# Coupled linear Schrödinger equations: Control and stabilization results

K. Bhandari$\,{}^{\dagger}$ and R. de A. Capistrano-
Filho$\,{}^{\ddagger}\,{}^{\star}$ and S. Majumdar$\,{}^{\S}$ and T. Y.
Tanaka$\,{}^{\P}$

###### Abstract.

This article presents some controllability and stabilization results for a
system of two coupled linear Schrödinger equations in the one-dimensional case
where the state components are interacting through the Kirchhoff boundary
conditions. Considering the system in a bounded domain, the null boundary
controllability result is shown. The result is achieved thanks to a new
Carleman estimate, which ensures a boundary observation. Additionally, this
boundary observation together with some trace estimates, helps us to use the
Gramian approach, with a suitable choice of feedback law, to prove that the
system under consideration decays exponentially to zero at least as fast as
the function $e^{-2\omega t}$ for some $\omega>0$.

###### Key words and phrases:

Boundary control, Kirchhoff conditions, coupled Schrödinger equations,
Carleman estimates, rapid stabilization

###### 2020 Mathematics Subject Classification:

35K20, 35Q55, 93B05, 93B07, 93B52

⋆Corresponding author<EMAIL_ADDRESS>
†Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 11567
Praha 1, Czech Republic;
<EMAIL_ADDRESS>
‡Departamento de Matemática, Universidade Federal de Pernambuco (UFPE),
50740-545, Recife (PE), Brazil<EMAIL_ADDRESS>
§Department of Mathematics, Indian Institute of Technology Bombay, Powai,
Mumbai 400076, India;
<EMAIL_ADDRESS>
¶Departamento de Matemática, Universidade Federal Rural de Pernambuco (UFRPE),
52171-900, Recife (PE), Brazil<EMAIL_ADDRESS>
##  1\. Introduction

This work is dedicated to the study of the boundary control problem and
stabilization issue of a linear system that appears modeling some problems in
the context of nonlinear optics. Precisely, our motivation comes from the
following system

(1.1) $\begin{cases}iu_{t}(t,x)+pu_{xx}(t,x)-\theta
u(t,x)+\bar{u}(t,x)v(t,x)=0,\quad t\geq 0,\ x\in\mathbb{R},\\\ i\sigma
v_{t}(t,x)+qv_{xx}(t,x)-\alpha v(t,x)+\frac{a}{2}u^{2}(t,x)=0,\\\
u(0,x)=u_{0}(x),\quad v(0,x)=v_{0}(x),\end{cases}$

where $u$ and $v$ are complex-valued functions and $\alpha,\theta$, and
$a:=1/\sigma$ are real numbers representing physical parameters of the system,
where $\sigma>0$ and $p,q=\pm 1$. Notice that the system (1.1) is given by the
nonlinear coupling of two dispersive equations of Schrödinger type through the
quadratic terms.

There are some physical meanings for the previous system, as mentioned before.
For example, the complex functions $u$ and $v$ represent amplitude packets of
an optical wave’s first and second harmonic, respectively. The values of
$\gamma_{1}$ and $\gamma_{2}$ depend on the signals provided between the
scattering/diffraction ratios, and the positive constant $\sigma$ measures the
scaling/diffraction magnitude indices. For details about this system, the
authors suggested the references [28, 13, 19], and the references therein.

Concerning the mathematical context, most of the work related to the system
(1.1) is devoted to proving the well-posedness of the Cauchy problem in
$\mathbb{R}^{n}$, for $n\in[1,6]$ or in a periodic framework $\mathbb{T}$. For
example, in [17] the authors showed the local well-posedness for the Cauchy
problem (1.1) on the spaces $L^{2}\left(\mathbb{R}^{n}\right)\times
L^{2}\left(\mathbb{R}^{n}\right)$ for $n\leq 4$ and
$H^{1}\left(\mathbb{R}^{n}\right)$ $\times H^{1}\left(\mathbb{R}^{n}\right)$
for $n\leq 6$. About qualitative properties of Cauchy problem (1.1), the case
where $\gamma_{1}=\gamma_{2}=1$ was studied in [4] for initial data
$u_{0},v_{0}$ in the same periodic Sobolev space $H^{s}(\mathbb{T})$. We also
cite that in [5], the author studied the well-posedness of the Cauchy problem
associated with the coupled Schrödinger equations with quadratic
nonlinearities. He obtained the local well-posedness for data in Sobolev
spaces with low regularity. Finally, the authors suggest the reference [24]
for the recent progress on nonlinear Schrödinger systems with quadratic
interactions.

In the context of the control theory, no author attempted to show
controllability results in bounded domains for the system (1.1). Due to this
fact, our motivation is to present the boundary control results for the linear
system associated with (1.1) posed in a bounded domain, giving a necessary
first step in the direction to prove the nonlinear results for the system
(1.1).

###  1.1. Problem setting

As mentioned before, our motivation in this work is to present, as a first
step, the control results to the linear Schrödinger system associated with
(1.1). More precisely, considering $T>0$ be any finite time and
$\Omega=(0,1)$, we define $Q_{T}:=(0,T)\times\Omega$ and
$\Sigma_{T}:=(0,T)\times\partial\Omega$. So, we will study the boundary
controllability of the following linearized system

(1.2) $\begin{cases}iu_{t}+\gamma_{1}u_{xx}-\alpha_{1}u=0&\text{ in }Q_{T},\\\
i\sigma v_{t}+\gamma_{2}v_{xx}-\alpha_{2}v=0&\text{ in }Q_{T},\\\
u(0,x)=u_{0}(x),\ v(0,x)=v_{0}(x)&\text{ in }\Omega,\end{cases}$

where the constants
$\sigma,\gamma_{1},\gamma_{2},\alpha_{1},\alpha_{2},\alpha$ are positive and
$(u_{0},v_{0})$ are given initial data in certain spaces which will be
specified later. We will consider the system (1.2) with the so-called
Kirchhoff boundary condition at the right spatial point $x=1$:

(1.3) $\begin{cases}u(t,1)=v(t,1)\ &\text{in }(0,T),\\\
\gamma_{1}u_{x}(t,1)+\frac{\gamma_{2}}{\sigma}v_{x}(t,1)+\alpha u(t,1)=0\
&\text{in }(0,T).\end{cases}$

Here, the boundary control $h\in L^{2}(0,T)$ acts either on the component $u$
or $v$ at the left spatial point $x=0$; to be more precise we set

(1.4a) $\displaystyle\text{either }\ \ u(t,0)=h(t),\ \ v(t,0)=0\quad$
$\displaystyle\text{ in }(0,T),$ (1.4b) $\displaystyle\text{or }\ \ u(t,0)=0,\
\ v(t,0)=h(t)\quad$ $\displaystyle\text{ in }(0,T).$

So, the first goal of this article is to answer the following null
controllability problem:

Problem $\mathcal{A}$: Given $T>0$, $(u_{0},v_{0})$ in a certain space ${X}$,
can one find an appropriate control input $h$ such that the corresponding
solution $(u,v)$ of (1.2) with boundary conditions (1.3)-(1.4b) (or
(1.3)-(1.4a)) satisfies

(1.5)
$(u(0,x),v(0,x))=(u_{0}(x),v_{0}(x))\quad\text{and}\quad(u(T,x),v(T,x))=(0,0),\
\ \forall x\in\Omega\ ?$

If we can positively answer the previous question, an interesting problem is
to study the boundary stabilization problem for the system (1.2) with boundary
conditions (1.3) and with a single boundary control force exerted on the
component $v$ (or $u$), namely (1.4b) (or (1.4a)). In this context, the second
main problem in this work treats the following stabilization issue:

Problem $\mathcal{B}$: Can we construct a stationary feedback law $h(t)$, of
the form ${F}_{\omega}(u(t,\cdot),v(t,\cdot))$, such that the solution of the
closed-loop system (1.2) with boundary condition (1.3)-(1.4b) (or
(1.3)-(1.4a)) decays exponentially to zero at any prescribed decay rate?

An important point to answer these questions is that we need to assume the
parameter $\sigma$, in the second equation of (1.2), be a positive number and
the choice of $\sigma$ will play a crucial role in deducing the required
controllability result, which will be discussed further up. Moreover, we
mention that now, the second problem will be called the rapid stabilization
problem.

Before giving details about the main results of the article and some important
facts, let us give a brief history of the control problem for the Schrödinger
type systems, as well as some references to the rapid stabilization issues of
partial differential equations (PDEs).

###  1.2. Literature review: Control theory

We will divide our state-of-the-art into three fronts. The first concerns some
(non-exhaustive) control results for a single Schrödinger equation. The second
part is dedicated to the control results obtained for systems like (1.2).
Finally, we will give a brief overview of the rapid stabilization problem for
some systems.

####  1.2.1. Control results for Schrödinger equation

It is well known that control properties for a single Schrödinger equation
have received a lot of attention in the last decades (see, e.g., [39, 23] for
an excellent review of the contributions up to 2014). There is an ongoing
effort to show new control results for this equation, and this effort is
giving significant progress for control properties. So, in this spirit, we can
cite [21, 30, 31, 32, 34, 35] and the references therein for control issues or
[8, 10, 22, 29, 36] and the references therein for Carleman estimates and
their applications to inverse problems.

####  1.2.2. Control results for coupled system

We are not aware of any results for systems where the coupling is given at the
Kirchhoff boundary condition for the Schrödinger type systems, as is our case.
However, concerning the coupled (internal) structure in the equation, there
are several results considering the cascade system for Schrödinger equations,
which we would like to mention.

We first mention that controllability results for systems of parabolic
equations are reviewed [3]. Concerning systems of hyperbolic equations, we
cite the works [1, 2, 14]. There and the references therein, the reader can
find results about the controllability of two coupled wave equations with only
one control, under the hypothesis of the geometric control condition. More
recently, in [33] a boundary controllability result is shown for a Schrödinger
cascade type system with periodic boundary conditions. This result is obtained
as a consequence of the controllability result for a cascade system of two
wave equations. We also refer to the work [26], where the authors studied the
null controllability of a linear system formed by two Schrödinger, controlling
only one of them using Carleman estimates. Finally, considering two uncoupled
wave equations with potentials on an interval, in [27], the authors
established a Carleman estimate for wave systems with simultaneous boundary
control, giving a boundary controllability result for uncoupled wave
equations.

####  1.2.3. Rapid stabilization of PDEs

For rapid stabilization issues, in recent years, some abstract methods [20,
37, 38] have been developed to obtain answers considering linear PDEs. The
method is based on the Gramian approach and the Riccati equations, and several
authors employed during the last few years this approach. For example, we can
cite, [11] for the KdV equation in a bounded domain, [9] for the KdV-KdV
equation with only one boundary feedback acting, [18] for one-dimensional
Schrödinger equation and of the beam and plate equations by moving or oblique
observations and, additionally, for vibrating strings and beams, we can refer
to [6].

We mention that this is only a small sample of the results concerning the
control and stabilization problem. We strongly encourage the reader to see the
previous references and the references therein.

###  1.3. Main results and structure of the article

We are now in a position to give comments on our main contribution to this
article. Consider, now on, the space

(1.6)
$\displaystyle\mathcal{H}=\big{\\{}(u_{1},u_{2})\in[H^{1}(\Omega)]^{2}\mid
u_{1}(0)=u_{2}(0)=0,\ u_{1}(1)=u_{2}(1)\big{\\}},$

as the natural space for belonging of the initial data associated with (1.2)
with the following norm

(1.7)
$\displaystyle\left\|(u_{1},u_{2})\right\|_{\mathcal{H}}=\bigg{(}\int_{\Omega}\left(|u_{1}^{\prime}(x)|^{2}+|u_{2}^{\prime}(x)|^{2}\right)dx\bigg{)}^{\frac{1}{2}}.$

and the associated inner product defined by

$\left<{(u_{1},u_{2})},{(v_{1},v_{2})}\right>_{\mathcal{H}}=\textnormal{Re}\int_{\Omega}u_{1}^{\prime}(x)\overline{v_{1}^{\prime}(x)}dx+\textnormal{Re}\int_{\Omega}u_{2}^{\prime}(x)\overline{v_{2}^{\prime}(x)}dx,$

for any $(u_{1},u_{2}),\,(v_{1},v_{2})\in\mathcal{H}$. Finally, we denote
$\mathcal{H}^{\prime}$ as the dual space of $\mathcal{H}$ with respect to the
pivot space $[L^{2}(0,1)]^{2}$.

The first result of our work gives the control problem for the system (1.2)
with boundary conditions (1.3)-(1.4b), that is when the control is acting on
the second component. Precisely, considering these boundary conditions, due to
a new Carleman estimate with boundary observation, the following result is
verified.

######  Theorem 1.1.

Consider

(1.8)
$\displaystyle\mathfrak{S}:=\left\\{\sigma>0\mid\sigma=\frac{\kappa\gamma_{2}}{\gamma_{1}},\
\ \kappa>3\right\\},$

where $\gamma_{1},\gamma_{2}>0$ are as appearing in (1.2). Let any $T>0$,
initial data $\left(u_{0},v_{0}\right)\in\mathcal{H}^{\prime}$ and parameters
$\gamma_{1},\gamma_{2},\alpha_{1},\alpha_{2},\alpha$ be given positive. Then,
for any $\sigma\in\mathfrak{S}$, there exists a control $h\in L^{2}(0,T)$ such
that the solution $(u,v)$ to the system (1.2) with boundary conditions
(1.3)-(1.4b) satisfies (1.5).

Since we can prove an observability inequality (for the associated adjoint
system), as mentioned before, naturally, the Problem $\mathcal{B}$ seems
reachable. The next result gives, for the coupled Schrödinger equation (1.2),
the following positive answer for the rapid stabilization problem.

######  Theorem 1.2.

Let any parameters $\gamma_{1},\gamma_{2},\alpha_{1},\alpha_{2},\alpha$ be
given positive. Then, for the same choices of $\sigma$ as in Theorem 1.1,
there exists a continuous linear map ${F}_{\omega}:\mathcal{H}\to\mathbb{C}$
and a positive constants $C$ and $\omega$, such that for every
$\left(u_{0},v_{0}\right)\in\mathcal{H}$, the solution $(u,v)$ of the closed-
loop system (1.2) with boundary conditions (1.3)-(1.4b), with
$h(t)={F}_{\omega}(u(t,\cdot),v(t,\cdot))$ satisfies

(1.9) $\|(u(t),v(t))\|_{\mathcal{H}}\leq Ce^{-2\omega
t}\left\|\left(u_{0},v_{0}\right)\right\|_{\mathcal{H}},\quad\forall t\geq 0.$

######  Remark 1.3.

In what concerns our main results, Theorems 1.1 and 1.2, the following remarks
are worth mentioning:

  * •

As usual in the literature the answer for the Problem $\mathcal{A}$, that is,
Theorem 1.1, is shown by using the Hilbert Uniqueness Method introduced by
Lions [25] and the classical duality theory of Dolecki and Russell [15]. For
that, it is essential to prove a suitable observability inequality with
boundary observation, and to do so, in the present article, we prove a new
Carleman estimate for the associated adjoint system to (1.2)-(1.3)-(1.4b).

  * •

Problem $\mathcal{B}$ is answered using a classical Gramian approach [20, 37,
38], giving the proof of Theorem 1.2.

  * •

We study the control and stabilization problems for the system (1.2) with
boundary conditions (1.3)-(1.4b), however, we point out that a similar
analysis can be performed if we instead consider (1.3)-(1.4a).

  * •

The choice of such $\sigma\in\mathfrak{S}$ in Theorem 1.1 is important to
deduce the required controllability result via Carleman estimate; for more
details, we refer to Remark 2.4.

Our work is composed of five parts, including the introduction. In Section 2,
the boundary controllability is considered. We obtain a new Carleman estimate
with boundary observation, which is the key point to prove the observability
inequality. So, with this observability in hand, the Theorem 1.1 is verified.
In Section 3, we recall Urquiza’s approach and use it to achieve the second
main result of the article, i.e., Theorem 1.2. Section 4 is devoted to
presenting some comments and open issues. Finally, in Appendix A, we present
an overview of the well-posedness results, for the direct and adjoint systems
associated with (1.2). Additionally, a key lemma, essential to prove the rapid
stabilization result, is proved.

##  2\. Boundary controllability

Let us first study the global null-controllability properties of the system
(1.2)-(1.3) when the control acts on the component $v$, that is precisely
(1.4b). The main tool is to establish a suitable Carleman estimate that yields
a proper observability inequality and utilizing that, we prove the required
controllability result for the concerned model.

######  Definition 2.1.

For a given $T>0$, the system (1.2) with boundary condition (1.3)-(1.4b) is
null controllable at time $T$ if for any given initial data
$(u_{0},v_{0})\in\mathcal{H}^{\prime}$, there exists a control function $h\in
L^{2}(0,T)$, such that solution $(u,v)$ to (1.2) satisfies (1.5).

Additionally, the solution by transposition of (1.2)-(1.3)-(1.4b) is given
below.

######  Definition 2.2.

Let $(u_{0},v_{0})\in\mathcal{H}^{\prime}$ and $h\in L^{2}(0,T)$. We say that
$(u,v)\in L^{\infty}(0,T;\mathcal{H}^{\prime})$ is solution of

(2.1) $\displaystyle\begin{cases}iu_{t}+\gamma_{1}u_{xx}-\alpha_{1}u=0&\text{
in }Q_{T},\\\ i\sigma v_{t}+\gamma_{2}v_{xx}-\alpha_{2}v=0&\text{ in
}Q_{T},\\\ u(t,0)=0&\text{ in }(0,T),\\\ v(t,0)=h(t)&\text{ in }(0,T),\\\
u(t,1)=v(t,1)\ &\text{ in }(0,T),\\\
\gamma_{1}u_{x}(t,1)+\frac{\gamma_{2}}{\sigma}v_{x}(t,1)+\alpha
u(t,1)=0&\text{ in }(0,T),\\\ u(0,x)=u_{0}(x),\ v(0,x)=v_{0}(x)&\text{ in
}\Omega,\end{cases}$

in the transposition sense if and only if

$\begin{split}\textnormal{Re}\int_{0}^{T}\langle(u(t),v(t)),(\overline{g}_{1},\overline{g}_{2})\rangle_{\mathcal{H}^{\prime},\mathcal{H}}dt=&\langle(u_{0},v_{0}),(\overline{\varphi_{1}(0,\cdot)},\overline{\varphi_{2}(0,\cdot)})\rangle_{\mathcal{H}^{\prime},\mathcal{H}}-\textnormal{Re}\int_{0}^{T}h(t)\overline{\varphi_{2,x}(t,0)}\,dt,\end{split}$

for every $(g_{1},g_{2})\in L^{1}(0,T;\mathcal{H})$ where
$(\varphi_{1},\varphi_{2})$ are the mild solution to the problem

(2.2)
$\begin{cases}i\varphi_{1,t}+\gamma_{1}\varphi_{1,xx}-\alpha_{1}\varphi_{1}=g_{1},&\text{
in }Q_{T},\\\
i\sigma\varphi_{2,t}+\gamma_{2}\varphi_{2,xx}-\alpha_{2}\varphi_{2}=g_{2},&\text{
in }Q_{T},\\\ \varphi_{1}(t,1)=\varphi_{2}(t,1),&\text{ in }(0,T),\\\
\gamma_{1}\varphi_{1,x}(t,1)+\frac{\gamma_{2}}{\sigma}\varphi_{2,x}(t,1)+\alpha\varphi_{1}(t,1)=0,&\text{
in }(0,T),\\\ \varphi_{1}(t,0)=0,\,\varphi_{2}(t,0)=0,&\text{ in
}(0,T),\end{cases}$

on the space $C([0,T];\mathcal{H})$, with
$(\varphi_{1}(T,\cdot),\varphi_{2}(T,\cdot))=(0,0)$ in $\Omega$.

We remark that the discussion of the system (2.2) is given in Appendix A.
Furthermore, the following result is verified.

######  Proposition 2.3.

Let $(u_{0},v_{0})\in\mathcal{H}^{\prime}$ and $h\in L^{2}(0,T)$. Then the
control system (2.1) has a unique solution $(u,v)$ in
$C([0,T];\mathcal{H}^{\prime}).$

###  2.1. Global Carleman estimate

With the previous definitions, we are in a position to obtain the global
Carleman estimate for the adjoint system associated with the system (1.2) with
boundary conditions (1.3)-(1.4b), namely,

(2.3)
$\begin{cases}i\varphi_{1,t}+\gamma_{1}\varphi_{1,xx}-\alpha_{1}\varphi_{1}=0,&\text{
in }Q_{T},\\\
i\sigma\varphi_{2,t}+\gamma_{2}\varphi_{2,xx}-\alpha_{2}\varphi_{2}=0,&\text{
in }Q_{T},\\\ \varphi_{1}(t,1)=\varphi_{2}(t,1),&\text{ in }(0,T),\\\
\gamma_{1}\varphi_{1,x}(t,1)+\frac{\gamma_{2}}{\sigma}\varphi_{2,x}(t,1)+\alpha\varphi_{1}(t,1)=0,&\text{
in }(0,T),\\\ \varphi_{1}(t,0)=0,\,\varphi_{2}(t,0)=0,&\text{ in }(0,T),\\\
\varphi_{1}(T,x)=\zeta_{1}(x),\ \varphi_{2}(T,x)=\zeta_{2}(x),&\text{ in
}\Omega,\end{cases}$

with $\zeta:=(\zeta_{1},\zeta_{2})\in\mathcal{H}$.

To this end, we introduce the space

$\begin{split}\mathcal{Q}:=&\Big{\\{}(\varphi_{1},\varphi_{2})\in[C^{2}(\overline{Q_{T}})]^{2}\mid\varphi_{1}(t,0)=\varphi_{2}(t,0)=0,\
\varphi_{1}(t,1)=\varphi_{2}(t,1),\\\
&\quad\qquad\qquad\qquad\gamma_{1}\varphi_{1,x}(t,1)+\frac{\gamma_{2}}{\sigma}\varphi_{2,x}(t,1)+\alpha\varphi_{1}(t,1)=0,\
\forall t\in[0,T]\Big{\\}}.\end{split}$

Now recall that $\sigma\in\mathfrak{S}$ (see (1.8)) and thus

(2.4) $\displaystyle\sigma=\frac{\kappa\gamma_{2}}{\gamma_{1}}\ \text{ for
some $\kappa>3$}.$

With this in hand, we consider the following auxiliary functions (motivated
from [7]):

(2.5) $\displaystyle\begin{cases}\beta_{j}(x)=2+c_{j}(x-1),\ \ \ j=1,2,\\\
c_{1}=1,\ c_{2}=-\kappa\,\text{ and thus }|c_{2}|>3.\end{cases}$

Therefore, $\beta_{j}\in C^{2}([0,1])$ and satisfy

$\beta_{2}\geq\beta_{1}>0,\ \ \text{in }\,[0,1],\ \
\beta_{1}(1)=\beta_{2}(1).$

Next, for any parameter $\lambda>1$, we introduce the following weight
functions:

(2.6)
$\displaystyle\xi_{j}(t,x)=\frac{e^{\lambda\beta_{j}(x)}}{t(T-t)},\quad\eta_{j}(t,x)=\frac{e^{2\lambda\left\|\beta_{j}\right\|_{\infty}}-e^{\lambda\beta_{j}(x)}}{t(T-t)},\quad\forall(t,x)\in
Q_{T},\ \ j=1,2.$

Note that

$\xi_{j},\,\eta_{j}>0\,\text{ for }j=1,2,\ \ \xi_{1}(t,1)=\xi_{2}(t,1),\ \
\eta_{1}(t,1)=\eta_{2}(t,1),$

since $\beta_{1}(1)=\beta_{2}(1)$. Concerning of the function $\xi_{j}$ and
$\eta_{j}$, we have the following behavior in $Q_{T}$:

(2.7) $\begin{cases}\xi_{j,x}=\lambda\xi_{j}c_{j},\ \
\eta_{j,x}=-\lambda\xi_{j}c_{j},\\\ \eta_{j,xx}=-\lambda^{2}c_{j}^{2}\xi_{j},\
\ \eta_{j,xxx}=-\lambda^{3}c_{j}^{3}\xi_{j},\\\ |\xi_{j,t}|\leq
CT\xi^{2}_{j},\ \ |\eta_{j,t}|\leq CT\xi^{2}_{j},\\\ |\eta_{j,xt}|\leq
C\lambda T\xi^{2}_{j},\ \ |\eta_{j,tt}|\leq
CT^{2}\xi^{3}_{j},\end{cases}\text{for $j=1,2$.}$

######  Remark 2.4.

We point out that the choice of $c_{2}$ in (2.5) and the value of
$\sigma\in\mathfrak{S}$ in (1.8) are crucial to obtain the Carleman estimate
(2.8). More precisely, those choices help us deal with some unusual boundary
integrals while proving the underlying Carleman estimate.

The main theorem of this subsection can be read as follows:

######  Theorem 2.5 (Carleman estimate).

Let the weight functions $\xi_{1},\xi_{2}$, $\eta_{1},\eta_{2}$ be chosen as
in (2.6) and $\sigma>0$ is taken as (2.4). Then, there exist constants
$\lambda_{0}>0$, $\mu_{0}>0$ and $C>0$, depending at most on
$\gamma_{1},\gamma_{2}$ and $c_{2}$, such that the following estimate holds
true

(2.8)
$\begin{split}s^{3}&\lambda^{4}\iint_{Q_{T}}\left(e^{-2s\eta_{1}}\xi^{3}_{1}|\varphi_{1}|^{2}+e^{-2s\eta_{2}}\xi^{3}_{2}|\varphi_{2}|^{2}\right)dxdt\\\
&+s\lambda^{2}\iint_{Q_{T}}\left(e^{-2s\eta_{1}}\xi_{1}|\varphi_{1,x}|^{2}+e^{-2s\eta_{2}}\xi_{2}|\varphi_{2,x}|^{2}\right)dxdt\\\
\leq&C\iint_{Q_{T}}(e^{-2s\eta_{1}}|L_{1}\varphi_{1}|^{2}+e^{-2s\eta_{2}}|L_{2}\varphi_{2}|^{2})dxdt\\\
&+Cs\lambda\int_{0}^{T}e^{-2s\eta_{2}(t,0)}\xi_{2}(t,0)|\varphi_{2,x}(t,0)|^{2}dt,\end{split}$

for every $\lambda\geq\lambda_{0}$, $s\geq s_{0}:=\mu_{0}(T+T^{2})$ and for
all $(\varphi_{1},\varphi_{2})\in\mathcal{Q}$, where

$L_{1}=i\partial_{t}+\gamma_{1}\partial_{xx}\quad\text{and}\quad
L_{2}=i\partial_{t}+\frac{\gamma_{2}}{\sigma}\partial_{xx}.$

To prove the above theorem, let us now define the following variables:

(2.9)
$\displaystyle\psi_{j}(t,x)=e^{-s\eta_{j}(t,x)}\varphi_{j}(t,x),\quad\forall(t,x)\in
Q_{T},\ \ \text{for }j=1,2,$

so that we have the following boundary conditions:

(2.10)
$\displaystyle\begin{cases}\psi_{j}(t,0)=0,\quad\psi_{1}(t,1)=\psi_{2}(t,1),\\\
\gamma_{1}\psi_{1,x}(t,1)+\frac{\gamma_{2}}{\sigma}\psi_{2,x}(t,1)=-\alpha\psi_{1}(t,1)+s\lambda\xi_{1}(t,1)\psi_{1}(t,1)\left(\gamma_{1}c_{1}+\frac{\gamma_{2}}{\sigma}c_{2}\right).\end{cases}$

We further denote

$F_{j}:=e^{-s\eta_{j}}L_{j}(\varphi_{j})=e^{-s\eta_{j}}L_{j}(e^{s\eta_{j}}\psi_{j}),$

where $L_{1}$ and $L_{2}$. So, with this, we find the following relations for
each $j=1,2$,

$\displaystyle
i(\psi_{j}e^{s\eta_{j}})_{t}=i\psi_{j,t}e^{s\eta_{j}}+is\psi_{j}e^{s\eta_{j}}\eta_{j,t},$
$\displaystyle(\psi_{j}e^{s\eta_{j}})_{x}=\psi_{j,x}e^{s\eta_{j}}+s\psi_{j}e^{s\eta_{j}}\eta_{j,x},$
$\displaystyle(\psi_{j}e^{s\eta_{j}})_{xx}=\psi_{j,xx}e^{s\eta_{j}}+2se^{s\eta_{j}}\psi_{j,x}\eta_{j,x}+s^{2}\psi_{j}e^{s\eta_{j}}|\eta_{j,x}|^{2}+s\psi_{j}e^{s\eta_{j}}\eta_{j,xx}.$

Using the previous relation, $F_{j}$ can be written as follows

(2.11) $\displaystyle M_{1}\psi_{j}+M_{2}\psi_{j}=F_{j},$

where

$\displaystyle\begin{cases}M_{1}\psi_{1}=2s\gamma_{1}\psi_{1,x}\eta_{1,x}+s\gamma_{1}\psi_{1}\eta_{1,xx}+is\psi_{1}\eta_{1,t},\\\
M_{2}\psi_{1}=i\psi_{1,t}+\gamma_{1}\psi_{1,xx}+s^{2}\gamma_{1}|\eta_{1,x}|^{2}\psi_{1},\end{cases}$

and

(2.12)
$\displaystyle\begin{cases}M_{1}\psi_{2}=2s\frac{\gamma_{2}}{\sigma}\psi_{2,x}\eta_{2,x}+s\frac{\gamma_{2}}{\sigma}\psi_{2}\eta_{2,xx}+is\psi_{2}\eta_{2,t},\\\
M_{2}\psi_{2}=i\psi_{2,t}+\frac{\gamma_{2}}{\sigma}\psi_{2,xx}+s^{2}\frac{\gamma_{2}}{\sigma}|\eta_{2,x}|^{2}\psi_{2}.\end{cases}$

Thus, we get from (2.11),

(2.13)
$\begin{split}\iint_{Q_{T}}\left(|M_{1}\psi_{j}|^{2}+|M_{2}\psi_{j}|^{2}\right)dxdt&+2\textnormal{Re}\iint_{Q_{T}}M_{1}\psi_{j}\overline{M_{2}\psi_{j}}dxdt\\\
&=\iint_{Q_{T}}|F_{j}|^{2}dxdt,\end{split}$

for $j=1,2$. Now, we are in a position to prove Theorem 2.5.

###### Proof of Theorem 2.5.

Our goal is to focus on the following inner product

$\textnormal{Re}\iint_{Q_{T}}M_{1}\psi_{j}\overline{M_{2}\psi_{j}}dxdt$

that contains $9$ terms. We will elaborately make the computations for $j=2$,
similarly, the computations can be done for $j=1$ as well.

Recall that the quantities $M_{1}\psi_{2}$ and $M_{2}\psi_{2}$ are given by
(2.12) and, for $j=2$, we have the relation (2.13). We further denote

$\textnormal{Re}\iint_{Q_{T}}M_{1}\psi_{2}\overline{M_{2}\psi_{2}}dxdt=\sum_{1\leq
k,l\leq 3}I_{kl},$

where all the terms $I_{kl}$ for $1\leq k,l\leq 3$ consists of the integral
term with the product involving the $k$-th term of $M_{1}\psi_{2}$ with the
$l$-th term of $M_{2}\psi_{2}$, and will be computed below. Now, we split the
proof into several steps.

$\bullet$ Step 1. Computations of the terms $I_{11}$, $I_{21}$ and $I_{32}$.

Let us start with $I_{11}$. Observe that

(2.14)
$\begin{split}I_{11}=&2s\frac{\gamma_{2}}{\sigma}\textnormal{Re}\iint_{Q_{T}}\psi_{2,x}\eta_{2,x}\overline{i\psi_{2,t}}dxdt\\\
=&2s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,x}\eta_{2,x}\overline{\psi_{2,t}}dxdt\\\
=&-2s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,x}\eta_{2,xt}\overline{\psi_{2}}dxdt-2s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,xt}\eta_{2,x}\overline{\psi_{2}}dxdt\\\
:=&J_{1}+J_{2},\end{split}$

where there is no boundary integral since

$\lim\limits_{t\to 0^{+}}\psi_{2}(t,\cdot)=\lim\limits_{t\to
T^{-}}\psi_{2}(t,\cdot)=0,$

thanks to the choices of weight functions (2.6).

Next, for the quantity $I_{21}$, we see that

(2.15)
$\begin{split}I_{21}=&s\frac{\gamma_{2}}{\sigma}\textnormal{Re}\iint_{Q_{T}}\psi_{2}\eta_{2,xx}\overline{i\psi_{2,t}}dxdt\\\
=&s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2}\eta_{2,xx}\overline{\psi_{2,t}}dxdt\\\
=&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,x}\eta_{2,x}\overline{\psi_{2,t}}dxdt-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2}\eta_{2,x}\overline{\psi_{2,tx}}dxdt\\\
&+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\psi_{2}(t,1)\overline{\psi_{2,t}(t,1)}\eta_{2,x}(t,1)dt\\\
=&s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,xt}\eta_{2,x}\overline{\psi_{2}}dxdt+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,x}\eta_{2,xt}\overline{\psi_{2}}dxdt\\\
&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2}\eta_{2,x}\overline{\psi_{2,xt}}dxdt+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\psi_{2}(t,1)\overline{\psi_{2,t}(t,1)}\eta_{2,x}(t,1)dt,\end{split}$

where we have applied integration by parts w.r.t. $x$ to the term involving
$\eta_{2,xx}$, then w.r.t. $t$ to the term involving $\overline{\psi_{2,t}}$
in $Q_{T}$, and using again the decay of $\eta_{2}$ at $t=0$ and $t=T$.

Now, thanks to the fact that
$-\textnormal{Im}(z)=\textnormal{Im}(\overline{z})$ (for any $z\in\mathbb{C}$)
in the third integral term in the last equality of (2.15), we get

(2.16)
$\begin{split}I_{21}=&2s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,xt}\eta_{2,x}\overline{\psi_{2}}dxdt+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,x}\eta_{2,xt}\overline{\psi_{2}}dxdt\\\
&+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\psi_{2}(t,1)\overline{\psi_{2,t}(t,1)}\eta_{2,x}(t,1)dt\\\
=&-J_{2}-\frac{J_{1}}{2}+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\psi_{2}(t,1)\overline{\psi_{2,t}(t,1)}\eta_{2,x}(t,1)dt.\end{split}$

Finally, the term $I_{32}$ is computed as follows.

(2.17)
$\begin{split}I_{32}=&s\frac{\gamma_{2}}{\sigma}\textnormal{Re}\iint_{Q_{T}}i\psi_{2}\eta_{2,t}\overline{\psi_{2,xx}}dxdt\\\
=&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\eta_{2,t}\psi_{2}\overline{\psi_{2,xx}}dxdt\\\
=&s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\left(\eta_{2,tx}\psi_{2}\overline{\psi_{2,x}}+\eta_{2,t}\psi_{2,x}\overline{\psi_{2,x}}\right)dxdt\\\
&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\eta_{2,t}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt\\\
=&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,x}\eta_{2,xt}\overline{\psi_{2}}dxdt\\\
&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\eta_{2,t}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt\\\
=&\frac{J_{1}}{2}-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\eta_{2,t}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt,\end{split}$

where we have used the fact that

$\frac{s\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\eta_{2,t}|\psi_{2,x}|^{2}dxdt=0$

since $\eta_{2,t}$ is real-valued function. Therefore, by adding (2.14),
(2.16) and (2.17), we get

(2.18)
$\begin{split}I_{11}+I_{21}+I_{32}=&J_{1}+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\psi_{2}(t,1)\overline{\psi_{2,t}(t,1)}\eta_{2,x}(t,1)dt\\\
&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\eta_{2,t}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt,\end{split}$

where $J_{1}$ satisfies

(2.19)
$\begin{split}|J_{1}|=&\left|2s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\iint_{Q_{T}}\psi_{2,x}\eta_{2,xt}\overline{\psi_{2}}dxdt\right|\\\
\leq&C\frac{\gamma_{2}}{\sigma}s\lambda
T\iint_{Q_{T}}\xi^{2}_{2}|\psi_{2}||\psi_{2,x}|dxdt\\\
\leq&\frac{C}{\sigma^{2}}s^{2}\lambda^{2}T\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt+CT\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt,\end{split}$

finishing step 1.

$\bullet$ Step 2. Computations of the terms $I_{12}$ and $I_{22}$.

Let us, for $I_{12}$, to perform by integration by parts w.r.t. space variable
to ensures that

$\begin{split}I_{12}=&2s\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\eta_{2,x}\psi_{2,x}\overline{\psi_{2,xx}}dxdt\\\
=&-2s\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\eta_{2,xx}|\psi_{2,x}|^{2}dxdt-2s\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\eta_{2,x}\psi_{2,xx}\overline{\psi_{2,x}}dxdt\\\
&+2s\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\eta_{2,x}(t,1)|\psi_{2,x}(t,1)|^{2}dt-2s\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\eta_{2,x}(t,0)|\psi_{2,x}(t,0)|^{2}dt.\end{split}$

Recalling the expressions of $\eta_{2,x}$ and $\eta_{2,xx}$ from (2.7), we get

(2.20)
$\begin{split}2I_{12}=&2s\lambda^{2}c_{2}^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt\\\
&-2s\lambda
c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,1)|\psi_{2,x}(t,1)|^{2}dt+2s\lambda
c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,0)|\psi_{2,x}(t,0)|^{2}dt,\end{split}$

where the boundary term $\psi_{2,x}(t,1)>0$ once we have $c_{2}<0$ (see
(2.5)).

The term $I_{22}$ can be computed as

$\begin{split}I_{22}=&s\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\eta_{2,xx}\psi_{2}\overline{\psi_{2,xx}}dxdt\\\
=&-s\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\eta_{2,xx}|\psi_{2,x}|^{2}dxdt-s\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\eta_{2,xxx}\psi_{2}\overline{\psi_{2,x}}dxdt\\\
&+s\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\int_{0}^{T}\eta_{2,xx}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt.\end{split}$

Again, using the expressions of $\eta_{2,xx}$ and $\eta_{2,xxx}$, thanks to
(2.7), we have that

(2.21)
$\begin{split}I_{22}=&s\lambda^{2}c_{2}^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt+s\lambda^{3}c_{2}^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\xi_{2}\psi_{2}\overline{\psi_{2,x}}dxdt\\\
&-s\lambda^{2}c_{2}^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\int_{0}^{T}\xi_{2}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt.\end{split}$

Now, by adding the final expressions of $I_{12}$ and $I_{22}$, i.e, (2.20) and
(2.21), yields that

(2.22)
$\begin{split}I_{12}+I_{22}=&2s\lambda^{2}c_{2}^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt+s\lambda^{3}c_{2}^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\xi_{2}\psi_{2}\overline{\psi_{2,x}}dxdt\\\
&-s\lambda
c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,1)|\psi_{2,x}(t,1)|^{2}dt+s\lambda
c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,0)|\psi_{2,x}(t,0)|^{2}dt\\\
&-s\lambda^{2}c_{2}^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\int_{0}^{T}\xi_{2}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt,\end{split}$

where the second term of the r.h.s. of (2.22), denoted by $J_{3}$, satisfies

(2.23)
$\begin{split}|J_{3}|=&\left|s\lambda^{3}c_{2}^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\xi_{2}\psi_{2}\overline{\psi_{2,x}}dxdt\right|\\\
\leq&\frac{C}{\epsilon}s\lambda^{4}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}|\psi_{2}|^{2}dxdt+\epsilon
s\lambda^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt\\\
\leq&\frac{C}{\epsilon}T^{4}s\lambda^{4}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt+\epsilon
s\lambda^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt,\end{split}$

for any $\epsilon>0$ small enough, where we have used the fact that
$\xi_{2}\leq CT^{4}\xi_{2}^{3}$.

So, at this point, it is worth mentioning that the boundary integral
consisting of $\psi_{2,x}(t,0)$ in (2.22) will lead to the observation term in
our final Carleman estimate (since we have exerted a controlled force on the
second component), finalizing the analysis in step 2.

$\bullet$ Step 3. Analysis of remaining terms $I_{13}$, $I_{23}$, $I_{31}$,
and $I_{33}$.

Let us first deal with the term $I_{13}$. Notice that the fact that
$2\textnormal{Re}(z)=z+\overline{z}$ (for any $z\in\mathbb{C}$), and then
integrating by parts w.r.t. $x$, gives us

$\begin{split}I_{13}=&2s^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\eta^{3}_{2,x}\psi_{2,x}\overline{\psi_{2}}dxdt\\\
=&s^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\eta^{3}_{2,x}\left(\overline{\psi_{2}}\psi_{2,x}+\psi_{2}\overline{\psi_{2,x}}\right)dxdt\\\
=&s^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\eta^{3}_{2,x}\left(\psi_{2}\overline{\psi_{2}}\right)_{x}dxdt\\\
=&-s^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}3\eta^{2}_{2,x}\eta_{2,xx}|\psi_{2}|^{2}dxdt+s^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\eta^{3}_{2,x}|\psi_{2}(t,1)|^{2}dt\\\
=&3s^{3}\lambda^{4}c_{2}^{4}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}^{3}|\psi_{2}|^{2}dxdt-s^{3}\lambda^{3}c_{2}^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}^{3}(t,1)|\psi_{2}(t,1)|^{2}dxdt,\end{split}$

where the boundary term has a positive sign since $c_{2}<0$.

Now, for the analysis of $I_{23}$ we just need to see that

$\begin{split}I_{23}=&s^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\iint_{Q_{T}}\eta_{2,xx}|\eta_{2,x}|^{2}|\psi_{2}|^{2}dxdt=-s^{3}\lambda^{4}c_{2}^{4}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt.\end{split}$

Let us see the term $I_{31}$. For this integral, using again the decay
properties of the weight function and integrating into the time variable, this
term can be written as

$\begin{split}I_{31}=&-s\textnormal{Re}\iint_{Q_{T}}\eta_{2,t}\psi_{2}\overline{\psi_{2,t}}dxdt\\\
=&-\frac{1}{2}s\iint_{Q_{T}}\eta_{2,t}\left(\psi_{2}\overline{\psi_{2,t}}+\overline{\psi_{2}}\psi_{2,t}\right)dxdt\\\
=&\frac{1}{2}s\iint_{Q_{T}}\eta_{2,tt}|\psi_{2}|^{2}dxdt,\end{split}$

which satisfies

$|I_{31}|\leq CsT^{2}\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt,$

using the boundness of $\eta_{2,tt}$ from (2.7).

For the last quantity $I_{33}$, since $\eta_{2,t}$ and $\eta_{2,x}$ are real
valued functions, we get that

$I_{33}=s^{3}\frac{\gamma_{2}}{\sigma}\textnormal{Re}\left(i\iint\eta_{2,t}|\eta_{2,x}|^{2}|\psi_{2}|^{2}dxdt\right)=0.$

Adding each peach of $I_{13}$, $I_{23}$, $I_{31}$ and $I_{33}$, we have

(2.24)
$\begin{split}I_{13}+I_{23}+I_{31}+I_{33}\geq&2s^{3}\lambda^{4}c_{2}^{4}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}^{3}|\psi_{2}|^{2}dxdt-s^{3}\lambda^{3}c_{2}^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
&-CsT^{2}\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt,\end{split}$

achieving the goal of step 3.

$\bullet$ Step 4. Finding an intermediate estimate.

With the previous steps 1,2 and 3 in hand, adding all the terms $I_{kl}$ for
$1\leq k,l\leq 3$, that is (2.18)–(2.19), (2.22)–(2.23), and (2.24), we can
get the following

(2.25) $\begin{split}\sum_{1\leq k,l\leq
3}I_{kl}\geq&2s^{3}\lambda^{4}c_{2}^{4}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}^{3}|\psi_{2}|^{2}dxdt+2s\lambda^{2}c_{2}^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt\\\
&-CsT^{2}\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt-\frac{C}{\sigma^{2}}s^{2}\lambda^{2}T\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt\\\
&-CT\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt-\frac{C}{\epsilon}T^{4}s\lambda^{4}\frac{\gamma_{2}^{2}}{\sigma^{2}}\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt\\\
&-\frac{\gamma_{2}^{2}}{\sigma^{2}}\epsilon
s\lambda^{2}\iint_{Q_{T}}\xi_{2}|\psi_{2,x}|^{2}dxdt-s^{3}\lambda^{3}c_{2}^{3}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
&-s\lambda
c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,1)|\psi_{2,x}(t,1)|^{2}dt+s\lambda
c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,0)|\psi_{2,x}(t,0)|^{2}dt\\\
&-s\lambda^{2}c_{2}^{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\textnormal{Re}\int_{0}^{T}\xi_{2}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt\\\
&+s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\psi_{2}(t,1)\overline{\psi_{2,t}(t,1)}\eta_{2,x}(t,1)dt\\\
&-s\frac{\gamma_{2}}{\sigma}\textnormal{Im}\int_{0}^{T}\eta_{2,t}(t,1)\psi_{2}(t,1)\overline{\psi_{2,x}(t,1)}dt.\end{split}$

Following a similar approach as the case of $\psi_{2}$, we can compute the
same for $\psi_{1}$, for this, let us denote

$\textnormal{Re}\iint_{Q_{T}}M_{1}\psi_{1}\overline{M_{2}\psi_{1}}dxdt=\sum_{1\leq
k,l\leq 3}H_{kl}$

In this case, one can find

(2.26) $\begin{split}\sum_{1\leq k,l\leq
3}H_{kl}\geq&2s^{3}\lambda^{4}c_{1}^{4}\gamma_{1}^{2}\iint_{Q_{T}}\xi_{1}^{3}|\psi_{1}|^{2}dxdt+2s\lambda^{2}c_{1}^{2}\gamma_{1}^{2}\iint_{Q_{T}}\xi_{1}|\psi_{1,x}|^{2}dxdt\\\
&-CsT^{2}\iint_{Q_{T}}\xi^{3}_{1}|\psi_{1}|^{2}dxdt-
Cs^{2}\lambda^{2}T\iint_{Q_{T}}\xi^{3}_{1}|\psi_{1}|^{2}dxdt\\\
&-CT\iint_{Q_{T}}\xi_{1}|\psi_{1,x}|^{2}dxdt-\frac{C}{\epsilon}T^{4}s\lambda^{4}\gamma_{1}^{2}\iint_{Q_{T}}\xi^{3}_{1}|\psi_{1}|^{2}dxdt\\\
&-\epsilon
s\lambda^{2}\iint_{Q_{T}}\xi_{1}|\psi_{1,x}|^{2}dxdt-s^{3}\lambda^{3}c_{1}^{3}\gamma_{1}^{2}\int_{0}^{T}\xi^{3}_{1}(t,1)|\psi_{1}(t,1)|^{2}dt\\\
&-s\lambda
c_{1}\gamma_{1}^{2}\int_{0}^{T}\xi_{1}(t,1)|\psi_{1,x}(t,1)|^{2}dt+s\lambda
c_{1}\gamma_{1}^{2}\int_{0}^{T}\xi_{1}(t,0)|\psi_{1,x}(t,0)|^{2}dt\\\
&-s\lambda^{2}c_{1}^{2}\gamma_{1}^{2}\textnormal{Re}\int_{0}^{T}\xi_{1}(t,1)\psi_{1}(t,1)\overline{\psi_{1,x}(t,1)}dt\\\
&+s\gamma_{1}\textnormal{Im}\int_{0}^{T}\psi_{1}(t,1)\overline{\psi_{1,t}(t,1)}\eta_{1,x}(t,1)dt\\\
&-s\gamma_{1}\textnormal{Im}\int_{0}^{T}\eta_{1,t}(t,1)\psi_{1}(t,1)\overline{\psi_{1,x}(t,1)}dt.\end{split}$

We are now in a position to combine the above estimate. For simplicity, we
denote $\nu_{1}=\gamma_{1}$ and $\nu_{2}=\frac{\gamma_{2}}{\sigma}.$ Choose
$\epsilon>0$ small enough in (2.25) and (2.26) so that, combining those
estimates, we have

(2.27)
$\begin{split}&\frac{1}{2}\sum_{j=1}^{2}\iint_{Q_{T}}\left(|M_{1}\psi_{j}|^{2}+|M_{2}\psi_{j}|^{2}\right)dxdt\\\
&+\sum_{j=1}^{2}2s^{3}\lambda^{4}c_{j}^{4}\nu_{j}^{2}\iint_{Q_{T}}\,\xi_{j}^{3}|\psi_{j}|^{2}dxdt+\sum_{j=1}^{2}2s\lambda^{2}\nu_{j}\iint_{Q_{T}}\xi_{j}|\psi_{j,x}|^{2}dxdt\\\
&-C\left(\frac{1}{\sigma^{2}}s^{2}\lambda^{2}T+T^{4}s\lambda^{4}\frac{\gamma^{2}}{\sigma^{2}}\right)\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt-
CT\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2,x}|^{2}dxdt\\\
&-C\left(s^{2}\lambda^{2}T+T^{4}s\lambda^{4}\gamma_{1}^{2}\right)\iint_{Q_{T}}\xi^{3}_{1}|\psi_{1}|^{2}dxdt-
CT\iint_{Q_{T}}\xi^{3}_{1}|\psi_{1,x}|^{2}dxdt\\\
&-s^{3}\lambda^{3}\sum_{j=1}^{2}c_{j}^{3}\nu_{j}^{2}\int_{0}^{T}\xi^{3}_{j}(t,1)|\psi_{j}(t,1)|^{2}dt-s\lambda\sum_{j=1}^{2}c_{j}\nu_{j}^{2}\int_{0}^{T}\xi_{j}(t,1)|\psi_{j,x}(t,1)|^{2}dt\\\
&+s\lambda\sum_{j=1}^{2}c_{j}\nu_{j}^{2}\int_{0}^{T}\xi_{j}(t,0)|\psi_{j,x}(t,0)|^{2}dt-s\lambda^{2}\sum_{j=1}^{2}c_{j}^{2}\nu_{j}^{2}\textnormal{Re}\int_{0}^{T}\xi_{j}(t,1)\psi_{j}(t,1)\overline{\psi_{j,x}(t,1)}dt\\\
&+s\sum_{j=1}^{2}\nu_{j}\textnormal{Im}\int_{0}^{T}\psi_{j}(t,1)\overline{\psi_{j,t}(t,1)}\eta_{j,x}(t,1)dt\\\
&-s\sum_{j=1}^{2}\nu_{j}\textnormal{Im}\int_{0}^{T}\eta_{j,t}(t,1)\psi_{j}(t,1)\overline{\psi_{j,x}(t,1)}dt\leq
C\sum_{j=1}^{2}\iint_{Q_{T}}|F_{j}|^{2}dxdt.\end{split}$

In the above, it is clear that there exist some positive constants
$\lambda_{0}$, $\mu_{0}$ such that if we choose $\lambda\geq\lambda_{0}$ and
$s\geq s_{0}:=\mu_{0}(T+T^{2})$, then the lower order integrals

$\displaystyle
C\left(\frac{1}{\sigma^{2}}s^{2}\lambda^{2}T+T^{4}s\lambda^{4}\frac{\gamma^{2}}{\sigma^{2}}\right)\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2}|^{2}dxdt,$
$\displaystyle
C\left(s^{2}\lambda^{2}T+T^{4}s\lambda^{4}\gamma_{1}^{2}\right)\iint_{Q_{T}}\xi^{3}_{1}|\psi_{1}|^{2}dxdt,$
$\displaystyle CT\iint_{Q_{T}}\xi^{3}_{2}|\psi_{2,x}|^{2}dxdt,$

and

$\displaystyle CT\iint_{Q_{T}}\xi^{3}_{1}|\psi_{1,x}|^{2}dxdt,$

can be absorbed by the corresponding leading integrals

$\displaystyle\sum_{j=1}^{2}2s^{3}\lambda^{4}c_{j}^{4}\nu_{j}^{2}\iint_{Q_{T}}\,\xi_{j}^{3}|\psi_{j}|^{2}dxdt\
\text{ and }\
\sum_{j=1}^{2}2s\lambda^{2}\nu_{j}\iint_{Q_{T}}\xi_{j}|\psi_{j,x}|^{2}dxdt.$

Now, it remains to find the proper estimates for the boundary integrals. We do
this in the following step.

$\bullet$ Step 5. Computing the boundary integrals.

We split this step into several parts. We hereby recall the fact that

$\xi_{1}(t,1)=\xi_{2}(t,1),\ \ \eta_{1}(t,1)=\eta_{2}(t,1),\ \
\psi_{1}(t,1)=\psi_{2}(t,1),$

and thus, from now onwards, we shall replace all the quantities
$\xi_{1}(t,1)$, $\eta_{1}(t,1)$ and $\psi_{1}(t,1)$ by $\xi_{2}(t,1)$,
$\eta_{2}(t,1)$ and $\psi_{2}(t,1)$ respectively. Moreover, we will denote the
six boundary terms in (LABEL:carleman_lemma) in the order presented above as
$B_{k}$ given by

$B_{k}=\sum_{j=1,2}B_{k,j},\quad 1\leq k\leq 6.$

Let us now give the details of each part.

– Part A: Computing $B_{1}:=B_{11}+B_{12}$.

First notice that, the term

$B_{12}=-s^{3}\lambda^{3}c^{3}_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt$

is positive, since $c_{2}<0.$ We need to absorb the negative term

$B_{11}=-s^{3}\lambda^{3}c^{3}_{1}\gamma_{1}^{2}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt$

by the term $B_{12}.$ To do that, since we have $c_{1}=1$ and, by (2.4) that

$\sigma=-\frac{c_{2}\gamma_{2}}{\gamma_{1}}$

we have, by adding $B_{11}$ and $B_{12}$, that

(2.28) $\displaystyle B_{1}$ $\displaystyle=B_{11}+B_{12}$
$\displaystyle=s^{3}\lambda^{3}\left(|c_{2}|^{3}\frac{\gamma^{2}_{2}}{\sigma^{2}}-\gamma_{1}^{2}\right)\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt$
$\displaystyle=s^{3}\lambda^{3}\gamma_{1}^{2}\left(|c_{2}|-1\right)\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt,$

which has a positive sign since by choice we have $|c_{2}|>3$.

– Part B: $B_{2}:=B_{21}+B_{22}$ estimate.

Now, on one hand, the integral

$B_{22}=-s\lambda
c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,1)|\psi_{2,x}(t,1)|^{2}dt$

is positive since $c_{2}<0$ and we shall use this term to absorb the negative
integral

$B_{21}=-s\lambda
c_{1}\gamma_{1}^{2}\int_{0}^{T}\xi_{2}(t,1)|\psi_{1,x}(t,1)|^{2}dt.$

To see that, thanks to the boundary conditions (2.10), we can ensure that

$\gamma_{1}\psi_{1,x}(t,1)=-\alpha\psi_{2}(t,1)-\frac{\gamma_{2}}{\sigma}\psi_{2,x}(t,1)+s\lambda\xi_{2}(t,1)\psi_{2}(t,1)\left(\gamma_{1}c_{1}+\frac{\gamma_{2}}{\sigma}c_{2}\right).$

Then, by using Young’s inequality, we obtain

(2.29) $\begin{split}B_{21}=&s\lambda
c_{1}\int_{0}^{T}\xi_{2}(t,1)|\gamma_{1}\psi_{1,x}(t,1)|^{2}dt\\\
\leq&3s\lambda
c_{1}\int_{0}^{T}\xi_{2}(t,1)\left|\frac{\gamma_{2}}{\sigma}\psi_{2,x}(t,1)\right|^{2}dt+3s\lambda
c_{1}\alpha^{2}\int_{0}^{T}\xi_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
&+3s^{3}\lambda^{3}c_{1}\left(\gamma_{1}c_{1}+\frac{\gamma_{2}}{\sigma}c_{2}\right)^{2}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
\leq&3s\lambda
c_{1}\int_{0}^{T}\xi_{2}(t,1)\left|\frac{\gamma_{2}}{\sigma}\psi_{2,x}(t,1)\right|^{2}dt+CT^{4}s\lambda
c_{1}\alpha^{2}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
&+3s^{3}\lambda^{3}c_{1}\left(\gamma_{1}c_{1}+\frac{\gamma_{2}}{\sigma}c_{2}\right)^{2}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt.\end{split}$

So, adding $B_{21}$ and $B_{22}$, the fact that $c_{1}=1$, and the choice of
$\sigma$ given by (2.4), yields that

(2.30) $\begin{split}B_{2}=&B_{21}+B_{22}\\\
\geq&\frac{s\lambda\gamma_{1}^{2}}{c_{2}^{2}}\left(|c_{2}|-3\right)\int_{0}^{T}\xi_{2}(t,1)\left|\psi_{2,x}(t,1)\right|^{2}dt-
CT^{4}s\lambda\alpha^{2}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt.\end{split}$

Note that, here we exactly need the assumption $|c_{2}|>3$ (in other words,
$\kappa>3$ in (2.4) since $c_{2}=-\kappa$) to make the first integral of
(2.30) positive.

– Part C: Computing each term of $B_{3}:=B_{31}+B_{32}$, that is, $B_{31}$ and
$B_{32}$.

Let us look at these terms separately. We have

(2.31) $\displaystyle B_{31}=s\lambda
c_{1}\gamma_{1}^{2}\int_{0}^{T}\xi_{1}(t,0)|\psi_{1,x}(t,0)|^{2}dt\geq 0,$

since $c_{1}=1$ and $\xi_{1}(t,0)>0.$ Now, using the expression (2.9), we find

$\psi_{2,x}(t,x)=e^{-s\eta_{2}(t,x)}\varphi_{2,x}(t,x)-se^{-s\eta_{2}(t,x)}\eta_{2,x}(t,x)\varphi_{2}(t,x).$

But $\varphi_{2}(t,0)=0$ and so,

$\psi_{2,x}(t,0)=e^{-s\eta_{2}(t,0)}\varphi_{2,x}(t,0).$

Therefore, we have

$\begin{split}|B_{32}|&=s\lambda\left|c_{2}\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,0)e^{-2s\eta_{2}(t,0)}|\varphi_{2,x}(t,0)|^{2}dt\right|\\\
&\leq\frac{s\lambda\gamma_{1}^{2}}{|c_{2}|}\int_{0}^{T}\xi_{2}(t,0)e^{-2s\eta_{2}(t,0)}|\varphi_{2,x}(t,0)|^{2}dt,\end{split}$

which is indeed the observation integral for our final Carleman estimate.

– Part D: Analysing the terms $B_{41}$ and $B_{42}$.

Next, we will compute the fourth boundary term as follows. First, observe that

$\displaystyle|B_{41}|=$
$\displaystyle\left|s\lambda^{2}c_{1}^{2}\gamma_{1}^{2}\textnormal{Re}\int_{0}^{T}\xi_{2}(t,1)\psi_{2}(t,1)\overline{\psi_{1,x}(t,1)}dt\right|$
$\displaystyle\leq$ $\displaystyle\epsilon
s\lambda\int_{0}^{T}\xi_{2}(t,1)|\gamma_{1}{\psi_{1,x}(t,1)}|^{2}dt+\frac{C}{\epsilon}s\lambda^{3}T^{4}\int_{0}^{T}\xi_{2}^{3}(t,1)|{\psi_{2}(t,1)}|^{2}dt,$

where $C=C(\gamma_{1})>0$. Here we have used Young’s inequality, the estimate
$\xi_{2}\leq CT^{4}\xi_{2}^{3}$, and the fact that $c_{1}=1$. In the above
estimate, we apply (2.29), so that one has

(2.32) $\begin{split}|B_{41}|\leq&3\epsilon
s\lambda\int_{0}^{T}\xi_{2}(t,1)\left|\frac{\gamma_{2}}{\sigma}{\psi_{2,x}(t,1)}\right|^{2}dt\\\
&+\left(C\epsilon
T^{4}\alpha^{2}s\lambda+\frac{C}{\epsilon}s\lambda^{3}T^{4}\right)\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
=&\frac{3\epsilon
s\lambda\gamma_{1}^{2}}{c_{2}^{2}}\int_{0}^{T}\xi_{2}(t,1)\left|\psi_{2,x}(t,1)\right|^{2}dt\\\
&+\left(C\epsilon
T^{4}\alpha^{2}s\lambda+\frac{C}{\epsilon}s\lambda^{3}T^{4}\right)\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt,\end{split}$

where we have used the choice of $\sigma$ as given by (2.4) and $c_{2}$ as
(2.5).

Similarly, one can estimate the term $B_{42}$ as follows:

(2.33) $\begin{split}|B_{42}|\leq&\epsilon
s\lambda|c_{2}|\frac{\gamma_{2}^{2}}{\sigma^{2}}\int_{0}^{T}\xi_{2}(t,1)|{\psi_{2,x}(t,1)}|^{2}dt+\frac{C}{\epsilon}s\lambda^{3}T^{4}\int_{0}^{T}\xi_{2}^{3}(t,1)|{\psi_{2}(t,1)}|^{2}dt\\\
=&\frac{\epsilon
s\lambda\gamma_{1}^{2}}{|c_{2}|}\int_{0}^{T}\xi_{2}(t,1)|{\psi_{2,x}(t,1)}|^{2}dt+\frac{C}{\epsilon}s\lambda^{3}T^{4}\int_{0}^{T}\xi_{2}^{3}(t,1)|{\psi_{2}(t,1)}|^{2}dt,\end{split}$

finalizing this part.

– Part E: Analysis of $B_{5}:=B_{51}+B_{52}$.

Let us look into the terms of $B_{5}.$ To do that, these terms can be viewed
as

(2.34) $\begin{split}B_{5}=&B_{51}+B_{52}\\\
=&s\sum_{j=1}^{2}\nu_{j}\textnormal{Im}\int_{0}^{T}\psi_{j}(t,1)\overline{\psi_{j,t}(t,1)}\eta_{j,x}(t,1)dt\\\
=&-s\lambda\left(c_{1}\gamma_{1}+c_{2}\frac{\gamma_{2}}{\sigma}\right)\textnormal{Im}\int_{0}^{T}\psi_{1}(t,1)\overline{\psi_{1,t}(t,1)}\xi_{1}(t,1)dt,\end{split}$

where we have used that $\eta_{j,x}=-\lambda\xi_{j}c_{j}.$

It is important to point out that this term is difficult to absorb in terms of
the leading terms because of the appearance time derivative term
$\overline{\psi_{1,t}(t,1)}$, and this is the main reason why we have chosen
the parameter $\sigma$ as in (1.8) (in other words, (2.4)) and $c_{2}=-\kappa$
in (2.5). Thanks to those choices, one readily has

$\left(c_{1}\gamma_{1}+\frac{c_{2}\gamma_{2}}{\sigma}\right)=0,$

once $c_{1}=1$, and this makes the quantity $B_{5}$ equal zero.

– Part F: Estimates to $B_{61}$ and $B_{62}$.

Finally, we compute the terms $B_{6}$. Using that

$|\eta_{j,t}|\leq
CT\xi^{2}_{j}\quad\text{and}\quad\sigma=\frac{\kappa\gamma_{2}}{\gamma_{1}}=-\frac{c_{2}\gamma_{2}}{\gamma_{1}},$

we get

(2.35) $\displaystyle|B_{62}|$ $\displaystyle\leq
CTs\frac{\gamma_{2}}{\sigma}\int_{0}^{T}|\psi_{2}(t,1)||{\psi_{2,x}(t,1)}||\xi_{2}(t,1)|^{2}dt$
$\displaystyle\leq\frac{\gamma^{2}_{1}}{c^{2}_{2}}\int_{0}^{T}\xi_{2}(t,1)\left|{\psi_{2,x}(t,1)}\right|^{2}dt+Cs^{2}T^{2}\int_{0}^{T}\xi_{2}^{3}(t,1)|{\psi_{2}(t,1)}|^{2}dt.$

Using the expression of $\gamma_{1}\psi_{1,x}(t,1)$ from (2.10), we further
get

(2.36) $\displaystyle|B_{61}|\leq$ $\displaystyle
CTs\gamma_{1}\int_{0}^{T}|\psi_{2}(t,1)||{\psi_{1,x}(t,1)}|\xi^{2}_{2}(t,1)dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\xi_{2}(t,1)|\gamma_{1}{\psi_{1,x}(t,1)}|^{2}dt+Cs^{2}T^{2}\int_{0}^{T}\xi_{2}^{3}(t,1)|{\psi_{2}(t,1)}|^{2}dt$
$\displaystyle\leq$
$\displaystyle\frac{3\gamma_{1}^{2}}{c_{2}^{2}}\int_{0}^{T}\xi_{2}(t,1)\left|{\psi_{2,x}(t,1)}\right|^{2}dt+C\left(\alpha^{2}T^{4}+s^{2}T^{2}\right)\int_{0}^{T}\xi_{2}^{3}(t,1)|{\psi_{2}(t,1)}|^{2}dt.$

Here, thanks to the quantity (2.29) we can find the estimate for the following
term

$\int_{0}^{T}\xi_{2}(t,1)|\gamma_{1}\psi_{1,x}(t,1)|^{2}dt.$

$\bullet$ Step 6. Getting rid of the lower-order boundary integrals.

Finally, thanks to the computation of the boundary terms (step 5), we can add
all the parts of this step to have the estimate for $B_{k}$ for $1\leq k\leq
6$. Putting together (2.28), (2.30), (2.31), (2.32), (2.33), (2.34), (2.35)
and (2.36), we find that

(2.37)
$\begin{split}\sum_{k=1}^{6}B_{k}\geq&s^{3}\lambda^{3}\gamma_{1}^{2}\left(|c_{2}|-1\right)\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
&+\frac{s\lambda\gamma^{2}_{1}}{c^{2}_{2}}\left(|c_{2}|-3\right)\int_{0}^{T}\xi_{2}(t,1)\left|\psi_{2,x}(t,1)\right|^{2}dt\\\
&-CT^{4}s\lambda\alpha^{2}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
&-\frac{\epsilon
s\lambda\gamma_{1}^{2}}{c_{2}^{2}}\left(3+|c_{2}|\right)\int_{0}^{T}\xi_{2}(t,1)\left|\psi_{2,x}(t,1)\right|^{2}dt\\\
&-\left(C\epsilon
T^{4}\alpha^{2}s\lambda+\frac{2C}{\epsilon}s\lambda^{3}T^{4}\right)\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dt\\\
&-\frac{4\gamma_{1}^{2}}{c_{2}^{2}}\int_{0}^{T}\xi_{2}(t,1)\left|\psi_{2,x}(t,1)\right|^{2}dt\\\
&-C\left(\alpha^{2}T^{4}+s^{2}T^{2}\right)\int_{0}^{T}\xi_{2}^{3}(t,1)|{\psi_{2}(t,1)}|^{2}dt\\\
&-\frac{s\lambda\gamma_{1}^{2}}{|c_{2}|}s\lambda\int_{0}^{T}\xi_{2}(t,0)e^{-2s\eta_{2}(t,0)}|\varphi_{2,x}(t,0)|^{2}dt.\end{split}$

In the above we fix $\epsilon>0$ small enough and choose
$\lambda\geq\lambda_{0}$ and $s\geq s_{0}=\mu_{0}(T+T^{2})$ (for
$\lambda_{0}>0$, $\mu_{0}>0$ large enough) so that all the lower order
boundary integrals can be absorbed by the first two leading integrals of
(2.37).

As a consequence, from (LABEL:carleman_lemma), one has

$\begin{split}&s^{3}\lambda^{4}\iint_{Q_{T}}\left(\xi^{3}_{1}|\psi_{1}|^{2}+\xi^{3}_{2}|\psi_{2}|^{2}\right)dxdt+s\lambda^{2}\iint_{Q_{T}}\left(\xi_{1}|\psi_{1,x}|^{2}+\xi^{3}_{2}|\psi_{2,x}|^{2}\right)dxdt\\\
&+s^{3}\lambda^{3}\int_{0}^{T}\xi^{3}_{2}(t,1)|\psi_{2}(t,1)|^{2}dxdt+s\lambda\int_{0}^{T}\xi_{2}(t,1)\left|\psi_{2,x}(t,1)\right|^{2}dt\\\
\leq&C\iint_{Q_{T}}(|F_{1}|^{2}+|F_{2}|^{2})dxdt+Cs\lambda\int_{0}^{T}e^{-2s\eta_{2}(t,0)}\xi_{2}(t,0)|\varphi_{2,x}(t,0)|^{2}dt,\end{split}$

for all $\lambda\geq\lambda_{0}$ and $s\geq s_{0}$. Therefore, the previous
inequality gives us the required Carleman estimate (2.8), and the proof of
Theorem 2.5 is complete. ∎

###  2.2. Observability inequality and its application

This section is devoted to proving a suitable observability inequality as a
consequence of the Carleman estimate (2.8), which is the key point to deduce
the null-controllability of the system (1.2).

######  Proposition 2.6.

For any $\zeta:=(\zeta_{1},\zeta_{2})\in\mathcal{H}$, the associated solution
of the adjoint system (2.3)

$\varphi:=(\varphi_{1},\varphi_{2})\in C([0,T];\mathcal{H})$

satisfies the following boundary observation

(2.38) $\|\varphi(0)\|^{2}_{\mathcal{H}}\leq
C\int_{0}^{T}|{\partial_{x}\varphi_{2}}(t,0)|^{2}dt.$

###### Proof.

Consider $\zeta\in\mathcal{H}$. Thanks to the Carleman inequality, given by
Theorem 2.5 to the solution $(\varphi_{1},\varphi_{2})$ of (2.3) with

$L_{1}\varphi_{1}=\alpha_{1}\varphi_{1}\quad\text{and}\quad
L_{2}\varphi_{2}=\frac{\alpha_{2}\varphi_{2}}{\sigma}\quad\left(\sigma=\frac{\kappa\gamma_{2}}{\gamma_{1}}\right),$

gives us that

(2.39)
$\begin{split}&s^{3}\lambda^{4}\iint_{Q_{T}}\left(e^{-2s\eta_{1}}\xi^{3}_{1}|\varphi_{1}|^{2}+e^{-2s\eta_{2}}\xi^{3}_{2}|\varphi_{2}|^{2}\right)dxdt\\\
&+s\lambda^{2}\iint_{Q_{T}}\left(e^{-2s\eta_{1}}\xi_{1}|\varphi_{1,x}|^{2}+e^{-2s\eta_{2}}\xi_{2}|\varphi_{2,x}|^{2}\right)dxdt\\\
\leq&C\iint_{Q_{T}}(e^{-2s\eta_{1}}|\varphi_{1}|^{2}+e^{-2s\eta_{2}}|\varphi_{2}|^{2})dxdt+Cs\lambda\int_{0}^{T}e^{-2s\eta_{2}(t,0)}\xi_{2}(t,0)|\varphi_{2,x}(t,0)|^{2}dt.\end{split}$

Now, using the fact that $1\leq 8T^{6}\xi^{3}_{j}$ for $j=1,2$, we can easily
absorb the first two integrals of the r.h.s. of (2.39) by the associated
leading integrals for any $s\geq CT^{2}$. Thus, we have the following
inequality from (2.39),

(2.40)
$\begin{split}s\lambda^{2}\iint_{Q_{T}}&\left(e^{-2s\eta_{1}}\xi_{1}|\varphi_{1,x}|^{2}+e^{-2s\eta_{2}}\xi_{2}|\varphi_{2,x}|^{2}\right)dxdt\\\
&\leq
Cs\lambda\int_{0}^{T}\xi_{2}(t,0)e^{-2s\eta_{2}(t,0)}|\varphi_{2,x}(t,0)|^{2}dt.\end{split}$

Moreover, for some $m,M>0$, we have the following relations

$\displaystyle e^{-2s\eta_{j}}\xi_{j}\geq m\ \text{ in
}(T/4,3T/4)\times(0,1),\quad\text{and}\quad e^{-2s\eta_{j}}\xi_{j}\leq M\
\text{ in }Q_{T},\ \ \text{for $j=1,2$},$

which together with (2.40), yields that

$\int_{T/4}^{3T/4}\int_{0}^{1}\left(|\varphi_{1,x}|^{2}+|\varphi_{2,x}|^{2}\right)dxdt\leq
C\int_{0}^{T}|\varphi_{2,x}(t,0)|^{2}dt.$

Then, from the estimate given by (A.3), one can deduce that

$\|\varphi(0)\|^{2}_{\mathcal{H}}\leq
C_{T}\int_{T/4}^{3T/4}\int_{0}^{1}\left(|\varphi_{1,x}|^{2}+|\varphi_{2,x}|^{2}\right)dxdt\leq
C\int_{0}^{T}|\varphi_{2,x}(t,0)|^{2}dt,$

and hence the required observability inequality (2.38) follows. ∎

We are now in a position to prove the first main result.

###### Proof of Theorem 1.1.

The proof follows the classical Hilbert uniqueness method introduced by Lions
[25]. Once we have the above observability inequality, then one can prove the
existence of a boundary control $h\in L^{2}(0,T)$ such that $(u,v)$ solutions
of (1.2) with boundary conditions (1.3)-(1.4b) satisfies (1.5). ∎

##  3\. Rapid exponential stabilization

This section is devoted to studying the boundary stabilization issues for
(1.2)-(1.3) with a single boundary control force acting on the component $v$,
namely (1.4b). More precisely, we construct a stationary feedback law $h(t)$,
of the form ${F}_{\omega}(u(t,\cdot),v(t,\cdot))$ such that the solution of
the closed-loop system decays exponentially to zero at any prescribed decay
rate.

The approach employed in this section was first introduced by Komornik in [20]
and has also been studied by Vaste [38] and Urquiza [37], which one will be
applied in our context and is the key argument of this section.

###  3.1. Gramian method

Let us consider the abstract control system

(3.1) $\begin{cases}\dot{y}(t)=\mathcal{A}y(t)+\mathcal{B}h(t),\quad
t\in(0,T),\\\ y(0)=y_{0},\end{cases}$

where $y(t)\in\mathcal{H},y_{0}\in\mathcal{H},h\in L^{2}(0,T)$, $\mathcal{B}$
is an unbounded operator from $\mathbb{C}$ to $\mathcal{H}$.
$\mathcal{A}:\mathcal{D(A)}\subset\mathcal{H}\to\mathcal{H}$ is an unbounded
operator and $\mathcal{D}(\mathcal{A})$ is dense in $\mathcal{H}.$ To employ
the method of Urquiza, one needs to take the following assumptions on the
operator $\mathcal{A}$ and $\mathcal{B}$

  * (H1)

The skew-adjoint operator $\mathcal{A}$ is an infinitesimal generator of a
strongly continuous group $\\{e^{t\mathcal{A}}\\}_{t\in\mathbb{R}}$ on
$\mathcal{H}$.

  * (H2)

The operator $\mathcal{B}:\mathbb{C}\to\mathcal{D}(\mathcal{A}^{*})^{\prime}$
is linear and continuous.

  * (H3)

(Regularity property) For every $T>0$ there exists $C(T)>0$ such that

$\int_{0}^{T}\left|\mathcal{B}^{*}e^{-t\mathcal{A}^{*}}y\right|^{2}dt\leq
C\left\|y\right\|^{2}_{\mathcal{H}},\quad\forall\,y\in\mathcal{D}(\mathcal{A}^{*}).$

  * (H4)

(Controllability property) There are two constants $T>0$ and $c(T)>0$ such
that

$\int_{0}^{T}\left|\mathcal{B}^{*}e^{-t\mathcal{A}^{*}}y\right|^{2}dt\geq
c\left\|y\right\|^{2}_{\mathcal{H}},\quad\forall\,y\in\mathcal{D}(\mathcal{A}^{*}).$

With these hypotheses in hand, the next result holds (for details, see [37],
Theorem $2.1$). Its proof mainly relies on general results about the algebraic
Riccati equation associated with the linear quadratic regulator problem (see
[16]).

######  Theorem 3.1.

Consider operators $\mathcal{A}$ and $\mathcal{B}$ under assumptions
(H1)-(H4). For any $\omega>0$, we have

  * (i)

The symmetric positive operator $\Lambda_{\omega}$ defined by

$\left<{\Lambda_{\omega}x},{z}\right>_{\mathcal{H}}=\int_{0}^{\infty}\left<{\mathcal{B}^{*}e^{-\tau(\mathcal{A}+\omega
I)^{*}}x},{\mathcal{B}^{*}e^{-\tau(\mathcal{A}+\omega
I)^{*}}z}\right>_{\mathbb{C}}d\tau,\,\forall\,x,z\in\mathcal{H}$

is coercive and is an isomorphism on $\mathcal{H}$.

  * (ii)

Let $F_{\omega}:=-\mathcal{B}^{*}\Lambda_{\omega}^{-1}$. The operator
$\mathcal{A}+\mathcal{B}F_{\omega}$ with
$\mathcal{D}(\mathcal{A}+\mathcal{B}F_{\omega})=\Lambda_{\omega}(\mathcal{D}(\mathcal{A}^{*}))$
is the infinitesimal generator of a strongly continuous semigroup on
$\mathcal{H}$.

  * (iii)

The closed-loop system (3.1) with $h=F_{\omega}(y)$ is exponentially stable,
that is,

$\left\|e^{t(\mathcal{A}+\mathcal{B}F_{\omega})}y\right\|_{\mathcal{H}}\leq
Ce^{\left(-2\omega+g(-\mathcal{A})\right)t}\left\|y\right\|_{\mathcal{H}},\forall
y\in\mathcal{H},$

where $C$ is a positive constant.

We will use Theorem 3.1 to prove the exponential stabilization of the coupled
Schrodinger equation (1.2)-(1.3)-(1.4b) with boundary feedback law. To apply
it, we need to verify all the assumptions (H1)–(H4) hold for our system (1.2).
Let us do it in the next subsection.

###  3.2. Verification of the hypotheses

It is well-known that a fundamental solution of the Schrödinger system can be
obtained by the Fourier expansion, see for instance [12, 18]. So, in this way,
considering the eigenvalues and the eigenfunctions that form an orthonormal
basis of $L^{2}(\Omega)$, we can define in $\mathcal{H}$ an inner product
similarly as in [9].

Notice that we can also find a representation by Fourier series for the
solutions of the system (1.2) (see e.g. [18], for one-dimensional Schrödinger
equation) and that the operator $\mathcal{A}$, defined by (A.1)-(A.2), is
skew-adjoint and hence generates an infinitesimal generator of a group
$\\{S(t)\\}_{t\in\mathbb{R}}$, thus (H1) follows. Also note that
$g(-\mathcal{A})=0$, where $g$ is the growth bound of the semigroup generated
by $\mathcal{A}$. Moreover, comparing the abstract system (3.1) with our
system (1.2), the control operator
$\mathcal{B}\in\mathcal{L}(\mathbb{C};\mathcal{D}(\mathcal{A}^{*})^{\prime})$
can be given as follows

$\left<{\mathcal{B}s},{(\phi_{1},\phi_{2})}\right>_{\mathcal{D}(\mathcal{A}^{*})^{\prime},\mathcal{D}(\mathcal{A}^{*})}=s\phi_{2}^{\prime}(0),\
\ s\in\mathbb{C},\ (\phi_{1},\phi_{2})\in\mathcal{D}(\mathcal{A}^{*}),$

and therefore, (H2) is verified. Additionally, note that the observability
inequality (2.38) gives directly (H4).

It remains for us to prove the hypothesis (H3), that is, to prove the trace
regularity. This hypothesis follows from the next proposition and is a
consequence of Lemma A.4.

######  Proposition 3.2.

For every $T>0$ there exists $C>0$ such that the following holds

(3.2) $\int_{0}^{T}|\varphi_{2,x}(t,0)|^{2}dt\leq
C\left\|(\zeta_{1},\zeta_{2})\right\|_{\mathcal{H}},$

for every solution $(\varphi_{1},\varphi_{2})$ of the adjoint problem (2.3)
with $(\zeta_{1},\zeta_{2})$ lies in sufficiently regular space.

###### Proof.

Choose $m\in C^{2}(0,1)$ such that $m(1)=0,m(0)>0$, and $m^{\prime}(1)=1$.
Thus, from (A.4), it follows that

(3.3)
$\begin{split}\frac{1}{2}\sum_{j=1}^{2}\nu_{j}&\int_{0}^{T}|\varphi_{j,x}(t,0)|^{2}m(0)dt\leq
C_{1}\left\|m\right\|_{L^{\infty}(0,1)}\bigg{[}\left\|(\varphi_{1}(T),\varphi_{2}(T))\right\|^{2}_{L^{2}(0,1)}\\\
&+\left\|(\varphi_{1}(T),\varphi_{2}(T))\right\|^{2}_{\mathcal{H}}+\left\|(\varphi_{1}(0),\varphi_{2}(0))\right\|^{2}_{L^{2}(0,1)}+\left\|(\varphi_{1}(0),\varphi_{2}(0))\right\|^{2}_{\mathcal{H}}\bigg{]}\\\
&+C_{2}\left\|m\right\|_{W^{1,\infty}(0,1)}\left(\int_{0}^{T}\left\|(\varphi_{1}(t),\varphi_{2}(t))\right\|^{2}_{\mathcal{H}}dt\right)\\\
&+C_{3}\left\|m\right\|_{W^{2,\infty}(0,1)}\left(\int_{0}^{T}\left\|(\varphi_{1}(t),\varphi_{2}(t))\right\|_{L^{2}(0,1)}\left\|(\varphi_{1}(t),\varphi_{2}(t))\right\|_{\mathcal{H}}dt\right).\end{split}$

Note that, using boundary conditions of (2.3), one can prove that the last
term of the identity (A.4) is negative. Moreover, using the classical
conservation of the energy for the Schrödinger equation we have:

$\displaystyle\left\|(\varphi_{1}(t),\varphi_{2}(t))\right\|_{L^{2}(0,1)}=\left\|(\varphi_{1}(0),\varphi_{2}(0))\right\|_{L^{2}(0,1)},\forall
t\in[0,T],$
$\displaystyle\left\|(\varphi_{1}(t),\varphi_{2}(t))\right\|_{\mathcal{H}}=\left\|(\varphi_{1}(0),\varphi_{2}(0))\right\|_{\mathcal{H}},\forall
t\in[0,T].$

Therefore from (3.3), we have the following result (3.2) when
$(\zeta_{1},\zeta_{2})$ is sufficiently smooth. Using a density argument we
get the result when $(\zeta_{1},\zeta_{2})\in\mathcal{H}$, showing the lemma,
and consequently (H3) is achieved. ∎

###  3.3. Proof of Theorem 1.2

In this section, we employ Urquiza’s method to construct the feedback law for
our system (1.2) to ensure the exponential decay (1.9).

To do that, let us consider $\Phi^{1}_{0}=(\varphi_{0},\psi_{0})$ and
$\Phi^{2}_{0}=(r_{0},s_{0})$ belonging of $\mathcal{H}$, and define the
following bilinear form

$a_{w}(\Phi^{1}_{0},\Phi^{2}_{0})=\int_{0}^{\infty}e^{-2\omega
t}\psi_{x}(t,0)\overline{s_{x}(t,0)}\,dt,$

where $\Phi^{1}=(\varphi,\psi)$ and $\Phi^{2}=(r,s)$ are the solutions of the
following systems respectively

$\begin{cases}i\varphi_{t}+\gamma_{1}\varphi_{xx}-\alpha_{1}\varphi=0,&\text{
in }Q_{T},\\\ i\sigma\psi_{t}+\gamma_{2}\psi_{xx}-\alpha_{2}\psi=0,&\text{ in
}Q_{T},\\\ \varphi(t,1)=\psi(t,1),&\text{ in }(0,T),\\\
\gamma_{1}\varphi_{x}(t,1)+\frac{\gamma_{2}}{\sigma}\psi_{x}(t,1)+\alpha\varphi(t,1)=0,&\text{
in }(0,T),\\\ \varphi(t,0)=0,\,\psi(t,0)=0,&\text{ in }(0,T),\\\
\varphi(T,x)=\varphi_{0}(x),\ \psi(T,x)=\psi_{0}(x),&\text{ in
}\Omega\end{cases}$

and

$\begin{cases}ir_{t}+\gamma_{1}r_{xx}-\alpha_{1}r=0,&\text{ in }Q_{T},\\\
i\sigma s_{t}+\gamma_{2}s_{xx}-\alpha_{2}s=0,&\text{ in }Q_{T},\\\
r(t,1)=s(t,1),&\text{ in }(0,T),\\\
\gamma_{1}r_{x}(t,1)+\frac{\gamma_{2}}{\sigma}s_{x}(t,1)+\alpha
r(t,1)=0,&\text{ in }(0,T),\\\ r(t,0)=0,\,s(t,0)=0,&\text{ in }(0,T),\\\
r(T,x)=r_{0}(x),\ s(T,x)=s_{0}(x),&\text{ in }\Omega.\end{cases}$

Let us define the operator
$\Lambda_{\omega}:\mathcal{H}\to\mathcal{H}^{\prime}$ satisfying the following

(3.4)
$\left<{\Lambda_{\omega}\Phi^{1}_{0}},{\Phi^{2}_{0}}\right>_{\mathcal{H}^{\prime},\mathcal{H}}=a_{\omega}(\Phi^{1}_{0},\Phi^{2}_{0}),\quad\forall\Phi^{1}_{0},\Phi^{2}_{0}\in\mathcal{H}.$

Next, we see that

$\begin{split}a_{w}(\Phi^{1}_{0},\Phi^{2}_{0})=&\int_{0}^{\infty}e^{-2\omega
t}\psi_{x}(t,0)\overline{s_{x}(t,0)}\,dt\\\ =&\int_{0}^{\infty}e^{-2\omega
t}{\mathcal{B}^{*}}\Phi^{1}(t)\,\,\overline{{\mathcal{B}^{*}}\Phi^{2}(t)}\,dt\\\
=&\int_{0}^{\infty}e^{-2\omega
t}\left({\mathcal{B}^{*}}S(T-t)^{*}S(-T)^{*}\Phi^{1}_{0}\right)\overline{\left({\mathcal{B}^{*}}S(T-t)^{*}S(-T)^{*}\Phi^{2}_{0}\right)}\,dt\\\
=&\int_{0}^{\infty}e^{-2\omega
t}\left({\mathcal{B}^{*}}S(-t)^{*}\Phi^{1}_{0}\right)\overline{\left({\mathcal{B}^{*}}S(-t)^{*}\Phi^{2}_{0}\right)}\,dt.\end{split}$

Therefore from (3.4), we have

$\left<{\Lambda_{\omega}\Phi^{1}_{0}},{\Phi^{2}_{0}}\right>_{\mathcal{H}^{\prime},\mathcal{H}}=\int_{0}^{\infty}e^{-2\omega
t}\left<{{\mathcal{B}^{*}}\mathbb{T}^{*}_{-t}\Phi^{1}_{0}},{\,{\mathcal{B}^{*}}\mathbb{T}^{*}_{-t}\Phi^{2}_{0}}\right>_{\mathbb{C}}\,dt.$

Thanks to Theorem 3.1, the operator $\Lambda_{\omega}$, defined by (3.4), is
coercive and isomorphism. Finally, let us define the operator
${F}_{\omega}:\mathcal{H}\to\mathbb{C}$ by

$\displaystyle{F}_{\omega}(\mathbf{z})=-\psi^{\prime}_{0}(0),$

where $\Phi^{1}_{0}=(\varphi_{0},\psi_{0})$ is the solution of the following
Lax-Milgram problem

$a_{\omega}(\Phi^{1}_{0},\Phi^{2}_{0})=\left<{\mathbf{z}},{\Phi^{2}_{0}}\right>,\,\forall\,\Phi^{2}_{0}\in\mathcal{H}.$

Hence, we obtain

$\left<{\Lambda_{\omega}\Phi^{1}_{0}},{\Phi^{2}_{0}}\right>_{\mathcal{H}^{\prime},\mathcal{H}}=\left<{\mathbf{z}},{\Phi^{2}_{0}}\right>,\forall\,\Phi^{2}_{0}\in\mathcal{H}.$

This gives $\Lambda_{\omega}\Phi^{1}_{0}=\mathbf{z}.$ It follows that
$\Phi^{1}_{0}=\Lambda_{\omega}^{-1}\mathbf{z}.$ Thus we have
${F}_{\omega}=-{\mathcal{B}}^{*}\Lambda_{\omega}^{-1}.$ Thanks to Theorem 3.1,
rapid exponential stabilization for the system (1.2) is established using the
feedback law $h(t)={F}_{\omega}(u(t,\cdot),v(t,\cdot))$. More precisely, we
get a positive constant $C$ such that the solution of (1.2) satisfies the
estimate (1.9), showing Theorem 1.2. ∎

##  4\. Further comments and open problems

In this paper, we considered the boundary control problems for coupled
Schrödinger equations through the Kirchhoff boundary conditions in a one-
dimensional case. The first result is obtained showing a new Carleman estimate
with boundary observation. Moreover, with this in hand, together with other
hypotheses over the operator $\mathcal{A}$, the second result ensures that the
solutions of the system decay exponentially with a decay rate of at least
$e^{-2\omega t}$.

Concerning Theorems 1.1 and 1.2 it is natural to ask whether they remain valid
in nonlinear framework (1.1). Due to the lack of regularity, we are not able
to extend our result for the nonlinear case yet, and this issue remains open.

Finally, we point out that in a forthcoming paper boundary conditions
different of (1.3)-(1.4a)/ (1.4b) will be considered, which will give a more
general view of the boundary control problems, at least, for the system (1.2).

## Statements and Declarations

### Acknowledgments

The work of Kuntal Bhandari was partially supported by the Czech-Korean
project GAČR/22-08633J. Capistrano-Filho was supported by CAPES grants numbers
88881.311964/2018-01 and 88881.520205/2020-01, CNPq grants numbers
307808/2021-1 and 401003/2022-1, MATHAMSUD grant 21-MATH-03 and Propesqi
(UFPE).

Part of this work was done while the second author visited Virginia Tech from
January to July 2023 and Centro de Modelamiento Matemático (Santiago-Chile) in
October 2023. The author thanks both institutions for their warm hospitality.

### Authors contributions:

Bhandari, Capistrano-Filho, Majumdar, and Tanaka work equality in
Conceptualization; formal analysis; investigation; writing–original draft;
writing–review and editing.

### Conflict of interest statement

This work does not have any conflicts of interest.

### Data availability statement

It does not apply to this article as no new data were created or analyzed in
this study.

##  Appendix A Auxiliary results

In this appendix, we briefly discuss the well-posedness of the control system
(1.2). Additionally, to show that the hypothesis (H3) of Urquiza’s approach is
verified, we need a key lemma which will be proved in the second part of this
appendix.

###  A.1. Well-posedness results

Consider the following operator associated with the control system (1.2),
given by

(A.1)
$\displaystyle\mathcal{A}=\begin{pmatrix}i\gamma_{1}\partial_{xx}-i\alpha_{1}\mathbb{I}_{d}&\mathbf{0}\\\
\\\
\mathbf{0}&i\frac{\gamma_{2}}{\sigma}\partial_{xx}-i\frac{\alpha_{2}}{\sigma}\mathbb{I}_{d}\end{pmatrix},$

with

(A.2)
$\begin{split}\mathcal{D}(\mathcal{A})=\big{\\{}(u_{1},u_{2})\in[H^{2}(\Omega)]^{2}\mid\,&u_{1}(0)=u_{2}(0)=0,\
u_{1}(1)=u_{2}(1),\\\
&\gamma_{1}u_{1,x}(1)+\frac{\gamma_{2}}{\sigma}u_{2,x}(1)+\alpha
u_{1}(1)=0\big{\\}}.\end{split}$

Let us consider the space (1.6) as the completion of
$\mathcal{D}(\mathcal{A})$ with the norm (1.7) and the associated inner
product in $\mathcal{H}$ is defined by (1.6).With this in hand, the first
result shows that the operator in consideration is dissipative.

######  Proposition A.1.

The operator $(\mathcal{A},\mathcal{D}(\mathcal{A}))$ generates a strongly
continuous unitary group in $[L^{2}(\Omega)]^{2}$.

###### Proof.

Let us consider $\mathbf{U}=(u,v)\in\mathcal{D}(\mathcal{A})$. A simple
computation gives that

$\begin{split}\textnormal{Re}\left<{\mathcal{A}\mathbf{U}},{\mathbf{U}}\right>_{[L^{2}(0,1)]^{2}}=&\textnormal{Re}\bigg{[}i\gamma_{1}\int_{0}^{1}u_{xx}\overline{u}dx-i\alpha_{1}\int_{0}^{1}|u|^{2}dx+i\frac{\gamma_{2}}{\sigma}\int_{0}^{1}v_{xx}\overline{v}dx-i\frac{\alpha_{2}}{\sigma}\int_{0}^{1}|u|^{2}dx\bigg{]}\\\
=&\textnormal{Re}\bigg{[}-i\gamma_{1}\int_{0}^{1}|u_{x}|^{2}dx-i\frac{\gamma_{2}}{\sigma}\int_{0}^{1}|v_{x}|^{2}dx+i\gamma_{1}u_{x}(1)\overline{u(1)}+i\frac{\gamma_{2}}{\sigma}v_{x}(1)\overline{v(1)}\bigg{]}\\\
=&\textnormal{Re}\bigg{[}i\gamma_{1}u_{x}(1)\overline{u(1)}+i\frac{\gamma_{2}}{\sigma}v_{x}(1)\overline{v(1)}\bigg{]}\\\
=&\textnormal{Re}\bigg{[}-i\alpha|u(1)|^{2}\bigg{]}=0.\end{split}$

By using semigroup theory, $\mathcal{A}$ generates a strongly continuous
unitary group on $[L^{2}(\Omega)]^{2}.$ Moreover it can be easily checked that
for all $(u,v)\in\mathcal{D}(\mathcal{A})$,

$\left<{\mathcal{A}(u,v)},{(u,v)}\right>=-\left<{(u,v)},{\mathcal{A}(u,v)}\right>$

and also $\mathcal{D}(\mathcal{A})=\mathcal{D}(\mathcal{A}^{*}).$ Therefore
$(\mathcal{A},\mathcal{D}(\mathcal{A}))$ is a skew adjoint operator. ∎

Note that, considering $(u_{0},v_{0})\in\mathcal{D(A)}$, then (1.2) has a
classical solution satisfying $(u,v)\in C([0,T];\mathcal{D(A)})\cap
C^{1}([0,T];[L^{2}(\Omega)]^{2})$, or, in more general way:

######  Proposition A.2.

Let $(u_{0},v_{0})\in[H_{0}^{s}(\Omega)]^{2}$, for $s\geq 0$. Then the linear
system (1.2) with boundary conditions (1.3)-(1.4a) (or (1.3)-(1.4b)), with
$h(t)=0,$ has a unique solution $(u,v)$ on the space
$C([0,T];[H_{0}^{s}(\Omega)]^{2})$.

###  A.2. Adjoint system

Let us remember that the adjoint associated with the system (1.2) with
(1.3)–(1.4a) or (1.3)–(1.4b), is given by (2.3), with given final data
$\zeta:=(\zeta_{1},\zeta_{2})$ from some suitable Hilbert space. Note that,
since $\mathcal{A}=-\mathcal{A}^{*}$ we have that
$\mathcal{D(A)}^{*}=\mathcal{D(A)}$, so, the following result shows the well-
posedness for the system (2.3).

######  Proposition A.3.

For given $\zeta:=(\zeta_{1},\zeta_{2})\in\mathcal{H}$, there exists a unique
solution $\varphi:=(\varphi_{1},\varphi_{2})\in C([0,T];\mathcal{H})$ to the
adjoint system (2.3) such that it satisfies

(A.3) $\displaystyle\|\varphi\|_{C([0,T];\mathcal{H})}\leq
C\|\zeta\|_{\mathcal{H}},$

for some constant $C>0$.

###### Proof.

The proof of well-posedness is the same as done in Proposition A.2. ∎

Additionally to that, if we consider $(g_{1},g_{2})\in
L^{1}(0,T;\mathcal{H})$, it can be shown that the solution
$(\varphi_{1},\varphi_{2})\in C([0,T],\mathcal{H})$ of (2.2) satisfies

$\left\|(\varphi_{1}(t),\varphi_{2}(t))\right\|_{\mathcal{H}}\leq
C||(g_{1},g_{2})||_{L^{1}(0,T;\mathcal{H})},$

###  A.3. Key lemma

This second part of this appendix is devoted to presenting an essential lemma
that is a key point in ensuring the hypothesis (H3) in Urquiza’s approach.

######  Lemma A.4.

First, consider a function $m\in C^{2}(0,1)$. Then the solution of (2.3)
satisfies the following identity:

(A.4)
$\begin{split}\frac{1}{2}\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}|\varphi_{j,x}(t,0)|^{2}m(0)dt=&\frac{1}{2}\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}|\varphi_{j,x}(t,1)|^{2}m(1)dt\\\
&-\frac{1}{2}\textnormal{Im}\left(\sum_{j=1}^{2}\int_{0}^{1}\overline{\varphi_{j}(T,x)}m(x)\varphi_{j,x}(T,x)dx\right)\\\
&+\frac{1}{2}\textnormal{Im}\left(\sum_{j=1}^{2}\int_{0}^{1}\overline{\varphi_{j}(0,x)}m(x)\varphi_{j,x}(0,x)dx\right)\\\
&+\frac{1}{2}\textnormal{Im}\left(\sum_{j=1}^{2}\int_{0}^{T}\varphi_{j,t}(t,1)m(1)\overline{\varphi_{j}(t,1)}dt\right)\\\
&-\frac{1}{2}\textnormal{Re}\left(\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}\int_{0}^{1}\varphi_{j,x}m^{\prime}(x)\overline{\varphi_{j,x}(t,x)}dxdt\right)\\\
&-\frac{1}{2}\textnormal{Re}\left(\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}\int_{0}^{1}\varphi_{j,x}m^{\prime\prime}(x)\overline{\varphi_{j}(t,x)}dxdt\right)\\\
&+\frac{1}{2}\textnormal{Re}\left(\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}\varphi_{j,x}(t,1)m^{\prime}(1)\overline{\varphi_{j}(t,1)}dt\right),\\\
&-\frac{1}{2}\textnormal{Re}\left(\sum_{j=1}^{2}\theta_{j}\int_{0}^{T}m(1)|{\varphi_{j}(t,1)}|^{2}dt\right),\end{split}$

where
$\nu_{1}=\gamma_{1},\nu_{2}=\frac{\gamma_{2}}{\sigma},\theta_{1}=\alpha_{1},\theta_{2}=\frac{\alpha_{2}}{\sigma}$.

###### Proof.

Multiply the equations (2.3) by
$(m\overline{\varphi_{1,x}}+\frac{1}{2}\overline{\varphi_{1}}m^{\prime})$ and
$(m\overline{\varphi_{1,x}}+\frac{1}{2}\overline{\varphi_{1}}m^{\prime})$
respectively and using integration by parts along with boundary conditions, we
have

$\begin{split}\textnormal{Re}\sum_{j=1}^{2}i\int_{0}^{T}&\int_{0}^{1}\varphi_{j,t}\left(\frac{1}{2}m^{\prime}(x)\overline{\varphi_{j}(t,x)}+m(x)\overline{\varphi_{j,x}(t,x)}\right)dxdt=\\\
&-\frac{1}{2}\textnormal{Im}\left(\sum_{j=1}^{2}\int_{0}^{1}\overline{\varphi_{j}(T,x)}m(x)\varphi_{j,x}(T,x)dx\right)\\\
&+\frac{1}{2}\textnormal{Im}\left(\sum_{j=1}^{2}\int_{0}^{1}\overline{\varphi_{j}(0,x)}m(x)\varphi_{j,x}(0,x)dx\right)\\\
&+\frac{1}{2}\textnormal{Im}\left(\sum_{j=1}^{2}\int_{0}^{1}\varphi_{j,t}(t,1)m(1)\overline{\varphi_{j}(t,1)}dt\right).\end{split}$

Similarly, we have from the second term of both equations:

$\begin{split}\textnormal{Re}\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}\int_{0}^{1}\varphi_{j,xx}&\left(\frac{1}{2}m^{\prime}(x)\overline{\varphi_{j}(t,x)}+m(x)\overline{\varphi_{j,x}(t,x)}\right)dxdt=\frac{1}{2}\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}|\varphi_{j,x}(t,1)|^{2}m(1)dt\\\
&-\frac{1}{2}\textnormal{Re}\left(\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}\int_{0}^{1}\varphi_{j,x}m^{\prime}(x)\overline{\varphi_{j,x}(t,x)}dxdt\right)\\\
&-\frac{1}{2}\textnormal{Re}\left(\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}\int_{0}^{1}\varphi_{j,x}m^{\prime\prime}(x)\overline{\varphi_{j}(t,x)}dxdt\right)\\\
&+\frac{1}{2}\textnormal{Re}\left(\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}\varphi_{j,x}(t,1)m^{\prime}(1)\overline{\varphi_{j}(t,1)}dt\right)\\\
&-\frac{1}{2}\left(\sum_{j=1}^{2}\nu_{j}\int_{0}^{T}m(0)|{\varphi_{j,x}(t,0)}|^{2}dt\right).\end{split}$

Also, we have

$\textnormal{Re}\sum_{j=1}^{2}\theta_{j}\int_{0}^{T}\int_{0}^{1}\varphi_{j}\left(\frac{1}{2}m^{\prime}(x)\overline{\varphi_{j}(t,x)}+m(x)\overline{\varphi_{j,x}(t,x)}\right)dxdt=\left(\sum_{j=1}^{2}\theta_{j}\int_{0}^{T}m(1)|{\varphi_{j}(t,1)}|^{2}dt\right).$

Putting together the previous relation (A.4) holds and the lemma is finished.
∎

## References

  * [1] F. Alabau-Boussouira, _Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDEs by a single control_ , Math. Control Signals Systems, 26 (2014), 1–46.
  * [2] F. Alabau-Boussouira and M. Léautaud, _Indirect controllability of locally coupled wave-type systems and applications_ , J. Math. Pures Appl., 99 (2013), 544–576.
  * [3] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, and L. de Teresa, _Recent results on the controllability of linear coupled parabolic problems: a survey_ , Math. Control Relat. Fields, 1:3 (2011), 267–306.
  * [4] J. Angulo and F. Linares, _Periodic pulses of coupled nonlinear Schrödinger equations in optics_ , Indiana Univ. Math. J. 56: 2 (2007), 847–878.
  * [5] I. I. Barbosa, _The Cauchy problem for nonlinear quadratic interactions of the Schrödinger type in one-dimensional space_ , Journal of Mathematical Physics, 7:59 (2018).
  * [6] A. Barhoumi, _Rapid pointwise stabilization of vibrating strings and beams_ , Bol. Soc. Paran. de Mat., 27:2 (2009), 43–59.
  * [7] K. Bhandari, F. Boyer, and V. Hernández-Santamaría, _Boundary null-controllability of 1-D coupled parabolic systems with Kirchhoff-type conditions_ , Mathematics of Control, Signals, and Systems, 33 (2021), 413–471.
  * [8] L. Baudouin and J.-P. Puel, _Uniqueness and stability in an inverse problem for the Schrödinger equation_ , Inverse Problems 18 (2001), 1537–1554.
  * [9] R. A. Capistrano-Filho, E. Cerpa and F. Gallego, _Rapid Exponential Stabilization of a Boussinesq System of KdV–KdV Type,_ Communications in Contemporary Mathematics, 25:03 (2023), 2150111.
  * [10] L. Cardoulis and P. Gaitan, _Simultaneous identification of diffusion coefficient and the potential for the Schrödinger operator with only one observation_ , Inverse Problems 26 (2010).
  * [11] E. Cerpa, E. Créépeau, _IRapid exponential stabilization for a linear Korteweg–deVries equation,_ Discrete Contin. Dyn. Syst., Ser. B, 11:3 (2009), 655–668
  * [12] J.-M. Coron, L. Gagnon, and M. Morancey, _Rapid stabilization of a linearized bilinear 1-D Schrödinger equation,_ Journal de Mathématiques Pures et Appliquées, 115 (2018), 24–73.
  * [13] R. DeSalvo, H. Vanherzeele, D. Hagan, M. Sheik-Bahae, G. Stegeman, and E. Van Stryland, _Self-focusing and self-defocusing by cascaded second-order effects in KTP_ , Opt. Lett. 17:1 (1992), 28–30.
  * [14] B. Dehman, J. Le-Rousseau, and M. Léautaud _Controllability of two coupled wave equations on a compact manifold_ , Arch. Ration. Mech. Anal., 211:1 (2014), 113–187.
  * [15] S. Dolecki and D.L. Russell, _A general theory of observation and control_ , SIAM J. Control Opt. 15 (1977), 185–220.
  * [16] F. Flandoli, I. Lasiecka and R. Triggiani, _Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler–Bernoulli boundary control problems_ , Ann. Mat. Pura Appl. 153 (1988), 307–382.
  * [17] N. Hayashi, T. Ozawa, and K. Tanaka, _On a system of nonlinear Schrödinger equations with quadratic interaction_ , Ann. Inst. Henri Poincare Non-Linear Anal., 30:4 (2013), 661–690.
  * [18] P. Jaming and V. Komornik, _Moving and oblique observations of beams and plates_ , Evolution Equations and Control Theory, 9:2 (2020), 447–468.
  * [19] Y. N. Karamzin and A. Sukhorukov, _Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: Mutual focusing of beams and limitation on the efficiency of optical frequency converters_ , JETP Lett. 20:11 (1974), 339–343.
  * [20] V. Komornik, _Rapid boundary stabilization of linear distributed systems_ , SIAM J. Control Optim. 35 (1997), 1591–1613.
  * [21] V. Komornik and P. Loreti, Fourier Series in Control Theory (Springer-Verlag, 2005).
  * [22] I. Lasiecka, R. Triggiani, and X. Zhang, _Carleman estimates at the $H^{1}(\Omega)$\- and $L^{2}(\Omega)$-level for nonconservative Schrödinger equations with unobserved Neumann B.C_, Arch. Inequal. Appl. 2 (2004), 215–338.
  * [23] C. Laurent, _Internal control of the Schrödinger equation_ , Math. Control Relat. Fields, 4:2 (2014), 161–186.
  * [24] C. Li and N. Hayashi, _Recent progress on nonlinear Schr¨odinger systems with quadratic interactions_ , Sci. World J. 2014 (2014), 214821.
  * [25] J.-L. Lions, _Exact controllability, stabilizability, and perturbations for distributed systems_ , Siam Rev. 30 (1988), 1–68.
  * [26] M. Lopez-Garcia, A. Mercado, and L. de Teresa, _Null controllability of a cascade system of Schrödinger equations_ , Electronic Journal of Differential Equations, 2016:74. (2016), 1–12.
  * [27] C. Louis-Rose and L. Tebou, _Carleman Estimates and Simultaneous Boundary Controllability of Uncoupled Wave Equations_ , Appl Math Optim 88:49 (2023).
  * [28] C. Menyuk, R. Schiek, and L. Torner, _Solitary waves due to $\xi(2)$:$\xi(2)$ cascading_, J. Opt. Soc. Am. B 11:12 (1994), 2434–2443.
  * [29] A. Mercado, A. Osses, and L. Rosier, _Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights_ , Inverse Problems 24 (2008), 1–18.
  * [30] L. Miller, _How violent are fast controls for Schrödinger and plate vibrations?_ , Arch. Rational Mech. Anal. 172 (2004), 429–456.
  * [31] K.-D. Phung, _Observability and controllability for Schrödinger equations_ , SIAM J. Control Optim. 40 (2001), 211–230.
  * [32] K. Ramdani, T. Takahashi, G. Tenenbaum, and M. Tucsnak, _A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator,_ J. Funct. Anal. 226 (2005), 193–229.
  * [33] L. Rosier and L. de Teresa, _Exact controllability of a cascade system of conservative equations_ , C. R. Math. Acad. Sci. Paris, Ser. I, 349 (2011), 291–296.
  * [34] L. Rosier and B.-Y. Zhang, _Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded domain_ , SIAM J. Control Optim. 48 (2009), 972–992.
  * [35] L. Rosier and B.-Y. Zhang, _Control and Stabilization of the Nonlinear Schrödinger Equation on Rectangles,_ Mathematical Models, and Methods in Applied Sciences, 12 (2010), 2293–2347.
  * [36] G. Yuan and M. Yamamoto, _Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality_ , Chinese Annals of Mathematics, 31:4 (2010), 555–578.
  * [37] J. M. Urquiza, _Rapid exponential feedback stabilization with unbounded control operators_ , SIAM J. Control Optim. 43:6 (2005), 2233–2244.
  * [38] A. Vest, _Rapid stabilization in a semigroup framework,_ SIAM J. Control Optim. 51:5 (2013), 4169–4188.
  * [39] E. Zuazua, _Remarks on the controllability of the Schrödinger equation, Quantum Control: Mathematical and Numerical Challenges_ , 193–211, CRM Proc. Lecture Notes, Vol. 33 (Amer. Math. Soc., 2003).