# Exact controllability and stabilization for linear dispersive PDE’s on the
two-dimensional torus

Francisco J. Vielma-Leal  Departamento de Matemática, Facultad de Ciencias
Naturales, Matemática y del Medio Ambiente, Universidad Tecnológica
Metropolitana, Santiago, Chile<EMAIL_ADDRESS>and  Ademir Pastor  IMECC-
UNICAMP, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brasil.
<EMAIL_ADDRESS>
###### Abstract.

The moment method is used to prove the exact controllability of a wide class
of bidimensional linear dispersive PDE’s posed on the two-dimensional torus
$\mathbb{T}^{2}.$ The control function is considered to be acting on a small
vertical and horizontal strip of the torus. Our results apply to several well-
known models including some bidimesional extensions of the Benajamin-Ono and
Korteweg-de Vries equations. As a by product, the exponential stabilizability
with any given decay rate is also established in $H^{s}_{p}(\mathbb{T}^{2}),$
with $s\geq 0,$ by constructing an appropriated feedback control law.

###### Key words and phrases:

Dispersive equations, Controllability, Stabilization, Benjamin-Ono equation,
Zakharov-Kuznetsov equation

###### 2020 Mathematics Subject Classification:

93B05, 93D15, 35Q53

##  1\. Introduction

The controllability and stabilizability for the linear Schrödinger equation on
higher dimensions have been intensively studied during the last years, see for
instance [9, 19, 23, 38, 46] and references therein. When the problem is posed
on a periodic domain, there are pioneering works on this issue developed by
the authors in [8, 21, 32] for the linear and nonlinear Schrodinger equations
in dimensions 2 and 3 (see also [22]). However, as far as we know, there are a
few works addressing the problems of exact controllability and asymptotic
stabilization for bidimensional linear dispersive-type equations on a periodic
setting. To the best of our knowledge, the only work dealing with this problem
for a different dispersive model is the recent one in [43], where the authors
study the internal controllability of a non-localized solution for the linear
and non-linear Kadomtsev-Petviashvili II equation.

As is well known, the first step to study the controllability of a nonlinear
equation is to understand the controllability of the corresponding linear
equation. So, our main goal in this paper is to investigate the control
properties of a quite general class of linear dispersive equations on the two-
dimensional torus $\mathbb{T}^{2}:=\mathbb{R}^{2}/(2\pi\mathbb{Z})^{2}.$ More
precisely, we are interested in the equation

$\partial_{t}u-\partial_{x}\mathcal{L}u=0,\;\;(x,y)\in\mathbb{T}^{2},\;t\in\mathbb{R},$
(1.1)

where $u\equiv u(x,y,t)$ denotes a real-valued function of three real
variables $x,y$ and $t,$ and $\mathcal{L}$ denotes a linear Fourier multiplier
operator. We assume that such multiplier $\mathcal{L}$ is of “order” $r-1,$
for some $r\in\mathbb{R},$ with $r\geq 1.$ This means that the symbol
$b:\mathbb{Z}^{2}\to\mathbb{R}$ satisfies

$\widehat{\mathcal{L}u}(\mathbf{k})=b(\mathbf{k})\widehat{u}(\mathbf{k}),\;\;\forall\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$
(1.2)

where $\widehat{u}$ stands for the (periodic) Fourier transform of $u$ (see
(2.2)), and

$|b(\mathbf{k})|\leq C|\mathbf{k}|^{r-1},$ (1.3)

for some positive constant $C$ and
$|\mathbf{k}|:=\sqrt{k_{1}^{2}+k_{2}^{2}}\geq N_{0},$ for some $N_{0}\geq 0.$
In view of the Parseval identity, it is easy to see that $\mathcal{L}$ is a
self-adjoint operator on $L_{p}^{2}(\mathbb{T}^{2})$ (see Section 2 for
notations) and commutes with derivatives.

There are several models that fit in the abstract form (1.1). For instance,
the bidimensional versions of the Benjamin-Ono (BO) and Korteweg-de Vries
(KdV) equations on a periodic setting. Specifically, the 2D Benjamin-Ono
($\mathcal{L}=\mathcal{H}^{(x)}\partial_{y}$) and the Zakharov-Kuznetsov
($\mathcal{L}=-\Delta$) equations, where $\mathcal{H}^{(x)}$ denotes the
Hilbert transform with respect to the $x$-variable and $\Delta$ is the
bidimensional Laplacian operator. More general equations can also be written
in the abstract form (1.1), for example, the Benjamin-Ono-Zakharov-Kuznetsov
($\mathcal{L}=\mathcal{H}^{(x)}\partial_{x}-\partial_{y}^{2}$) and the
dispersion generalized BOZK ($\mathcal{L}=D^{\alpha}_{x}-\partial_{y}^{2}$)
equations, where $D^{\alpha}_{x}$ is defined for $\alpha>0.$ As we will see
below, all these equations may be treated in a single way.

As usual, the idea to study the controllability of (1.1) is to add a forcing
term $f\equiv f(x,y,t)$ as a control input. So, we shall consider the
following non-homogeneous initial-value problem (IVP):

$\partial_{t}u-\partial_{x}\mathcal{L}u=f,\;\;\;u(x,y,0)=u_{0}(x,y),\;\;\;(x,y)\in\mathbb{T}^{2},\;t\in\mathbb{R},$
(1.4)

for some suitable control $f$. Here, we will assume that $f$ is acting on a
small set composed by the union of a vertical and a horizontal strip; this
means, $f$ is assumed to be supported on a set of the form
$\left((\omega_{1}\times\mathbb{T})\cup(\omega_{2}\times\mathbb{T})\right)\subset\mathbb{T}^{2},$
where $\omega_{1}$ and $\omega_{2}$ are small open intervals in $\mathbb{T}$
(see Figure 1).

Figure 1. Region where the control $f$ is acting.

Note that (1.1) conserves the total mass, that is, the quantity

$\int_{\mathbb{T}^{2}}u(x,y,t)\;dxdy$

is conserved by any solution of (1.1). In order to keep the mass conserved in
the control system (1.4), we demand the function $f$ to satisfy

$\int_{\mathbb{T}^{2}}f(x,y,t)\;dxdy=0,\;\;\forall t\in\mathbb{R}.$ (1.5)

In this regard, we consider the control $f$ of the form $Gh$, where $h$ is a
function defined in $\mathbb{T}^{2}\times[0,T]$ and the operator
$G:H^{s}_{p}(\mathbb{T}^{2})\to H^{s}_{p}(\mathbb{T}^{2}),$ $s\geq 0,$ is
defined in the following way: let $g_{1}$ and $g_{2}$ be non-negative real-
valued functions in $C^{\infty}(\mathbb{T})$ such that

$2\pi\widehat{g_{1}}(0)=\int_{\mathbb{T}}g_{1}(x)dx=1,$ (1.6)
$2\pi\widehat{g_{2}}(0)=\int_{\mathbb{T}}g_{2}(y)dy=1.$ (1.7)

Assume $\text{supp}\;g_{1}=\overline{\omega_{1}}\subset\mathbb{T}$ and
$\text{supp}\;g_{2}=\overline{\omega_{2}}\subset\mathbb{T},$ where
$\omega_{1}=\\{x\in\mathbb{T}:g_{1}(x)>0\\}$ and
$\omega_{2}=\\{y\in\mathbb{T}:g_{2}(y)>0\\}$ are open intervals. Now, we
define the operator $G$ as

$G(\phi)(x,y):=g_{2}(y)\;G_{1}(\phi)(x,y)+g_{1}(x)\;G_{2}(\phi)(x,y),\;\;\phi\in
H^{s}_{p}(\mathbb{T}^{2}),\;s\geq 0,$ (1.8)

where

$G_{1}(\phi)(x,y)=\frac{1}{2\pi}\phi(x,y)-\frac{1}{(2\pi)^{2}}\;\int_{0}^{2\pi}\phi(x^{\prime},y)dx^{\prime},$
(1.9)

and

$G_{2}(\phi)(x,y)=\frac{1}{2\pi}\phi(x,y)-\frac{1}{(2\pi)^{2}}\;\int_{0}^{2\pi}\phi(x,y^{\prime})dy^{\prime}.$
(1.10)

It is easy to see that $G$ is linear and self-adjoint as an operator from
$L^{2}_{p}(\mathbb{T}^{2})$ into $L^{2}_{p}(\mathbb{T}^{2}).$ In addition, it
is bounded in $H_{p}^{s}(\mathbb{T}^{2})$, that is, there exists a constant
$C$ depending only on $s,g_{1}$, and $g_{2}$ such that

$\|G(\phi)\|_{H_{p}^{s}(\mathbb{T}^{2})}\leq
C\|\phi\|_{H_{p}^{s}(\mathbb{T}^{2})}.$ (1.11)

Since we are setting $f=Gh$, the function $h$ can now be considered as the new
control function and for each $t\in[0,T]$ we have that (1.5) holds.

Next, we specify the problems that we address in this work, which are
fundamental in control theory:

Exact controllability problem: Let $s\geq 0$ and $T>0$ be given. Assume
$u_{0}$ and $u_{1}$ belong to $H^{s}_{p}(\mathbb{T}^{2})$ with
$\widehat{u_{0}}(0,0)=\widehat{u_{1}}(0,0).$ Can one find a control input $h$
such that the unique solution of the initial-valued problem (IVP)

$\begin{cases}\partial_{t}u-\partial_{x}\mathcal{L}u=G(h),\;\;(x,y)\in\mathbb{T}^{2},\;t\in\mathbb{R},\\\
u(x,y,0)=u_{0}(x,y),\end{cases}$ (1.12)

is defined until time $T$ and satisfies $u(x,y,T)=u_{1}(x,y),$ for all
$(x,y)\in\mathbb{T}^{2}$?

Asymptotic stabilizability problem: Let $s\geq 0$ and $u_{0}\in
H^{s}_{p}(\mathbb{T}^{2})$ be given. Can one define a feedback control law
$f=G(Ku)$ for some linear operator $K,$ such that the resulting closed-loop
system

$\begin{cases}\partial_{t}u-\partial_{x}\mathcal{L}u=G(Ku),\;\;(x,y)\in\mathbb{T}^{2},\;t\in\mathbb{R},&\\\
u(x,y,0)=u_{0}(x,y),\end{cases}$ (1.13)

is globally well-defined and asymptotically stable to an equilibrium point as
$t\to+\infty$?

Let us now describe our results. First, we state a result regarding
controllability of equation (1.12). Similar to the criteria in [47], the
following results directly link the problem of controllability with some
specific properties of the eigenvalues associated to the operator
$\partial_{x}\mathcal{L}.$ To derive our first result, we assume that
$\partial_{x}\mathcal{L}$ has a countable number of eigenvalues which, except
on the coordinates axes, have finite multiplicity. Specifically, we will
assume the following hypothesis hold:

  * $(H1)$

$\partial_{x}\mathcal{L}\psi_{\mathbf{k}}=i\lambda_{\mathbf{k}}\psi_{\mathbf{k}},$
where $\psi_{\mathbf{k}}$ is defined in (2.3) and
$\lambda_{\mathbf{k}}=k_{1}b(\mathbf{k}),$ for all
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}.$

The eigenvalues in the sequence
$\\{i\lambda_{\mathbf{k}}\\}_{\mathbf{k}\in\mathbb{Z}^{2}}$ are not
necessarily distinct and we need to distinguish simple and multiple
eigenvalues. Therefore, for each $\mathbf{k}^{\prime}\in\mathbb{Z}^{2},$ we
set
$I(\mathbf{k}^{\prime}):=\\{\mathbf{k}\in\mathbb{Z}^{2}:\lambda_{\mathbf{k}}=\lambda_{\mathbf{k}^{\prime}}\\}$
and $m(\mathbf{k}^{\prime}):=\\#I(\mathbf{k}^{\prime}),$ where
$\\#I(\mathbf{k}^{\prime})$ denotes the number of elements in
$I(\mathbf{k}^{\prime}).$ In particular, $m(\mathbf{k}^{\prime})=1$ if
$i\lambda_{\mathbf{k}^{\prime}}$ is a simple eigenvalue.

  * $(H2)$

For any $\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$ $\lambda_{\mathbf{k}}$ is
even in the first variable and odd in the second one, it means,
$\lambda_{(k_{1},k_{2})}=\lambda_{(-k_{1},k_{2})},$ and
$\lambda_{(k_{1},-k_{2})}=-\lambda_{(k_{1},k_{2})}.$ Furthermore, for any
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$ with $k_{1}\neq 0,$ we have that
the unique entire solution $j_{2}\in\mathbb{Z}$ of equation
$\lambda_{(k_{1},j_{2})}=\lambda_{(k_{1},k_{2})}$ is $j_{2}=k_{2}.$ Also, for
any $\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$ with $k_{2}\neq 0,$ the
unique entire solutions of equation
$\lambda_{(j_{1},k_{2})}=\lambda_{(k_{1},k_{2})}$ are $j_{1}=\pm k_{1}.$

Assumptions $(H1)$ and $(H2)$ allows the eigenvalues $i\lambda_{\mathbf{k}}$
to have infinite multiplicity on the coordinate axes. Also, it may occur that
$m(\mathbf{k}^{\prime})\to\infty$ as $\lambda_{\mathbf{k}^{\prime}}\to\infty$
(see Subsection 6.2 where a particular example involving the 2D-BO equation is
given). Both properties make the problem of exact controllability associated
to (1.4) a great challenge and require us to find strong control functions $f$
acting not only on a small open subset of $\mathbb{T}^{2}$ but on small
subsets of vertical and horizontal strips of the 2-torus (see Figure 1).

If we count only the distinct eigenvalues, we obtain a countable (maximal) set
$\mathbb{J}\subset\mathbb{Z}^{2}$ and a sequence
$\\{\lambda_{\mathbf{k}}\\}_{\mathbf{k}\in\mathbb{J}},$ with the property that
$\lambda_{\mathbf{k}}\neq\lambda_{\mathbf{k}^{\prime}}$ for any
$\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{J}$ with
$\mathbf{k}\neq\mathbf{k}^{\prime}.$ Now we are able to state our first
result.

######  Theorem 1.1.

Let $s\geq 0$ and assume $(H1)$ and $(H2).$ Suppose that

$\begin{split}\gamma&:=\inf_{\begin{subarray}{c}\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{J}\\\
\mathbf{k}\neq\mathbf{k}^{\prime}\end{subarray}}|\lambda_{\mathbf{k}}-\lambda_{\mathbf{k}^{\prime}}|>0\end{split}$
(1.14)

and define

$\begin{split}\gamma^{\prime}&:=\underset{S\subset\mathbb{J}}{\sup}\;\underset{\mathbf{k}\neq\mathbf{k}^{\prime}}{\underset{\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{J}\backslash
S}{\inf}}|\lambda_{\mathbf{k}}-\lambda_{\mathbf{k}^{\prime}}|,\end{split}$
(1.15)

where $S$ runs over all finite subsets of $\mathbb{J}.$ Then, for any
$T>\frac{2\pi}{\gamma^{\prime}}$ and for each $u_{0},u_{1}\in
H_{p}^{s}(\mathbb{T}^{2})$ with $\widehat{u_{0}}(0,0)=\widehat{u_{1}}(0,0),$
there exists a function $h\in L^{2}([0,T];H^{s}_{p}(\mathbb{T}^{2}))$ such
that the unique solution $u$ of the non-homogeneous system

$\begin{cases}u\in C\left([0,T];H^{s}_{p}(\mathbb{T}^{2})\right),&\\\
\partial_{t}u=\partial_{x}\mathcal{L}u+G(h)(t)\in
H^{s-r}_{p}(\mathbb{T}^{2}),\;\;t\in(0,T),&\\\ u(0)=u_{0}.&\end{cases}$ (1.16)

satisfies $u(T)=u_{1}.$ Furthermore,

$\|h\|_{L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))}\leq\nu\left(\|u_{0}\|_{H^{s}_{p}(\mathbb{T}^{2})}+\|u_{1}\|_{H^{s}_{p}(\mathbb{T}^{2})}\right)$
(1.17)

for some positive constant $\nu\equiv\nu(s,g_{1},T).$

Using symmetry, we can replace $(H2)$ by the following hypothesis and still
have a similar result of exact controllability for system (1.12):

  * $(H3)$

For any $\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$ $\lambda_{\mathbf{k}}$ is
odd in the first variable and even in the second one, that is,
$\lambda_{(-k_{1},k_{2})}=-\lambda_{(k_{1},k_{2})}$ and
$\lambda_{(k_{1},k_{2})}=\lambda_{(k_{1},-k_{2})}.$ Furthermore, For any
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$ with $k_{1}\neq 0$ we have the
unique entire solutions $j_{2}\in\mathbb{Z}$ of equation
$\lambda_{(k_{1},j_{2})}=\lambda_{(k_{1},k_{2})}$ are $j_{2}=\pm k_{2}.$
Moreover, for any $\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$ with $k_{2}\neq
0$ the unique entire solution $j_{1}\in\mathbb{Z}$ of equation
$\lambda_{(j_{1},k_{2})}=\lambda_{(k_{1},k_{2})}$ is $j_{1}=k_{1}.$

In this case, we denote by $\mathbb{I}$ the (maximal) subset of
$\mathbb{Z}^{2}$ such that the sequence
$\\{\lambda_{\mathbf{k}}\\}_{\mathbf{k}\in\mathbb{I}}$ have the property
$\lambda_{\mathbf{k}}\neq\lambda_{\mathbf{k}^{\prime}}$ for any
$\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{I}$ with
$\mathbf{k}\neq\mathbf{k}^{\prime}.$ We now estate our second result.

######  Theorem 1.2.

Let $s\geq 0$ and assume $(H1)$ and $(H3).$ Suppose that

$\begin{split}\gamma&:=\inf_{\begin{subarray}{c}\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{I}\\\
\mathbf{k}\neq\mathbf{k}^{\prime}\end{subarray}}|\lambda_{\mathbf{k}}-\lambda_{\mathbf{k}^{\prime}}|>0\end{split}$
(1.18)

and define

$\begin{split}\gamma^{\prime}&:=\underset{S\subset\mathbb{I}}{\sup}\;\underset{\mathbf{k}\neq\mathbf{k}^{\prime}}{\underset{\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{I}\backslash
S}{\inf}}|\lambda_{\mathbf{k}}-\lambda_{\mathbf{k}^{\prime}}|,\end{split}$
(1.19)

where $S$ runs over all finite subsets of $\mathbb{I}.$ Then, for any
$T>\frac{2\pi}{\gamma^{\prime}}$ the same conclusions of Theorem 1.1 hold.

The idea to prove Theorems 1.1 and 1.2 is to use the moment method (see, for
instance, [40, 47]). Combined with a generalization of Ingham’s theorem (see
[17]), the construction of the function $h$ reduces in analyzing the solutions
of an algebraic equation or system of equations.

######  Remark 1.3.

Note if $\lambda_{\mathbf{k}}\in\mathbb{Z}$, for all $\mathbf{k}\in\mathbb{J}$
(or $\mathbb{I}$), then we always have $\gamma,\gamma^{\prime}\geq 1$. This
situation occurs, for instance, when $\mathcal{L}$ is a differential operator.
See Section 6 for some examples.

Attention is now turned to our stabilization result. Choosing an appropriate
linear bounded operator $K$ one is able to show that the resulting closed-loop
system is exponentially stable with an arbitrary exponential decay rate. More
precisely,

######  Theorem 1.4.

Let $g_{1},\;g_{2}$ be as in (1.6)-(1.7) and let $s\geq 0,$ and $\lambda>0$ be
given. Under the assumptions of Theorem 1.1 or Theorem 1.2, there exists a
bounded linear operator $K_{\lambda}$ from $H_{p}^{s}(\mathbb{T}^{2})$ to
$H_{p}^{s}(\mathbb{T}^{2})$ such that the unique solution $u$ of the closed-
loop system

$\begin{cases}u\in C([0,+\infty);H^{s}_{p}(\mathbb{T}^{2})),\hbox{}\\\
\partial_{t}u(t)=\partial_{x}\mathcal{L}u(t)+GK_{\lambda}u(t)\in
H_{p}^{s-r}(\mathbb{T}^{2}),\quad t>0,\\\ u(0)=u_{0}\in
H_{p}^{s}(\mathbb{T}^{2}),\end{cases}$ (1.20)

satisfies

$\|u(\cdot,t)-\widehat{u_{0}}(0,0)\|_{H_{p}^{s}(\mathbb{T}^{2})}\leq
Me^{-\lambda t}\|u_{0}-\widehat{u_{0}}(0,0)\|_{H_{p}^{s}(\mathbb{T}^{2})},$

for all $t\geq 0,$ and some positive constant $M=M(g_{1},g_{2},\lambda,s).$

######  Remark 1.5.

Using a simple feedback control law $Ku=-G^{\ast},$ where $G^{\ast}$ denotes
the adjoint operator of $G$, we can prove that the closed-loop system (1.13)
is exponentially stable in $L^{2}_{p}(\mathbb{T}^{2})$ for some exponential
decay rate by using similar arguments as in [37, Theorem 5.4] (see also [25,
41]).

The paper is organized as follows. In Section 2 we introduce the basic
notation and review some definitions related to periodic functions. In Section
3 we just prove the well-posedness of our associated IVPs. Theorems 1.1 and
1.2 is then proved in Section 4. Section 5 is dedicated to prove the
stabilization result. Finally, in Section 6 we apply our results to prove the
exact controllability and exponential stabilization for some well known
models.

##  2\. Preliminaries

In this section we introduce some basic notation and summarize some important
results related with the theory of distributions on the two-dimensional torus
(2-torus, for short). We will use $\mathbf{k}$ and $\mathbf{j}$ for generic
points $(k_{1},k_{2})$ and $(j_{1},j_{2})$ in $\mathbb{Z}^{2}$. For multi-
indices $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2})$ and
$\boldsymbol{\beta}=(\beta_{1},\beta_{2})$ in $\mathbb{N}^{2}$ we say that
$\boldsymbol{\beta}\leq\boldsymbol{\alpha}$ if and only if
$\beta_{i}\leq\alpha_{i},\;i=1,2.$ Also, we define
$\boldsymbol{\alpha}!=\alpha_{1}!\alpha_{2}!$ and
$|\boldsymbol{\alpha}|=\alpha_{1}+\alpha_{2}.$ Given two vectors
$\mathbf{x}=(x_{1},y_{1})$ and $\mathbf{x}^{\prime}=(x_{2},y_{2})$ in
$\mathbb{R}^{2},$ $\mathbf{x}\cdot\mathbf{x}^{\prime}=x_{1}x_{2}+y_{1}y_{2},$
denotes the usual inner product. Also, $|\mathbf{x}|$ denotes the usual
Euclidean norm of $\mathbf{x}.$

###  2.1. Distributions on the 2-torus

Here we recall some aspects of the Fourier analysis on the torus as well as
some properties of the periodic distributions. The details may be found in
[11, Chapter 3]. The space of test functions on the 2-torus is the space
$C^{\infty}(\mathbb{T}^{2})$ of all $C^{\infty}$ functions that are
$2\pi$-periodic in every coordinate. The topology generated by the family of
semi-norms $\sup|D^{\mathbf{\boldsymbol{\alpha}}}f(\mathbf{x})|$ allows one to
see $C^{\infty}(\mathbb{T}^{2})$ as a locally convex topological space. We
denote by $\mathcal{G}(\mathbb{Z}^{2})$ the space of the _rapidly decreasing
sequences on $\mathbb{Z}^{2}$_ of all complex-valued sequences
$\mu=\left\\{\mu_{\mathbf{k}}\right\\}_{\mathbf{k}\in\mathbb{Z}^{2}}$ such
that

$\sum_{\mathbf{k}\in\mathbb{Z}^{2}}|\mathbf{k}^{\boldsymbol{\alpha}}|\;|\mu_{\mathbf{k}}|<\infty,\;\;\text{for
all multi-index}\;\boldsymbol{\alpha}\in\mathbb{N}^{2}.$ (2.1)

Recall that
$\mathbf{k}^{\boldsymbol{\alpha}}=(k_{1},k_{2})^{(\alpha_{1},\alpha_{2})}=k_{1}^{\alpha_{1}}k_{2}^{\alpha_{2}}$.
The space $\mathcal{G}(\mathbb{Z}^{2})$ is a Hausdorff locally convex
topological space with the topology induced by the family of semi-norms

$\|\mu\|_{\infty,\boldsymbol{\alpha}}:=\sup\limits_{\mathbf{k}\in\mathbb{Z}^{2}}\left(|\mathbf{k}^{\boldsymbol{\alpha}}|\;|\mu_{\mathbf{k}}|\right),$

where $\boldsymbol{\alpha}$ ranges over all multi-indices in $\mathbb{N}^{2}.$
The dual spaces of $C^{\infty}(\mathbb{T}^{2})$ and
$\mathcal{G}(\mathbb{Z}^{2})$ under these topologies are denoted by
$\mathcal{D}^{\prime}(\mathbb{T}^{2})$ (the space of all _distributions on
$\mathbb{T}^{2}$_) and $\mathcal{G}^{\prime}(\mathbb{Z}^{2})$ (the space of
_sequences with slow growth on $\mathbb{Z}^{2}$_), respectively.
$C^{\infty}(\mathbb{T}^{2})$ is dense in
$\mathcal{D}^{\prime}(\mathbb{T}^{2})$ and we can define the usual operations
of differentiation, translation, reflection, convolution and multiplication.
The Fourier transform of $u\in\mathcal{D}^{\prime}(\mathbb{T}^{2})$ is the
sequence
$\left\\{\widehat{u}(\mathbf{k})\right\\}_{\mathbf{k}\in\mathbb{Z}^{2}}$
defined as

$\widehat{u}(\mathbf{k})=\frac{1}{(2\pi)^{2}}\left\langle
u,e^{-i\mathbf{k}\cdot\mathbf{x}}\right\rangle,\;\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},\;\;\mathbf{x}=(x,y)\in\mathbb{T}^{2},$
(2.2)

where $\left\langle\cdot,\cdot\right\rangle$ denotes the pairing between
$\mathcal{D}^{\prime}(\mathbb{T}^{2})$ and $C^{\infty}(\mathbb{T}^{2}).$ The
map
${}^{\wedge}:\mathcal{D}^{\prime}(\mathbb{T}^{2})\to\mathcal{G}^{\prime}(\mathbb{Z}^{2})$
is a linear bijection with inverse
${}^{\vee}:\mathcal{G}^{\prime}(\mathbb{Z}^{2})\to\mathcal{D}^{\prime}(\mathbb{T}^{2})$
(the inverse Fourier transform) defined by

$\eta=\left\\{\eta_{\mathbf{k}}\right\\}_{\mathbf{k}\in\mathbb{Z}^{2}}\mapsto\eta^{\vee}(\mathbf{x}):=\sum\limits_{\mathbf{k}\in\mathbb{Z}^{2}}\eta_{\mathbf{k}}\;e^{i\mathbf{k}\cdot\mathbf{x}},\;\;\forall\;\mathbf{x}=(x,y)\in\mathbb{T}^{2},$

where the series converges in the sense of
$\mathcal{D}^{\prime}(\mathbb{T}^{2})$.

Recall that $L^{2}_{p}(\mathbb{T}^{2})$ (the standard space of square
integrable $2\pi$-periodic functions) with complex inner product

$\left(u,v\right)_{L^{2}_{p}(\mathbb{T}^{2})}=\int\limits_{\mathbb{T}^{2}}u(x,y)\overline{v(x,y)}dxdy$

is a Hilbert space. If $u\in L^{2}_{p}(\mathbb{T}^{2})$ then

$\widehat{u}(\mathbf{k})=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}e^{-i\mathbf{k}\cdot\mathbf{x}}u(x,y)dxdy.$

###  2.2. Sobolev spaces and Fourier series

(See [45, Chapter 3 $\&$ 4]) In this subsection we will introduce the so-
called Sobolev spaces of $L^{2}$-type on the 2-torus. Given $s\in\mathbb{R}$,
the (periodic) Sobolev space of order $s$ is defined as

$H^{s}_{p}(\mathbb{T}^{2}):=\left\\{\displaystyle{u=\sum_{\mathbf{k}\in\mathbb{Z}^{2}}\widehat{u}(\mathbf{k})\;e^{i\mathbf{k}\cdot\mathbf{x}}\in\mathcal{D}^{\prime}(\mathbb{T}^{2})}\Bigl{|}\|u\|_{H^{s}_{p}(\mathbb{T})}^{2}:=(2\pi)^{2}\sum_{\mathbf{k}\in\mathbb{Z}^{2}}(1+|\mathbf{k}|)^{2s}|\widehat{u}(\mathbf{k})|^{2}<\infty\right\\}.$

The space $H^{s}_{p}(\mathbb{T}^{2})$ is a Hilbert space endowed with the
inner product

$\displaystyle{(u\,,\,v)_{H^{s}_{p}(\mathbb{T}^{2})}=(2\pi)^{2}\sum_{\mathbf{k}\in\mathbb{Z}^{2}}(1+|\mathbf{k}|)^{2s}\widehat{u}(\mathbf{k})\;\overline{\widehat{v}(\mathbf{k})},\;\;v\in
H^{s}_{p}(\mathbb{T}^{2}),\;u\in H^{s}_{p}(\mathbb{T}^{2}).}$

For any $s\in\mathbb{R}$, $(H^{s}_{p}(\mathbb{T}^{2}))^{\prime}$, the
topological dual of $H^{s}_{p}(\mathbb{T}^{2})$, is isometrically isomorphic
to $H^{-s}_{p}(\mathbb{T}^{2})$, where the duality is implemented by the
pairing

$\displaystyle{\langle u,v\rangle_{H^{-s}_{p}(\mathbb{T}^{2})\times
H_{p}^{s}(\mathbb{T}^{2})}=(2\pi)^{2}\sum_{\mathbf{k}\in\mathbb{Z}^{2}}\widehat{u}(\mathbf{k})\;\overline{\widehat{v}(\mathbf{k})}},\;\;\text{for
all}\;v\in H^{s}_{p}(\mathbb{T}^{2}),\;u\in H^{-s}_{p}(\mathbb{T}^{2}).$

If $s_{1},\;s_{2}\in\mathbb{R}$ with $s_{1}\geq s_{2}$ then
$H_{p}^{s_{1}}(\mathbb{T}^{2})\hookrightarrow H_{p}^{s_{2}}(\mathbb{T}^{2}),$
where the embedding is dense. Also $H_{p}^{0}(\mathbb{T}^{2})$ is
isometrically isomorphic to $L^{2}_{p}(\mathbb{T}^{2})$.

It is well-known that any distribution $u\in H^{s}_{p}(\mathbb{T}^{2})$,
$s\in\mathbb{R}$, may be written as

$u=2\pi\sum_{\mathbf{k}\in\mathbb{Z}^{2}}\widehat{u}(\mathbf{k})\psi_{\mathbf{k}}=\lim\limits_{l\to\infty}S_{l}(u),$

where the limit is taken in the sense of
$\mathcal{D}^{\prime}(\mathbb{T}^{2}),$ $S_{l}(u)$ is the $l$-th partial sum
of the Fourier series associated to $u,$ defined by

$S_{l}(u)(\mathbf{x}):=2\pi\sum\limits_{\begin{subarray}{c}\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}\\\
|\mathbf{k}|\leq l\end{subarray}}\widehat{u}(\mathbf{k})\psi_{\mathbf{k}},$

and $\left\\{\psi_{\mathbf{k}}\right\\}_{\mathbf{k}\in\mathbb{Z}^{2}}$ is a
complete orthonormal sequence in $L^{2}_{p}(\mathbb{T}^{2})$ (see [11, Chapter
3 $\&$ 3.1]) formed by the complex-valued functions

$\psi_{\mathbf{k}}(\mathbf{x})=\frac{1}{2\pi}e^{i\mathbf{k}\cdot\mathbf{x}},\;\;\mathbf{k}\in\mathbb{Z}^{2},\;\mathbf{x}=(x,y)\in\mathbb{T}^{2}.$
(2.3)

##  3\. Well-posedness

Before establishing our main result concerning the exact controllability a we
need a well-posedness theory associated with (1.1). The following result
states the well-posedness with initial data in $H^{s}_{p}(\mathbb{T}^{2}),$
$s\in\mathbb{R}$.

######  Theorem 3.1.

Assume $\mathcal{L}$ satisfies (1.2)-(1.3). Then for any and $u_{0}\in
H_{p}^{s}(\mathbb{T}^{2})$, $s\in\mathbb{R}$, the IVP

$\begin{cases}u\in C(\mathbb{R};H_{p}^{s}(\mathbb{T}^{2})),&\\\
\partial_{t}u=\partial_{x}\mathcal{L}u\in
H^{s-r}_{p}(\mathbb{T}^{2}),\;\;\;t\in\mathbb{R},&\\\ u(0)=u_{0},\end{cases}$
(3.1)

has a unique solution.

###### Proof.

This follows from the standard semigroup theory. For the sake of completeness
we bring some steps (see for e.g. [3, 13, 36] for more details). We consider
$\partial_{x}\mathcal{L}:D(\partial_{x}\mathcal{L})\subseteq
H^{s-r}(\mathbb{T}^{2})\to H^{s-r}(\mathbb{T}^{2})$, where
$D(\partial_{x}\mathcal{L})=H^{s}(\mathbb{T}^{2})$. Note that
$D(\partial_{x}\mathcal{L})$ is dense in $H^{s-r}(\mathbb{T}^{2})$ and, using
the definition of $\partial_{x}\mathcal{L}$ and the properties of the Fourier
transform, we see that it is skew-adjoint, that is, for any $\varphi,\psi\in
H^{s}(\mathbb{T}^{2})$,

$\begin{split}\left(\partial_{x}\mathcal{L}\varphi,\psi\right)_{H^{s-r}(\mathbb{T}^{2})}=-\left(\varphi,\partial_{x}\mathcal{L}\psi\right)_{H^{s-r}(\mathbb{T}^{2})}.\end{split}$

Hence, by Stone’s theorem it follows that $\partial_{x}\mathcal{L}$ generates
a strongly continuous unitary group of contractions, say,
$\left\\{U(t)\right\\}_{t\in\mathbb{R}}$ on $H^{s-r}(\mathbb{T}^{2})$. Theorem
3.2.3 in [3] now implies the desired result. ∎

######  Remark 3.2.

Using the Fourier transform, we may deduce that the unique solution given in
Theorem 3.1 satisfies

$\widehat{u(t)}(\mathbf{k})=e^{ik_{1}b(\mathbf{k})t}\widehat{u_{0}}(\mathbf{k}),\;\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$

or by taking the inverse Fourier transform,

$u(t)=\left(e^{ik_{1}b(\mathbf{k})t}\widehat{u_{0}}(\mathbf{k})\right)^{\vee},\;\;t\in\mathbb{R}.$

It means that

$u(\mathbf{x},t)=\sum_{\mathbf{k}\in\mathbb{Z}^{2}}e^{ik_{1}b(\mathbf{k})t}\widehat{u_{0}}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}},\;\;\mathbf{x}=(x,y)\in\mathbb{T}^{2},$

must be the unique solution of IVP (1.1) with initial data $u_{0}\in
H^{s}_{p}(\mathbb{T}^{2}),$ where the series converges in the sense of
$\mathcal{D}^{\prime}(\mathbb{T}^{2})$. In particular, the unitary group
$U(t)$ is given by

$t\mapsto
U(t)\varphi:=e^{\partial_{x}\mathcal{L}t}\varphi=\left(e^{ik_{1}b(\mathbf{k})t}\widehat{\varphi}(\mathbf{k})\right)^{\vee},\;\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$

in such way that the solution of IVP (1.1) with initial data $u_{0}\in
H^{s}_{p}(\mathbb{T}^{2}),$ becomes $u(t)=U(t)u_{0},$ for any
$t\in\mathbb{R}.$ Also, recall that according to [3, Corollary 3.2.6] the
adjoint operator $U(t)^{\ast}$ of $U(t)$ is linear bounded and satisfies
$U(t)^{*}=U(-t)$, for all $t\in\mathbb{R}.$

Next, we deal with the well-posedness of the non-homogeneous linear problem
(1.12). The following lemma is needed.

######  Lemma 3.3.

Let $s\geq 0$ and $G$ be defined as in (1.8). Given any $T>0$, the operator

$G:L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))\longrightarrow
L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))$

is linear and bounded.

###### Proof.

It is clear that $G$ is linear. In addition, for any $h\in
L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))$, it follows from (1.11) that

$\begin{split}\|Gh\|^{2}_{L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))}&=\int_{0}^{T}\|Gh(t)\|^{2}_{H_{p}^{s}(\mathbb{T}^{2})}dt\\\
&\leq C^{2}\int_{0}^{T}\|h(t)\|^{2}_{H_{p}^{s}(\mathbb{T}^{2})}dt\\\
&=C^{2}\|h\|^{2}_{L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))},\end{split}$

which yields the desired. ∎

######  Theorem 3.4.

Let $T>0,$ $s\geq 0,$ $u_{0}\in H_{p}^{s}(\mathbb{T}),$ and $h\in
L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2})).$ Then there exists a unique (mild)
solution $u\in C([0,T];H_{p}^{s}(\mathbb{T}^{2}))$ of the IVP (1.12).

###### Proof.

As in the proof of Lemma 3.3 we infer that $Gh\in
L^{1}([0,T];H_{p}^{s}(\mathbb{T}^{2})).$ Corollary 2.2 and Definition 2.3 in
[36, Chapter 4] imply that

$u(t)=U(t)u_{0}+\int_{0}^{t}U(t-t^{\prime})Gh(t^{\prime})dt^{\prime}$

is the unique (mild) solution of

$\begin{cases}u\in C\left([0,T];H^{s}_{p}(\mathbb{T}^{2})\right),&\\\
\partial_{t}u=\partial_{x}\mathcal{L}u+Gh(t)\in
H^{s-r}_{p}(\mathbb{T}^{2}),\;\;t\in(0,T),&\\\ u(0)=u_{0}\in
H_{p}^{s}(\mathbb{T}^{2}),\end{cases}$

which in turn provides the solution of (1.12). ∎

##  4\. Exact controllability results

This section is devoted to prove Theorems 1.1 and 1.2, as an application of
the classical moment method (see [40]). Before starting with the results, note
that by replacing $u_{1}$ by $u_{1}-U(T)u_{0}$ if necessary, we may assume
without loss of generality that $u_{0}=0.$ Consequently, in view of our
assumptions, we may assume $\widehat{u_{1}}(0,0)=\widehat{u_{0}}(0,0)=0.$

Let us start by writing the terminal estate $u_{1}\in
H_{p}^{s}(\mathbb{T}^{2})$ as

$u_{1}(\mathbf{x})=\sum\limits_{\mathbf{k}\in\mathbb{Z}^{2}}\widehat{u_{1}}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}}=2\pi\sum\limits_{\mathbf{k}\in\mathbb{Z}^{2}}\widehat{u_{1}}(\mathbf{k})\;\psi_{\mathbf{k}}(\mathbf{x}),$
(4.1)

where the series converges in the distributional sense and $\psi_{\mathbf{k}}$
is defined as in (2.3). Next result characterizes the exact controllability of
the linear non-homogeneous system (1.16). The idea of the proof is similar to
that of [37, Lemma 4.1], passing to the frequency space when necessary; so we
omit the details.

######  Lemma 4.1.

Let $s\geq 0$ and $T>0$ be given. Assume $u_{1}\in H_{p}^{s}(\mathbb{T}^{2})$
with $\widehat{u_{1}}(0,0)=0.$ Then, there exists $h\in
L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))$ such that the solution of the IVP
(1.16) with initial data $u_{0}=0$ satisfies $u(T)=u_{1}$ if and only if

$\int_{0}^{T}\left\langle
Gh(\cdot,\cdot,t),\varphi(\cdot,\cdot,t)\right\rangle_{H^{s}_{p}(\mathbb{T}^{2})\times(H^{s}_{p}(\mathbb{T}^{2}))^{\prime}}dt=\left\langle
u_{1}(\cdot,\cdot),\varphi_{0}(\cdot,\cdot)\right\rangle_{H^{s}_{p}(\mathbb{T}^{2})\times(H^{s}_{p}(\mathbb{T}^{2}))^{\prime}},$
(4.2)

for any $\varphi_{0}\in(H^{s}_{p}(\mathbb{T}^{2}))^{\prime}$, where $\varphi$
is the solution of the adjoint problem

$\begin{cases}\varphi\in
C\left([0,T]:\left(H_{p}^{s}(\mathbb{T}^{2})\right)^{\prime}\right),\\\
\partial_{t}\varphi=\partial_{x}\mathcal{L}\varphi\in
H_{p}^{-s-r}(\mathbb{T}^{2}),\quad t>0,\\\ \varphi(T)=\varphi_{0}.\end{cases}$
(4.3)

The following characterization to show the existence of control for the linear
system (1.16) (with initial data $u_{0}=0$) is a direct consequence of Lemma
4.1. It provides a method to find the control function $h$ explicitly. For a
proof in a very similar situation, we refer the reader to [47, Lemma 4.3].

######  Corollary 4.2 (Moment Equation).

Let $s\geq 0$ and $T>0$ be given. If $u_{1}$ is written as in (4.1) and
satisfies $\widehat{u_{1}}(0,0)=0,$ then the solution of (1.16) with initial
data $u_{0}=0$ satisfies $u(T)=u_{1}$ if and only if there exists $h\in
L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))$ such that

$\int_{0}^{T}\left(Gh(\cdot,\cdot,t),e^{-i\lambda_{\mathbf{k}}(T-t)}\psi_{\mathbf{k}}(\cdot,\cdot)\right)_{L^{2}(\mathbb{T}^{2})}dt=2\pi\widehat{u_{1}}(\mathbf{k}),\;\;\forall\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$
(4.4)

where $\lambda_{\mathbf{k}}=k_{1}b(\mathbf{k}).$

With Corollary 4.2 in hand, we see that in order to prove Theorem 1.1 (for
instance) we only need to construct a control function $h$ satisfying relation
(4.4). Before that, we need two additional results. The first one, gives some
properties of how the operators $G_{1}$ and $G_{2}$ behave at complex
exponential functions.

######  Lemma 4.3.

For $G_{1}$ and $G_{2}$ as in (1.9)-(1.10), define

$m^{j,k}_{1}:=\widehat{G_{1}(e^{ijx})}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}G_{1}(e^{ijx})e^{-ikx}\;dx,\;\;\;\;j,k\in\mathbb{Z},$
(4.5)
$m^{j,k}_{2}:=\widehat{G_{2}(e^{ijy})}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}G_{2}(e^{ijy})e^{-iky}\;dy,\;\;\;\;j,k\in\mathbb{Z}.$
(4.6)

Then

  * (i)

$m_{n}^{j,0}=0,$ for all $j\in\mathbb{Z}$ and $n=1,2;$

  * (ii)

$m_{n}^{0,k}=0$, for all $k\in\mathbb{Z}$ and $n=1,2;$

  * (iii)

$m_{n}^{0,0}=0$, for $n=1,2;$

  * (iv)

If $j,k\in\mathbb{Z}$ with $j\neq 0$ and $k\neq 0,$ then

$m_{n}^{j,k}=\begin{cases}\displaystyle{\frac{1}{2\pi}},&\text{if}\;j=k,\\\
0,&\text{if}\;j\neq k,\end{cases}$ (4.7)

for $n=1,2.$

###### Proof.

The proof follows by direct calculations. ∎

The second result, gives the existence of a biorthogonal basis with respect to
$\\{e^{-i\lambda_{\mathbf{k}}t}\\}_{\mathbf{k}\in\mathbb{J}}$.

######  Lemma 4.4.

Let
$H:=\overline{\text{span}\\{e^{-i\lambda_{\mathbf{k}}t}:\mathbf{k}\in\mathbb{J}\\}}$
in $L^{2}([0,T]).$ There exists a unique basis
$\\{q_{\mathbf{k}}\\}_{\mathbf{k}\in\mathbb{J}}\subset H$ such that

$(e^{-i\lambda_{\mathbf{k}}t}\;,\;q_{\mathbf{k}^{\prime}})_{H}=\int_{0}^{T}e^{-i\lambda_{\mathbf{k}}t}\overline{q_{\mathbf{k}^{\prime}}}(t)\;dt=\delta_{\mathbf{k}\mathbf{k}^{\prime}},\;\;\;\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{J},$
(4.8)

where $\delta_{\mathbf{k}\mathbf{k}^{\prime}}$ represents the Kronecker delta.

###### Proof.

The proof is quite well-known by now (see, for instance, [25, Theorem 1.3] or
[47, Thorem 1.3]). The main idea is to use Ingham’s Theorem (see [17, Theorem
4.6, pag. 67]) to show that
$\\{e^{-i\lambda_{\mathbf{k}}t}\\}_{\mathbf{k}\in\mathbb{J}}$ is a Riesz basis
of $H$. Then, the existence of the biorthogonal basis follows from the
standard theory in Hilbert spaces (see [12]). ∎

We are now able to show the main results of this section regarding exact
controllability.

###### Proof of Theorem 1.1.

We prove this Theorem in three steps. Recall, we are assuming
$\widehat{u_{1}}(0,0)=0$ in (4.1).

Step 1. Construction of $h.$

Let $\\{q_{\mathbf{k}}\\}_{\mathbf{k}\in\mathbb{J}}$ be the sequence obtained
in Lemma 4.4. As a first step we will extend the definition of
$q_{\mathbf{k}}$ for all $\mathbf{k}\in\mathbb{Z}^{2}$. We do this following
the rule: given $\mathbf{k}\in\mathbb{Z}^{2}$ we know that there exists
$\mathbf{k}^{\prime}\in\mathbb{J}$ such that
$\lambda_{\mathbf{k}}=\lambda_{\mathbf{k}^{\prime}},$ so we define

$q_{\mathbf{k}}(t):=q_{\mathbf{k}^{\prime}}(t),\quad t\in[0,T].$ (4.9)

The control function $h$ is now defined as

$h(x,y,t)=\sum_{\textbf{j}\in\mathbb{Z}^{2}}h_{\textbf{j}}\;\overline{q_{\textbf{j}}(t)}\;\psi_{\textbf{j}}(x,y),\;\;(x,y)\in\mathbb{T}^{2},$
(4.10)

for suitable coefficients $h_{\textbf{j}}$’s to be determined later by using
the Moment equation. Therefore, we note that for any
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ the left-hand side of (4.4) can be
rewritten as

$\begin{split}I:=&\int\limits_{0}^{T}\left(Gh(x,y,t),e^{-i\lambda_{\mathbf{k}}(T-t)}\psi_{\mathbf{k}}(x,y)\right)_{L^{2}(\mathbb{T}^{2})}dt\\\
&=\int\limits_{0}^{T}\left(\sum_{\textbf{j}\in\mathbb{Z}^{2}}h_{\textbf{j}}\overline{q_{\textbf{j}}(t)}G(\psi_{\textbf{j}})(x,y,t),e^{-i\lambda_{\mathbf{k}}(T-t)}\psi_{\mathbf{k}}(x,y)\right)_{L^{2}_{p}(\mathbb{T}^{2})}dt\\\
&=\sum_{\textbf{j}\in\mathbb{Z}^{2}}h_{\textbf{j}}\int\limits_{0}^{T}\overline{q_{\textbf{j}}(t)}\;e^{i\lambda_{\mathbf{k}}(T-t)}\;dt\left(G(\psi_{\textbf{j}})(x,y),\;\psi_{\mathbf{k}}(x,y)\right)_{L^{2}(\mathbb{T}^{2})}\\\
&=\sum_{\textbf{j}\in\mathbb{Z}^{2}}h_{\textbf{j}}\;e^{i\lambda_{\mathbf{k}}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{\mathbf{k}}t}\;\overline{q_{\textbf{j}}(t)}\;dt\right)\left(G(\psi_{\textbf{j}})(x,y),\;\psi_{\mathbf{k}}(x,y)\right)_{L^{2}(\mathbb{T}^{2})},\end{split}$

where

$\begin{split}\left(G(\psi_{\textbf{j}})(x,y),\;\psi_{\mathbf{k}}(x,y)\right)_{L^{2}(\mathbb{T}^{2})}&=\left(g_{2}(y)\frac{e^{ij_{2}y}}{2\pi}G_{1}(e^{ij_{1}x}),\;\frac{e^{ik_{1}x}e^{ik_{2}y}}{2\pi}\right)_{L^{2}(\mathbb{T}^{2})}\\\
&\quad+\left(g_{1}(x)\frac{e^{ij_{1}x}}{2\pi}G_{2}(e^{ij_{2}y}),\;\frac{e^{ik_{1}x}e^{ik_{2}y}}{2\pi}\right)_{L^{2}(\mathbb{T}^{2})}\\\
&=\widehat{g_{2}(y)}(k_{2}-j_{2})\left(\frac{1}{2\pi}\int_{0}^{2\pi}G_{1}(e^{ij_{1}x})e^{-ik_{1}x}dx\right)\\\
&\quad+\widehat{g_{1}(x)}(k_{1}-j_{1})\left(\frac{1}{2\pi}\int_{0}^{2\pi}G_{2}(e^{ij_{2}y})e^{-ik_{2}y}dy\right)\\\
&=\widehat{g_{2}(y)}(k_{2}-j_{2})\;m_{1}^{j_{1},k_{1}}+\widehat{g_{1}(x)}(k_{1}-j_{1})\;m_{2}^{j_{2},k_{2}}.\end{split}$

Hence,

$\begin{split}I&=\sum_{\textbf{j}\in\mathbb{Z}^{2}}h_{\textbf{j}}\;e^{i\lambda_{\mathbf{k}}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{\mathbf{k}}t}\;\overline{q_{\textbf{j}}(t)}\;dt\right)\widehat{g_{2}(y)}(k_{2}-j_{2})\;m_{1}^{j_{1},k_{1}}\\\
&\quad+\sum_{\textbf{j}\in\mathbb{Z}^{2}}h_{\textbf{j}}\;e^{i\lambda_{\mathbf{k}}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{\mathbf{k}}t}\;\overline{q_{\textbf{j}}(t)}\;dt\right)\widehat{g_{1}(x)}(k_{1}-j_{1})\;m_{2}^{j_{2},k_{2}},\end{split}$
(4.11)

with $m_{n}^{j_{n},k_{n}}$ defined as in (4.5)-(4.6) for $n=1,2.$

Step 2. Construction of the coefficients $h_{\textbf{j}}$.

First of all, note that in order to prove the first part of the theorem,
identity (4.11), Lemma 4.3 (ii), and Corollary 4.2 yield that it suffices to
choose $h_{\textbf{j}}$’s such that

$\begin{split}2\pi\widehat{u_{1}}(\mathbf{k})&=\sum_{\textbf{j}\in\mathbb{Z}^{\ast}\times\mathbb{Z}}h_{\textbf{j}}\;e^{i\lambda_{\mathbf{k}}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{\mathbf{k}}t}\;\overline{q_{\textbf{j}}(t)}\;dt\right)\widehat{g_{2}(y)}(k_{2}-j_{2})\;m_{1}^{j_{1},k_{1}}\\\
&\quad+\sum_{\textbf{j}\in\mathbb{Z}\times\mathbb{Z}^{\ast}}h_{\textbf{j}}\;e^{i\lambda_{\mathbf{k}}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{\mathbf{k}}t}\;\overline{q_{\textbf{j}}(t)}\;dt\right)\widehat{g_{1}(x)}(k_{1}-j_{1})\;m_{2}^{j_{2},k_{2}},\end{split}$
(4.12)

for all $\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}.$ Recall that
$\mathbb{Z}^{*}=\mathbb{Z}\setminus\\{0\\}$.

We will now show that we may indeed choose $h_{\textbf{j}}$’s satisfying
(4.12). To see this, first observe that, since $\widehat{u_{1}}(0,0)=0$, part
(i) in Lemma 4.3 implies that (4.12) holds for $(k_{1},k_{2})=(0,0)$
independently of $h_{\textbf{j}}$’s. In particular, we may choose
$h_{(0,0)}=0$. Next, from Lemma 4.3 (i)-(iv), if
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ with $k_{1}\neq 0$ and $k_{2}=0,$
we see that (4.12) reduces to

$\begin{split}2\pi\widehat{u_{1}}(k_{1},0)&=\sum_{\textbf{j}\in\mathbb{Z}^{\ast}\times\mathbb{Z}}h_{\textbf{j}}\;e^{i\lambda_{(k_{1},0)}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{(k_{1},0)}t}\;\overline{q_{\textbf{j}}(t)}\;dt\right)\widehat{g_{2}(y)}(-j_{2})\;m_{1}^{j_{1},k_{1}}\\\
&=\sum_{j_{2}\in\mathbb{Z}}h_{(k_{1},j_{2})}\;e^{i\lambda_{(k_{1},0)}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{(k_{1},0)}t}\;\overline{q_{(k_{1},j_{2})}(t)}\;dt\right)\widehat{g_{2}(y)}(-j_{2})\;m_{1}^{k_{1},k_{1}}.\end{split}$
(4.13)

According to (4.8)-(4.9), all terms of the series on the right-hand side of
(4.13) is zero, except for those $j_{2}\in\mathbb{Z}$ such that

$\lambda_{(k_{1},j_{2})}=\lambda_{(k_{1},0)}.$

In view of hypothesis $(H2),$ this holds only for $j_{2}=0.$ Hence,

$\begin{split}2\pi\widehat{u_{1}}(k_{1},0)&=h_{(k_{1},0)}\;e^{i\lambda_{(k_{1},0)}T}\widehat{g_{2}(y)}(0)\;\frac{1}{2\pi}\end{split}$
(4.14)

and, in view of (1.7),

$\begin{split}h_{(k_{1},0)}&=(2\pi)^{3}\widehat{u_{1}}(k_{1},0)e^{-i\lambda_{(k_{1},0)}T}.\end{split}$
(4.15)

Similarly, from Lemma (4.3) (i)-(iv), if
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ with $k_{1}=0$ and $k_{2}\neq 0,$
we see that (4.12) reduces to

$\begin{split}2\pi\widehat{u_{1}}(0,k_{2})&=\sum_{j_{1}\in\mathbb{Z}}h_{(j_{1},k_{2})}\;e^{i\lambda_{(0,k_{2})}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{(0,k_{2})}t}\;\overline{q_{(j_{1},k_{2})}(t)}\;dt\right)\widehat{g_{1}(x)}(-j_{1})\;m_{2}^{k_{2},k_{2}}.\end{split}$
(4.16)

In view of hypothesis $(H2),$ we see that
$\lambda_{(j_{1},k_{2})}=\lambda_{(0,k_{2})}$ only for $j_{1}=\pm 0.$ Hence,
from (4.8)-(4.9), and (1.6) we deduce that

$\begin{split}2\pi\widehat{u_{1}}(0,k_{2})&=h_{(0,k_{2})}\;e^{i\lambda_{(0,k_{2})}T}\widehat{g_{1}(x)}(0)\;\frac{1}{2\pi}\end{split}$
(4.17)

or, equivalently,

$\begin{split}h_{(0,k_{2})}&=(2\pi)^{3}\widehat{u_{1}}(0,k_{2})e^{-i\lambda_{(0,k_{2})}T}.\end{split}$
(4.18)

Finally, if $\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ with $k_{1}\neq 0$ and
$k_{2}\neq 0,$ we get from (4.12) and Lemma (4.3) (iv), that

$\begin{split}2\pi\widehat{u_{1}}(k_{1},k_{2})&=\sum_{j_{2}\in\mathbb{Z}}h_{(k_{1},j_{2})}\;e^{i\lambda_{(k_{1},k_{2})}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{(k_{1},k_{2})}t}\;\overline{q_{(k_{1},j_{2})}(t)}\;dt\right)\widehat{g_{2}(y)}(k_{2}-j_{2})\;m_{1}^{k_{1},k_{1}}\\\
&\quad+\sum_{j_{1}\in\mathbb{Z}}h_{(j_{1},k_{2})}\;e^{i\lambda_{(k_{1},k_{2})}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{(k_{1},k_{2})}t}\;\overline{q_{(j_{1},k_{2})}(t)}\;dt\right)\widehat{g_{1}(x)}(k_{1}-j_{1})\;m_{2}^{k_{2},k_{2}}.\end{split}$
(4.19)

According to (4.8)-(4.9), all terms in the sums are zero, except the ones
where the entire variables $j_{1},j_{2}$ solve the following equations:

$\lambda_{(k_{1},j_{2})}=\lambda_{(k_{1},k_{2})},$
$\lambda_{(j_{1},k_{2})}=\lambda_{(k_{1},k_{2})}.$

In view of hypothesis $(H2),$ we have the solutions $j_{2}=k_{2}$ and
$j_{1}=\pm k_{1},$ respectively. Hence,

$\begin{split}2\pi\widehat{u_{1}}(k_{1},k_{2})&=h_{(k_{1},k_{2})}\;e^{i\lambda_{(k_{1},k_{2})}T}\widehat{g_{2}(y)}(0)\;\frac{1}{2\pi}+h_{(k_{1},k_{2})}\;e^{i\lambda_{(k_{1},k_{2})}T}\widehat{g_{1}(x)}(0)\;\frac{1}{2\pi}\\\
&\quad+h_{(-k_{1},k_{2})}\;e^{i\lambda_{(k_{1},k_{2})}T}\widehat{g_{1}(x)}(2k_{1})\;\frac{1}{2\pi},\end{split}$
(4.20)

or, equivalently,

$\begin{split}(2\pi)^{2}e^{-i\lambda_{(k_{1},k_{2})}T}\widehat{u_{1}}(k_{1},k_{2})&=h_{(k_{1},k_{2})}\;\left(\widehat{g_{2}(y)}(0)+\widehat{g_{1}(x)}(0)\right)+h_{(-k_{1},k_{2})}\;\widehat{g_{1}(x)}(2k_{1}).\end{split}$
(4.21)

Note that in the left-hand side of (4.21) appear the coefficients
$h_{(k_{1},k_{2})}$ and $h_{(-k_{1},k_{2})}$; so, in order to determine them
we will couple (4.21) with another equation. To do so, observe that

$\begin{split}2\pi\widehat{u_{1}}(-k_{1},k_{2})&=\sum_{j_{2}\in\mathbb{Z}}h_{(-k_{1},j_{2})}\;e^{i\lambda_{(-k_{1},k_{2})}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{(-k_{1},k_{2})}t}\;\overline{q_{(-k_{1},j_{2})}(t)}\;dt\right)\widehat{g_{2}(y)}(k_{2}-j_{2})\;m_{1}^{-k_{1},-k_{1}}\\\
&\quad+\sum_{j_{1}\in\mathbb{Z}}h_{(j_{1},k_{2})}\;e^{i\lambda_{(-k_{1},k_{2})}T}\left(\int\limits_{0}^{T}e^{-i\lambda_{(-k_{1},k_{2})}t}\;\overline{q_{(j_{1},k_{2})}(t)}\;dt\right)\widehat{g_{1}(x)}(-k_{1}-j_{1})\;m_{2}^{k_{2},k_{2}}.\end{split}$

As before, from (4.8)-(4.9), all terms in the sums are zero, except the ones
for which $j_{1},j_{2}$ solves

$\lambda_{(-k_{1},j_{2})}=\lambda_{(-k_{1},k_{2})},$
$\lambda_{(j_{1},k_{2})}=\lambda_{(-k_{1},k_{2})}.$

In view of hypothesis $(H2)$ again, we must have $j_{2}=k_{2}$ and $j_{1}=\mp
k_{1}$. Hence,

$\begin{split}2\pi\widehat{u_{1}}(-k_{1},k_{2})&=h_{(-k_{1},k_{2})}\;e^{i\lambda_{(-k_{1},k_{2})}T}\widehat{g_{2}(y)}(0)\;\frac{1}{2\pi}+h_{(-k_{1},k_{2})}\;e^{i\lambda_{(-k_{1},k_{2})}T}\widehat{g_{1}(x)}(0)\;\frac{1}{2\pi}\\\
&\quad+h_{(k_{1},k_{2})}\;e^{i\lambda_{(-k_{1},k_{2})}T}\widehat{g_{1}(x)}(-2k_{1})\;\frac{1}{2\pi},\end{split}$

or, which is the same,

$\begin{split}(2\pi)^{2}e^{-i\lambda_{(-k_{1},k_{2})}T}\widehat{u_{1}}(-k_{1},k_{2})&=h_{(-k_{1},k_{2})}\;\left(\widehat{g_{2}(y)}(0)+\widehat{g_{1}(x)}(0)\right)+h_{(k_{1},k_{2})}\;\widehat{g_{1}(x)}(-2k_{1}).\end{split}$
(4.22)

It follows from (4.21) and (4.22) that we must solve the linear system

$\begin{cases}h_{(k_{1},k_{2})}\;\left(\widehat{g_{2}(y)}(0)+\widehat{g_{1}(x)}(0)\right)+h_{(-k_{1},k_{2})}\;\widehat{g_{1}(x)}(2k_{1})=(2\pi)^{2}e^{-i\lambda_{(k_{1},k_{2})}T}\widehat{u_{1}}(k_{1},k_{2}),&\\\
h_{(-k_{1},k_{2})}\;\left(\widehat{g_{2}(y)}(0)+\widehat{g_{1}(x)}(0)\right)+h_{(k_{1},k_{2})}\;\widehat{g_{1}(x)}(-2k_{1})=(2\pi)^{2}e^{-i\lambda_{(-k_{1},k_{2})}T}\widehat{u_{1}}(-k_{1},k_{2}).&\end{cases}$

To see that such a system has a (unique) solution, using (1.6)-(1.7), we may
write it as

$\begin{pmatrix}\displaystyle{\frac{1}{\pi}}&\widehat{g_{1}(x)}(2k_{1})\\\
\widehat{g_{1}(x)}(-2k_{1})&\displaystyle{\frac{1}{\pi}}\end{pmatrix}\begin{pmatrix}h_{(k_{1},k_{2})}\\\
h_{(-k_{1},k_{2})}\end{pmatrix}=(2\pi)^{2}\begin{pmatrix}e^{-i\lambda_{(k_{1},k_{2})}T}\widehat{u_{1}}(k_{1},k_{2})\\\
e^{-i\lambda_{(-k_{1},k_{2})}T}\widehat{u_{1}}(-k_{1},k_{2})\end{pmatrix}.$

If we set

$M:=\begin{pmatrix}\displaystyle{\frac{1}{\pi}}&\widehat{g_{1}(x)}(2k_{1})\\\
\widehat{g_{1}(x)}(-2k_{1})&\displaystyle{\frac{1}{\pi}}\end{pmatrix},$

then

$\text{det}(M)=\displaystyle{\frac{1}{\pi^{2}}}-\widehat{g_{1}(x)}(2k_{1})\;\overline{\widehat{g_{1}(x)}(2k_{1})}=\displaystyle{\frac{1}{\pi^{2}}}-\left|\widehat{g_{1}(x)}(2k_{1})\right|^{2}=:d_{k_{1}}.$

Now, observe that since $g_{1}$ is a non-negative function, from (1.6), we
deduce,

$\left|\widehat{g_{1}(x)}(2k_{1})\right|\leq\frac{1}{2\pi}\int_{0}^{2\pi}g_{1}(x)dx=\frac{1}{2\pi}.$

Hence,

$d_{k_{1}}\geq\frac{1}{\pi^{2}}-\frac{1}{4\pi^{2}}=\frac{3}{4\pi^{2}},\;\forall
k_{1}\in\mathbb{Z},$

and the matrix $M$ is invertible with

$M^{-1}=\begin{pmatrix}\dfrac{1}{\pi
d_{k_{1}}}&-\dfrac{\widehat{g_{1}(x)}(2k_{1})}{d_{k_{1}}}\\\ \\\
-\dfrac{\widehat{g_{1}(x)}(-2k_{1})}{d_{k_{1}}}&\dfrac{1}{\pi
d_{k_{1}}}\end{pmatrix}.$ (4.23)

This implies that the above system has a solution and, in addition, there
exists a constant $D$, independent of $k_{1}\in\mathbb{Z}^{\ast},$ such that

$\|M^{-1}\|\leq D,$

where $\|M^{-1}\|$ is the Euclidean norm of the matrix $M^{-1}.$

Step 3. The function $h$ defined by (4.10) with $h_{(0,0)}=0,$ $h_{(k_{1},0)}$
given by (4.15), $h_{(0,k_{2})}$ given by (4.18), and $h_{(k_{1},k_{2})}$
given as the solution of

$\begin{pmatrix}h_{(k_{1},k_{2})}\\\
h_{(-k_{1},k_{2})}\end{pmatrix}=M^{-1}\begin{pmatrix}(2\pi)^{2}e^{-i\lambda_{(k_{1},k_{2})}T}\widehat{u_{1}}(k_{1},k_{2})\\\
(2\pi)^{2}e^{-i\lambda_{(-k_{1},k_{2})}T}\widehat{u_{1}}(-k_{1},k_{2})\end{pmatrix},\;\;\text{for
all}\;(k_{1},k_{2})\in\mathbb{Z}^{\ast}\times\mathbb{Z}^{\ast}.$ (4.24)

belongs to $L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))$.

Indeed, recall from Lemma 4.4 that
$\\{q_{\mathbf{k}}\\}_{\mathbf{k}\in\mathbb{J}}$ is a Riesz basis for $H$.
Thus, from [12, Theorem 7.13] and the definition of $q_{\mathbf{k}}$,
$\mathbf{k}\in\mathbb{Z}^{2}$ it follows that
$\\{q_{\mathbf{k}}\\}_{\mathbf{k}\in\mathbb{Z}^{2}}$ is a bounded sequence in
$L^{2}([0,T])$. Hence, in view of (4.10), we deduce the existence of a
positive constant $C$ such that

$\displaystyle\begin{split}\|h\|^{2}_{L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))}&=\sum_{\mathbf{k}\in\mathbb{Z}^{2}}(1+|\mathbf{k}|)^{2s}|h_{\mathbf{k}}|^{2}\int_{0}^{T}|q_{\mathbf{k}}(t)|^{2}\;dt\\\
&\leq
C\sum_{\mathbf{k}\in\mathbb{Z}^{2}}(1+|\mathbf{k}|)^{2s}|h_{\mathbf{k}}|^{2}\\\
&=C\sum_{k_{1}\in\mathbb{Z}^{\ast}}(1+|(k_{1},0)|)^{2s}|h_{(k_{1},0)}|^{2}\\\
&\quad+C\sum_{k_{2}\in\mathbb{Z}^{\ast}}(1+|(0,k_{2})|)^{2s}|h_{(0,k_{2})}|^{2}\\\
&\quad+C\sum_{(k_{1},k_{2})\in\mathbb{Z}^{\ast}\times\mathbb{Z}^{\ast}}(1+|(k_{1},k_{2})|)^{2s}|h_{(k_{1},k_{2})}|^{2}.\\\
\end{split}$ (4.25)

From (4.24), we infer that

$|h_{(k_{1},k_{2})}|^{2}\leq\|M^{-1}\|^{2}(2\pi)^{4}\left(|\widehat{u_{1}}(k_{1},k_{2})|^{2}+|\widehat{u_{1}}(-k_{1},k_{2})|^{2}\right),\;\forall(k_{1},k_{2})\in\mathbb{Z}^{\ast}\times\mathbb{Z}^{\ast}.$

Therefore,

$\begin{split}(1+|(k_{1},k_{2})|)^{2s}|h_{(k_{1},k_{2})}|^{2}&\leq
D^{2}(2\pi)^{4}(1+|(k_{1},k_{2})|)^{2s}|\widehat{u_{1}}(k_{1},k_{2})|^{2}\\\
&\quad+D^{2}(2\pi)^{4}|(1+|(-k_{1},k_{2})|)^{2s}|\widehat{u_{1}}(-k_{1},k_{2})|^{2},\end{split}$
(4.26)

for all $(k_{1},k_{2})\in\mathbb{Z}^{\ast}\times\mathbb{Z}^{\ast}.$ Thus,
identities (4.15), (4.18) and (4.26) imply

$\begin{split}\|h\|^{2}_{L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))}&\leq
C(2\pi)^{6}\sum_{k_{1}\in\mathbb{Z}^{\ast}}(1+|(k_{1},0)|)^{2s}|\widehat{u_{1}}(k_{1},0)|^{2}\\\
&\quad+C(2\pi)^{6}\sum_{k_{2}\in\mathbb{Z}^{\ast}}(1+|(0,k_{2})|)^{2s}|\widehat{u_{1}}(0,k_{2})|^{2}\\\
&\quad+2C(2\pi)^{4}D^{2}\sum_{(k_{1},k_{2})\in\mathbb{Z}^{\ast}\times\mathbb{Z}^{\ast}}(1+|(k_{1},k_{2})|)^{2s}|\widehat{u_{1}}(k_{1},k_{2})|^{2}\\\
&\leq\nu^{2}\sum_{\mathbf{k}\in\mathbb{Z}^{2}}(1+|\mathbf{k}|)^{2s}|\widehat{u_{1}}(\mathbf{k})|^{2},\end{split}$
(4.27)

where $\nu^{2}=3\max\\{C(2\pi)^{6},2C(2\pi)^{4}D^{2}\\}.$ Since $u_{1}\in
H^{s}_{p}(\mathbb{T}^{2})$ the above series converges. In addition (1.17)
holds (recall $u_{0}=0$). This completes the proof of the theorem. ∎

The proof of Theorem 1.2 is very similar with minor modifications, so we omit
the details.

######  Corollary 4.5.

Assume $s\geq 0$. Under the assumptions of Theorem 1.1 or Theorem 1.2, for any
$T>\frac{2\pi}{\gamma^{\prime}}$, there exists a unique bounded linear
operator

$\begin{cases}\Phi:H_{p}^{s}(\mathbb{T}^{2})\times
H_{p}^{s}(\mathbb{T}^{2})\longrightarrow
L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))&\\\
(u_{0},u_{1})\longmapsto\Phi(u_{0},u_{1})=:h\end{cases}$

such that

$u_{1}=U(T)u_{0}+\int_{0}^{T}U(T-s)(G(\Phi(u_{0},u_{1})))(\cdot,\cdot,s)ds$

and

$\|\Phi(u_{0},u_{1})\|_{L^{2}([0,T];H_{p}^{s}(\mathbb{T}^{2}))}\leq\nu\left(\|u_{0}\|_{H_{p}^{s}(\mathbb{T}^{2})}+\|u_{1}\|_{H_{p}^{s}(\mathbb{T}^{2})}\right),$

for some positive constant $\nu.$

######  Remark 4.6.

The constant $\nu$ in Corollary 4.5 depends only on $s,\;g_{1}$ and $T$ (resp.
$s,\;g_{2}$ and $T$) under assumptions of Theorem 1.1 (resp. Theorem 1.2).

######  Corollary 4.7.

Let $s\geq 0.$ Under assumptions of Theorem 1.1 or Theorem 1.2, for any
$T>\frac{2\pi}{\gamma^{\prime}}$ there exists $\delta>0$ such that

$\int_{0}^{T}\left\|G^{\ast}U(-t)^{\ast}\phi\right\|^{2}_{H_{p}^{s}(\mathbb{T}^{2})}(t)dt\geq\delta^{2}\|\phi\|^{2}_{H_{p}^{s}(\mathbb{T}^{2})},\;\forall\phi\in
H_{p}^{s}(\mathbb{T}^{2}),$ (4.28)

where the constant $\delta$ depends only on $s,g_{1},$ and $T$ (resp.
$s,g_{2},$ and $T$) under assumptions of Theorem 1.1 (resp. Theorem 1.2).

###### Proof.

This is a consequence of the Hilbert Uniqueness Method (HUM) due to J.-L.
Lions [27]. Actually, as is well known, the exact controllability is
equivalent to the observability inequality (4.28). See for instance [28,
Theorem 2.3] or [39, Theorem 2.4]. ∎

######  Remark 4.8.

If $\gamma^{\prime}=+\infty,$ then Corollaries 4.5 and 4.7 are valid for any
time $T>0.$

##  5\. Stabilization Results

In this section we prove the exponential stabilization result stated in
Theorem 1.4. First, we show if $K$ is a bounded operator in
$H_{p}^{s}(\mathbb{T}^{2})$ then system (1.13) is globally well-posed in
$H_{p}^{s}(\mathbb{T}^{2}),$ $s\geq 0.$

######  Theorem 5.1.

Let $u_{0}\in H_{p}^{s}(\mathbb{T}^{2}),$ $s\geq 0.$ Then the IVP (1.13) has a
unique (mild) solution

$u\in C([0,\infty);H_{p}^{s}(\mathbb{T}^{2})).$

###### Proof.

We know, from Theorem 3.1, that operator $\partial_{x}\mathcal{L}$, with
domain $H^{s+r}_{p}(\mathbb{T}^{2})$ is the infinitesimal generator of a
unitary group in $H^{s}_{p}(\mathbb{T}^{2})$. Hence, it also generates of a
$C_{0}$-semigroup $\left\\{U(t)\right\\}_{t\geq 0}$. We also know that $GK$ is
a bounded linear operator on $H^{s}_{p}(\mathbb{T}^{2}).$ From the semigroup
theory (see [36, page 76]), we get that operator $\partial_{x}\mathcal{L}+GK,$
which is a perturbation of $\partial_{x}\mathcal{L}$ by a bounded linear
operator, is the infinitesimal generator of a $C_{0}$-semigroup, say,
$\left\\{T(t)\right\\}_{t\geq 0}$ on $H^{s}_{p}(\mathbb{T}^{2}).$
Consequently, (1.13) has a unique mild solution. ∎

###### Proof of Theorem 1.4.

The well-posedness of IVP (1.20) is given by Theorem 5.1. Then, Theorem 1.4 is
a direct consequence of Corollary 4.7 and the classical principle: Exact
controllability implies exponential stabilizability for conservative control
systems (see [28, Theorem 2.3-2.4] and [44, Theorem 2.1]). Actually, according
to [44] one can choose

$K_{\lambda}=-G^{\ast}D_{T,\lambda}^{-1},$

where, for some $T>\frac{2\pi}{\gamma^{\prime}},$

$D_{T,\lambda}\phi=\int_{0}^{T}e^{-2\lambda\tau}U(-\tau)GG^{\ast}U(-\tau)^{\ast}\phi\;d\tau,\;\;\;\forall\phi\in
H_{p}^{s}(\mathbb{T}^{2}),$ (5.1)

and $U(t)$ is the $C_{0}$-semigroup generated by $\partial_{x}\mathcal{L}.$ ∎

##  6\. Applications

In many situation, internal waves arise due to the gravitational effects, at
the interface of two layers in a stratified fluid. Several theoretical models
exist which govern the evolution of long internal waves with small amplitudes
in such cases. When the height of the heavier fluid is much larger than that
of the upper layer, the motion is described by the Benjamin-Ono equation (BO)
[2, 35]:

$\partial_{t}u-\mathcal{H}\partial^{2}_{x}u+u\partial_{x}u=0,\;\;x\in\mathbb{R},\;t>0,$
(6.1)

where $\mathcal{H}$ denotes the Hilbert transform. Equation (6.1) may also be
viewed as a general model for the propagation of weakly nonlinear long waves
incorporating the lowest-order effects of nonlinearity and non-local
dispersion and it turns out to be important in many others physical situations
(see, for instance, [7, 14, 29]).

On the other hand, when the total depth of the a fluid is very small, the
motion is governed by the Korteweg-de Vries (KdV) equation

$\partial_{t}u+\partial^{3}_{x}u+u\partial_{x}u=0,\;\;x\in\mathbb{R},\;t>0,$
(6.2)

derived in [18] as a model for the propagation of long one dimensional surface
gravity waves with small amplitude in a shallow channel of water. The KdV
equation has a very rich structure from the mathematical point of view and it
has also been derived in several other physical context (see, for instance,
[1]).

In both situations above, when transversal effects must also be considered,
the resulting equations are bidimensional. Hence, in this section, we present
some particular examples of bidimensional dispersive PDE’s, where the general
control theory developed in this work can be applied to their linear
counterpart.

###  6.1. The Zakharov-Kuznetsov (ZK) equation:

One of the most accepted generalization of the KdV equation in two dimensions
is the Zakharov-Kuznetsov (ZK) equation:

$\partial_{t}u+\partial_{x}\Delta
u+u\partial_{x}u=0,\;\;(x,y)\in\mathbb{R}^{2},\;t>0,$ (6.3)

where $\Delta$ denotes the bidimensional Laplacian, that is,
$\Delta=\partial_{x}^{2}+\partial_{y}^{2}.$ Equation (6.3) models ion-acoustic
waves propagating in a low-pressure magnetized plasma. It was derived in [48]
where the existence and stability for circularly symmetric soliton solutions
were established. Questions of local well-posedness for (6.3) in the Sobolev
spaces $H^{s}(\mathbb{R}^{2})$ may be found, for instance in, [10], [16],
[24], [34]. The initial-value problem posed on the two dimensional torus was
studied in [26]. In addition, in [33] the authors addressed the exact
controllability of the linear ZK equation on a rectangle with a left Dirichlet
boundary control by using the flatness approach.

Here we address the exact controllability associated with the linear equation

$\begin{cases}\partial_{t}u+\partial_{x}\Delta
u=Gh,\;\;\;(x,y)\in\mathbb{T}^{2},\;t>0,\\\ u(x,y,0)=u_{0}(x,y).\end{cases}$
(6.4)

In order to set (6.4) as in (1.4), we define $\mathcal{L}=-\Delta$ so that

$b(\mathbf{k}):=|\mathbf{k}|^{2}=k_{1}^{2}+k_{2}^{2},\;\;\text{for
all}\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$

and the eigenvalues of $\partial_{x}\mathcal{L}$ are $i\lambda_{\mathbf{k}}$
with

$\lambda_{\mathbf{k}}:=k_{1}b(\mathbf{k})=k_{1}(k_{1}^{2}+k_{2}^{2}).$

Clearly,

$|b(\mathbf{k})|\leq|\mathbf{k}|^{2},\;\;\text{for
all}\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$

and (1.3) holds.

Also, it is easy to check that $(H3)$ holds and the value of $\gamma$ in
(1.18) is equal to $1.$ Note that, if $n\in\mathbb{N},$ then
$\lambda_{(2^{2n},0)}\to\infty,$ $\lambda_{(1,2^{3n})}\to\infty,$ as
$n\to\infty$ and $\lambda_{(2^{2n},0)}\neq\lambda_{(1,2^{3n})}$ with

$\left|\lambda_{(2^{2n},0)}-\lambda_{(1,2^{3n})}\right|=|2^{6n}-1-2^{6n}|=1.$

Hence, $\gamma^{\prime}$ defined in (1.19) is also equal to $1.$

Applying Theorem 1.2 we conclude that system (6.4) is exactly controllable in
any time $T>2\pi$ in the Sobolev space $H_{p}^{s}(\mathbb{T}^{2})$ with $s\geq
0,$ where the control function $h$ is given by (4.10). Also, Theorem 1.4 holds
and the system (6.4) is exponentially stabilizable with any decay rate
$\lambda>0.$

###  6.2. The 2D Benjamin-Ono (2D-BO) equation:

In this subsection, we consider a two-dimensional extension of the BO
equation, which reads as

$\partial_{t}u-\mathcal{H}^{(x)}\partial^{2}_{xy}u+u\partial_{y}u=0,\;\;(x,y)\in\mathbb{R}^{2},\;t>0,$
(6.5)

where $\mathcal{H}^{(x)}$ denotes the Hilbert transform with respect to the
$x$-variable, that is, via Fourier transform,

$\widehat{\mathcal{H}^{(x)}u}(\xi,\eta)=-i\;\text{sng}(\xi)\widehat{u}(\xi,\eta),\qquad(\xi,\eta)\in\mathbb{R}^{2}.$

From the mathematical point of view, local and global well-posedness for (6.5)
have been studied in [30] and [31].

The control equation associated to the linear part of (6.5) on the periodic
setting reads as follows:

$\partial_{t}u-\mathcal{H}^{(x)}\partial^{2}_{xy}u=Gh,\;\;u(x,y,0)=u_{0}(x,y),\;\;\;(x,y)\in\mathbb{T}^{2},\;t>0.$
(6.6)

In this case the operator $\mathcal{L}$ takes the form
$\mathcal{L}=\mathcal{H}^{(x)}\partial_{y},$ where the Hilbert transform
$\mathcal{H}^{(x)}$ in the frequency space is given by

$\widehat{\mathcal{H}^{(x)}u}(k_{1},k_{2}):=-i\;\text{sng}(k_{1})\widehat{u}(k_{1},k_{2}),\;\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}.$

Therefore,

$b(\mathbf{k})=k_{2}\;\text{sgn}(k_{1}),$

and the eigenvalues of operator $\partial_{x}\mathcal{L}$ have the form
$i\lambda_{\mathbf{k}}$ with

$\lambda_{\mathbf{k}}:=k_{1}b(\mathbf{k})=|k_{1}|k_{2},\;\;\;\mathbf{k}\in\mathbb{Z}^{2}.$
(6.7)

In what follows we shall show that Theorems 1.1, and 1.4 can be applied to
prove that (6.6) is exactly controllable in any time $T>2\pi,$ and
exponentially stabilizable with any given decay rate in the Sobolev space
$H_{p}^{s}(\mathbb{T}^{2})$, $s\geq 0.$ Indeed, first of all note that

$|b(\mathbf{k})|\leq|\mathbf{k}|,\;\;\;\mathbf{k}\in\mathbb{Z}^{2},$

and (1.3) is true with $r=2$. From (6.7) it is clear that $(H2)$ holds.
Additionally, for any $\mathbf{k},\mathbf{k}^{\prime}\in\mathbb{J},$

$\left|\lambda_{\mathbf{k}}-\lambda_{\mathbf{k}^{\prime}}\right|=\Big{|}|k_{1}|k_{2}-|k^{\prime}_{1}|k^{\prime}_{2}\Big{|}\geq
1.$

Also, note that, if $k_{1}\to\infty$ then $\lambda_{(k_{1}+1,1)}\to\infty,$
$\lambda_{(k_{1},1)}\to\infty,$ and
$\lambda_{(k_{1}+1,1)}\neq\lambda_{(k_{1},1)}$ with

$\left|\lambda_{(k_{1}+1,1)}-\lambda_{(k_{1},1)}\right|=\left||k_{1}+1|-|k_{1}|\right|=1.$

Therefore, $\gamma$ and $\gamma^{\prime}$ defined respectively by (1.14) and
(1.15) are, in this case, equal to 1. The result follows as desired.

###  6.3. The Benjamin-Ono-Zakharov-Kuznetsov (BOZK) equation:

Another model that may be seen as a two-dimensional extension of the BO
equation is the so called BOZK equation:

$\partial_{t}u-\mathcal{H}^{(x)}\partial_{x}^{2}u+\partial_{x}\partial_{y}^{2}u+u\partial_{x}u=0,\;\;\;(x,y)\in\mathbb{R}^{2},\;t>0.$
(6.8)

The equation in (6.8) was introduced in [20] [15], and it has applications to
electromigration in thin nanoconductors on a dielectric substrate. Local and
global well-posedness for the Cauchy problem associated with (6.8) in Sobolev
spaces was studied, for instance, in [4], [5], and [42].

In this subsection we investigate the control and stabilization properties of
linear BOZK equation:

$\partial_{t}u-\mathcal{H}^{(x)}\partial_{x}^{2}u+\partial_{x}\partial_{y}^{2}u=Gh,\;\;u(x,y,0)=u_{0}(x,y),\;\;\;(x,y)\in\mathbb{T}^{2},\;t>0.$
(6.9)

Here, we consider the operator $\mathcal{L}$ defined in (1.4) as
$\mathcal{L}:=\mathcal{H}^{(x)}\partial_{x}-\partial_{y}^{2}.$ Therefore,

$b(\mathbf{k})=|k_{1}|+k_{2}^{2},\;\;\;\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2},$

and

$|b(\mathbf{k})|=|k_{1}|+k_{2}^{2}\leq|\mathbf{k}|+|\mathbf{k}|^{2}\leq
2|\mathbf{k}|^{2},\;\;\text{for all}\;\mathbf{k}\in\mathbb{Z}^{2}.$

The eigenvalues of operator $\partial_{x}\mathcal{L}$ are
$i\lambda_{\mathbf{k}}$ with

$\lambda_{\mathbf{k}}:=k_{1}b(\mathbf{k})=k_{1}(|k_{1}|+k_{2}^{2}).$

Next, we shall verify that (H3) holds. We easily check that
$\lambda_{(k_{1},-k_{2})}=\lambda_{(k_{1},k_{2})}$ and
$\lambda_{(-k_{1},k_{2})}=-\lambda_{(k_{1},k_{2})}$ for all
$(k_{1},k_{2})\in\mathbb{Z}^{2}$. Also, for $(k_{1},k_{2})\in\mathbb{Z}^{2}$,
with $k_{1}\neq 0$, it is clear that the unique entire solutions of the
$j_{2}$-equation

$\lambda_{(k_{1},j_{2})}=\lambda_{(k_{1},k_{2})}\iff
k_{1}j_{2}^{2}=k_{1}k_{2}^{2},$

are $j_{2}=\pm k_{2}.$ On the other hand, for
$(k_{1},k_{2})\in\mathbb{Z}^{2}$, with $k_{2}\neq 0,$ we analyze the entire
solutions of the $j_{1}$-equation

$\lambda_{(j_{1},k_{2})}=\lambda_{(k_{1},k_{2})}\iff
j_{1}(|j_{1}|+k_{2}^{2})=k_{1}(|k_{1}|+k_{2}^{2}),$

which can be rewritten in the following form

$j_{1}|j_{1}|-k_{1}|k_{1}|+k_{2}^{2}(j_{1}-k_{1})=0.$ (6.10)

It is enough to assume $k_{1}\neq 0,$ because the unique solution of (6.10)
with $k_{1}=0$ is clearly $j_{1}=0$ and the desired result follows.
Immediately, we observe that $j_{1}$ and $k_{1}$ should share the same sign.
To see this, it suffices to note that the expression on left-hand side in
(6.10) is strictly positive if $j_{1}\geq 0$ and $k_{1}<0$ and strictly
negative when $j_{1}\leq 0$ and $k_{1}>0.$ Therefore, to solve equation (6.10)
with $k_{1}>0,$ we may assume $j_{1}\geq 0$ to see that it is equivalent to

$(j_{1}-k_{1})(j_{1}+k_{1}+k_{2}^{2})=0,$

from which we obtain that the unique entire solution is $j_{1}=k_{1}.$
Similarly, when $k_{1}<0,$ equation (6.10) is equivalent to

$(j_{1}-k_{1})(-j_{1}-k_{1}+k_{2}^{2})=0,$

and again the unique entire solution is $j_{1}=k_{1}.$ Consequently, $(H3)$
holds.

Finally, we note that $\gamma$ given by (1.18) is equal to 1. In addition, by
taking $\lambda_{(k_{1},0)}$ and $\lambda_{(1,k_{1})}$ for any
$0<k_{1}\in\mathbb{Z}$ we easily verify that $\lambda_{(k_{1},0)}\to\infty,$
$\lambda_{(1,k_{1})}\to\infty$ as $k_{1}\to\infty$ and
$\lambda_{(k_{1},0)}\neq\lambda_{(1,k_{1})}$ with

$|\lambda_{(k_{1},0)}-\lambda_{(1,k_{1})}|=1.$

from which we infer $\gamma^{\prime}=1$ (see (1.19)). Thus, we can apply
Theorems 1.2 and 1.4 to deduce that system (6.9) is exactly controllable in
any $T>2\pi,$ and exponentially stabilizable with any decay rate in the
Sobolev space $H_{p}^{s}(\mathbb{T}^{2})$, $s\geq 0.$

###  6.4. The dispersion generalized Benjamin-Ono-Zakharov-Kuznetsov (dgBOZK)
equation:

To finish our applications, we shall consider the dgBOZK equation

$\partial_{t}u-D_{x}^{\alpha}\partial_{x}u+\partial_{x}\partial_{y}^{2}u+u\partial_{x}u=0,\;\;\;\;(x,y)\in\mathbb{R}^{2},\;t>0,$
(6.11)

where $\alpha>0$ and $D_{x}^{\alpha}$ is defined via Fourier transform as
$\widehat{D_{x}^{\alpha}u}(\xi,\eta)=|\xi|^{\alpha}\widehat{u}(\xi,\eta).$ In
the case $\alpha\in(1,2)$, equation (6.11) may be seen as an interpolation
between the ZK and BOZK equations in the sense that in the limiting cases
$\alpha=2$ and $\alpha=1$, (6.11) reduces to ZK and BOZK equations,
respectively. The interested reader will find some local and global well-
posedness results for the associated Cauchy problem in [6] and [42].

As in the earlier examples, here we study the control problem for the linear
dgBOZK equation:

$\partial_{t}u-D_{x}^{\alpha}\partial_{x}u+\partial_{x}\partial_{y}^{2}u=Gh,\;\;u(x,y,0)=u_{0}(x,y),\;\;(x,y)\in\mathbb{T}^{2},\;t>0,$
(6.12)

where $\alpha>0$ and $D_{x}^{\alpha}$ is now defined as
$\widehat{D_{x}^{\alpha}u}(\mathbf{k})=|k_{1}|^{\alpha}\widehat{u}(\mathbf{k}).$
Thus, the operator $\mathcal{L}$ reads as
$\mathcal{L}:=D_{x}^{\alpha}-\partial_{y}^{2},$ so that

$b(\mathbf{k})=|k_{1}|^{\alpha}+k_{2}^{2},$

and

$|b(\mathbf{k})|\leq|k_{1}|^{\alpha}+k_{2}^{2}\leq|\mathbf{k}|^{\alpha}+|\mathbf{k}|^{2}\leq\begin{cases}2|\mathbf{k}|^{2},&\text{if}\;0<\alpha<2,\\\
2|\mathbf{k}|^{\alpha},&\text{if}\;\alpha\geq 2,\end{cases}$

which means that (1.3) holds. The eigenvalues of $\partial_{x}\mathcal{L}$ are
$i\lambda_{\mathbf{k}}$ with

$\lambda_{\mathbf{k}}:=k_{1}b(\mathbf{k})=k_{1}(|k_{1}|^{\alpha}+k_{2}^{2}).$

Let us check that (H3) also holds here. Indeed, clearly the eigenvalues are
even in the second variable and odd in the first one. Also, for any
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ given with $k_{1}\neq 0,$ it is
easy to show that the unique entire solutions of the $j_{2}$-equation

$\lambda_{(k_{1},j_{2})}=\lambda_{(k_{1},k_{2})}\iff
k_{1}(|k_{1}|^{\alpha}+j_{2}^{2})=k_{1}(|k_{1}|^{\alpha}+k_{2}^{2}),$

are $j_{2}=\pm k_{2}.$ On the other hand, if
$\mathbf{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ is such that $k_{2}\neq 0,$ we now
analyze the entire solutions of the $j_{1}$-equation

$\begin{split}\lambda_{(j_{1},k_{2})}=\lambda_{(k_{1},k_{2})}&\iff
j_{1}(|j_{1}|^{\alpha}+k_{2}^{2})=k_{1}(|k_{1}|^{\alpha}+k_{2}^{2}),\end{split}$

which can be rewritten as

$j_{1}|j_{1}|^{\alpha}-k_{1}|k_{1}|^{\alpha}+k_{2}^{2}(j_{1}-k_{1})=0.$ (6.13)

Similar to the analysis for the BOZK equation, we may assume $k_{1}\neq 0$ and
observe that $j_{1},$ $k_{1}$ share the same sign. Without loss of generality,
let us assume $k_{1}>0$ (the case $k_{1}<0$ being similar). Therefore, we may
assume $j_{1}\geq 0$ and (6.13) is equivalent to

$\begin{split}j_{1}^{\alpha+1}-k_{1}^{\alpha+1}+k_{2}^{2}(j_{1}-k_{1})=0.\end{split}$
(6.14)

Recall we want to show that (6.14) has no other solution than $j_{1}=k_{1}$.
Assume by contradiction the existence of another solution, say, with
$j_{1}>k_{1}$. Then, from the Mean Value Theorem, for some $\theta$ between
$k_{1}$ and $j_{1}$, we have

$(\alpha+1)\theta^{\alpha}(j_{1}-k_{1})+k_{2}^{2}(j_{1}-k_{1})=0$

or

$(j_{1}-k_{1})[(\alpha+1)\theta^{\alpha}+k_{2}^{2}]=0.$

Since the expression between brackets is positive, this last identity is
clearly a contradiction. This shows hypothesis $(H3)$ holds.

Hence, Theorems 1.2 and 1.4 also apply in this case and we conclude that
system (6.12) is exactly controllable at any time $T>2\pi$ and exponentially
stabilizable with any decay rate.

### Acknowledgment

This paper was written when the first author had a postdoctoral position at
IMECC-UNICAMP, whose hospitality he gratefully acknowledges. He also
acknowledges the financial support from FAPESP/Brazil grant 2020/14226-4. The
second author is partially supported by CNPq/Brazil grant 303762/2019-5 and
FAPESP/Brazil grant 2019/02512-5.

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