N(\Omega)} < \rho$.

We consider the map
\begin{align*}
F: \overline{B_R}[C^0([0,T],H^2_N(\Omega))] & \to \overline{B_R}[C^0([0,T],H^2_N(\Omega))] \\
\zeta & \mto \xi,
\end{align*}
where $\xi:=F(\zeta)$ is the solution of
$$\begin{cases}
(i\partial_t-\mathcal{A}) \xi = \Phi(\zeta) + v (1+\zeta), & (t,x,y) \in (0,T)\times\Omega, \\
\partial_\nu \xi(t,x,y)=0, & (t,x,y) \in (0,T)\times\partial\Omega, \\
\xi(0,x,y)=\zeta_0(x,y), & (x,y) \in \Omega. \\
\end{cases}$$
The end of the proof consists in applying the Banach fixed point theorem to the map~$F$.

\subsubsection*{Step~3}
We prove that $F$ takes values in $\overline{B_R}[C^0([0,T],H^2_N)]$.

Let $\zeta \in \overline{B_R}[C^0([0,T],H^2_N)]$. We introduce the solution $\mu_3$ of
$$\begin{cases}
(-\Delta+1) \mu_3(t,x_1,x_2) = (-\Delta+1)(v \zeta)(t,x_1,x_2), & (t,x_1,x_2) \in (0,T)\times\Omega, \\
\partial_\nu \mu_3(t,x_1,x_2) =0, & (t,x_1,x_2) \in (0,T)\times\partial \Omega
\end{cases}$$
and the functions
$$\mu_1:=v + v\zeta - \mu_3, \quad \quad \mu_2:=\mu_3 + \Phi(\zeta).$$
By Step~1, $(-\Delta+1)(v \zeta) \in L^2((0,T)\times\Omega)$ and thus $\mu_3 \in L^2((0,T),H^2_N(\Omega))$.
Therefore~$\mu_1$ and $\mu_2$ satisfy the assumptions of Theorem \ref{thm:Puel_Neumann} and \hbox{$\xi\!:=\!F(\zeta)\!\in\!C^0([0,T],H^2_N(\Omega))$}.
By Proposition \ref{Prop_mu1} and \eqref{def:R_rho_delta}
\begin{align*}
\|\xi\|_{L^\infty((0,T),H^2_N)}
& \leqslant \|\zeta_0\|_{H^2_N} + C \|\partial_{\nu} \mu_1\|_{L^2((0,T)\times\Gamma_c)}\\
&\hspace*{3.5cm} + \int_0^T \big(\|\mu_3(s)\|_{H^2_N} + \|\Phi[\zeta(s)]\|_{H^2_N} \big)\,ds \\
& \leqslant \frac{R}{2} + C \|g(1+\zeta)\|_{L^2((0,T)\times\Gamma_c)} \\
&\hspace*{3.5cm}+ \sqrt{T}\, \| (-\Delta+1)(v \zeta) \|_{L^2((0,T)\times\Omega)} + T C_1 R^2 \\
& \leqslant \frac{3R}{4} + C \delta (1+C_2) + \sqrt{T}\, C_0 C' \delta R \\
& \leqslant R.
\end{align*}

\subsubsection*{Step~4}
We prove that $F$ is a contraction.
Let $\zeta, \widetilde{\zeta} \in \overline{B_R}[C^0([0,T],H^2_N)]$. By Proposition \ref{Prop_mu1} and \eqref{def:R_rho_delta},\enlargethispage{8pt}%
\begin{align*}
\|\xi-\widetilde{\xi}\|_{L^\infty((0,T),H^2_N)}&
\leqslant C \|\partial_{\nu} (\mu_1-\widetilde{\mu}_1) \|_{L^2((0,T)\times\Gamma_c)}\\
& \hspace*{.8cm}+
\int_0^T \big(\|(\mu_3-\widetilde{\mu}_3)(s)\|_{H^2_N} + \|\Phi(\zeta(s))-\Phi(\widetilde{\zeta}(s))\|_{H^2_N} \big)\,ds \\
&\hspace*{-.7cm}\leqslant C \|g(\zeta-\widetilde{\zeta})\|_{L^2((0,T)\times\Gamma_c)} + \sqrt{T}\, \| (-\Delta+1)[v (\zeta-\widetilde{\zeta})] \|_{L^2((0,T)\times\Omega)}\\
&\hspace*{3.4cm}+ T C_3 R \|\zeta-\widetilde{\zeta}\|_{L^\infty((0,T),H^2_N)} \\
&\hspace*{-.7cm}\leqslant \big(C \delta C_2 + C_0 C' \delta + T C_3 R \big) \|\zeta-\widetilde{\zeta}\|_{L^\infty((0,T),H^2_N)} \\
&\hspace*{-.7cm}\leqslant \frac{1}{2} \|\zeta-\widetilde{\zeta}\|_{L^\infty((0,T),H^2_N)}.\qedhere
\end{align*}
\end{proof}

\subsection{$C^1$-regularity of the end-point map}
\label{Subsec:C1_NL}

The end-point map is defined by
\begin{equation} \label{def:thetaT}
\begin{array}{lrclcl}
\Theta_T: & [\mathcal{S} \cap H^2_{N}(\Omega,\mathbb{C})] & \times & L^2((0,T)\times\partial\Omega,\mathbb{R}) & \dpl\to & [\mathcal{S} \cap H^2_{N}(\Omega,\mathbb{C})]^2 \\[3pt]
& (\psi_0 &, & g) & \dpl\mto & (\psi(0),\psi(T)),
\end{array}
\end{equation}
where $\psi$ is the solution of \eqref{NL_syst_toy}.
The goal of this section is to state the following result, which is a consequence of the estimate \eqref{Puel_estimee_NL}.

\begin{Prop} \label{Prop:EPM_C1}
Let $T>0$ and $\rho, \delta >0$ be as in Proposition \ref{Prop:WP}.
The end-point map $\Theta_T$ defined by \eqref{def:thetaT} is $C^1$ on
$$\big\{ (\psi_0,g) \in H^2_N \times L^2((0,T)\times\partial\Omega) \sep \|\psi_0-\psi_{\rref}(0)\|_{H^2_N} < \rho \text{ and } \|g\|_{L^2((0,T)\times\partial \Omega)}<\delta \big\}$$
and\vspace*{-5pt}
$$d\Theta_T(\psi_{\rref}(0),0)\cdot(\Psi_0,G)=\Big(\Psi_0,\frac{e^{-i\sfrac{\epsilon T }{\sqrt{\pi L}} }}{\sqrt{\pi L}} \Psi(T)\Big),$$
where $\Psi$ is the solution of
\begin{equation} \label{Lin_syst_toy}
\begin{cases}
(i\partial_t-\mathcal{A}) \Psi = V, & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu \Psi(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\Psi(0,x,y)=\Psi_0(x), & x \in \Omega, \\
(-\Delta+1) V=0, & (t,x) \in (0,T)\times\Omega, \\
\partial_\nu V(t,x)=G(t,x)1_{\Gamma_c}(x), & (t,x) \in (0,T)\times\partial\Omega.
\end{cases}
\end{equation}
\end{Prop}

\subsection{Controllability of the linearized system}
\label{Subsec:Linearized_NL}

Let
$$T_\mathcal{S}:=\left\{ \varphi \in L^2(\Omega,\mathbb{C}) \sep \Re \left(\int_{\Omega} \varphi(x,y) \,dx dy\right)=0 \right\}.$$
The goal of this section is the proof of the following result.

\begin{Prop} \label{Prop:Lin_cont}
Let $T>0$. There exists a continuous map
\begin{align*}
L: \big(T_\mathcal{S} \cap H^2_{N}(\Omega,\mathbb{C})\big)^2 & \to L^2((0,T)\times\partial\Omega) \\
(\Psi_0,\Psi_f) & \mto G
\end{align*}
such that, for every $\big(\Psi_0,\Psi_f) \in (T_\mathcal{S} \cap H^2_{N}(\Omega,\mathbb{C})\big)^2$
the solution of \eqref{Lin_syst_toy} satisfies $\Psi(T)=\nobreak\Psi_f$.
\end{Prop}

\subsubsection{Hilbert Uniqueness Method}

The proof of Proposition \ref{Prop:Lin_cont} requires the following observability result, where $\mathcal{B}$ is defined by
$$D(\mathcal{B}):=H^2_N(\Omega,\mathbb{C}), \quad \quad
\mathcal{B} \phi := - \Delta \phi + i \frac{2 \epsilon}{\pi L} (-\Delta_N+1)^{-1}(\Im(\phi)).$$

\begin{Prop} \label{Prop:Obs_phi}
Let $T>0$. There exists $\mathcal{C}>0$ such that, for every $\phi_T \in L^2(\Omega)$,
the solution of
\begin{equation} \label{Adj_NL}
\begin{cases}
(i\partial_t-\mathcal{B}) \phi = 0, \quad & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} \phi(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\phi(T,x)=\phi_T(x), & x \in \Omega,
\end{cases}
\end{equation}
satisfies
\begin{equation} \label{Obs_ineq_NL}
\|\phi_T\|_{L^2(\Omega)} \leqslant \mathcal{C} \|\Im(\phi)\|_{L^2((0,T)\times\Gamma_c)}.
\end{equation}
\end{Prop}

Note that the solution of \eqref{Adj_NL} can be computed explicitly (as in the case $\epsilon=0$) because $\mathcal{B}$ preserves $\mathbb{C} \xi_{p,n}$.
The boundary condition in \eqref{Adj_NL} has to be understood in the sense of the semi-group.
The proof of Proposition \ref{Prop:Obs_phi} relies on two intermediate results:
\begin{itemize}
\item the weak observability of \eqref{Adj_NL}, proved in Section \ref{subsubs:weak_obs_NL} below,
\item a unique continuation property, proved in Section \ref{subsub:UC_NL}.
\end{itemize}
Note that the trace $\Im(\phi)|_{\partial \Omega}$ is well-defined in $L^2((0,T)\times\partial\Omega)$ (see Lemma \ref{Lem:Trace_L2} below).

\begin{proof}[Proof of Proposition \ref{Prop:Lin_cont}, assuming Proposition \ref{Prop:Obs_phi}]
We want to prove the existence of a continuous right inverse for the continuous operator
\begin{align*}
F: L^2((0,T)\times\Gamma_c,\mathbb{R}) & \to T_\mathcal{S} \\
G & \mto (-\Delta+1)\Psi(T),
\end{align*}
where $\Psi$ is the solution of \eqref{Lin_syst_toy} with $\Psi_0=0$.
By \cite[Th.\, II.10 \& II.19]{Brezis_An_Fonc}, it is equivalent to prove the existence of a constant $\mathcal{C}>0$ such that
$$\|\phi_T\|_{L^2(\Omega)} \leqslant \mathcal{C} \|F^*(\phi_T)\|_{L^2((0,T)\times\Gamma_c)}, \quad \forall \phi_T \in T_{\mathcal{S}}.$$
Thus, to deduce Proposition \ref{Prop:Lin_cont} from Proposition \ref{Prop:Obs_phi}, it is sufficient to prove that
$$F^*(\phi_T) = - \Im(\phi)_{|\Gamma_c}, \quad \forall \phi_T \in T_{\mathcal{S}},$$
where $\phi$ is associated to $\phi_T$ by \eqref{Adj_NL}. The trace $\Im(\phi)_{|\Gamma_c}$ is defined as an extension, thus,
it is sufficient to prove that
\begin{equation} \label{calcul_adjoint_NL}
F^*(\phi_T) = - \Im(\phi)_{|\Gamma_c}, \quad \forall \phi_T \in T_{\mathcal{S}} \cap C^\infty_c(\Omega).
\end{equation}
Let $\phi_T \in T_{\mathcal{S}} \cap C^\infty_c(\Omega)$.
Then $\phi \in C^\infty([0,T],H^s_N(\Omega))$ for every $s>0$, thus $\widetilde{\phi}:=(-\Delta+1)\phi \in C^\infty([0,T],H^s_N(\Omega))$ for every $s>0$ and
$$\begin{cases}
(i\partial_t+\mathcal{B}) \widetilde{\phi} = 0, \quad & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} \widetilde{\phi}(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\widetilde{\phi}(T,x)=(-\Delta+1)\phi_T(x), & x \in \Omega,
\end{cases}$$
because $\mathcal{B}$ and $(-\Delta+1)$ commute.
The functions $\Psi$ and $\widetilde{\phi}$ are solutions in the sense of the semi-group, thus
\begin{align*}
\Re \langle \Psi(T),\widetilde{\phi}(T) \rangle & = \Im \bigg(\int_0^T \int_{\Omega} V(t,x) \overline{\widetilde{\phi}(t,x)} \,dx \,dt \bigg)\\
& = \Im \bigg(\int_0^T \int_{\Omega} V(t,x) \overline{(-\Delta+1)\phi(t,x)} \,dx \,dt \bigg).
\end{align*}
Therefore,
\begin{align*}
\Re \langle F(G), \phi_T \rangle
& = \Re \langle (-\Delta+1)\Psi(T), \phi_T \rangle \\
& = \Re \langle \Psi(T), (-\Delta+1) \phi_T \rangle \quad\text{ because } \Psi(T), \phi_T \in H^2_N(\Omega) \\
& = \Im \bigg(\int_0^T \int_{\Omega} V(t,x) \overline{(-\Delta+1)\phi(t,x)} \,dx \,dt \bigg) \\
& = - \int_0^T \int_{\Gamma_c} G(t,x) \Im(\phi(t,x)) d\sigma(x) \,dt.
\end{align*}
This proves \eqref{calcul_adjoint_NL}.
\end{proof}

\begin{Lem} \label{Lem:Trace_L2}
Let $T>0$. There exists $C=C(T)>0$ such that, for every $\phi_0 \in L^2(\Omega)$,
the solution $\phi \in C^0([0,T],L^2(\Omega))$ of \eqref{Adj_NL} satisfies
$$\int_0^T \int_{\partial \Omega} |\phi(t,x)|^2 d\sigma(x) \,dt \leqslant C \|\phi_0\|_{L^2(\Omega)}^2.$$
\end{Lem}

\skpt
\begin{proof}[Proof of Lemma \ref{Lem:Trace_L2}]

\subsubsection*{Step~1}
Proof when $\epsilon=0$. The following equality holds in $C^0([0,T],L^2(\Omega))$
$$\phi(t,x_1,x_2)=\sum_{p,n=0}^\infty \langle \phi_0, \xi_{p,n} \rangle e^{-i\left[ p^2 + \left(\sfrac{n\pi}{L}\right)^2 \right] t} \xi_{p,n}(x_1,x_2).$$
Using \eqref{def:xi_pn} and the orthogonality of $(\zeta_p)_{p \in \mathbb{N}}$ in $L^2(0,\pi)$, we get
\begin{align*}
\int_0^T\hspace*{-2mm} \int_0^\pi \hspace*{-1mm}|\phi(t,x_1,0)|^2 \,dx_1 \,dt
& = \int_0^T\hspace*{-2mm} \int_0^\pi \bigg| \sum_{p=0}^\infty
\bigg(\sum_{n=0}^\infty \langle \phi_0, \xi_{p,n} \rangle e^{-i\left[ p^2 + \left(\sfrac{n\pi}{L}\right)^2 \right] t} \bigg) \zeta_p(x_1) \bigg|^2 \,dx_1 \,dt
\\ & = \int_0^T \sum_{p=0}^\infty
\bigg| \sum_{n=0}^\infty \langle \phi_0, \xi_{p,n} \rangle e^{-i\left[ p^2 + \left(\sfrac{n\pi}{L}\right)^2 \right] t} \bigg|^2 \,dt
\\ & = \sum_{p=0}^\infty \int_0^T
\bigg| \sum_{n=0}^\infty \langle \phi_0, \xi_{p,n} \rangle e^{-i \left(\sfrac{n\pi}{L}\right)^2 t} \bigg|^2 \,dt
\\ & \leqslant \sum_{p=0}^\infty C_\mathrm{Ing} \sum_{n=0}^\infty | \langle \phi_0, \xi_{p,n} \rangle|^2
= C_\mathrm{Ing} \|\phi_0\|_{L^2(\Omega)}^2,
\end{align*}
where $C_\mathrm{Ing}(T)>0$ is the Ingham constant associated to the family $\big(e^{-i \left(\sfrac{n\pi}{L}\right)^2 t} \big)_{n \in \mathbb{N}}$
in $L^2(0,T)$ (see \cite{haraux}).

\subsubsection*{Step~2}
Proof when $\epsilon \neq 0$. The following equality holds in $L^2(\Omega)$.
$$\phi(t)=e^{-i\Delta_N t} \phi_0 + \frac{2\epsilon}{\pi L} \int_0^t e^{-i \Delta_N(t-s)} (-\Delta_N+1)^{-1}(\Im(\phi(s)))\,ds, \quad \forall t \in [0,T].$$
By Step 1, the trace on $\partial \Omega$ of $e^{-i\Delta_N t} \phi_0$ has an $L^2((0,T)\times\nobreak\partial\Omega)$-norm bounded by $\sqrt{C_\mathrm{Ing}}\, \|\phi_0\|_{L^2(\Omega)}$.
The second term of the right-hand side belongs to $C^0([0,T],H^2_N(\Omega))$ and its $L^\infty((0,T),H^2_N(\Omega))$-norm is bounded~by
$$\frac{2\epsilon}{\pi L} \int_0^T \|\phi(s)\|_{L^2(\Omega)} = \frac{2\epsilon T}{\pi L}\, \|\phi_0\|_{L^2(\Omega)}.$$
Thus, this term as a trace on $\partial \Omega$ with an $L^2((0,T)\times\partial\Omega)$-norm that satisfies a similar estimate.
\end{proof}

\subsubsection{Weak observability}
\label{subsubs:weak_obs_NL}

The goal of this section is the proof of the following result.

\begin{Prop} \label{Prop:Weak_Obs_NL}
Let $T>0$. There exists $\mathcal{C}'=\mathcal{C}'(T)>0$ such that, for every $\phi_T \in L^2(\Omega)$,
the solution of \eqref{Adj_NL} satisfies
\begin{equation} \label{Weak_Obs_NL}
\|\phi_T\|_{L^2(\Omega)} \leqslant \mathcal{C}' \big(\|\Im(\phi)\|_{L^2((0,T)\times\Gamma_c)} + \|\phi_T\|_{H^{-2}(\Omega)} \big).
\end{equation}
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop:Weak_Obs_NL}]
We recall Tenenbaum and Tucsnak's result \hbox{\cite[Th.\,1.1]{Tenenbaum}}:
for every $T>0$ and every non-empty open subset $\Gamma$ of $\partial\Omega$,
there exists \hbox{$\mathcal{C}_0\!=\!\mathcal{C}_0(T,\Gamma)\!>\!0$} such that,
for every $\phi_T \in L^2(\Omega)$, the solution $\widetilde{\phi}$ of
\begin{equation} \label{Schro_phi_tilde}
\begin{cases}
(i\partial_t + \Delta) \widetilde{\phi}(t,x)=0, \quad & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} \widetilde{\phi}(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\widetilde{\phi}(T,x)=\phi_T(x), & x \in \Omega,
\end{cases}
\end{equation}
satisfies
\begin{equation} \label{Tenen_Tucs}
\|\phi_T\|_{L^2(\Omega)} \leqslant \mathcal{C}_0 \| \widetilde{\phi} \|_{L^2((0,T)\times\Gamma)}.
\end{equation}

\subsubsection*{Step~1}
We prove that, for every $T>0$, there exists $\mathcal{C}_1=\mathcal{C}_1(T)>0$ such that
for every $\phi_T \in L^2(\Omega)$, the solution $\widetilde{\phi}$ of \eqref{Schro_phi_tilde}
satisfies
\begin{equation} \label{Step1_NL}
\|\phi_T\|_{L^2(\Omega)} \leqslant \mathcal{C}_1 \big(\|\Im(\widetilde{\phi})\|_{L^2((0,T)\times\Gamma_c)} + \|\phi_T\|_{H^{-1}(\Omega)} \big).
\end{equation}
Let $T>0$ and $\rho \in C^\infty_c(\mathbb{R},\mathbb{R}^+)$ be such that $\rho \equiv 1$ on $(T/3,2T/3)$ and $\Supp(\rho) \subset (0,T)$.
Let $\phi_T \in L^2(\Omega)$. We have
\begin{align*}
\|\phi_T\|_{L^2(\Omega)}^2
& \leqslant \mathcal{C}_0^2 \int_{T/3}^{2T/3} \int_{\Gamma_c} |\widetilde{\phi}(t,x)|^2 d\sigma(x) \,dt \quad \text{ with } \mathcal{C}_0=\mathcal{C}_0(T/3) \text{ by } \eqref{Tenen_Tucs} \\
& \leqslant \mathcal{C}_0^2 \int_{\mathbb{R}} \int_{\Gamma_c} |\rho(t) \widetilde{\phi}(t,x)|^2 d\sigma(x) \,dt \\
& \leqslant \frac{\mathcal{C}_0^2}{2} \int_{\mathbb{R}} \int_{\Gamma_c} \Big(|\rho(t) \widetilde{\phi}(t,x)|^2 + |\rho(t) \overline{\widetilde{\phi}(t,x)}|^2 \Big) d\sigma(x) \,dt \\
& \leqslant \frac{\mathcal{C}_0^2}{2} \int_{\mathbb{R}} \int_{\Gamma_c} \big(|\rho(t) \Im(\widetilde{\phi}(t,x))|^2 + 2 \Re(\rho(t)^2 \widetilde{\phi}(t,x)^2) \big) d\sigma(x) \,dt.
\end{align*}
In order to get \eqref{Step1_NL}, it suffices to prove the existence of $C>0$ such that
$$\int_{\mathbb{R}} \int_{\Gamma_c} \Re[ \rho(t)^2 \widetilde{\phi}(t,x)^2 ] d\sigma(x) \,dt \leqslant C \|\phi_T\|_{H^{-1}(\Omega)}^2.$$
The function $\rho^2$ belongs to $C^\infty_c(\mathbb{R},\mathbb{R})$, thus, for every $N \in \mathbb{N}^*$, there exists $C_N>0$ such that
$$\left| \int_{\mathbb{R}} \rho(t)^2 e^{-i \omega t} \,dt \right| \leqslant \frac{C_N}{(1+\omega)^N}, \quad \forall \omega>0.$$
We have
\begin{align*}
\bigg| \int_{\mathbb{R}} \int_{\Gamma_c} \rho(t)^2 \widetilde{\phi}&(t,x)^2 d\sigma(x) \,dt \bigg| \\
&= \bigg| \sum_{j,J \in \mathbb{N}^*} \phi_{T,j} \phi_{T,J} \left(\int_{\mathbb{R}} \rho(t)^2 e^{-(\lambda_j+\lambda_J)t} \,dt \right)
\left(\int_{\Gamma_c} \varphi_j(x) \varphi_J(x) d\sigma(x) \right) \bigg| \\
&\leqslant
\sum_{j,J \in \mathbb{N}^*} |\phi_{T,j} \phi_{T,J}| \,\frac{C_N}{(1+\lambda_j+\lambda_J)^N}\, C \lambda_j^\alpha \lambda_J^\alpha \quad \text{ for some } C,\alpha>0\\
&\leqslant
C C_N \bigg(\sum_{j \in \mathbb{N}^*} |\phi_{T,j} | \frac{\lambda_j^\alpha}{(1+\lambda_j)^N} \bigg)^{2}
\phantom{\int_{\Gamma_c} \varphi_j(x) \varphi_J(x) d\sigma(x) \bigg|} \\
&\leqslant
C C_N \bigg(\sum_{j \in \mathbb{N}^*} \frac{|\phi_{T,j} |^2}{1+\lambda_j} \bigg)^2
\bigg(\sum_{j \in \mathbb{N}^*} \frac{\lambda_j^{2\alpha}(1+\lambda_j)}{(1+\lambda_j)^{2N}} \bigg)^2 \\
&\leqslant
C \|\phi_T\|_{H^{-1}(\Omega)}^2 \quad \text{ for } N \text{ large enough.}
\end{align*}

\subsubsection*{Step~2}
Conclusion. Working by contradiction,
we assume the existence of a sequence $(\phi_T^n)_{n \in \mathbb{N}^*}$ such that
the associated solutions $\phi^n$ of \eqref{Adj_NL} satisfy
\begin{equation} \label{absurd_NL}
1 = \|\phi_T^n\|_{L^2(\Omega)} > n \big(\|\Im(\phi^n)\|_{L^2((0,T)\times\Gamma_c)} + \|\phi_T^n\|_{H^{-2}(\Omega)} \big).
\end{equation}
We introduce the solutions $\widetilde{\phi}^n$ of \eqref{Schro_phi_tilde} associated to final condition
$\widetilde{\phi}^n(T)=\phi_T^n$. Then $\xi^n:= \phi^n - \widetilde{\phi}^n$ solves
$$\begin{cases}
(i\partial_t + \Delta) \xi^n = i \dfrac{2\epsilon}{\pi L} (-\Delta_N+1)^{-1} \Im(\phi^n), \quad & (t,x) \in (0,T)\times\Omega, \\
\partial_{\nu} \xi^n(t,x)=0, & (t,x) \in (0,T)\times\partial\Omega, \\
\xi^n(T,x)=0. & x \in \Omega,
\end{cases}$$
From \eqref{absurd_NL}, we deduce that
\[
\phi_T^n \underset{n \to \infty}{\to} 0\quad\text{in }H^{-2}(\Omega).
\]
Moreover this sequence is bounded in $L^2(\Omega)$, thus, by interpolation
\[
\phi_T^n \underset{n \to \infty}{\to} 0\quad\text{in }H^{-1}(\Omega).
\]
Therefore,
\begin{align*}
(-\Delta_N+1)^{-1} \Im(\phi^n) &\underset{n \to \infty}{\to}0\quad\text{ in }L^\infty((0,T),H^{1}(\Omega))\\
\tag*{and}
\xi^n &\underset{n \to \infty}{\to} 0\text{ in }L^\infty((0,T),H^{1}(\Omega)).
\end{align*}
As a consequence $\Im(\xi^n)_{|\Gamma_c} \underset{n \to \infty}{\to} 0$ in $L^\infty((0,T) \times \Gamma_c)$.
Moreover, $\Im(\phi^n)_{|\Gamma_c} \underset{n \to \infty}{\to} 0$ in $L^2((0,T)\times \Gamma_c)$ by \eqref{absurd_NL},
thus $\Im(\widetilde{\phi^n})_{|\Gamma_c} \underset{n \to \infty}{\to} 0$ in $L^2((0,T)\times \Gamma_c)$.
From Step~1, we get
$$1 = \|\phi_T^n\|_{L^2(\Omega)} \leqslant \mathcal{C}_1 \big(\|\Im(\widetilde{\phi}^n)\|_{L^2((0,T)\times\Gamma_c)} + \|\phi_T^n\|_{H^{-1}(\Omega)} \big)
\underset{n \to \infty}{\to} 0,$$
which is a contradiction.
\end{proof}

\subsubsection{Unique continuation}
\label{subsub:UC_NL}

The goal of this section is to prove the following result.

\begin{Prop} \label{Prop:UC_NL}
We assume that $\epsilon$ satisfies \eqref{hyp:epsilon}.
Let $T>0$, $\Gamma_c$ be an open subset of $\partial \Omega$ and
$\phi_T \in T_\mathcal{S} \setminus \{0\}$ and $\phi$ be the solution of \eqref{Adj_NL}.
Then $\Im(\phi)_{|\Gamma_c}$ does not identically vanish on $(0,T)\times\Gamma_c$.
\end{Prop}

\begin{proof}[Proof of Proposition \ref{Prop:UC_NL}]
To simplify, we assume that $\Gamma_c=(a,b) \times \{0\}$, where \hbox{$0<a<b<1$}. We introduce
$$N_T:=\left\{ \phi_0 \in T_{\mathcal{S}} \sep \Im(\phi) =0 \text{ on } (0,T)\times\Gamma_c \right\},$$
where $\phi(t):=S(t)\phi_0$ solves \eqref{Adj_NL} with initial condition at $t=0$: $\phi(0)=\phi_0$.
$N_T$ is an $\mathbb{R}$-vector subspace of $L^2(\Omega)$. Working by contradiction, we assume that $N_T \neq \{0\}$.

\subsubsection*{Step~1}
We prove that $N_T$ is stable by $(-i\mathcal{B})$ and has finite dimension.
Let $\phi_0 \in\nobreak N_T$.
Since $\phi_0 \in L^2(\Omega)$, then $\phi_\epsilon:=\psfrac{S(\epsilon)\phi_0-\phi_0}{\epsilon}$ is bounded in $H^{-2}_N(\Omega)$ when $[\epsilon \to 0]$.
Moreover, for $\epsilon<T/2$, $\phi_\epsilon \in N_{T/2}$.
Applying \eqref{Weak_Obs_NL} to $\phi_\epsilon$, we get
$$\| \phi_\epsilon \|_{L^2(\Omega)} \leqslant \mathcal{C}\left(T/2\right) \|\phi_\epsilon\|_{H^{-2}_N(\Omega)}, \quad \forall \epsilon \in (0,T/2).$$
Thus $\phi_\epsilon$ is bounded in $L^2(\Omega)$ when $[\epsilon \to 0]$.
Therefore $\phi_0 \in H^2_N(\Omega)$. Indeed, by Fatou's lemma,
\begin{align*}
\|\phi_0\|_{H^2_N(\Omega)}^2 = \sum_{k=1}^\infty | \lambda_k \langle \phi_0,\varphi_k\rangle |^2  &= \sum_{k=1}^\infty \liminf_{\epsilon \to 0} \Big| \frac{e^{-i\lambda_k \epsilon} - 1}{\epsilon} \langle \phi_0,\varphi_k\rangle \Big|^2 \\
& \leqslant \liminf_{\epsilon \to 0} \sum_{k=1}^\infty \Big| \frac{e^{-i\lambda_k \epsilon} - 1}{\epsilon} \langle \phi_0,\varphi_k\rangle \Big|^2 \\
& \leqslant \liminf_{\epsilon \to 0} \|\phi_\epsilon\|_{L^2(\Omega)}^2 < \infty.
\end{align*}
Let $T' \in (0,T)$. The map
\begin{align*}
\mathcal{J}_{T'}: L^2(\Omega) & \to L^2((0,T')\times\Gamma_c) \\
\phi_0 & \mto \Im(\phi)\big{|}_{\Gamma_c}
\end{align*}
is continuous (see Lemma \ref{Lem:Trace_L2}).
Moreover, $\mathcal{J}_{T'}(\phi_\epsilon)=0$ for every $\epsilon \in (0,T-T')$
and $\phi_\epsilon {\to} -i\mathcal{B}\phi_0$ in $L^2(\Omega)$ when $[\epsilon \to 0]$
thus $\mathcal{J}_{T'}(-i\mathcal{B}\phi_0)=0$. This holds for every $T' \in (0,T)$,
therefore $-i\mathcal{B}\phi_0 \in N_T$.
We have proved that $N_T$ is stable by $-i\mathcal{B}$ and only contains smooth functions in $D\left[ \mathcal{B}^s \right]$, $\forall s\in \mathbb{N}^*$.
Applying estimate \eqref{Weak_Obs_NL} to $(-i\mathcal{B}\phi_0)$, we~get
$$\| \mathcal{B}\phi_0 \|_{L^2(\Omega)} \leqslant \mathcal{C}' \| \mathcal{B}\phi_0 \|_{H^{-2}(\Omega)}.$$
This shows that the unit ball of $N_T$ is compact (for the $L^2(\Omega)$-topology), thus $N_T$ has finite dimension, by the Riesz theorem.

\subsubsection*{Step~2}
We prove that $N_T$ contains a finite sum of functions $\xi_{p,n}$.
The finite dimen\-sional $\mathbb{R}$-vector space $N_T$ is stable by the operator $-i\mathcal{B}$,
thus there exists $\alpha, \beta \in \mathbb{R}$ such that
$$\text{Ker}[ P(-i\mathcal{B}) ] \cap N_T \neq \{0\}, \quad \text{ where } P(X)=X^2+\alpha X+\beta.$$
Let $\phi_0 \in \text{Ker}[ P(-i\mathcal{B}) ] \cap N_T$. Let $a_{p,n}, b_{p,n} \in \mathbb{R}$ be such that
$$\langle \phi_0,\xi_{p,n} \rangle = a_{p,n}+ib_{p,n}, \quad \forall p,n \in \mathbb{N}.$$
Note that $a_{0,0}=0$ because $\phi_0 \in T_{\mathcal{S}}$.
Explicit computations show that the relation $P(-i\mathcal{B}) \phi_0=0$ reads
$$\sum_{p,n=0}^\infty \Big(
P[-i\lambda_{p,n}] a_{p,n}
+ i P\Big[ -i\Big(\lambda_{p,n}+\frac{2\epsilon}{\pi L(\lambda_{p,n}+1)} \Big)\Big] b_{p,n}
\Big) \xi_{p,n}=0,$$
where $\lambda_{p,n}:=p^2 + \left(\sfrac{n\pi}{L} \right)^2$. Thus
$$ P[-i\lambda_{p,n}] a_{p,n}
= P\Big[ -i\Big(\lambda_{p,n}+\frac{2\epsilon}{\pi L(\lambda_{p,n}+1)} \Big)\Big] b_{p,n} =0, \quad
\forall p,n \in \mathbb{N}.$$
The polynomial $P$ has at most two roots and the two quantities
\[
\lambda_{p,n}\quad\text{and}\quad\lambda_{p,n}+\sfrac{2\epsilon}{\pi L(\lambda_{p,n}+1)}
\]
diverge when $\|(n,p)\| \to \infty$,
thus only a finite number of coefficients $\langle \phi_0, \xi_{p,n} \rangle$ can be different from zero.

\subsubsection*{Step~3}
Conclusion. Let $N \in \mathbb{N}$ be such that $\phi_0=\sum_{n,p=0}^N \langle \phi_0,\xi_{p,n} \rangle \xi_{n,p}$.

\subsubsection*{Step~3.1}
Assumption \eqref{hyp:epsilon} implies that $\lambda_{p,n}+\sfrac{2\epsilon}{\pi L(\lambda_{p,n}+1)} >0$ for every $n,p \in\nobreak \mathbb{N}$.
Explicit computations show that
$$\Im[\phi(t,x_1,0)]=\sum_{p,n=0}^N \left(
b_{p,n} \cos(\omega_{p,n}t)
- \gamma_{p,n} a_{p,n} \sin(\omega_{p,n}t)
\rangle) \right) \zeta_p(x_1),$$
where
\begin{align*}
\omega_{p,n}&:=\sqrt{\lambda_{p,n}\Big(\lambda_{p,n}+\frac{2\epsilon}{\pi L(\lambda_{p,n}+1)} \Big)}, \quad \forall p,n \in \mathbb{N}\,,\\
\gamma_{p,n}&:=
\sqrt{\frac{\lambda_{n,p}}{\lambda_{p,n}+\sfrac{2\epsilon}{\pi L(\lambda_{p,n}+1)}}}\,.
\end{align*}
The continuous function $(t,x) \mto \Im[\phi(t,x_1,0)]$ vanishes on $(0,T)\times(a,b)$ because $\phi_0 \in N_T$.
In particular, for every $t \in [0,T]$, the function $x_1 \mto \Im[\phi(t,x_1,0)]$ vanishes on $(a,b)$.
But the functions $(x_1 \mto \cos(px_1))_{0 \leqslant p \leqslant N}$ are linearly independent on $(a,b)$, thus\vspace*{-5pt}
$$\sum_{n=0}^N \big(
b_{p,n} \cos(\omega_{p,n}t) - \gamma_{p,n} a_{p,n} \sin(\omega_{p,n}t)
\big) =0,\quad \forall t \in (0,T), 1 \leqslant p \leqslant N.$$
We notice that Assumption \eqref{hyp:epsilon} on $\epsilon$ imply
$$\omega_{p,n}^2 \neq \omega_{p,m}^2, \quad \forall p, n, m \in \mathbb{N} \text{ such that } n \neq m \text{ and } (p,n), (p,m)\neq(0,0),$$
where\vspace*{-5pt}
$$\omega_{p,n}^2 := \big(p^2 + \left(\sfrac{n\pi}{L}\right)^2 \big)\Big(p^2 + \left(\sfrac{n\pi}{L}\right)^2 + \frac{2\epsilon}{\pi L \left(p^2 + \sfrac{n\pi}{L} + 1 \right)} \Big),
\quad \forall p,n,m \in \mathbb{N}.$$
Under Assumption \eqref{hyp:epsilon}, the map $f(s):=s\left(s+\sfrac{2\epsilon}{\pi L(s+1)}\right)$ is strictly increasing on $[m,\infty)$ and thus the above property holds.

So, the frequencies $\{ \omega_{p,n} \sep 0 \leqslant n \leqslant N\}$ are all different for every $p \in \mathbb{N}$.
Thus, the previous relations imply that $b_{p,n}=a_{p,n}=0$ for every $0 \leqslant p,n \leqslant N$,
\ie $\phi_0 =0$, which is a contradiction.
\end{proof}

\subsubsection{Observability}

Proposition \ref{Prop:Obs_phi} follows from Propositions \ref{Prop:Weak_Obs_NL} and \ref{Prop:UC_NL},
by working as in the proof of Proposition \ref{Prop:Obs_eig}.

\subsubsection{Controllability of the nonlinear PDE}
\label{Subsec:TIL_NL}

The proof of Theorem \ref{Main_Thm_loc} consists in applying the inverse mapping theorem to
the end-point map $\Theta$ defined in \eqref{def:thetaT}, as in Section \ref{subsec:control_HF_nl}.

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