# Control of the Schrödinger equation in $\mathbb{R}^{3}$: The critical case

Pablo Braz e Silva  Departamento de Matemática, Universidade Federal de
Pernambuco, S/N Cidade Universitária, 50740-545, Recife (PE), Brazil
<EMAIL_ADDRESS>,  Roberto de A. Capistrano–Filho
<EMAIL_ADDRESS>,  Jackellyny Dassy do Nascimento Carvalho
<EMAIL_ADDRESS>and  David dos Santos Ferreira  Institut Élie Cartan
de Lorraine, UMR CNRS 7502, équipe SPHINX, INRIA, Université de Lorraine,
F-54506 Vandoeuvre-les-Nancy Cedex, France<EMAIL_ADDRESS>
###### Abstract.

This article deals with the $\dot{H}^{1}$–level exact controllability for the
defocusing critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$.
Firstly, we show the problem under consideration to be well-posed using
Strichartz estimates. Moreover, through the Hilbert uniqueness method, we
prove the linear Schrödinger equation to be controllable. Finally, we use a
perturbation argument and show local exact controllability for the critical
nonlinear Schrödinger equation.

###### Key words and phrases:

NLS system, Critical case, Internal controllability, HUM

###### 2020 Mathematics Subject Classification:

93B05, 93B07, 35Q55, 49K40

∗Corresponding author<EMAIL_ADDRESS>
Braz e Silva was partially supported by CAPES/MATH-AMSUD #8881.520205/2020-01,
CAPES/PRINT #88887.311962/2018-00, CAPES/COFECUB #88887.879175/2023-00, CNPq,
Brazil #421573/2023-6, #305233/2021-1, #308758/2018-8, #432387/2018-8.
Capistrano–Filho was partially supported by CAPES/MATH-AMSUD
#8881.520205/2020-01, CAPES/PRINT #88887.311962/2018-00, CAPES/COFECUB
#88887.879175/2023-00, CNPq, Brazil, #421573/2023-6, #307808/2021-1 and
#401003/2022-1 and Propesqi (UFPE). Carvalho was partially supported by CNPq,
CAPES-MATHAMSUD #8881.520205/2020-01 and FACEPE BFD-0014-1.01/23.

##  1\. Introduction

###  1.1. Addressed issues and review of the literature

We consider the $\dot{H}^{1}(\mathbb{R}^{3})$ exact controllability for the
defocusing critical nonlinear Schrödinger equation (C-NLS)

(1.1) $\left\\{\begin{array}[]{lr}i\partial_{t}u+\Delta u-|u|^{4}u=f(x,t),\
\mbox{on }\mathbb{R}^{3}\times[0,+\infty),\\\
u(x,0)=u_{0}(x)\in\dot{H}^{1}(\mathbb{R}^{3}),\end{array}\right.$

where $u=u(x,t)$ is a complex-valued function of two variables
$x\in\mathbb{R}^{3}$ and $t\in\mathbb{R}$, the subscripts denote the
corresponding partial derivatives, while the function $f(x,t)$ is a control
input. We are mainly concerned with the following exact control problem for
system (1.1).

Control problem: Let $T>0$ be given. For any given
$u_{0},u_{T}\in\dot{H}^{1}(\mathbb{R}^{3})$, can one find a control $f(x,t)$
such that system (1.1) admits a solution $u\in
C\left([0,T];\dot{H}^{1}(\mathbb{R}^{3})\right)$ satisfying $u(x,T)=u_{T}(x)$
in $\mathbb{R}^{3}$?

Control properties of Schrödinger equations have received a lot of attention
in the last decades. For example, regarding control issues, one may see [12,
19, 20, 21] and the references therein. As for Carleman estimates and
applications to inverse problems, we cite [2, 4, 5, 16, 18, 27] and the
references therein. An excellent review of the contributions up to 2003 is in
[29].

Let us detail some recent results. The results due to Illner et al. [8, 9]
considered internal controllability of the nonlinear Schrödinger equation
posed on a finite interval $(-\pi,\pi)$

(1.2) $\begin{cases}i\partial_{t}v+\partial_{xx}v+\lambda|v|^{2}v=f(x,t),\quad
x\in(-\pi,\pi),\\\ v(-\pi,t)=v(\pi,t),\text{ }\text{
}\partial_{x}v(-\pi,t)=\partial_{x}v(\pi,t),\end{cases}$

where the forcing function $f=f(x,t)$, supported in a sub-interval of
$(-\pi,\pi)$, is a control input. They showed that system (1.2) is locally
exactly controllable in the space

$H^{1}_{p}(-\pi,\pi):=\\{v\in H^{1}(-\pi,\pi):v(-\pi)=v(\pi)\\}.$

Later, Lange and Teismann [13] considered the internal control of the same
nonlinear Schrödinger equation in (1.2) posed on a finite interval but with
the homogeneous Dirichlet boundary conditions

$v(-\pi,t)=v(\pi,t)=0,$

and showed that this is locally exactly controllable in the space
$H^{1}_{0}(0,\pi)$ around a special ground state of the equation (see also
[14] for internal controllability of the nonlinear Schrödinger equation posed
on a finite interval).

In [24], Rosier and Zhang considered the nonlinear Schrödinger equation

(1.3) $i\partial_{t}u+\Delta u+\lambda|u|^{2}u=0$

posed on a bounded domain $K$ in $\mathbb{R}^{n}$ with both Dirichlet boundary
conditions and Neumann boundary conditions. They showed that if either

$s>\frac{n}{2}$

or

$0\leq s<\frac{n}{2},\text{ }\text{ with }1\leq n<2+2s$

or

$s=0,1\text{ }\text{ with }n=2,$

then both systems, with Dirichlet and Neumann conditions, when the control
inputs are acting on the whole boundary of $K$, are locally exactly
controllable in the classical Sobolev space $H^{s}(K)$ around any smooth
solution of the Schrödinger equation.

In [25], the authors extend the results of Rosier and Zhang [23]. More
precisely, they assume that the spatial variable belongs to the rectangle

$\mathcal{R}=(0,l_{1})\times\cdot\cdot\cdot\times(0,l_{n})$

and investigate the control properties of the semi-linear Schrödinger equation

$i\partial_{t}u+\Delta u+\lambda|u|^{\alpha}u=ia(x)h(x,t),\text{ }\text{
}x\in\mathbb{T}^{n}\text{ }\text{ }t\in(0,T),$

where $\lambda\in\mathbb{R}$ and $\alpha\in 2\mathbb{N}^{\ast}$ by combining
new linear controllability results in the spaces $H^{s}(\mathcal{R})$ with
Bourgain analysis. In this case, the geometric control condition is not
required (see [25] for more details).

Finally, considering a $2d$-compact Riemann manifold $M$ without boundary,
Dehman et al. [7] studied internal control and stabilization of nonlinear
Schrödinger equations

$i\partial_{t}w+\Delta w-|w|^{2}w=f(x,t),\text{ }x\in M.$

They showed, in particular, that the system is semi-globally exactly
controllable and semi-globally exponentially stabilizable in the space
$H^{1}(M)$ assuming that both the geometric control condition and a unique
continuation condition are satisfied (see [7] for more details). The
interesting work [3] extended this one: The authors studied global
controllability and stabilization properties for the fractional Schrödinger
equation on $d-$dimensional compact Riemannian manifolds without boundary
$(M,g)$. They used microlocal analysis to show the propagation of regularity
which, together with the geometric control condition and a unique continuation
property, allowed them to prove global control results.

###  1.2. Main results

As discussed above, most of the available results at the moment are for the
classical Schrödinger equation (1.3) in different domains and considering the
control inputs acting on the whole boundary. However, in the critical case,
namely, system (1.1), the internal control problem remains open.

With the aim of presenting the first answer to the control problem stated at
the beginning of the section, let us consider the control system

(1.4) $\left\\{\begin{array}[]{lr}i\partial_{t}u+\Delta
u-|u|^{4}u=\varphi(x)h(x,t),\ \mbox{on }\mathbb{R}^{3}\times[0,+\infty),\\\
u(x,0)=u_{0}(x)\in\dot{H}^{1}(\mathbb{R}^{3}),\end{array}\right.$

where the function $\varphi\in C^{\infty}(\mathbb{R}^{3},[0,1])$ and satisfies

(1.5) $\varphi(x)=\left\\{\begin{array}[]{lr}0,\ if\ |x|\leq R,\\\ 1,\ if\
|x|\geq R+1,\end{array}\right.$

for some $R>0$ large enough. Our result below gives a first answer in this
direction.

######  Theorem 1.1 (Local exact controllability).

Let $T>0$ be given. There exists $\delta>0$ such that for any $u_{0}$ and
$\tilde{u}_{0}$ in $H^{1}(\mathbb{R}^{3})$ satisfying
$\|u_{0}\|_{H^{1}}\leq\delta$ and $\|\tilde{u}_{0}\|_{H^{1}}\leq\delta,$ one
can find $h(x,t)\in C(\mathbb{R};H^{1}(\mathbb{R}^{3}))$ such that system
(1.4) admits a solution $u\in C([0,T];{H}^{1}(\mathbb{R}^{3}))$ satisfying
$u(T)=\tilde{u}_{0}.$

In our arguments, Strichartz-type inequalities play a fundamental role giving
the well-posedness of system (1.4). In addition, as a first step, we have used
a Carleman estimate to obtain the following linear control result.

######  Theorem 1.2.

For every initial data $u_{0}\in H^{1}(\mathbb{R}^{3})$ and every $T>0$, there
exists a control $h(x,t)\in C(\mathbb{R};H^{1}(\mathbb{R}^{3}))$ with support
in $\mathbb{R}\times(\mathbb{R}^{3}\backslash B_{R}(0))$, $R>0$, such that the
unique solution of the linear system associated to (1.4) satisfies
$u(T,\cdot)=0$.

With Theorem 1.2 in hand, a perturbation argument ensures that we can get a
local exact controllability result for the critical Schrödinger equation
(1.4), so giving Theorem 1.1.

###  1.3. Structure of this work

We finish our introduction by giving an outline of this work. It is divided as
follows:

– In Section 2, we give auxiliary results that play an important role in
establishing our control result. Precisely, we present a review of the Cauchy
problem for the Schrödinger equation.

– In Section 3, we present the proof of our main results. Firstly, we show the
exact controllability around the null trajectories for the linear system
associated with (1.4), proving Theorem 1.2. Then, through a perturbation
argument, we show Theorem 1.1.

– Finally, we discuss some future perspectives in Section 4.

##  2\. A review of the Cauchy problem

###  2.1. Smoothing

For the sake of completeness, we discuss the smoothing properties of the
linear Schrödinger equation,

(2.1) $\begin{cases}i\partial_{t}u+\Delta
u=0,&x\in\mathbb{R}^{3},t\in\mathbb{R},\\\
u(x,0)=\psi(x),&x\in\mathbb{R}^{3},\end{cases}$

which will play an important role in establishing the exact controllability
for the defocusing critical nonlinear Schrödinger equation (1.4). To this end,
for $j\in\\{1,2,3\\}$, let $P_{j}$ be the differential operator on
$\mathbb{R}^{4}$ defined by

(2.2) $P_{j}v(t,x)=(x_{j}+2it\partial_{j})v(t,x).$

For a multi-index $\alpha$, define the differential operator $P_{\alpha}$ on
$\mathbb{R}^{4}$ by

$P_{\alpha}=\prod_{j=1}^{3}P_{j}^{\alpha_{j}}.$

Additionally, for $x\in\mathbb{R}^{3}$, consider

$x^{\alpha}=\prod_{j=1}^{3}x_{j}^{\alpha_{j}}.$

For any smooth function $u(t,x)$, one has

$P_{j}u(t,x)=2ite^{i\frac{|x|^{2}}{4t}}\frac{\partial}{\partial
x_{j}}\big{(}e^{-i\frac{|x|^{2}}{4t}}u(t,x)\big{)}.$

Indeed, note that

$\displaystyle 2ite^{i\frac{|x|^{2}}{4t}}\frac{\partial}{\partial
x_{j}}\big{(}e^{-i\frac{|x|^{2}}{4t}}u\big{)}$ $\displaystyle=$
$\displaystyle-2ite^{i\frac{|x|^{2}}{4t}}\frac{2ix_{j}}{4t}e^{-i\frac{|x|^{2}}{4t}}u(t,x)+2ite^{i\frac{|x|^{2}}{4t}}e^{-i\frac{|x|^{2}}{4t}}\frac{\partial}{\partial
x_{j}}u(t,x)$ $\displaystyle=$ $\displaystyle
x_{j}u(t,x)+2it\frac{\partial}{\partial x_{j}}u(t,x).$

Hence,

$P_{\alpha}u(t,x)=(2it)^{|\alpha|}e^{i\frac{|x|^{2}}{4t}}D^{\alpha}\big{(}e^{-i\frac{|x|^{2}}{4t}}u(t,x)\big{)}.$

On the other hand, we easily obtain

$[P_{j},i\partial_{t}+\Delta]=0.$

Thus, considering $u\in C(\mathbb{R},H^{1}(\mathbb{R}^{3}))$ to be any
solution of the linear Schrödinger equation (2.1), one has that $P_{j}u$ and
$P_{\alpha}u$ is also a solution.

Now, with the previous analysis in hand and taking into account the relation
(2.2), we present the next result, which gives a local smoothing property for
the linear Schrödinger equation (2.1).

######  Proposition 2.1.

Let $\alpha$ be a multi-index and $T>0$ be given. Let $\psi\in
H^{1}(\mathbb{R}^{3})$ be such that $x^{\alpha}\psi\in H^{1}(\mathbb{R}^{3})$.
Then, the corresponding solution $u$ of the IVP (2.1) satisfies

$P_{\alpha}u\in C(\mathbb{R};H^{1}(\mathbb{R}^{3}))$

and there exists a constant $C$ depending only on $T$ and $\alpha$ such that

$\|P_{\alpha}u\|_{H^{1}(\mathbb{R}^{3})}\leq
C\|x^{\alpha}\psi\|_{H^{1}(\mathbb{R}^{3})}$

holds for any $t\in[-T,T]$. In particular, if $\psi\in H^{1}(\mathbb{R}^{3})$
has compact support, then $u$ is infinitely smooth everywhere except at $t=0$.

###### Proof.

A standard density argument assures that it is sufficient to prove the result
for $\psi\in\mathcal{S}(\mathbb{R}^{3})$. To this end, assume, first, that
$|\alpha|=1$, so that $P_{\alpha}=P_{j}$ for some $j\in\\{1,2,3\\}$. Now, note
that

$\|u(t)\|_{H^{1}(\mathbb{R}^{3})}=\|\psi\|_{H^{1}(\mathbb{R}^{3})},$

for any $t\in[-T,T]$. Let $u^{j}(t,x)=P_{j}u(t,x)$. Applying the operator
$P_{j}$ to (2.1) yields

$\begin{cases}i\partial_{t}u^{j}+\Delta u^{j}=0,\\\
u^{j}(0,x)=x_{j}\psi,\end{cases}$

due to the fact that $P_{j}u(0,x)=x_{j}u(0,x)$. Since

$u^{j}(t)=e^{it\Delta}(x_{j}\psi),$

we get that

$\|u^{j}(t)\|_{H(\mathbb{R}^{3})}\leq C\|x_{j}\psi\|_{H^{1}(\mathbb{R}^{3})}.$

The general case $(|\alpha|>1)$ follows by induction. ∎

###  2.2. Existence

Here, we are interested in studying the $\dot{H}^{1}$ critical, defocusing,
Cauchy problem for the nonlinear Schrödinger equation (C-NLS)

(2.3) $\left\\{\begin{array}[]{lr}i\partial_{t}u+\Delta u-|u|^{4}u=g,\ \
(t,x)\in\mathbb{R}\times\mathbb{R}^{3},\\\
u(0)=u_{0}\in\dot{H}^{1}(\mathbb{R}^{3}),\end{array}\right.$

where $g(t,x)=g\in L^{\infty}_{loc}(\mathbb{R},H^{1}(\mathbb{R}^{3}))$. Let us
first present some definitions used throughout the paper.

######  Definition 1.

Let s $\in\mathbb{R}$. The homogeneous Sobolev space
$\dot{H}^{s}\left(\mathbb{R}^{d}\right)$ is the space of tempered
distributions $u$ over $\mathbb{R}^{d}$ which have Fourier transform belonging
to $L_{loc}^{1}\left(\mathbb{R}^{d}\right)$ and satisfying

$\|u\|_{\dot{H}^{s}}^{2}\stackrel{{\scriptstyle\text{ def
}}}{{=}}\int_{\mathbb{R}^{d}}|\xi|^{2s}|\widehat{u}(\xi)|^{2}d\xi<\infty.$

We note that the spaces $\dot{H}^{s}$ and $\dot{H}^{s^{\prime}}$ are not
comparable for inclusion.

######  Definition 2.

A pair $(q,r)$ is called $L^{2}$-admissible if $r\in[2,6)$ and q satisfies

$\frac{2}{q}+\frac{3}{r}=\frac{3}{2}.$

Such a pair is called $H^{1}$-admissible if $r\in[6,+\infty)$ and q satisfies

$\frac{2}{q}+\frac{3}{r}=\frac{1}{2}.$

With these definitions in hand, we present two results that are paramount to
prove that the Cauchy problem (2.3) is well-posed. The first one is Strichartz
estimates and the second one is a standard Sobolev embedding. These results
can be found in [6, 10].

######  Lemma 2.2.

Let $(q,r)$ be a $L^{2}$-admissible pair. We have

(2.4) $\|e^{it\Delta}h\|_{L^{q}_{t}L^{r}_{x}}\leq c\|h\|_{L^{2}},$ (2.5)
$\Bigg{\|}\int_{-\infty}^{+\infty}e^{i(t-\tau)\Delta}g\
d\tau\Bigg{\|}_{L^{q}_{t}L^{r}_{x}}+\Bigg{\|}\int_{0}^{t}e^{i(t-\tau)\Delta}g\
d\tau\Bigg{\|}_{L^{q}_{t}L^{r}_{x}}\leq
c\|g\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}_{x}},$

and

$\Bigg{\|}\int_{-\infty}^{+\infty}e^{it\Delta}g(\tau)\
d\tau\Bigg{\|}_{L^{2}_{x}}\leq C\|g\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}_{x}}.$

Additionally, we have

(2.6) $\Bigg{\|}\int_{-\infty}^{+\infty}e^{i(t-\tau)\Delta}g(\tau)\
d\tau\Bigg{\|}_{L^{q}_{t}L^{r}_{x}}\leq
C\|g\|_{L^{m^{\prime}}_{t}L^{n^{\prime}}_{x}}$

where $(q,r)$, $(m,n)$ are any $L^{2}$-admissible pair, wich is an
generalization of (2.5).

######  Lemma 2.3.

For $v\in C_{0}^{\infty}(\mathbb{R}\times\mathbb{R}^{3})$, we have

$\|v\|_{L^{10}_{t}L^{10}_{x}}\leq C\|\nabla
v\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}.$

For an interval $I$, define the norms $S(I),$ $W(I)$ and $Z(I)$ by

$\|u\|_{S(I)}=\|u\|_{L^{10}(I;L^{10}(\mathbb{R}^{3}))},\ \
\|u\|_{Z(I)}=\|u\|_{L^{10}(I;L^{\frac{30}{13}}(\mathbb{R}^{3}))}\ \mbox{and}\
\ \
\|u\|_{W(I)}=\|u\|_{L^{\frac{10}{3}}(I;L^{\frac{10}{3}}(\mathbb{R}^{3}))}.$

The following theorem gives us the solution to problem (2.3).

######  Theorem 2.4.

Let $u_{0}\in\dot{H}^{1}(\mathbb{R}^{3})$. If $\|u_{0}\|_{\dot{H}^{1}}$ is
small enough, then there exists an interval $I$ and a unique solution $u(t,x)$
of problem (2.3) in $I\times\mathbb{R}^{3}$, with $u\in
C(I;\dot{H}^{1}(\mathbb{R}^{3}))$, satisfying

$\|\nabla u\|_{W(I)}<\infty,\ \|u\|_{S(I)}<\infty\mbox{ and }\|\nabla
u\|_{Z(I)}<\infty.$

###### Proof.

We follow the ideas from [11]. Assume, without loss of generality, that
$t_{0}=0$. Observe that the Cauchy problem (2.3) is equivalent to the integral
equation (Duhamel’s formula)

$u(t)=e^{it\Delta}u_{0}-\int_{0}^{t}e^{i(t-\tau)\Delta}[|u|^{4}u+g]\ d\tau.$

Define

$|||u|||=\sup_{t\in I}\|u(t)\|_{L^{2}}+\sup_{t\in I}\|\nabla
u(t)\|_{L^{2}}+\|u\|_{S(I)}+\|\nabla u\|_{W(I)}+\|\nabla u\|_{Z(I)}.$

For $R>0$ to be conveniently chosen later on, consider the set

$B_{R}:=\Big{\\{}u(t,x)\text{ on }I\times\mathbb{R}^{3}:|||u|||\leq
R\Big{\\}}.$

We want to show that the operator $\Phi_{u_{0}}:B_{R}\longrightarrow B_{R}$
defined by

$\Phi_{u_{0}}(u)=e^{it\Delta}u_{0}-\int_{0}^{t}e^{i(t-\tau)\Delta}[|u|^{4}u+g]\
d\tau$

has a fixed point for $R$ small enough. To this end, first, observe that

$\displaystyle\|\Phi_{u_{0}}(u)\|_{L^{2}_{x}}$ $\displaystyle\leq$
$\displaystyle\|e^{it\Delta}u_{0}\|_{L^{2}_{x}}+\Big{\|}\int_{0}^{t}\nabla
e^{i(t-\tau)\Delta}[|u|^{4}u+g]\ d\tau\Big{\|}_{L^{2}_{x}}$
$\displaystyle\leq$
$\displaystyle\|u_{0}\|_{L^{2}}+C\||u|^{4}u\|_{L^{1}_{t}L^{2}_{x}}+\|g\|_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\leq$ $\displaystyle
C\|u_{0}\|_{\dot{H}^{1}}+C\|u\|_{S(I)}^{5}+C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}$

and

$\displaystyle\|\nabla\Phi_{u_{0}}(u)\|_{L^{2}_{x}}$ $\displaystyle\leq$
$\displaystyle\|\nabla
e^{it\Delta}u_{0}\|_{L^{2}_{x}}+\Big{\|}\int_{0}^{t}\nabla
e^{i(t-\tau)\Delta}[u^{5}+g]\ d\tau\Big{\|}_{L^{2}_{x}}$ $\displaystyle\leq$
$\displaystyle\|\nabla
u_{0}\|_{L^{2}}+C\|\nabla|u|^{4}u\|_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}+\|\nabla
g\|_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle
C\|u_{0}\|_{\dot{H}^{1}}+C\|u\|_{S(I)}^{4}\|\nabla
u\|_{W(I)}+C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}.$

Choosing the length of $I$ small enough such that
$C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}\leq C\|u_{0}\|_{\dot{H}^{1}}$, we have

$\displaystyle\|\Phi_{u_{0}}(u)\|_{L^{2}_{x}}+\|\nabla\Phi_{u_{0}}(u)\|_{L^{2}_{x}}$
$\displaystyle\leq$ $\displaystyle 2C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}.$

Secondly, notice that

$\displaystyle\|\nabla\Phi_{u_{0}}(u)\|_{W(I)}$ $\displaystyle\leq$
$\displaystyle\|\nabla e^{it\Delta}u_{0}\|_{W(I)}+\Big{\|}\int_{0}^{t}\nabla
e^{i(t-\tau)\Delta}[u^{5}+g]\ d\tau\Big{\|}_{W(I)}$ $\displaystyle\leq$
$\displaystyle\|\nabla
u_{0}\|_{L^{2}}+C\|\nabla|u|^{4}u\|_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}+\|\nabla
g\|_{L^{1}_{t}L^{2}_{x}}.$

Due to Hölder’s inequality with $p=\frac{7}{4}$ and $q=\frac{7}{3}$, we get

$\|\nabla|u|^{4}u\|_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}\leq
C\|u\|^{4}_{S(I)}\|\nabla u\|_{W(I)}.$

Thus

$\displaystyle\|\nabla\Phi_{u_{0}}(u)\|_{W(I)}$ $\displaystyle\leq$
$\displaystyle C\Big{(}\|\nabla u_{0}\|_{L^{2}}+\|u\|^{4}_{S(I)}\|\nabla
u\|_{W(I)}+\|\nabla g\|_{L^{1}_{t}L^{2}_{x}}\Big{)}$ $\displaystyle\leq$
$\displaystyle
C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}+C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}.$

Choosing the length of $I$ small enough such that
$C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}\leq C\|u_{0}\|_{\dot{H}^{1}}$, one gets

$\displaystyle\|\nabla\Phi_{u_{0}}(u)\|_{W(I)}$ $\displaystyle\leq$
$\displaystyle 2C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}.$

On other hand, using estimate (2.4) with $q=10$ and $r=30/13$, due to the
inequality (2.6) with the same $q,r$ and $m^{\prime}=2$ and
$n^{\prime}=\frac{6}{5}$, Hölder’s inequality gives

$\displaystyle\|\nabla\Phi_{u_{0}}(u)\|_{Z(I)}$ $\displaystyle\leq$
$\displaystyle\|\nabla e^{it\Delta}u_{0}\|_{Z(I)}+\Big{\|}\int_{0}^{t}\nabla
e^{i(t-\tau)\Delta}[u^{5}+g]\ d\tau\Big{\|}_{Z(I)}$ $\displaystyle\leq$
$\displaystyle\|\nabla
u_{0}\|_{L^{2}}+C\|\nabla|u|^{4}u\|_{L^{2}_{t}L^{\frac{6}{5}}_{x}}+C\|\nabla
g\|_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle\|\nabla
u_{0}\|_{L^{2}}+C\|\nabla u\|_{Z(I)}\|u\|^{4}_{S(I)}+C\|\nabla
g\|_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle
C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}+C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}$
$\displaystyle\leq$ $\displaystyle 2C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}$

since $C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}\leq C\|u_{0}\|_{\dot{H}^{1}}$
Finally,

$\displaystyle\|\Phi_{u_{0}}(u)\|_{S(I)}$ $\displaystyle\leq$
$\displaystyle\|\nabla\Phi_{u_{0}}(u)\|_{Z(I)}$ $\displaystyle\leq$
$\displaystyle\|\nabla e^{it\Delta}u_{0}\|_{Z(I)}+\Big{\|}\int_{0}^{t}\nabla
e^{i(t-\tau)\Delta}[u^{5}+g]\ d\tau\Big{\|}_{Z(I)}$ $\displaystyle\leq$
$\displaystyle\|\nabla u_{0}\|_{L^{2}}+C\|\nabla
u\|_{Z(I)}\|u\|^{4}_{S(I)}+C\|\nabla g\|_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\leq$ $\displaystyle C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}+C\|\nabla
g\|_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle
C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}+C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}$
$\displaystyle\leq$ $\displaystyle 2C\|u_{0}\|_{\dot{H}^{1}}+CR^{5},$

since $C_{I}\|g\|_{L^{\infty}_{t}H^{1}_{x}}\leq C\|u_{0}\|_{\dot{H}^{1}}$.
Summing up, we get

$\displaystyle|||\Phi_{u_{0}}(u)|||$ $\displaystyle\leq$ $\displaystyle
2C\|u_{0}\|_{\dot{H}^{1}}+CR^{5}\leq R$

as long as $\|u_{0}\|_{\dot{H}^{1}}\leq\frac{R}{2C}-\frac{R^{5}}{2}$, where
$R<(1/C)^{1/4}$.

Next, denoting $f(u)=|u|^{4}u$, we get

$\displaystyle\|\Phi_{u_{0}}(u)-\Phi_{u_{0}}(v)\|_{L^{2}_{x}}$
$\displaystyle\leq$ $\displaystyle C\|f(u)-f(v)\|_{L^{1}_{t}L^{2}_{x}}$
$\displaystyle\leq$ $\displaystyle
C\|u-v\|_{S(I)}\big{(}\|u\|^{4}_{S(I)}+\|v\|^{4}_{S(I)}\big{)},$
$\displaystyle\|\nabla\Phi_{u_{0}}(u)-\nabla\Phi_{u_{0}}(v)\|_{L^{2}_{x}}$
$\displaystyle\leq$ $\displaystyle C\|\nabla f(u)-\nabla
f(v)\|_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}$ $\displaystyle\leq$
$\displaystyle C\Biggl{(}\Big{\|}|u|^{4}|\nabla u-\nabla
v|\Big{\|}_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}+\Big{\|}|u-v||u|^{3}|\nabla
v|\Big{\|}_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}$ $\displaystyle\mbox{
}+\Big{\|}|u-v||v|^{3}|\nabla
v|\Big{\|}_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}\Biggr{)}$
$\displaystyle\leq$ $\displaystyle C\Biggl{(}\|u\|_{S(I)}^{4}\|\nabla u-\nabla
v\|_{W(I)}+\|u-v\|_{S(I)}\|\nabla v\|_{W(I)}\|u\|^{3}_{S(I)}+$
$\displaystyle\mbox{ }+\|u-v\|_{S(I)}\|\nabla
v\|_{W(I)}\|v\|^{3}_{S(I)}\Biggr{)}$ $\displaystyle\leq$ $\displaystyle
CR^{4}\|u-v\|_{S(I)}+CR^{4}\|\nabla u-\nabla v\|_{W(I)},$

and

$\displaystyle\|\nabla\Phi_{u_{0}}(u)-\nabla\Phi_{u_{0}}(v)\|_{W(I)}$
$\displaystyle\leq$ $\displaystyle C\|\nabla f(u)-\nabla
f(v)\|_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}$ $\displaystyle\leq$
$\displaystyle C\Biggl{(}\Big{\|}|u|^{4}|\nabla u-\nabla
v|\Big{\|}_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}+\Big{\|}|u-v||u|^{3}|\nabla
v|\Big{\|}_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}$ $\displaystyle\mbox{
}+\Big{\|}|u-v||v|^{3}|\nabla
v|\Big{\|}_{L^{\frac{10}{7}}_{t}L^{\frac{10}{7}}_{x}}\Biggr{)}$
$\displaystyle\leq$ $\displaystyle C\Biggl{(}\|u\|_{S(I)}^{4}\|\nabla u-\nabla
v\|_{W(I)}+\|u-v\|_{S(I)}\|\nabla v\|_{W(I)}\|u\|^{3}_{S(I)}+$
$\displaystyle\mbox{ }+\|u-v\|_{S(I)}\|\nabla
v\|_{W(I)}\|v\|^{3}_{S(I)}\Biggr{)}$ $\displaystyle\leq$ $\displaystyle
CR^{4}\|u-v\|_{S(I)}+CR^{4}\|\nabla u-\nabla v\|_{W(I)}.$

Following the same reasoning as before,

$\displaystyle\|\nabla\Phi_{u_{0}}(u)-\nabla\Phi_{u_{0}}(v)\|_{Z(I)}$
$\displaystyle\leq$ $\displaystyle C\Biggl{(}\|u\|_{S(I)}^{4}\|\nabla u-\nabla
v\|_{Z(I)}+\|u-v\|_{S(I)}\|\nabla v\|_{Z(I)}\|u\|^{3}_{S(I)}+$
$\displaystyle\mbox{ }+\|u-v\|_{S(I)}\|\nabla
v\|_{Z(I)}\|v\|^{3}_{S(I)}\Biggr{)}$ $\displaystyle\leq$ $\displaystyle
CR^{4}\|\nabla u-\nabla v\|_{Z(I)}+CR^{4}\|u-v\|_{S(I)}.$

Moreover, by Sobolev’s embedding,

$\|\Phi_{u_{0}}(u)-\Phi_{u_{0}}(v)\|_{S(I)}\leq\|\nabla\Phi_{u_{0}}(u)-\nabla\Phi_{u_{0}}(v)\|_{Z(I)}\leq
CR^{4}\|\nabla u-\nabla v\|_{Z(I)}+CR^{4}\|u-v\|_{S(I)}.$

Summing up yields

$\displaystyle|||\Phi_{u_{0}}(u)-\Phi_{u_{0}}(v)|||$ $\displaystyle\leq$
$\displaystyle CR^{4}\|\nabla u-\nabla
v\|_{Z(I)}+CR^{4}\|u-v\|_{S(I)}+CR^{4}\|\nabla u-\nabla v\|_{W(I)}$
$\displaystyle\leq$ $\displaystyle CR^{4}\|\nabla u-\nabla
v\|_{Z(I)}+CR^{4}\|u-v\|_{S(I)}+CR^{4}\|\nabla u-\nabla v\|_{W(I)}$
$\displaystyle\mbox{ }+CR^{4}\sup_{t\in I}\|\nabla u(t)-\nabla
v(t)\|_{L^{2}_{x}}+CR^{4}\sup_{t\in I}\|u(t)-v(t)\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle CR^{4}|||u-v|||$

Thus, if $R>0$ is such that $CR^{4}<1$, then $\Phi_{u_{0}}$ is a contraction
in $B_{R}$ and, therefore, has a unique fixed point, i.e., problem (2.3) has a
local solution defined on a maximal interval $[0,T]$. ∎

######  Remark 1.

Observe that it is possible to use the energy estimates to get global
existence, that is, the solution $u=u(x,t)$ of problem (2.3) is globally well-
defined in time. To verify this, first consider the energy defined by

$E(t)=\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}+\frac{1}{6}\int_{\mathbb{R}^{3}}|u|^{6},$

which is conserved if $g=0$. Multiplying equation (2.3) by
$\partial_{t}\bar{u}$ and integrating, we have

$\begin{split}E(t)\leq&E(0)-\mathrm{Re}\int_{0}^{t}\int_{\mathbb{R}^{3}}g\partial_{t}\overline{u}\
dxdt\\\ \leq&E(0)-\mathrm{Re}\int_{0}^{t}\int_{\mathbb{R}^{3}}g(i\Delta
u-i|u|^{4}u-ig)\ dxdt\\\ \leq&E(0)+C\int_{0}^{t}\|\nabla
g(\tau)\|_{L^{2}}\|\nabla u(\tau)\|_{L^{2}}\ d\tau\\\
&+C\int_{0}^{t}\|g(\tau)\|_{L^{6}}\|u(\tau)^{5}\|_{L^{\frac{6}{5}}}\
d\tau+\int_{0}^{t}\|g(\tau)\|^{2}_{L^{2}}\ d\tau\\\
\leq&E(0)+C\int_{0}^{t}\|\nabla g(\tau)\|_{L^{2}}\sqrt{E(\tau)}\
d\tau+C\int_{0}^{t}\|g(\tau)\|_{L^{6}}(E(\tau))^{\frac{5}{6}}\
d\tau+\|g\|^{2}_{L^{2}([0,T]\times\mathbb{R}^{3})}\\\
\leq&E(0)+C\int_{0}^{t}\|g(\tau)\|_{H^{1}}\sqrt{E(\tau)}\
d\tau+C\int_{0}^{t}\|g(\tau)\|_{H^{1}}(E(\tau))^{\frac{5}{6}}\
d\tau+\|g\|^{2}_{L^{2}([0,T]\times\mathbb{R}^{3})}.\end{split}$

Choosing $\tau$ such that $E(\tau)>k$, we have

$\begin{split}E(t)\leq&E(0)+C\int_{0}^{t}\|g(\tau)\|_{H^{1}}(E(\tau))^{-\frac{1}{3}}(E(\tau))^{\frac{5}{6}}\
d\tau\\\ &+C\int_{0}^{t}\|g(\tau)\|_{H^{1}}(E(\tau))^{\frac{5}{6}}\
d\tau+\|g\|^{2}_{L^{2}([0,T]\times\mathbb{R}^{3})}\\\
\leq&E(0)+C\int_{0}^{t}\|g(\tau)\|_{H^{1}}(E(\tau))^{\frac{5}{6}}\
d\tau+\|g\|^{2}_{L^{2}([0,T]\times\mathbb{R}^{3})}\\\
\leq&E(0)+C\int_{0}^{t}\|g(\tau)\|_{H^{1}}\big{(}1+(E(\tau))^{\frac{5}{6}}\big{)}\
d\tau+\|g\|^{2}_{L^{2}([0,T]\times\mathbb{R}^{3})}.\end{split}$

Therefore,

$\displaystyle\max_{0\leq t\leq T}E(\tau)$ $\displaystyle\leq$ $\displaystyle
E(0)+C\Big{(}1+\max_{0\leq\tau\leq
t}(E(\tau))^{\frac{5}{6}}\Big{)}\|g\|_{L^{1}([0,T];H^{1}(\mathbb{R}^{3}))}+\|g\|^{2}_{L^{2}([0,T]\times\mathbb{R}^{3})}.$

So, finally, it follows that

$E(t)\leq
C\Big{(}1+E(0)^{6}+\|g\|^{10}_{L^{2}([0,T]\times\mathbb{R}^{3})}+\|g\|^{6}_{L^{1}([0,T];H^{1}(\mathbb{R}^{3}))}\Big{)}.$

So, if $g\in L^{\infty}_{loc}(\mathbb{R},H^{1}(\mathbb{R}^{3}))$, then the
energy is bounded, which implies global existence in
$\dot{H}^{1}(\mathbb{R}^{3})$. Now, for the $L^{2}$–energy (or mass), define
the following quantity

$\overline{E}(t)=\frac{1}{2}\|u(t)\|^{2}_{L^{2}}.$

Multiplying equation (2.3) by $\overline{u}$, taking the imaginary part and
integrating by parts yields

$\displaystyle\frac{1}{2}\|u(t)\|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\|u(0)\|^{2}_{L^{2}}+\mathrm{Im}\int_{0}^{t}\int_{\mathbb{R}^{3}}g\cdot\overline{u}\
dxdt$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\|u(0)\|^{2}_{L^{2}}+\int_{0}^{t}\int_{\mathbb{R}^{3}}|g\cdot\overline{u}|\
dxdt$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\|u(0)\|^{2}_{L^{2}}+\int_{0}^{t}\|g(\tau)\|_{L^{2}}\|u(\tau)\|_{L^{2}}dt.$

So

$\displaystyle\overline{E}(t)$ $\displaystyle\leq$
$\displaystyle\overline{E}(0)+\int_{0}^{t}\|g(\tau)\|_{L^{2}}\|u(\tau)\|_{L^{2}}dt$
$\displaystyle\leq$
$\displaystyle\overline{E}(0)+C\int_{0}^{t}\|g(\tau)\|_{L^{2}}\sqrt{\overline{E}(\tau)}dt.$

This implies that the $L^{2}$–energy is bounded if $g\in
L^{\infty}_{loc}(\mathbb{R},H^{1}(\mathbb{R}^{3}))$ and yields global
existence in ${L}^{2}(\mathbb{R}^{3})$.

##  3\. Controllability for the critical nonlinear Schrödinger equation

In this section, we prove Theorem 1.1 using a duality strategy which reduces
the controllability problem (1.4) to prove an observability inequality, the
so-called “Hilbert Uniqueness Method” [17], for the solutions of the linear
system

(3.1) $\left\\{\begin{array}[]{lr}i\partial_{t}u+\Delta u=\varphi(x)h(x,t),\ \
x\in\mathbb{R}^{3},\ t\in(0,T),\\\ u(0)=u_{0},\end{array}\right.$

where $\varphi=\varphi(x)$ is defined by (1.5).

###  3.1. Linear Schrödinger equation: Exact controllability

We prove the following:

######  Theorem 3.1.

For every initial data $u_{0}\in H^{1}(\mathbb{R}^{3})$ and every $T>0$, there
exist $R>0$ and a control $h(x,t)\in C(\mathbb{R},H^{1}(\mathbb{R}^{3}))$ with
support in $\mathbb{R}\times(\mathbb{R}^{3}\backslash B_{R}(0))$ such that the
unique solution of system (3.1) satisfies $u(T,\cdot)=0$.

The exact controllability of system (3.1) follows from the observability
inequality, namely,

(3.2) $\|v_{0}\|^{2}_{H^{-1}}\leq C\int_{0}^{T}\|\varphi v(t)\|^{2}_{H^{-1}}\
dt,$

where $v(x,t)$ is a solution to the adjoint system associated to (3.1)

(3.3) $\left\\{\begin{array}[]{lr}i\partial_{t}v+\Delta v=0\ in\
\mathbb{R}\times\mathbb{R}^{3},\\\ v(0)=v_{0}\in
H^{-1}(\mathbb{R}^{3}).\end{array}\right.$

The observability inequality (3.2) is given by the following result.

######  Proposition 3.2.

Let $\varphi$ be a $C^{\infty}$ real function on $\mathbb{R}^{3}$ as in (1.5).
Then, for every $T>0$, there exists a constant $C=C(T)>0$ such that inequality
(3.2) holds for every solution $v(t,x)$ of system (3.3).

###### Proof.

We will split the proof into several steps.

First step: $H^{1}$–observability.

######  Lemma 3.3.

Consider the system

(3.4) $\left\\{\begin{array}[]{lr}i\partial_{t}w+\Delta w=0,\ \
x\in\mathbb{R}^{3},\ t\in(0,T),\\\ w(0)=w_{0}\in
H^{1}(\mathbb{R}^{3}).\end{array}\right.$

There exists a constant $C>0$ such that for each $w_{0}\in
H^{1}(\mathbb{R}^{3})$, the solution $w(t)$ of (3.4) satisfies

(3.5) $\|w_{0}\|^{2}_{H^{1}(\mathbb{R}^{3})}\leq C\int_{0}^{T}\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt.$

###### Proof.

Let $q\in C_{0}^{\infty}(\mathbb{R}^{3})$ such that

$q(x)=\left\\{\begin{array}[]{lr}x,\ if\ |x|\leq R+2,\\\ 0,\ if\ |x|\geq
R+3.\end{array}\right.$

Multiplying the equation in (3.4) by
$q\cdot\nabla\overline{w}+\frac{1}{2}\overline{w}(div_{x}q)$, taking the real
part and integrating by parts gives

(3.6)
$\begin{split}\frac{1}{2}\mathrm{Im}\int_{\mathbb{R}^{3}}(wq\cdot\nabla\overline{w})\
dx\Bigg{\arrowvert}_{0}^{T}&+\frac{1}{2}\mathrm{Re}\int_{0}^{T}\int_{\mathbb{R}^{3}}w\nabla(div_{x}q)\cdot\nabla\overline{w}\
dxdt\\\
&+\mathrm{Re}\int_{0}^{T}\int_{\mathbb{R}^{3}}\sum_{j,k=1}^{3}\Big{(}\frac{\partial
q_{k}}{\partial x_{j}}\frac{\partial\overline{w}}{\partial
x_{k}}\frac{\partial w}{\partial x_{j}}\Big{)}\ dxdt=0,\end{split}$

where we have used the fact that the function $q(x)$ has compact support.
Notice that (3.4) is forward and backward well-posed in
$H^{1}(\mathbb{R}^{3})$, so for any $t_{0}\in[0,T],$ there exists a constant
$c>0$ such that

(3.7) $\|w(t_{0})\|^{2}_{H^{1}(\mathbb{R}^{3})}\leq
c\int_{0}^{T}\|w(t_{0})\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt=c\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt.$

Thus, it follows from (3.6) and (3.7) that

$\displaystyle\int_{0}^{T}\int_{B_{R+2}(0)}|\nabla w|^{2}\ dxdt$
$\displaystyle\leq$ $\displaystyle
C_{\varepsilon}\Bigg{(}\int_{0}^{T}\int_{B_{R+3}(0)\backslash
B_{R+2}(0)}|\nabla w|^{2}\ dxdt+\int_{0}^{T}\|w(t)\|_{L^{2}(\mathbb{R}^{3})}\
dt\Bigg{)}$ $\displaystyle\mbox{
}+\varepsilon\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt,$

for any $\varepsilon>0$ and some constant $C_{\varepsilon}>0$. We also have

$\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\leq C\Bigg{(}\int_{B_{R+2}(0)}|\nabla
w|^{2}\ dx+\|\varphi w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\Bigg{)}.$

Indeed, observe that

$\displaystyle\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}$ $\displaystyle=$
$\displaystyle\|w(t)\|^{2}_{H^{1}(B_{R+1}(0))}+\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3}\backslash
B_{R+1}(0))}$ $\displaystyle=$
$\displaystyle\|w(t)\|^{2}_{H^{1}(B_{R+1}(0))}+\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3}\backslash B_{R+1}(0))}$ $\displaystyle\leq$
$\displaystyle\|w(t)\|^{2}_{H^{1}(B_{R+2}(0))}+\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}$ $\displaystyle\leq$ $\displaystyle\|\nabla
w(t)\|^{2}_{L^{2}(B_{R+2}(0))}+\|\varphi w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})},$

showing the claim. Moreover, if $\varepsilon$ is small enough, we obtain

(3.8) $\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt\leq
C\Bigg{(}\int_{0}^{T}\|\varphi w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt+\int_{0}^{T}\|w(t)\|^{2}_{L^{2}(\mathbb{R}^{3})}\Bigg{)}.$

Note that

$\displaystyle\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt$
$\displaystyle\leq$ $\displaystyle C\Bigg{(}\int_{0}^{T}\|\nabla
w(t)\|^{2}_{L^{2}(B_{R+2}(0))}\ dt+\int_{0}^{T}\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt\Bigg{)}$ $\displaystyle\leq$
$\displaystyle C_{\varepsilon}\Bigg{(}\int_{0}^{T}\int_{B_{R+3}(0)\backslash
B_{R+2}(0)}|\nabla w|^{2}\ dxdt+\int_{0}^{T}\|w(t)\|_{L^{2}(\mathbb{R}^{3})}\
dt$ $\displaystyle\mbox{ }+\int_{0}^{T}\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt\Bigg{)}+\varepsilon\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt$
$\displaystyle\leq$ $\displaystyle
C_{\varepsilon}\Bigg{(}\int_{0}^{T}\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}dt+\int_{0}^{T}\|w(t)\|_{L^{2}(\mathbb{R}^{3})}\
dt$ $\displaystyle\mbox{ }+\int_{0}^{T}\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt\Bigg{)}+\varepsilon\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt$
$\displaystyle\leq$ $\displaystyle
C_{\varepsilon}\Bigg{(}\int_{0}^{T}\|\varphi
w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}dt+\int_{0}^{T}\|w(t)\|_{L^{2}(\mathbb{R}^{3})}\
dt\Bigg{)}$ $\displaystyle\mbox{
}+\varepsilon\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt.$

So, it remains to show

(3.9) $\int_{0}^{T}\|w(t)\|_{L^{2}(\mathbb{R}^{3})}\ dt\leq
c\int_{0}^{T}\|\varphi w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}dt$

to achieve Lemma 3.3. To this end, let us argue by contradiction, that is,
suppose that (3.9) does not hold. If this is the case, there exists a sequence
$\\{w_{n,0}\\}$ in $H^{1}(\mathbb{R}^{3})$ such that the corresponding
sequence of solutions $\\{w_{n}\\}$ of (3.4) satisfies

(3.10) $1=\int_{0}^{T}\|w_{n}(t)\|_{L^{2}(\mathbb{R}^{3})}\ dt\geq
n\int_{0}^{T}\|\varphi w_{n}(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}dt,\ \ n=1,2,...$

Due to inequalities (3.8) and (3.10), we get

$\int_{0}^{T}\|w_{n}(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt\leq
C\Bigg{(}\int_{0}^{T}\|\varphi w_{n}(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt+\int_{0}^{T}\|w_{n}(t)\|^{2}_{L^{2}(\mathbb{R}^{3})}\Bigg{)}\leq C,$

and so the sequence $\\{w_{n}\\}$ is bounded in
$L^{2}\big{(}(0,T);H^{1}(\mathbb{R}^{3})\big{)}$. Hence, by inequality (3.7),
the sequence $\\{w_{n}(0)=w_{n,0}\\}_{n\in\mathbb{N}}$ is bounded in
$H^{1}(\mathbb{R}^{3})$. Extracting a subsequence and still denoting it by
$\\{w_{n,0}\\}$, we may assume that

$w_{n,0}\rightharpoonup w_{0}\mbox{ weakly in }H^{1}(\mathbb{R}^{3})$

and

$w_{n}\rightharpoonup w\mbox{ weakly in
}L^{2}\big{(}(0,T);H^{1}(\mathbb{R}^{3})\big{)},$

where $w\in C\big{(}[0,T];H^{1}(\mathbb{R}^{3})\big{)}$ is a solution of
(3.4). By inequality (3.10), $\varphi w_{n}\rightarrow 0$ in
$L^{2}\big{(}(0,T);H^{1}(\mathbb{R}^{3})\big{)}$ strongly. Since $\varphi
w_{n}\rightharpoonup 0$ in $L^{2}\big{(}(0,T);H^{1}(\mathbb{R}^{3})\big{)}$
weakly, we conclude that $\varphi w\equiv 0$ on $(0,T)\times\mathbb{R}^{3}$.
Therefore,

$w\equiv 0,\ \ |x|>R+1,\ t\in(0,T).$

According to Proposition 2.1, one has $w\in
C^{\infty}(\mathbb{R}^{3}\times(0,T))$. Now, we are in a position to use the
unique continuation property for the Schrödinger equation showed in [18] to
conclude that

$w\equiv 0\mbox{ on }\mathbb{R}^{3}\times(0,T).$

Since $\varphi w_{n}\rightarrow 0$ strongly in
$L^{2}\big{(}(0,T);H^{1}(\mathbb{R}^{3})\big{)}$, we get

(3.11) $w_{n}\rightarrow 0\mbox{ strongly in
}L^{2}\big{(}(0,T);H^{1}(\mathbb{R}^{3}\backslash B_{R+1}(0))\big{)}.$

On the other hand, taking into account (3.4) and (3.8), we have

$\displaystyle\int_{0}^{T}\|w_{n}(t)\|^{2}_{H^{1}(B_{R+1}(0))}\ dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\|w_{n}(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\ dt$
$\displaystyle\leq$ $\displaystyle C\Big{(}\int_{0}^{T}\|\varphi
w_{n}(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt+\int_{0}^{T}\|w_{n}(t)\|^{2}_{L^{2}(\mathbb{R}^{3})}\ dt\Big{)},$
$\displaystyle\int_{0}^{T}\|\partial_{t}w_{n}(t)\|^{2}_{H^{-1}(B_{R+1}(0))}\
dt$ $\displaystyle=$ $\displaystyle\int_{0}^{T}\|-\Delta
w_{n}(t)\|^{2}_{H^{-1}(B_{R+1}(0))}\ dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\|(1-\Delta)w_{n}(t)-w_{n}(t)\|^{2}_{H^{-1}(B_{R+1}(0))}\
dt$ $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\|(1-\Delta)w_{n}(t)\|^{2}_{H^{-1}(B_{R+1}(0))}\ dt$
$\displaystyle\mbox{
}+2\int_{0}^{T}\|(1-\Delta)w_{n}(t)\|_{H^{-1}(B_{R+1}(0))}\|w_{n}(t)\|_{H^{-1}(B_{R+1}(0))}\
dt$ $\displaystyle\mbox{ }+\int_{0}^{T}\|w_{n}(t)\|^{2}_{H^{-1}(B_{R+1}(0))}\
dt$ $\displaystyle\leq$ $\displaystyle
C\int_{0}^{T}\|w_{n}(t)\|^{2}_{H^{1}(B_{R+1}(0))}$

and

$\begin{split}\int_{0}^{T}\|\nabla w_{n}(t)\|^{2}_{H^{-1}(B_{R+1}(0))}\
dt\leq&C\int_{0}^{T}\|\nabla w_{n}(t)\|^{2}_{L^{2}(B_{R+1}(0))}\ dt\\\
\leq&C\int_{0}^{T}\|\nabla w_{n}(t)\|^{2}_{L^{2}(\mathbb{R}^{3}))}\ dt\\\
\leq&C\int_{0}^{T}\|w_{n}(t)\|^{2}_{H^{1}(\mathbb{R}^{3}))}\ dt.\end{split}$

Therefore,

$w_{n}\mbox{ is bounded in }L^{2}\big{(}(0,T);H^{1}(B_{R+1}(0))\big{)}\cap
H^{1}\big{(}(0,T);H^{-1}(B_{R+1}(0))\big{)}.$

Due to Aubin’s lemma (see [26]) and the convergence (3.11), we conclude that
for a subsequence, still denoted by $\\{w_{n}\\}$,

$w_{n}\rightarrow w=0\mbox{ strongly in
}L^{2}\big{(}(0,T);L^{2}(\mathbb{R}^{3})\big{)}$

which contradicts (3.10). So, the estimate (3.5) follows from (3.7), (3.8) and
(3.9) as follows

$\begin{split}\|w(0)\|^{2}_{H^{1}(\mathbb{R}^{3})}\leq&C\int_{0}^{T}\|w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt\\\ \leq&C\Big{(}\int_{0}^{T}\|\varphi w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt+\int_{0}^{T}\|w(t)\|^{2}_{L^{2}(\mathbb{R}^{3})}\ dt\Big{)}\\\
\leq&C\int_{0}^{T}\|\varphi w(t)\|^{2}_{H^{1}(\mathbb{R}^{3})}\
dt,\end{split}$

showing the lemma. ∎

Second step: Weak observability inequality.

We prove a bound which is weaker than the observability inequality (3.2).

######  Lemma 3.4.

Let $v$ be the solution of system (3.3) with $v_{0}\in
H^{-1}(\mathbb{R}^{3})$. Then,

(3.12) $\|v_{0}\|^{2}_{H^{-1}}\leq C\Biggr{(}\int_{0}^{T}\|\varphi
v(t)\|^{2}_{H^{-1}}\ dt+\|(1-\varphi(x/2))v_{0}\|^{2}_{H^{-2}}\Biggr{)}.$

###### Proof.

Again, let us argue by contradiction. If inequality (3.12) is not verified,
there exists a sequence $\\{v_{n}\\}$ of solutions to problem (3.3) in
$C([0,T];H^{-1}(\mathbb{R}^{3}))$ such that

(3.13) $1=\|v_{n}(0)\|^{2}_{H^{-1}}\geq n\Biggr{(}\int_{0}^{T}\|\varphi
v_{n}(t)\|^{2}_{H^{-1}}\
dt+\|(1-\varphi(x/2))v_{n}(0)\|^{2}_{H^{-2}}\Biggr{)}.$

Up to a subsequence, we may assume that

$v_{n}\rightharpoonup v\ \mbox{ in }\
L^{\infty}((0,T);H^{-1}(\mathbb{R}^{3}))\mbox{ weak*}$

and

(3.14) $v_{n}(0)\rightharpoonup v(0)\mbox{ in }H^{-1}(\mathbb{R}^{3})\mbox{
weak, }$

where $v\in C([0,T];H^{-1}(\mathbb{R}^{3}))$ is a solution of problem (3.3).
By inequality (3.13),

$\varphi v_{n}\rightarrow 0\text{ (strongly) in
}L^{2}((0,T);H^{-1}(\mathbb{R}^{3})).$

Since

$\varphi v_{n}\rightharpoonup\varphi v\text{ in
$L^{\infty}((0,T);H^{-1}(\mathbb{R}^{3}))$ weak*,}$

we conclude that $\varphi v\equiv 0$. Therefore, $v(t,x)=0$ for $|x|>R+1$ and
$t\in(0,T)$. So, using the unique continuation property as in Step 1, we get
that $v\equiv 0$. In particular, $v(0)=0$.

Now, we claim that

(3.15) $\|\varphi(x/2)v_{n}(0)\|^{2}_{H^{-2}}\leq C\int_{0}^{T}\|\varphi
v_{n}(t)\|^{2}_{H^{-1}}\ dt.$

To prove (3.15), introduce the function
$\tilde{v}_{n}(x,t)=\varphi(x/2)v_{n}(x,t)$ which satisfies

$i\partial_{t}\tilde{v}_{n}+\Delta\tilde{v}_{n}=f_{n}$

where $f_{n}=[\Delta\varphi(x/2)]v_{n}+2\nabla\varphi(x/2)\nabla v_{n}$. Then,
the fact that $\operatorname{{supp}}[\varphi(x/2)]\subset\\{\varphi=1\\}$
yields

$\begin{split}\|\tilde{v}_{n}(0)\|^{2}_{H^{-2}(\mathbb{R}^{3})}\leq&c\Biggr{(}\int_{0}^{T}\|\tilde{v}_{n}(t)\|^{2}_{H^{-2}(\mathbb{R}^{3})}\
dt+\int_{0}^{T}\|f_{n}(t)\|^{2}_{H^{-2}(\mathbb{R}^{3})}\ dt\Biggr{)}\\\
\leq&c\int_{0}^{T}\|\varphi
v_{n}(t)\|^{2}_{H^{-1}(\mathbb{R}^{3})},\end{split}$

giving (3.15). Now, using (3.13), one has

$\displaystyle\|v_{n}(0)\|^{2}_{H^{-2}}$ $\displaystyle\leq$ $\displaystyle
2\Big{(}\|\varphi(x/2)v_{n}(0)\|^{2}_{H^{-2}}+\|(1-\varphi(x/2))v_{n}(0)\|^{2}_{H^{-2}}\Big{)}$
$\displaystyle\leq$ $\displaystyle c\int_{0}^{T}\|\varphi
v_{n}(t)\|^{2}_{H^{-1}}\
dt+2\|(1-\varphi(x/2))v_{n}(0)\|^{2}_{H^{-2}}\rightarrow 0,$

that is,

(3.16) $v_{n}(0)\rightarrow 0\mbox{ strongly in }H^{-2}(\mathbb{R}^{3}).$

Let $w_{n}=(1-\Delta)^{-1}v_{n}$. Then $w_{n}\in
C([0,T];H^{1}(\mathbb{R}^{3}))$ is a solution of the equation (3.3). By the
convergences (3.14) and (3.16), we can ensure that

$w_{n}(0)\rightharpoonup 0\mbox{ in }H^{1}(\mathbb{R}^{3})\mbox{ weakly }$

and

(3.17) $w_{n}\rightarrow 0\mbox{ in }C([0,T];L^{2}(\mathbb{R}^{3}))\mbox{
strongly. }$

Now, split $\varphi w_{n}$ as

$\varphi w_{n}=(1-\Delta)^{-1}(\varphi
v_{n})-(1-\Delta)^{-1}[\varphi,(1-\Delta)]w_{n}.$

Observe that the operator $[\varphi,(1-\Delta)]$ maps $L^{2}(\mathbb{R}^{3})$
continuously into $H^{-1}(\mathbb{R}^{3})$. So, due to the convergence (3.17),
we get that

(3.18) $(1-\Delta)^{-1}[\varphi,(1-\Delta)]w_{n}\rightarrow 0\mbox{ in
}C([0,T];H^{1}(\mathbb{R}^{3})).$

On the other hand, by (3.13),

(3.19) $(1-\Delta)^{-1}(\varphi v_{n})\rightarrow 0\mbox{ in
}L^{2}((0,T);H^{1}(\mathbb{R}^{3})).$

Therefore, by the convergences (3.18) and (3.19) above, it follows that

$\varphi w_{n}\rightarrow 0\mbox{ in }L^{2}((0,T);H^{1}(\mathbb{R}^{3})).$

Since $w_{n}$ satisfies (3.3), using Lemma 3.3, more precisely, the
observability inequality (3.5), we conclude that

$w_{n}(0)\rightarrow 0\mbox{ in }H^{1}(\mathbb{R}^{3})\mbox{ strongly, }$

and so

$v_{n}(0)\rightarrow 0\mbox{ in }H^{-1}(\mathbb{R}^{3})\mbox{ strongly,}$

which is a contradiction to the fact that $\|v_{n}(0)\|^{2}_{H^{-1}}=1$, for
all $n$. This finishes the proof. ∎

Third step: Proof of the observability inequality (3.2).

If (3.2) is false, then there exists a sequence $\\{v_{n}\\}$ of solutions to
(3.3) in $C([0,T];H^{-1}(\mathbb{R}^{3}))$ such that

(3.20) $1=\|v_{n}(0)\|^{2}_{H^{-1}}\geq n\int_{0}^{T}\|\varphi
v_{n}(t)\|^{2}_{H^{-1}}\ dt,\ \forall n\geq 0.$

Extracting a subsequence, still denoted by the same indexes, we have that

$v_{n}\rightharpoonup v\ \mbox{ in }\
L^{\infty}((0,T);H^{-1}(\mathbb{R}^{3}))\mbox{ weak*}$

and

$v_{n}(0)\rightharpoonup v(0)\mbox{ in }H^{-1}(\mathbb{R}^{3})\mbox{ weak, }$

for some solution $v\in C([0,T];H^{-1}(\mathbb{R}^{3}))$ of the system (3.3).
Note that

$\varphi v_{n}\rightharpoonup\varphi v\text{ in
$L^{\infty}(0,T;H^{-1}(\mathbb{R}^{3}))$ weak*}$

and this, combined with (3.20) ($\varphi v_{n}\rightarrow 0$ in
$L^{2}\big{(}(0,T);H^{-1}(\mathbb{R}^{3})\big{)}$), yields $\varphi v\equiv 0$
and, hence, $v\equiv 0$ for $|x|>R+1$, $t\in(0,T)$. So, by the unique
continuation property as in Step 2, we deduce that $v\equiv 0$ on
$\mathbb{R}^{3}\times(0,T)$. On the other hand, the sequence
${(1-\varphi(x/2))v_{n}(0)}$ is bounded in $H^{-1}(\mathbb{R}^{3})$ and has
compact support contained in $B_{2R+2}(0)$. Therefore, extracting a
subsequence, we may assume that it converges strongly in
$H^{-2}(\mathbb{R}^{3})$. Moreover, its limit is necessarily $0$ since

${(1-\varphi(x/2))v_{n}(0)}\rightharpoonup 0\text{ in
$H^{-2}(\mathbb{R}^{3})$}.$

Using inequality (3.12), we conclude that $\|v_{n}(0)\|_{H^{-1}}\rightarrow
0$, which contradicts (3.20). This proves the desired observability inequality
(3.3) and finishes the proof of Proposition 3.2. ∎

###### Proof of Theorem 3.1.

We prove Theorem 3.1 using Hilbert’s uniqueness method. First, note that since
the Schrödinger equation (3.1) is backward well-posed, we may assume that
$u(T)=0$ without loss of generality.

Now, consider the two systems

(3.21) $\left\\{\begin{array}[]{lr}i\partial_{t}u+\Delta u=\varphi(x)h(x,t)\
in\ [0,T]\times\mathbb{R}^{3},\\\ u(T)=0,\end{array}\right.$

with $\varphi(x)$ satisfying (1.5), and

$\left\\{\begin{array}[]{lr}i\partial_{t}v+\Delta v=0\ in\
[0,T]\times\mathbb{R}^{3},\\\ v(0)=v_{0}\in
H^{-1}(\mathbb{R}^{3}).\end{array}\right.$

Multiplying the equation of the first system by $\overline{v}$ and integrating
by parts, we obtain

$i\int_{\mathbb{R}^{3}}\Big{[}\overline{v}(T)u(T)-\overline{v_{0}}u(0)\Big{]}\
dx=\int_{0}^{T}\int_{\mathbb{R}^{3}}\varphi(x)h(x,t)\overline{v(x,t)}\ dxdt.$

Hence, taking $L^{2}(\mathbb{R}^{3})$ as pivot space, one has

(3.22) $\langle
v_{0},-iu_{0}\rangle=\int_{0}^{T}\langle\varphi(x)v,h(t)\rangle\ dt,$

where $\langle\cdot,\cdot\rangle$ denotes the duality pairing between
$H^{-1}\left(\mathbb{R}^{3}\right)$ and $H^{1}\left(\mathbb{R}^{3}\right)$.
Consider the continuous map $\Lambda:H^{-1}(\mathbb{R}^{3})\rightarrow
H^{1}(\mathbb{R}^{3})$ defined by $\Lambda v=(v,\cdot)_{1}.$ Given any
$v_{0}\in H^{-1}\left(\mathbb{R}^{3}\right)$, let $h(t)=\Lambda^{-1}(\varphi
v(t))\left(h\in C\left([0,T];H^{1}\left(\mathbb{R}^{3}\right)\right)\right)$
and let $u$ be the corresponding solution of system (3.21). Finally, set
$\Gamma\left(v_{0}\right)=-iu(\cdot,0)$. Then, we have

$\left\langle
v_{0},\Gamma\left(v_{0}\right)\right\rangle=\int_{0}^{T}\|\varphi
v(t)\|_{H^{-1}\left(\mathbb{R}^{3}\right)}^{2}dt\geqslant
c\left\|v_{0}\right\|_{H^{-1}\left(\mathbb{R}^{3}\right)}^{2},$

in view of the observability inequality (3.2) and identity (3.22). It follows
from the Lax-Milgram theorem that $\Gamma$ defines an isomorphism, and this
concludes the proof of Theorem 3.1. ∎

###  3.2. Nonlinear system: Proof of Theorem 1.1

The proof is based on a perturbation argument due to E. Zuazua [28]. To use
it, consider the following two Schrödinger systems with initial data in
$H^{-1}$ and null initial data, namely

$\left\\{\begin{array}[]{lr}i\partial_{t}\Phi+\Delta\Phi=0\mbox{ on}\
[0,T]\times\mathbb{R}^{3},\\\ \Phi(0)=\Phi_{0}\in
H^{-1}(\mathbb{R}^{3}),\end{array}\right.$

and

(3.23) $\left\\{\begin{array}[]{lr}i\partial_{t}u+\Delta
u-|u|^{4}u=A\Phi\mbox{ on}\ [0,T]\times\mathbb{R}^{3},\\\
u(T)=0,\end{array}\right.$

respectively, where $A$ is defined as in Theorem 3.1 by

$A\Phi:=\Lambda^{-1}(\varphi(x)\Phi).$

Now, define the operator

$\displaystyle\mathcal{L}:H^{-1}(\mathbb{R}^{3})$ $\displaystyle\rightarrow
H^{1}(\mathbb{R}^{3})$ $\displaystyle\Phi_{0}$
$\displaystyle\mapsto\mathcal{L}\Phi_{0}=u_{0}=u(0).$

The goal is then to show that $\mathcal{L}$ is onto in a small neighborhood of
the origin of $H^{1}(\mathbb{R}^{3})$. To this end, split $u$ as $u=v+\Psi$,
where is $\Psi$ a solution of

$\left\\{\begin{array}[]{lr}i\partial_{t}\Psi+\Delta\Psi=A\Phi\ in\
[0,T]\times\mathbb{R}^{3},\\\ \Psi(T)=0,\end{array}\right.$

and $v$ is a solution of

(3.24) $\left\\{\begin{array}[]{lr}i\partial_{t}v+\Delta v=|u|^{4}u\ in\
[0,T]\times\mathbb{R}^{3},\\\ v(T)=0.\end{array}\right.$

Clearly $u,v$ and $\Psi$ belong to $C([0,T],H^{1}(\mathbb{R}^{3}))\cap
L^{10}([0,T];L^{10}(\mathbb{R}^{3}))$ and $u(0)=v(0)+\Psi(0)$. We can write

$\mathcal{L}\Phi_{0}=\mathcal{J}\Phi_{0}+\Gamma\Phi_{0},$

where $\mathcal{J}\Phi_{0}=v_{0}$. Observe that $\mathcal{L}\Phi_{0}=u_{0}$,
or equivalently, $\Phi_{0}=-\Gamma^{-1}\mathcal{J}\Phi_{0}+\Gamma^{-1}u_{0}$.

Now, define the operator

$\displaystyle\mathcal{B}:H^{-1}(\mathbb{R}^{3})$ $\displaystyle\rightarrow
H^{-1}(\mathbb{R}^{3})$ $\displaystyle\Phi_{0}$
$\displaystyle\rightarrow\mathcal{B}\Phi_{0}=\Gamma^{-1}\mathcal{J}\Phi_{0}+\Gamma^{-1}u_{0},$

where we are taking into account that $\Gamma$ is the linear control
isomorphism between $H^{-1}$ and $H^{1}$, due to Theorem 3.1.

Now, the goal is to prove that $\mathcal{B}$ has a fixed point near the origin
of $H^{-1}(\mathbb{R}^{3})$. More precisely, let us prove that if
$\|u_{0}\|_{H^{1}}$ is small enough, then $\mathcal{B}$ is a contraction on a
small ball $B_{R}$ of $H^{-1}(\mathbb{R}^{3})$. We may assume $T<1$ and we
will denote by $C>0$ any constant that may have its numerical value changed
line by line. Since $\Gamma$ is an isomorphism, we have

(3.25)
$\begin{split}\|\mathcal{B}\Phi_{0}\|_{H^{-1}}\leq&\|\Gamma^{-1}\mathcal{J}\Phi_{0}\|_{H^{-1}}+\|\Gamma^{-1}u_{0}\|_{H^{-1}}\\\
\leq&C\left(\|\mathcal{J}\Phi_{0}\|_{H^{1}}+\|u_{0}\|_{H^{1}}\right)\\\
\leq&C\left(\|v(0)\|_{H^{1}}+\|u_{0}\|_{H^{1}}\right).\end{split}$

Claim 1: There exists $C>0$ such that

(3.26) $\|v(0)\|_{H^{1}}\leq C\|\nabla
u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}.$

Indeed, note that due to the classical energy estimate for system (3.24),
Strichartz estimates (see Lemma 2.2) and a Sobolev embedding (see Lemma 2.3),
we have

$\displaystyle\|v(0)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\|v(T)\|_{L^{2}}+\Big{\|}\int_{0}^{t}e^{i(t-\tau)\Delta}|u|^{4}u\
d\tau\Big{\|}_{L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle
C\|u^{5}\|_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle
C\|u\|^{5}_{L^{10}_{t}L^{10}_{x}}$ $\displaystyle\leq$ $\displaystyle
C\|\nabla u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}$

and

$\displaystyle\|\nabla v(0)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\|\nabla
v(T)\|_{L^{2}}+\Big{\|}\int_{0}^{t}e^{i(t-\tau)\Delta}\nabla|u|^{4}u\
d\tau\Big{\|}_{L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle C\|\nabla
u\|_{L^{{10}}_{t}L^{\frac{30}{13}}_{x}}\|u\|^{4}_{L^{10}_{t}L^{10}_{x}}$
$\displaystyle\leq$ $\displaystyle\|\nabla
u\|^{5}_{L^{{10}}_{t}L^{\frac{30}{13}}_{x}}.$

Thus,

$\|v(0)\|^{2}_{L^{2}}\leq C\|\nabla
u\|^{10}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}$

and

$\|\nabla v(0)\|^{2}_{L^{2}}\leq C\|\nabla
u\|^{10}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}.$

Summing up, we have (3.26), thus showing Claim 1.

Claim 2: There exists $C>0$ such that

(3.27) $\|\nabla u\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\leq
C\|\Phi_{0}\|_{H^{-1}}.$

In fact, applying Lemma 2.2 to system (3.23), one gets

$\displaystyle\|\nabla u\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}$
$\displaystyle\leq$ $\displaystyle\|\nabla u(T)\|_{L^{2}}+C\|\nabla
u\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\|u\|^{4}_{L^{10}_{t}L^{10}_{x}}+C\|\nabla
A\Phi\|_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\leq$ $\displaystyle
C\left(\|\nabla
u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+\|A\Phi\|_{L^{2}_{t}H^{1}_{x}}\right).$

Note that, using the fact that $\Lambda$ is an isomorphism, we get

$\|A\Phi\|_{H^{1}}=\|\Lambda^{-1}(\varphi\Phi)\|_{H^{1}}\leq
C\|\varphi\Phi\|_{H^{-1}}$

or, equivalently,

$\|A\Phi\|_{L^{2}H^{1}}\leq\Big{(}\int_{0}^{T}\|\varphi\Phi\|^{2}_{H^{-1}}\
dt\Big{)}^{\frac{1}{2}}.$

Then, the duality (3.22) yields

$\displaystyle\|\nabla u\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+C\Big{(}\int_{0}^{T}\|\varphi\Phi\|^{2}_{H^{-1}}\
dt\Big{)}^{\frac{1}{2}}$ $\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+C\Big{(}\langle\Gamma\Phi_{0},\Phi_{0}\rangle\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+C\Big{(}\|\Gamma\Phi_{0}\|_{H^{1}}\|\Phi_{0}\|_{H^{-1}}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+C\Big{(}\|\Phi_{0}\|^{2}_{H^{-1}}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{5}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+C\|\Phi_{0}\|_{H^{-1}}.$

Using a bootstrap argument, taking $\|\Phi_{0}\|_{H^{-1}}\leq R$ with $R$
small enough, we get (3.27), showing Claim 2.

Note that putting together inequalities (3.26) and (3.27) into (3.25), we
conclude

$\displaystyle\|B\Phi_{0}\|_{H^{-1}}$ $\displaystyle\leq$ $\displaystyle
C\Big{(}\|v(0)\|_{H^{1}}+\|u_{0}\|_{H^{1}}\Big{)}$ $\displaystyle\leq$
$\displaystyle C\Big{(}\|\Phi_{0}\|^{5}_{H^{-1}}+\|u_{0}\|_{H^{1}}\Big{)}.$

Then, choosing $R$ small enough and $\|u_{0}\|_{H^{1}}\leq\frac{R}{2C}$, we
get

$\|B\Phi_{0}\|_{H^{-1}}\leq R$

and, therefore, $\mathcal{B}$ reproduces the ball $B_{R}$ of
$H^{-1}(\mathbb{R})$.

Finally, we prove that $\mathcal{B}$ is a contraction map. To do this, let us
study the systems

$\left\\{\begin{array}[]{lr}i\partial_{t}(u_{1}-u_{2})+\Delta(u_{1}-u_{2})-|u_{1}|^{4}u_{1}+|u_{2}|^{4}u_{2}=A(\Phi^{1}-\Phi^{2}),\\\
(u_{1}-u_{2})(T)=0,\end{array}\right.$

and

(3.28)
$\left\\{\begin{array}[]{lr}i\partial_{t}(v_{1}-v_{2})+\Delta(v_{1}-v_{2})=|u_{1}|^{4}u_{1}-|u_{2}|^{4}u_{2},\\\
(v_{1}-v_{2})(T)=0.\end{array}\right.$

As before, we also have

(3.29) $\|\mathcal{B}\Phi_{0}^{1}-\mathcal{B}\Phi_{0}^{2}\|_{H^{-1}}\leq
C\|v_{1}(0)-v_{2}(0)\|_{H^{1}}.$

We now estimate of $v_{1}(0)-v_{2}(0)$ in the $H^{1}$-norm. First, applying
the Strichatz estimates (see Lemma 2.2) to the system (3.28) yields that

$\begin{split}\|v_{1}(0)-v_{2}(0)\|_{L^{2}}\leq&\Big{\|}\int_{0}^{t}e^{i(t-\tau)\Delta}(|u_{1}|^{4}u_{1}-|u_{2}|^{4}u_{2})\
d\tau\Big{\|}_{L^{2}_{x}}\\\
\leq&C\|u_{1}^{5}-u_{2}^{5}\|_{L^{1}_{t}L^{2}_{x}}\\\
\leq&C\|u_{1}-u_{2}\|_{L^{5}_{t}L^{10}_{x}}\Big{(}\|u_{1}\|^{4}_{L^{5}_{t}L^{10}_{x}}+\|u_{2}\|^{4}_{L^{5}_{t}L^{10}_{x}}\Big{)}\\\
\leq&C\|u_{1}-u_{2}\|_{L^{10}_{t}L^{10}_{x}}\Big{(}\|u_{1}\|^{4}_{L^{10}_{t}L^{10}_{x}}+\|u_{2}\|^{4}_{L^{10}_{t}L^{10}_{x}}\Big{)}\\\
\leq&C\|\nabla u_{1}-\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\Big{(}\|\nabla
u_{1}\|^{4}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+\|\nabla
u_{2}\|^{4}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\Big{)}\\\ \leq&CR^{4}\|\nabla
u_{1}-\nabla u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\end{split}$

and

$\begin{split}\|\nabla v_{1}(0)-\nabla
v_{2}(0)\|_{L^{2}}\leq&\Big{\|}\int_{0}^{t}\nabla
e^{i(t-\tau)\Delta}(|u_{1}|^{4}u_{1}-|u_{2}|^{4}u_{2})\
d\tau\Big{\|}_{L^{2}_{x}}\\\
\leq&\|\nabla(|u_{1}|^{4}u_{1}-|u_{2}|^{4}u_{2})\|_{L^{2}_{t}L^{\frac{6}{5}}_{x}}\\\
\leq&C\|\nabla(u_{1}-u_{2})\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\|u_{1}\|^{4}_{L^{10}_{t}L^{10}_{x}}\\\
\leq&C\|\nabla(u_{1}-u_{2})\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\|\nabla
u_{1}\|^{4}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\\\ &+C\|\nabla u_{1}-\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\left(\|\nabla
u_{1}\|^{3}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\|\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\right.\\\ &\left.+\|\nabla
u_{2}\|^{3}_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\|\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\right)\\\
\leq&CR^{4}\|\nabla(u_{1}-u_{2})\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+CR^{4}\|\nabla(u_{1}-u_{2})\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\\\
\leq&CR^{4}\|\nabla(u_{1}-u_{2})\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}.\end{split}$

Thus,

$\|v_{1}(0)-v_{2}(0)\|_{L^{2}}^{2}\leq CR^{8}\|\nabla u_{1}-\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}^{2}$

and

$\|\nabla v_{1}(0)-\nabla v_{2}(0)\|^{2}_{L^{2}}\leq CR^{8}\|\nabla
u_{1}-\nabla u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}^{2}.$

These bounds together give us the $H^{1}$-estimate

$\|v_{1}(0)-v_{2}(0)\|_{H^{1}}\leq CR^{4}\|\nabla u_{1}-\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}.$

Now, let us bound the right-hand side of this inequality. To this end, first
notice that

$\displaystyle\|\nabla(u_{1}-u_{2})\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}$
$\displaystyle\leq$
$\displaystyle\|\nabla(|u_{1}|^{4}u_{1}-|u_{2}|^{4}u_{2})\|_{L^{2}_{t}L^{\frac{6}{5}}_{x}}+\|\nabla
A(\Phi^{1}-\Phi^{2})\|_{L^{1}_{t}L^{2}_{x}}$ $\displaystyle\leq$
$\displaystyle CR^{4}\|\nabla u_{1}-\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+C\|A(\Phi^{1}-\Phi^{2})\|_{L^{2}_{t}H^{1}_{x}}$
$\displaystyle\leq$ $\displaystyle CR^{4}\|\nabla u_{1}-\nabla
u_{2}\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}+C\|\Phi_{0}^{1}-\Phi_{0}^{2}\|_{H^{-1}}.$

So, choosing $R>0$ small enough, we get

$\|\nabla(u_{1}-u_{2})\|_{L^{10}_{t}L^{\frac{30}{13}}_{x}}\leq
C\|\Phi_{0}^{1}-\Phi_{0}^{2}\|_{H^{-1}}.$

Therefore,

(3.30)
$\begin{split}\|v_{1}(0)-v_{2}(0)\|_{H^{1}}=&\Bigg{(}\|v_{1}(0)-v_{2}(0)\|_{L^{2}}^{2}+\|\nabla
v_{1}(0)-\nabla v_{2}(0)\|^{2}_{L^{2}}\Bigg{)}^{\frac{1}{2}}\\\
\leq&CR^{4}\|\Phi_{0}^{1}-\Phi_{0}^{2}\|_{H^{-1}}.\end{split}$

Finally, we get by inequalities (3.29) and (3.30) that

$\displaystyle\|\mathcal{B}\Phi_{0}^{1}-\mathcal{B}\Phi_{0}^{2}\|_{H^{-1}}$
$\displaystyle\leq$ $\displaystyle C\|v_{1}(0)-v_{2}(0)\|_{H^{1}}$
$\displaystyle\leq$ $\displaystyle
CR^{4}\|\Phi_{0}^{1}-\Phi_{0}^{2}\|_{H^{-1}},$

concluding that $\mathcal{B}$ is a contraction on a small ball $B_{R}$ of
$H^{-1}$. This completes the proof of Theorem 1.1.∎

##  4\. Comments and open issues

In this work, we have studied the local exact controllability of the critical
C-NLS in dimension $3$. Considering the defocusing critical nonlinear
Schrödinger equation (1.4), we showed this system to be exact controllable in
$\dot{H}^{1}(\mathbb{R}^{3})$–level, that is, it satisfies
$u(T)=\tilde{u}_{0}$ for $u_{0}$ and $\tilde{u}_{0}$ in
$\dot{H}^{1}(\mathbb{R}^{3})$ satisfying $\|u_{0}\|_{H^{1}}\leq\delta$,
$\|\tilde{u}_{0}\|_{H^{1}}\leq\delta$, and $h(x,t)\in
C([0,T];H^{1}(\mathbb{R}^{3}))$. Concerning our main results, Theorems 1.1 and
1.2, the following remarks are worth mentioning.

  * •

The main tools to achieve these results were

– Strichartz estimates for the Schrödinger operator [6, 10], which give the
well-posedness theory for system (1.4);

– A unique continuation property for the linear system associated with (1.4),
which follows by the Carleman estimate showed in [18] (see also [15]);

– A perturbation argument, as presented in [28], which allows one to extend
the result for the nonlinear system (1.4).

  * •

For the $3d$–case, Laurent [15] showed large time global internal
controllability for the nonlinear Schrödinger equation on some compact
manifolds of dimension $3$. The results therein were obtained for the
nonlinearity $|u|^{2}u$ instead of $|u|^{4}u$. In this sense, our result
completes the local control result given by the author for the critical case
(1.4) in $\mathbb{R}^{3}$.

  * •

Since we are working on the whole space $\mathbb{R}^{3}$ and we have a control
taking the form $\varphi(x)h(x,t)$ in the system (1.4) (remember the
definition of $\varphi$ in (1.5)), the geometric control condition (see, for
instance, [1, 22]) is easily satisfied.

Thus, our work gives the first step to understanding control problems for the
C-NLS in dimension $3$, and, therefore, some important open issues appear. Let
us present them below.

  * ($\mathcal{A})$

Stabilization problem: Can one find a feedback control law $f=Ku$ so that the
resulting closed-loop system

$i\partial_{t}u+\Delta
u-|u|^{4}u=Ku,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{3},$

is asymptotically stable at an equilibrium point as $t\rightarrow+\infty$?

  * ($\mathcal{B})$

Global control problem: If the answer to the previous question is positive,
another natural issue is to obtain a global control result for the system
(1.4), i.e., a control result for large data. This would be a consequence of
the global stabilization result together with the local control result shown
in Theorem 1.1.

We hope to address these problems in a forthcoming paper.

## Acknowledgment

This work is part of Carvalho’s Ph.D. thesis at the Department of Mathematics
of the Universidade Federal de Pernambuco. It was mostly done while the third
author was visiting Université de Lorraine. The third author thanks the host
institution for its warm hospitality.

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