\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 74, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/74\hfil Null controllability]
{Null controllability of a cascade system of Schr\"odinger equations}

\author[M. L\'{o}pez-Garc\'ia, A. Mercado, L. de Teresa \hfil EJDE-2016/74\hfilneg]
{Marcos L\'opez-Garc\'ia, Alberto Mercado, Luz de Teresa}

\address{Marcos L\'opez-Garc\'ia \newline
Instituto de Matem\'aticas-Unidad Cuernavaca,
Universidad Nacional Aut\'onoma de M\'exico,
Apdo. Postal 273-3, Cuernavaca Mor. CP 62251, M\'exico}
\email{flopez@matem.unam.mx}

\address{Alberto Mercado \newline
Departamento de Matem\'atica,
Universidad T\'ecnica Federico Santa Mar\'ia,
Casilla 110-V, Valpara\'iso, Chile}
\email{alberto.mercado@usm.cl}

\address{Luz de Teresa \newline
Instituto de Matem\'aticas,
Universidad Nacional Aut\'onoma de M\'exico,
Circuito Exterior, CU, 04510 M\'exico}
\email{ldeteresa@im.unam.mx}

\thanks{Submitted August 7, 2015. Published March 17, 2016.}
\subjclass[2010]{93B05, 93C20}
\keywords{Null controllability; coupled equations; Schr\"odinger equation;
\hfill\break\indent  non geometrical condition; Carleman inequality}

\begin{abstract}
 This article presents a control problem for a cascade system of two linear
 $N$-dimensional Schr\"odinger equations. We address the problem of null
 controllability by means of a control supported in a region not satisfying
 the classical geometrical control condition. The proof is based on the
 application of a Carleman estimate with degenerate weights to each one
 of the equations  and a careful analysis of the system  in order to prove
 null controllability with only one control force.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

The controllability of coupled systems of PDE's has been intensely
studied in recent years.  In particular,  very interesting problems
arise when  there are less controls than equations.

Null controllability  results  for systems of parabolic equations are  reviewed
in the survey \cite{survey}.
About the  systems of hyperbolic  equations, we can mention
\cite{Alabau,Alabau-Leautaud,DLL:14}, where the
controllability of two coupled wave equations is proved with only one control,
under the hypothesis of the geometric control condition. In
\cite{Alabau,Alabau-Leautaud}, the authors show that as a consequence, the
same result is valid for a system of two Schr\"odinger equations.
A boundary controllability result is proved in \cite{deTeresa-Rosier}
for a cascade system of Schr\"odinger equations with periodic boundary conditions,
also as a consequence of the controllability result for a cascade system
of two wave equations.
In this article we are interested in the null controllability of a linear
system formed by two Schr\"odinger equations, controlling  only one of them.

The controllability of (scalar)  Schr\"odinger equations has been intensively
studied in recent  years.
In \cite{Leb} a  general result about this problem was obtained:
the author  proved  that
if the  wave equation is controllable at some time $T_0>0$ from controls supported
in a subset of the domain, then  the Schr\"odinger equation
is controllable with controls supported in the given region, for any $T>0$.

For the wave equation, boundary controllability  at time $T>0$
is equivalent (see \cite{BLR}) to the fact that the zone of control meets every
ray of the geometric optic in the domain in a smaller time than $T$.
This is called the \emph{geometric control condition}.
In the case of the Schr\"odinger equation, this is no longer true:
for some particular domains the equation  is  controllable by mean of controls
acting in some open subset of the boundary
that do not satisfies the geometric control condition
(\cite{BuZw,Rosier-Zhang,Tucs,Ten}).
In \cite{Zua}  a comprehensive  review of related results is presented.
It remains an open problem to find general controllability results
for the Schr\"odinger equation with weaker geometric conditions.

A very important  tool  to prove controllability of evolution PDE's is given
by Carleman estimates. To our knowledge the first paper  to derive a global
Carleman estimate for Schr\"odinger operators is \cite{Zhang}, where was
proved the exact controllability of a plate equation.
Carleman estimates are also used to study a related problem: the stability
of the inverse problem of retrieving a given  coefficient in an equation,
from observations of a trace of the solution.
In \cite{Bau-Puel} a Carleman estimate for Schr\"odinger operators  is proved,
with observations on a subset of the boundary satisfying the geometric control
condition. This implies  the stability of the stated inverse problem.
Very interesting results are proved in \cite{LTZ} for non conservative
Schr\"odinger equations in both cases:
with observations on a set satisfying the geometrical condition and not
satisfying it. In \cite{MOR}, some Carleman inequalities with an observation set
not satisfying  the geometric condition  are proved,
and then  the stability of an inverse problem for a space-dependent
coefficient is obtained.
The results of \cite{MOR} also imply the controllability of a scalar
Schr\"odinger equation by means of an $H^{-1}$ internal control acting
in an open set not satisfying the geometric control condition.
See Remark \ref{ControlMOR}.

The main objective of this article is to use the Carleman estimates
from \cite{MOR} in order to prove the controllability of a coupled system
of two Schr\"odinger equations  from an open subset of the domain
which does not satisfy the geometric control condition.
We are able to prove the result with a  control  acting in only one
of the equations.
As far as we know it is the first distributed null controllability
result for coupled Schr\"odinger equations that is not a consequence
of a similar result for the wave equation.

Let $\Omega \subset \mathbb{R}^{N}$ be a bounded, open set with $C^2$
boundary, $N\geq 1$. Let $\omega $ and $\mathcal{O}$ be two nonempty open
subsets of $\Omega$. For $T>0$ we set $Q=\Omega \times (0,T)$,
and $\Sigma =\partial \Omega \times (0,T) $. We consider the
following cascade system of Schr\"odinger equations:
\begin{equation}
\begin{gathered}
ip_{t}+\Delta p=h_{\omega}  \quad \text{in \ }Q, \\
iu_{t}+\Delta u=p\rho _{\mathcal{O}} \quad \text{in  }Q, \\
p=0,  \quad u=0  \quad \text{on  }\Sigma , \\
p(x,0) =p^0(x), \quad u(x,0) =u^0(x)  \quad \text{in  }\Omega ,
\end{gathered}  \label{main}
\end{equation}
where $p^0,u^0$ are given, $h_{\omega }$ is a control with support in
$\omega \times (0,T) $ and $\rho _{\mathcal{O}}$ is a regular
approximation of the characteristic function $1_{\mathcal{O}}$ of the set
$\mathcal{O}$.

In this work we analyze the null controllability of the cascade system
\eqref{main}  with one interior control $h_{\omega }$, i.e. we give conditions
on $T$, $\omega$, and $\mathcal{O}$ such that for every
$(p^0,u^0)$ in $L^2(\Omega)^2$ there exists a control $h_{\omega }$ with support
in $\omega \times (0,T)$ such that the corresponding solution
of \eqref{main}  satisfies
\begin{equation}
p(x,T) =0, \quad u(x,T) =0\quad \text{in }\Omega.
\label{null}
\end{equation}

Throughout this article, we  denote $X=D(-\Delta) = H^2\cap H_0^1(\Omega)$
endowed with the usual norm
$$
\| v\| _{X }=\Big( \int_{\Omega }|
\Delta v| ^2dx\Big) ^{1/2}\quad \text{for all }v\in X.
$$

Here and throughout the paper  $n(x)$  denotes  the unitary exterior
normal vector at $x \in \partial \Omega$,
$e_1$ means the unitary vector $(1, 0, \ldots , 0) \in \mathbb{R}^N$,
and $x_1$ is the first component of $x\in \mathbb{R}^N$.

To state the hypothesis on the domain and the observability region, let
 $[a,d]$  be the  $x_1$-projection of $\overline{\Omega}$.
We shall assume that there exists an open set
$\widehat \omega \subset \omega \cap \mathcal{O}$ with
$\operatorname{dist}(\partial\widehat \omega \cap \Omega,
\partial \omega \cap \Omega)\ge \alpha>0$
and real numbers $b,c$ with $a \leq  b < c \leq d$
such that
\begin{gather}
([b,c] \times \mathbb{R}^{N-1})  \cap \Omega \subset  \widehat \omega,
\label{H0bis}\\
n(x)\cdot e_1=0\quad \text{for all }x\in \partial \widehat
\omega \cap \partial \Omega.
\label{H1bis}
\end{gather}
Also we assume that there exists  a function $\psi \in C^{4}( [a,d])$ satisfying
\begin{equation}\label{H2bis}
\begin{gathered}
 \psi '\neq 0 \text{ in }[a,b]\cup \lbrack c,d],\quad
  \psi '(x_1)n(x)\cdot e_1\leq 0 \text{ for all } x\in \partial \Omega,
 \\
 | \psi '| ^2+\psi ''\geq 0  \text{ in } [a,b]\cup \lbrack c,d],
 \quad
 \psi \geq \frac{2}{3}\| \psi \| _{L^{\infty }(a,d)} \text{ in } (a,d).
 \end{gathered}
 \end{equation}
Our main result reads as follows:

\begin{theorem} \label{Control}
Suppose that there exists an open set
$ \widehat \omega \subset \omega \cap  \mathcal{O}$ satisfying
\eqref{H0bis}-\eqref{H1bis} and a function $\psi \in C^4([a,d])$
satisfying the hypotheses \eqref{H2bis}.
Then for each $(p^0,u^0)\in L^2(\Omega )^2$ there
exists a control $h_\omega\in L^2( 0,T;X ') $ supported in
$\omega \times (0,T)$ such that the
solution $(p,u)$ of system  \eqref{main} satisfies $(p(T),u(T))=(0,0)$.
\end{theorem}


\begin{remark} \rm
Hypotheses  \eqref{H0bis}-\eqref{H2bis} are  satisfied if  $\Omega$
is an ``stadium"  (see Figure \ref{fig1}), and
 $\widehat \omega$ is a ``strip" in $\mathbb{R}^N$, that is
$\widehat \omega = ( (b - \varepsilon, c + \varepsilon) \times \mathbb{R}^{N-1} )
 \cap \Omega$ for  some $\varepsilon > 0$ ($\widehat \omega $ is the shaded
region in Figure \ref{fig1}),
 and $\psi$ is given by
$$
\psi(x) = \begin{cases}
 x - a_1,  & x \in [a, b], \\
 d_1 -  x,  & x \in [c, d],   \\
\rho(x), &   x \in [b,c],
 \end{cases}
$$
 where $a_1 < a$ and $d< d_1$, and $\rho$  is a suitable function.
Note that \eqref{H2bis} are fulfilled
if $a_1$ and $d_1$ are chosen  such that  $a- a_1$  and $d_1-d$  are large enough.
\end{remark}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1}  % Dibujo5.jpg
\caption{Observation region in a stadium $\Omega \subset \mathbb{R}^2$.}
\end{center}
\label{fig1}
\end{figure}


\begin{remark} \label{rmk1} \rm
For any $h\in L^2( 0,T;X') $ and any
$(p^0,u^0)\in L^2(\Omega )^2 \subset (X')^2$,
 the cascade system \eqref{main} has exactly one solution $( p,u) $ (in
the sense specified in Section \ref{preli}), with
$(p,u) \in    C ([ 0,T] ;  X' )^2$,
provided that $\rho _{\mathcal{O}}\in C^2( \overline{\Omega })$.
\end{remark}

The proof of Theorem \ref{Control} is based on the existence of a constant
$C>0$\ such that the observability inequality
\begin{equation}
\| z(0)\| _{L^2}^2+\| q(0)\| _{L^2}^2\leq
C\int_0^T\int_{\Omega }| \Delta ( \rho _{\omega }q)
| ^2\,dx\,dt  \label{obsine}
\end{equation}
holds for any solution of the adjoint system
\begin{equation}
\begin{gathered}
iz_{t}+\Delta z=0 \quad \text{in  }Q, \\
iq_{t}+\Delta q=z\rho _{\mathcal{O}} \quad \text{in  }Q, \\
z=0, \quad q=0  \quad \text{on  }\Sigma , \\
z( x,T) =z^0(x), \quad q( x,T) =q^0(x)  \quad \text{in }\Omega ,
\end{gathered}  \label{adjmain}
\end{equation}
associated to $( z^0,q^0) \in X^2$.
To have the appropriate regularity and support of the control, in
\eqref{obsine} we consider a function $\rho _{\omega }\in C^2(
\overline{\Omega }) $, such that $\rho _{\omega }(x) =0$
for all $x\in \Omega \backslash \omega $ and $\rho _{\omega }(x) =1$ for all
$x$ in a large part of $\omega$; in Section \ref{obsection} we give
the precise details.

The rest of this article is organized as follows: in Section 2 we state
the functional framework where we will state the controllability problems.
Section 3 is devoted to prove the observability inequality \eqref{obsine}.
Finally,  Theorem \ref{Control} will be proved in  Section 4.

\section{Well posedness} \label{preli}

In this section we recall some existence and regularity results for the
Schr\"odinger equation. These results can be found in \cite{caze}.
From now on, $C$ stands for a generic positive constant depending only on
$\Omega $, $T$, $\omega $ and $\mathcal{O}$, which can take different
values from line to line.

Let $k\in L^2 ( 0,T; X)$ and $v^0\in X$. Then the solution $v$ of the linear
problem
\begin{equation}
\begin{gathered}
iv_{t}+\Delta v = k \quad \text{in  }Q, \\
v = 0 \quad \text{on  }\Sigma , \\
v(x,0)  = v^0(x) \quad \text{in }\Omega ,
\end{gathered}  \label{schrogen}
\end{equation}
satisfies $v\in C ([ 0,T];X  )$.
Moreover, there exists $C>0$ such that
\[
\| v \|_{L^\infty(0,T; X) }
\leq C( \| v^0\| _{X}  + \|k\| _{L^2( 0,T; X ) }) .
\]

When $k\in C ( [ 0,T] ,L^2( \Omega ) )$
 the corresponding solution satisfies
$v\in C( [ 0,T], X ) \cap C^{1}( [ 0,T],L^2( \Omega ) ) $.

We need to solve \eqref{schrogen} with $k\in L^2( 0,T; X ') $
 and $v^0\in L^2(\Omega ) $. Under this assumption on $k$ and
considering $p^0\in L^2( \Omega ) $, the solution by transposition of
\begin{equation}
\begin{gathered}
ip_{t}+\Delta p=k \quad \text{in }Q, \\
p=0 \quad \text{on }\Sigma , \\
p(x,0) =p^0(x) \quad \text{in }\Omega ,
\end{gathered}  \label{verygen}
\end{equation}
is, by definition, the unique function $p\in L^2( 0,T;X') $ satisfying
\begin{equation}
\int_0^T\langle p( t) ,g( t) \rangle dt
=\int_0^T\langle k( t) ,\varphi _{g}( \cdot,t) \rangle dt
+ i ( p^0,\varphi _{g}( \cdot ,0)) _{L^2( \Omega ) } \label{schrotrans}
\end{equation}
for all $g\in L^2( 0,T; X )$,
where $\langle \cdot ,\cdot \rangle $ represents the duality
between $X$ and $X'$, and
for each $g\in L^2( 0,T; X ) $ we have
denoted by $\varphi _{g}$ the solution to the corresponding adjoint system
\begin{equation}
\begin{gathered}
i\varphi _{t}+\Delta \varphi =g \quad \text{in }Q, \\
\varphi =0 \quad \text{on }\Sigma , \\
\varphi ( x,T) =0 \quad  \text{in }\Omega .
\end{gathered}  \label{adjtrans}
\end{equation}
Note that  the solution $\varphi _{g}$ of \eqref{adjtrans} satisfies
\begin{gather*}
\varphi _{g} \in   L^2( 0,T; X  ) \cap C([0,T];L^2(\Omega)), \\
\| \varphi _{g}\| _{L^2( 0,T; X ) }+\| \varphi _{g}( 0) \| _{L^2( \Omega
) }\leq C\| g\| _{L^2( 0,T; X ) }.
\end{gather*}
Since $ X $ is a reflexive space, we have
$L^2( 0,T; X ) '=L^2( 0,T;  X')$.
Hence \eqref{schrotrans} makes sense, and we conclude that there exists
a unique $p \in L^2(0,T; X')$ solution of \eqref{verygen}, which satisfies
\begin{equation*}
\| p\| _{L^2( 0,T; X  ') }\leq C( \| k\| _{L^2( 0,T; X') }
+\| p^0\| _{L^2( \Omega) }) .
\end{equation*}
By energy estimates and density arguments we obtain that in fact
$p \in C([0,T]; X')$.

\section{Observability inequality} \label{obsection}

In this section the observability inequality \eqref{obsine}
for the adjoint system \eqref{adjmain}  will be proved.
We  use a Carleman estimate proved in  \cite{MOR}.
To state the result, we introduce the following notation.
For the function $\psi$ given by \eqref{H2bis}, we set
$C_{\psi }=2\| \psi \| _{L^{\infty }( \Omega )}$ and we define the
auxiliary functions
\begin{equation*}
\theta ( x,t) :=\frac{e^{\lambda \psi ( x_1) }}{t(T-t) },\quad
\varphi (x,t) :=\frac{e^{\lambda C_{\psi }}-e^{\lambda \psi ( x_1) }}{t( T-t) },
\quad\text{for all }( x,t) \in \Omega \times (0,T) ,
\end{equation*}
for  $\lambda > 0$.

The following Carleman inequality for the Schr\"odinger equation
is a particular case of \cite[Corollary 3.3]{MOR}.

\begin{proposition} \label{PropMOR}
Let us define $\widetilde \omega = ((b,c)\times \mathbb{R}^{N-1})\cap \Omega$
(see \eqref{H0bis}), take $\psi \in C^{4}( \mathbb{R}) $ be a function
satisfying  \eqref{H2bis}.
Then there exists a constant $C>0$\ such that for
$f\in C^{2,1}( \overline{\Omega }\times [ 0,T] )$, with $f=0$ on $\Sigma $,
it holds
\begin{equation} \label{inemor}
\begin{aligned}
&\int_Q[ \theta | \frac{\partial f}{\partial x_1}|^2+\theta ^3| f| ^2]
e^{-2s\varphi }\,dx\,dt \\
&\leq C\Big( \int_Q| f_{t} + i \Delta f| ^2e^{-2s\varphi }\,dx\,dt
 + \int_0^T\int_{\widetilde\omega }[ \theta | \frac{\partial f}{\partial x_1}| ^2
 +\theta ^3| f| ^2] e^{-2s\varphi }\,dx\,dt\Big).
\end{aligned}
\end{equation}
\end{proposition}

\begin{remark} \label{rmk3} \rm
The result in \cite{MOR} is more general than  Proposition \ref{PropMOR},
because it does  not ask the weight function to depend only in one variable.
 Here we use such a  function  to  estimate one of the observations
of the two equations of the system, and to obtain
the controllability with only one control (see Proposition \ref{Carleman1Obs}).
\end{remark}

\begin{remark} \label{ControlMOR} \rm
It is not difficult to see that Proposition \ref{PropMOR} implies the
observability inequality
 \begin{equation}\label{eq1}
\| q( 0) \| _{L^2( \Omega ) }\leq
C\int_0^T \int_{\widetilde\omega }( | q| ^2+| \nabla q| ^2) \,dx\,dt
\end{equation}
for all $ q^0  \in H_0^{1}( \Omega )$, where $q$ solves  the equation
\begin{equation}\label{cr}
\begin{gathered}
i q_{t} + \Delta q =0 \quad \text{in }\Omega,  \\
q=0 \quad \text{on }\Sigma,\\
q( 0)  = q^0  \quad  \text{in }  \Omega.
\end{gathered}
\end{equation}
In fact, inequality \eqref{eq1} follows from \eqref{inemor} (with $f=q$ solution
to  \eqref{cr}) and the fact that
 $$
\int_Q\theta ^3| q| ^2e^{-2s\varphi }\,dx\,dt\geq
C_{T}\int_{T/4}^{3T/4}\int_{\Omega }| q| ^2\,dx\,dt
=TC_{T}/2\| q( 0) \| _{L^2( \Omega )}.
$$
From \eqref{eq1} we have a  controllability result: for every
$u^0  \in L^2(\Omega)$ there exists
a control $h\in L^2( 0,T;H^{-1}( \Omega )) $ supported on $\widetilde\omega$
such that the solution of the equation
\begin{gather*}
i u_{t} + \Delta u  = h \quad \text{in }\Omega,\\
u = 0 \quad \text{on }\Sigma,\\
u( 0)  = u^0 \quad \text{in }  \Omega,
\end{gather*}
satisfies $u(T) =0$.
\end{remark}

As we said, the proof of Theorem \ref{Control} depends on an observability
inequality for the adjoint system \eqref{adjmain}. The result
is the following.

\begin{proposition} \label{Carleman1Obs}
Assume the hypothesis of Theorem \ref{Control}.
There exists a constant $C>0$ such that
\begin{equation}  \label{OBS}
\| z(0)\| _{L^2}^2+\| q(0)\| _{L^2}^2
\leq C\int_0^T\int_{\Omega }| \Delta ( \rho _{\omega} q)| ^2\,dx\,dt
\end{equation}
for all $z^0,q^0\in H^2\cap H_0^{1}( \Omega ) $, where
$(z,q)$ is the solution of \eqref{adjmain},
and $\rho _{\omega}$ is a cut-off function supported in ${\omega}$.
\end{proposition}

\begin{remark} \label{rmk5} \rm
Every solution of \eqref{adjmain}  with $z^0,q^0\in H^2\cap H_0^{1}( \Omega ) $
satisfies
\begin{equation*}
z,q\in C( [ 0,T] ;X) \cap C^{1}( [ 0,T] ;L^2( \Omega ) ) .
\end{equation*}
\end{remark}

\begin{proof}[Proof of Proposition \ref{Carleman1Obs}]
We will deal with regular solutions, getting the final result by standard
density arguments.
 Let $\widehat \omega$ be the open satisfying hypothesis \eqref{H0bis}
and \eqref{H1bis}.
We claim that there exists  a constant $C$ such that,
for every $\varepsilon >0$, the following estimates hold for solutions
of the adjoint system \eqref{adjmain}.
\smallskip

\noindent\textbf{Claim 1.}
\begin{equation} \label{c1}
\begin{aligned}
&\int_0^T\int_{ \widetilde \omega}\theta ^3|
z| ^2e^{-2s\varphi }\,dx\,dt \\
&\leq C \varepsilon^{-1} \int_0^T\int_{\widehat{\omega}}( | q|
^2+| \nabla q| ^2) \,dx\,dt
+\varepsilon \int_Q[ \theta ^3| z| ^2+\theta
| \frac{\partial z}{\partial x_1}| ^2] e^{-2s\varphi}\,dx\,dt.
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Claim 2.}
\begin{equation} \label{c2}
\begin{aligned}
&\int_0^T\int_{ \widetilde \omega}\theta |
 \frac{\partial z}{\partial x_1}| ^2e^{-2s\varphi }\,dx\,dt\\
&\leq C\varepsilon^{-1} \int_0^T\int_{\widehat{\omega}}[ | \nabla
q| ^2+| \nabla ( \frac{\partial q}{\partial x_1})
| ^2] \,dx\,dt
+\varepsilon \int_Q\theta | \frac{\partial z}{\partial x_1}
| ^2e^{-2s\varphi }\,dx\,dt.
\end{aligned}
\end{equation}

Next we apply \eqref{inemor} to $\widetilde{z}( t) =iz(T-t) $ and
$\widetilde{q}( t) =q( T-t) $  to obtain
\begin{equation} \label{A}
\int_Q[ \theta ^3| z| ^2+\theta |
\frac{\partial z}{\partial x_1}| ^2] e^{-2s\varphi }\,dx\,dt
\leq C\int_0^T\int_{ \widetilde \omega}[ \theta
^3| z| ^2+\theta | \frac{\partial z}{\partial x_1}
| ^2] e^{-2s\varphi }\,dx\,dt,
\end{equation}
and
\begin{equation} \label{B}
\begin{aligned}
&\int_Q\theta ^3| q| ^2e^{-2s\varphi }\,dx\,dt \\
&\leq C\Big( \int_0^T\int_{ \widetilde \omega}[ \theta
^3| q| ^2+\theta | \frac{\partial q}{\partial x_1}
| ^2] e^{-2s\varphi }\,dx\,dt
 +\int_0^T\int_{\Omega }\rho _{\mathcal{O}
}^2| z| ^2e^{-2s\varphi }\,dx\,dt\Big) .
\end{aligned}
\end{equation}
Taking $\varepsilon$ small enough in  \eqref{c1}, \eqref{c2}
and combining with \eqref{A} we obtain
\begin{equation}
\int_Q[ \theta ^3| z| ^2+\theta | \frac{
\partial z}{\partial x_1}| ^2] e^{-2s\varphi }\,dx\,dt
\leq C\int_0^T\int_{\widehat{\omega}}[ | q|
^2+| \nabla q| ^2+| \nabla ( \frac{\partial q}{
\partial x_1}) | ^2] \,dx\,dt,  \label{prev}
\end{equation}
whereas a combination with \eqref{B} leads to
\begin{equation}
\int_Q\theta ^3| q| ^2e^{-2s\varphi }\,dx\,dt
\leq C\int_0^T\int_{\widehat{\omega}}[ | q|^2+| \nabla q| ^2] \,dx\,dt
+C\int_Q| z|^2\,dx\,dt,  \label{qine}
\end{equation}
for some constant $C>0$.

We know that $-i\Delta $\ generates a group of isometries
 $( \mathcal{T}( t) ) _{t\in \mathbb{R}}$ on
$L^2( \Omega )$ so $z( t) =\mathcal{T}( t-T) z^0$, and \eqref{prev}
yields
\begin{equation}
\begin{aligned}
\| z(0)\| _{L^2}^2
&= \frac{2}{T}\int_{T/4}^{3T/4}\int_{\Omega }| z| ^2\,dx\,dt   \\
&\leq C\int_Q\theta ^3| z| ^2e^{-2s\varphi }\,dx\,dt\\
&\leq C\int_0^T\int_{\widehat{\omega}}[ |q| ^2+| \nabla q| ^2
+| \nabla ( \frac{\partial q}{\partial x_1}) | ^2] \,dx\,dt.
\end{aligned} \label{iso}
\end{equation}
Multiplying the second  equation in \eqref{adjmain}  by $i\overline{q}$ and
integrating with respect to space we obtain
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\int_{\Omega }| q( x,T-t) |^2dx
-\operatorname{Re}\Big( i\int_{\Omega }| \nabla q( x,T-t)
| ^2dx\Big)
=\operatorname{Re}\int_{\Omega }i\rho _{\mathcal{O}}(x) ( z\overline{q})
( x,T-t) dx;
\end{equation*}
therefore
\begin{equation*}
\frac{d}{dt}\int_{\Omega }| q( x,T-t) | ^2dx
\leq \int_{\Omega }| q( x,T-t) | ^2dx+\int_{\Omega}| z( x,T-t) | ^2dx
\end{equation*}
and the Gronwall inequality implies
\begin{equation} \label{C}
\begin{aligned}
\| q(0)\| _{L^2}^2
&\leq e^{T-t}\int_{\Omega }| q(x,T-t) | ^2dx
 +\int_{t}^Te^{T-s}\int_{\Omega }| z(x,T-s) | ^2dxds   \\
&\leq e^T\int_{\Omega }| q( x,T-t) |^2dx
 +e^T\int_Q| z| ^2\,dx\,dt.
\end{aligned}
\end{equation}
Integrating \eqref{C} with respect to time on $[ T/4,3T/4]$, and using
\eqref{qine} we obtain
\begin{equation} \label{D}
\begin{aligned}
\| q(0)\| _{L^2}^2
&\leq \frac{2e^T}{T}\int_{T/4}^{3T/4}
\int_{\Omega }| q| ^2\,dx\,dt+Te^T\| z(0)\|_{L^2}^2   \\
&\leq C\int_Q\theta ^3| q| ^2e^{-2s\varphi}\,dx\,dt
 +Te^T\| z(0)\| _{L^2}^2  \\
&\leq C\int_0^T\int_{\widehat{\omega}}[ |
q| ^2+| \nabla q| ^2] \,dx\,dt
+C\| z(0)\|_{L^2( \Omega ) }^2.
\end{aligned}
\end{equation}
From  \eqref{iso} and \eqref{D} we obtain
\begin{align*}
\| ( q(0),z(0)) \| _{( L^2( \Omega )) ^2}^2
&\leq C\int_0^T\int_{\widehat{\omega}}
[ | \nabla q| ^2+| q| ^2+| \nabla (
\frac{\partial q}{\partial x_1}) | ^2] \,dx\,dt \\
&\leq C\int_0^T\int_{\Omega }| \Delta ( \rho _{\omega}q) | ^2\,dx\,dt
\end{align*}
where $\rho_\omega$ is a cut-off function with support in $\overline{\omega}$
and such that $\rho_\omega =1$ in $\widehat \omega$.
This completes the proof of Proposition \ref{Carleman1Obs},
and it only  remains to prove the two claims.

We consider a function
$\sigma =\sigma ( x_1) \in C_{c}^{\infty }( \mathbb{R})$
such that $\sigma \equiv 1$ in $[b,c]$ and
 \begin{equation} \label{supp}
\operatorname{supp}(\sigma) \subset P_1( \widehat{\omega}),
\end{equation}
 where $P_1$ is the canonical projection onto the $x_1$-axis.
Also we set
\[
\eta(t)=t^{-1}(T-t)^{-1}.
\]


\noindent\textbf{Proof of Claim 1.}
Recall that $z$ satisfies the homogeneous Schr\"odinger equation and
$\theta ^{m}e^{-2s\varphi }$ vanishes at $t=0$, $t=T$ for every $m\geq 0$.
 Multiplying the second equation in \eqref{adjmain} by
$\sigma \theta^3 \bar z e^{- 2s\varphi}$  and integrating in $Q$ we obtain
\begin{equation}\label{e:z}
\int_0^T\int_{  \mathcal O}\sigma\theta ^3|z| ^2e^{-2s\varphi }\,dx\,dt
 =\int_0^T\int_{ \Omega}\sigma \theta ^3e^{-2s\varphi }\overline{z}
( i q_{t}+ \Delta q) \,dx\,dt.
\end{equation}
Integrating by parts the right hand side of \eqref{e:z}, having in mind
\eqref{supp} and using  that $\sigma  \bar z = \sigma q = 0$
 on $\partial \widehat \omega$, we obtain
\begin{equation}
\begin{aligned}
&\int_0^T\int_{\Omega}\sigma \theta
^3e^{-2s\varphi }\overline{z}( i q_{t}+ \Delta q) \,dx\,dt\\
&= \int_0^T\int_{\widehat{\omega}}q\nabla ( \sigma
\theta ^3e^{-2s\varphi }) \cdot \nabla \overline{z}-\overline{z}
\nabla ( \sigma \theta ^3e^{-2s\varphi }) \cdot \nabla q\,dx\,dt  \\
&\quad - i \int_0^T\int_{\widehat{\omega}}\sigma ( \theta
^3e^{-2s\varphi }) _{t}\overline{z}q\,dx\,dt.
\end{aligned}\label{E}
\end{equation}
Straightforward computations show that
\begin{gather*}
( \theta ^3e^{-2s\varphi }) _{t}
=\theta ^3e^{-2s\varphi }( 3-2s\varphi ) \frac{2t-T}{t( T-t) }, \\
\nabla ( \sigma \theta ^3e^{-2s\varphi })
 = \theta^3e^{-2s\varphi }( 3\lambda \sigma \psi '+\sigma '
 +2s\lambda \sigma \theta \psi ') e_1.
\end{gather*}
Note that $\theta \leq C\eta $ on $\overline{\Omega }$. Therefore,
\begin{gather*}
| ( \theta ^3e^{-2s\varphi }) _{t}|
\leq C\theta^{3/2}\eta ^{7/2}e^{-2s\varphi }, \\
| \nabla ( \sigma \theta ^3e^{-2s\varphi }) |
\leq C\theta ^{1/2}\eta ^{7/2}e^{-2s\varphi }.
\end{gather*}
Hence, taking into account that $|e^{-2s\varphi} \eta^m| \leq C$
for any $m >0$, Claim 1 follows from \eqref{e:z},
\eqref{E} and the Cauchy-Schwarz inequality $| ab|
\leq \varepsilon a^2+\frac{1}{4\varepsilon }b^2$, $\varepsilon >0$.
\smallskip

\noindent\textbf{Proof of Claim 2.}
The Green identity implies
\begin{align*}
&\int_{\widehat{\omega}}\Delta ( \sigma \theta
e^{-2s\varphi }\frac{\partial \overline{q}}{\partial x_1}) \frac{
\partial z}{\partial x_1}dx \\
&= \int_{\widehat{\omega}}\sigma \theta e^{-2s\varphi }\frac{\partial
\overline{q}}{\partial x_1}\Delta ( \frac{\partial z}{\partial x_1}) dx
+ \int_{\partial \widehat{\omega}}\frac{
\partial z}{\partial x_1}\frac{\partial }{\partial \nu }( \sigma
\theta e^{-2s\varphi }\frac{\partial \overline{q}}{\partial x_1}) dS\\
&\quad - \int_{\partial \widehat{\omega}}\sigma \theta e^{-2s\varphi }
\frac{\partial \overline{q}}{\partial x_1}\frac{\partial }{\partial \nu }
( \frac{\partial z}{\partial x_1}) dS,
\end{align*}
and then, since $\frac{\partial q}{\partial x_1}=\frac{\partial z}{\partial x_1}=0$
on $\partial \widehat{\omega}\cap \partial \Omega $ we obtain that
\begin{align*}
&\int_0^T\int_{ \widetilde \omega}\theta | \frac{
\partial z}{\partial x_1}| ^2e^{-2s\varphi }\,dx\,dt \\
&\leq \int_0^T\int_{\widehat{\omega}}\sigma \theta
e^{-2s\varphi }\frac{\partial z}{\partial x_1}\frac{\partial }{\partial
x_1}( \overline{i q_{t}  + \Delta q}) \,dx\,dt \\
&= + i \int_0^T\int_{\widehat{\omega}}[ \sigma (\theta e^{-2s\varphi }) _{t}
+i \Delta ( \sigma \theta e^{-2s\varphi }) ]
\frac{\partial z}{\partial x_1}\frac{\partial \overline{q}}{\partial x_1}\,dx\,dt \\
&\quad -2 \int_0^T\int_{\widehat{\omega}}[ \nabla (
\sigma \theta e^{-2s\varphi }) \cdot \nabla ( \frac{\partial
\overline{q}}{\partial x_1}) ] \frac{\partial z}{\partial x_1
}\,dx\,dt.
\end{align*}
From these estimates and taking into account that
\begin{gather*}
| ( \theta e^{-2s\varphi }) _{t}| + | \Delta
( \sigma \theta e^{-2s\varphi }) |
\leq  C\theta ^{1/2}\eta ^{5/2}e^{-2s\varphi }, \\
| \nabla ( \sigma \theta e^{-2s\varphi }) |
\leq C\theta ^{1/2}\eta ^{3/2}e^{-2s\varphi },
\end{gather*}
we obtain the proof of Claim 2.
\end{proof}

\section{Proof of the main result}

In this section we will deduce Theorem \ref{Control} from Proposition
\ref{Carleman1Obs}.

\begin{proof}[Proof of Theorem \ref{Control}]
Note that the observability inequality for $(z^0, q^0)\in X^2 $
implies that   we can define a norm
\[
\|(z^0,q^0)\|^2_V=\int_0^T\int_{\Omega }| \Delta ( \rho _{\omega }q)
| ^2\,dx\,dt
\]
 where $q$ is the solution  to
\begin{gather*}
iz_{t}+\Delta z=0 \quad \text{in  }Q, \\
iq_{t}+\Delta q=z\rho _{\mathcal{O}} \quad \text{in  }Q, \\
z=0, q=0  \quad \text{on }\Sigma , \\
z( x,T) =z^0(x), \quad q( x,T) =q^0(x)  \quad \text{in }\Omega .
\end{gather*}
Indeed,   $\|(z^0,q^0)\|_V$ is clearly a semi-norm and the observability
inequality implies that $\|(z^0,q^0)\|_V=0$
takes to $(z(0), q(0))=0$. Using the conservation of energy for $z$
we obtain easily that $z^0=0$ and then $q^0=0$.
We set  the space $V$ as the completion of  $X^2$ with this norm.
For $(z^0,q^0)\in V$ and given fixed initial data $(p^0,u^0)\in L^2(\Omega)^2$
we define the functional
$$
J(z^0, q^0)=\frac12\int_0^T\int_\Omega | \Delta ( \rho _{\omega }q)
| ^2\,dx\,dt  + \int_\Omega q(0)p^0+ \int_\Omega z(0)u^0.
$$
Standard arguments show that $J$ is continuous, convex and, thanks
to the observability inequality, coercive. This implies that $J$ reaches
a minimum at a point $(\widehat z^0,\widehat q^0)\in V$.
At this point the following optimality condition holds
\begin{equation}\label{optcond}
\int_0^T\int_\Omega \Delta ( \rho _{\omega }\widehat q)\Delta
( \rho _{\omega }q)+ \int_\Omega q(0)p^0+ \int_\Omega z(0)u^0=0
\end{equation} for any
$(z^0, q^0)\in V$.
Now we propose as control $h_\omega = \Delta^2(\rho_\omega \widehat q)\rho_\omega$.
Note that the corresponding solution to \eqref{main} satisfies $(p(T), u(T))=0$.
In fact, \eqref{optcond} is valid for any $(z^0,q^0)\in X^2$ and $(z,q)$
the corresponding solution to \eqref{adjmain}. So taking the duality
product of \eqref{main} by $(z,q)$ we obtain
\begin{align*}
0&=\int_0^T\int_\Omega \Delta ( \rho _{\omega }\widehat q)
 \Delta ( \rho _{\omega }q)+ \int_\Omega q(0)p^0+ \int_\Omega z(0)u^0\\
&= \langle p(T),q^0  \rangle_{X',X} + \langle u(T),z^0 \rangle_{X',X}
\end{align*}
and the null controllability result is proved.
\end{proof}

\section{Concluding remarks}

It would be interesting to find similar results in the case of three or more
coupled Schr\"odinger equations.
In the case of parabolic equations there are various results.
In particular in \cite{GdT} the case of \emph{cascade} system of
 parabolic equations  was treated by a repeated argument of Carleman
inequalities and local energy estimates. For the case of Schr\"odinger
equations this is not the case, at least when the control set do not satisfy
the geometric condition. Let us consider the case of three coupled
Schr\"odinger equations in cascade. The adjoint system will be of the form
\begin{gather*}
iz_{t}+\Delta z=0 \quad \text{in  }Q, \\
iq_{t}+\Delta q=z\rho _{\mathcal{O}} \quad \text{in  }Q, \\
iv_{t}+\Delta v=q\rho _{\mathcal{O}} \quad \text{in  }Q,\\
z=0, \quad q=0, \quad v=0  \quad  \text{on  }\Sigma , \\
z( x,T) =z^0(x),\quad  q( x,T) =q^0(x),\quad  v( x,T) =v^0(x) \quad \text{in }\Omega .
\end{gather*}
Using the arguments in this article it is not difficult to see that the
following observability inequality holds
\begin{equation}\label{obsfin}
\| ( z(0), q(0), v(0)) \| _{( L^2( \Omega )) ^3}^2
\le C\int_0^T\int_{\Omega }| \Delta ( \rho _{\omega}q) | ^2
+ |\nabla ( \rho _{\omega }v)|^2 \,dx\,dt.
\end{equation}
This inequality implies that the system
\begin{gather*}
ip_{t}+\Delta p=h^1_\omega \quad \text{in  }Q, \\
iu_{t}+\Delta u=p\rho _{\mathcal{O}} + h^2_\omega \quad \text{in  }Q, \\
iw_{t}+\Delta w=u\rho _{\mathcal{O}} \quad \text{in }Q,\\
p=0, u=0, w=0  \quad \text{on }\Sigma , \\
p(x,0) =p^0(x), \quad u(x,0) =u^0(x),\quad
 w(x,0) =w^0(x) \quad \text{in }\Omega ,
\end{gather*}
with controls $h^1_\omega \in L^2(0,T; X')$ and
$h^2_\omega \in L^2(0,T; H^{-1}(\Omega))$ is null controllable.
However  to control  only in the first equation it would be necessary
to eliminate the term in $q$ in the observability inequality \eqref{obsfin}.
It seems to be an open  problem that cannot be treated with the techniques
used in this article. As far as we know this problem is still open even
in the case in which $\omega$ satisfies the geometric condition.

\subsection*{Acknowledgments}
This research was partially supported by  PAPIIT-IN101013, UNAM M\'exico.
The second author is partially supported by FONDECYT (Chile) grant 1120610, 
Basal CMM U. de Chile and ANILLO ACT1106.
The second author would like to thank the hospitality of the
Instituto de Matem\'aticas, UNAM,  where the research was accomplished.



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\end{document}