# Observability of the linear Zakharov–Kuznetsov equation

Roberto de A. Capistrano–Filho*  Departamento de Matemática, Universidade
Federal de Pernambuco, S/N Cidade Universitária, 50740-545, Recife (PE) Brazil
<EMAIL_ADDRESS>,  Vilmos Komornik  College of Mathematics and
Computational Science, Shenzhen University, Shenzhen 518060, People’s Republic
of China and Département de mathématique, Université de Strasbourg, 7 rue René
Descartes, 67084 Strasbourg Cedex, France<EMAIL_ADDRESS>and
Ademir F. Pazoto  Instituto de Matemática, Universidade Federal do Rio de
Janeiro, C.P. 68530 - Cidade Universitária - Ilha do Fundão, 21945-970, Rio de
Janeiro (RJ), Brazil<EMAIL_ADDRESS>
###### Abstract.

We study the linear Zakharov–Kuznetsov equation with periodic boundary
conditions. Employing some tools from the nonharmonic Fourier series we obtain
several internal observability theorems. Then we prove various exact
controllability and rapid uniform stabilization results by applying a duality
principle and a general feedback construction. The method presented here
introduces a new insight into the control of dispersive equations on two-
dimensional cases and may be adapted to more general equations.

###### Key words and phrases:

KdV type equations, internal observability, controllability, feedback
stabilization, nonharmonic analysis, Diophantine analysis

###### 2020 Mathematics Subject Classification:

Primary: 35Q53 Secondary: 93B07, 93B05, 93B52

*Corresponding author<EMAIL_ADDRESS>
##  1\. Introduction

The Zakharov–Kuznetsov equation

$u_{t}+u_{x}+u_{xxx}+u_{xyy}+uu_{x}=0$

was originally derived by Zakharov and Kuznetsov [32] in three dimensions to
describe weakly magnetized ion-acoustic waves in a strongly magnetized plasma.
The two-dimensional equation still has some physical meaning, modeling the
behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold
ions and hot isothermal electrons in the presence of a uniform magnetic field
(see [25, 26] for the details and [18] for rigorous mathematical treatment.)

In this paper, we consider the linearized Zakharov–Kuznetsov (ZK) equation

(1.1) $u_{t}+u_{x}+u_{xxx}+u_{xyy}=0$

on the two-dimensional torus $\mathbb{T}\times\mathbb{T}$, where $u(x,y,t)$
denotes a real-valued function of three real variables $x$, $y$ and $t$, and
$\mathbb{T}$ denotes the unit circle. As a higher-dimensional analog of the
Korteweg-de Vries equation it has also been extensively studied and the body
of literature is huge. In the following, we deliver an overview of the well-
posedness and the controllability theory in two dimensions.

###  1.1. Historical background

The Zakharov–Kuznetsov equation and its linearized version are special cases
of a broad class of dispersive equations for which the well-posedness theory
associated with the pure initial-value problem posed on the whole plane has
been intensively investigated. In this respect, we can mention the works [6,
12, 20, 24] and the references therein. As for the periodic case, the authors
of the pioneering work [19] proved that the initial value problem for the
Zakharov–Kuznetsov equation is locally well-posed in the Sobolev spaces
$H^{s}(\mathbb{T}\times\mathbb{T})$ for $s>5/3$. Their main tool was a variant
of Strichartz’s estimate. By using a nonlinear Loomis–Whitney type inequality,
the assumption $s>5/3$ was weakened to $s>1$ in [13]. More recently, it was
shown by Osawa [27] that the two-dimensional Zakharov–Kuznetsov equation is
also locally well-posed in $H^{s}(\mathbb{R}\times\mathbb{T})$, for
$9/10<s<1$.

By contrast, the mathematical theory pertaining to the study of
controllability properties is considerably less advanced, especially in what
concerns the two-dimensional linear dispersive-type equations on a periodic
setting. To our knowledge, the first result on the subject was obtained by
Rivas and Sun in [29], where the authors studied the internal controllability
of the Kadomtsev–Petviashvili II equation. More precisely, by applying the
Hilbert Uniqueness Method due to Lions [21] and the techniques of
semiclassical and microlocal analysis, they proved the controllability in
$L^{2}(\mathbb{T}\times\mathbb{T})$ from a vertical strip. Additionally, a
negative result for the controllability in $L^{2}(\mathbb{T}\times\mathbb{T})$
from a horizontal strip is also shown.

Recently, Pastor and Vielma Leal [28] applied the moment method [30] to
establish the controllability of several two-dimensional linear dispersive
equations posed on the two-dimensional torus in the framework of Sobolev
spaces $H^{s}(\mathbb{T}\times\mathbb{T})$ with $s\geq 0$. To obtain the
results, the author assumes that the control has a specific form and acts on
the union of a narrow vertical and a narrow horizontal strip. As an
application of the controllability results, they built an appropriate linear
bounded operator to show that the resulting closed-loop system is
exponentially stable with an arbitrary exponential decay rate. We emphasize
that this is a direct consequence of the Hilbert Uniqueness Method mentioned
above and the classical principle: Exact controllability implies exponential
stabilizability for conservative control systems [22, 31]. Their analysis also
covers the linearized equation (1.1). In this context, it is worth mentioning
the work [4] where Capistrano-Filho et al investigated the nonlinear
Kawahara–Kadomtsev–Petviashvili II equation with a damping mechanism
$a(x,y)u$, acting as a feedback-damping mechanism, and an anti-damping
mechanism, preventing energy decrease. In addition, Rosier and Mo Chen
addressed in [23] the exact controllability of the linear equation (1.1) on a
rectangle with a left Dirichlet boundary control by using the flatness
approach.

###  1.2. Problem setting

In this paper, we are mainly concerned with the internal observability,
controllability, and rapid stabilization of the linear ZK equation with
$2\pi$-periodic boundary condition. More precisely, given a subset
$\Omega\subset\mathbb{T}\times\mathbb{T}$ and a Hilbert space $\mathcal{H}$ of
initial data, we study the following questions:

Problem $\mathcal{A}$: Given $T>0$, is it possible to recover, at least
theoretically, the initial data $u_{0}\in\mathcal{H}$ from the observation of
the solution of the homogeneous system

(1.2)
$\begin{cases}&u_{t}+u_{x}+u_{xxx}+u_{xyy}=0\quad\text{in}\quad\mathbb{T}\times\mathbb{T}\times\mathbb{R},\\\
&u(0)=u_{0}\quad\text{in}\quad\mathbb{T}\times\mathbb{T}\end{cases}$

on $\Omega\times(0,T)$?

Here and in the sequel we often consider $u$ as a function of $t$ with values
in a space of functions of $(x,y)$, and we write $u(t)$ instead of
$u(\cdot,\cdot,t)$ for brevity.

Problem $\mathcal{B}$: Given $T>0$ and $z_{0},z_{1}\in\mathcal{H}$, can we
find an appropriate control input $v$ supported on $\Omega\times(0,T)$ such
that the corresponding solution $z$ of the system

(1.3)
$\begin{cases}&z_{t}+z_{x}+z_{xxx}+z_{xyy}=v\quad\text{in}\quad\mathbb{T}\times\mathbb{T}\times\mathbb{R},\\\
&z(0)=z_{0}\quad\text{in}\quad\mathbb{T}\times\mathbb{T}\end{cases}$

satisfies the final condition $z(T)=z_{1}$?

Problem $\mathcal{C}$: Given $\omega>0$ (arbitrarily large), can we construct
a feedback law $v=F_{\omega}z$ such that the solutions of the resulting
closed-loop system (1.3) decays exponentially to zero at least as fast as the
function $e^{-\omega t}$?

We emphasize that we are interested in finding “small” observation and control
domains $\Omega$. Problem $\mathcal{A}$ will be answered by employing diverse
tools from nonharmonic analysis and Diophantine approximations, and specific
geometric constructions. Based on the observability results, Problem
$\mathcal{B}$ will be solved by applying the observation-control duality
theory of Dolecki and Russell [5] in the set-up of Lions [21]. Finally,
Problem $\mathcal{C}$ will be solved by applying a variant of the Hilbert
Uniqueness Method, developed in [14].

###  1.3. Notation and main results

To state our results we introduce some notations. Given any two real numbers
$r$ and $s$, we define the Hilbert space $H^{r,s}(\mathbb{T}\times\mathbb{T})$
as the completion of the vector space spanned by the functions

$(x,y)\mapsto e^{i(mx+ny)}$

with respect to the Euclidean norm

$\left\lVert\sum_{m,n\in\mathbb{Z}}c_{m,n}e^{i(mx+ny)}\right\rVert_{r,s}:=\left(\sum_{m,n\in\mathbb{Z}}\left((1+m^{2})^{r}+(1+n^{2})^{s}\right)\left\lvert
c_{m,n}\right\rvert^{2}\right)^{1/2}.$

Note that
$L^{2}(\mathbb{T}\times\mathbb{T})=H^{0,0}(\mathbb{T}\times\mathbb{T})$.

We will also need a special subspace of $H^{r,s}(\mathbb{T}\times\mathbb{T})$,
namely the orthogonal complement of the functions $u(x,y)$ depending only on
the variable $y$. It will be denoted by
$H^{r,s}_{x}(\mathbb{T}\times\mathbb{T})$, and we write
$L^{2}_{x}(\mathbb{T}\times\mathbb{T})$ instead of
$H^{0,0}_{x}(\mathbb{T}\times\mathbb{T})$. Finally, as usual, we will identify
the $L^{2}$ spaces occurring in this paper with their duals.

We start by investigating the observability of the homogeneous system (1.2).
Our first result states that the initial data may be uniquely determined by
the observation of the solutions on any fixed vertical segment during an
arbitrarily small time interval (see the figure below). More precisely, we
have the following result:

######  Theorem 1.1.

For any fixed $x_{0}\in\mathbb{T}$ and bounded interval $I$ of length $|I|>0$,
there exist two constants $C_{1},C_{2}>0$ such that all solutions of (1.2)
with $u_{0}\in L^{2}(\mathbb{T}\times\mathbb{T})$ satisfy the following
estimates:

(1.4) $C_{1}\left\lVert
u_{0}\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}^{2}\leq\int_{I}\int_{\mathbb{T}}\left\lvert
u(x_{0},y,t)\right\rvert^{2}\ dy\ dt\leq C_{2}\left\lVert
u_{0}\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}^{2}.$

We point out that Theorem 1.1 is not valid for horizontal segments instead of
vertical ones, but we have positive results by observing the solutions on _at
least two_ horizontal segments, and by considering special initial data (see
the figure below). The result, in this context, is the following one:

######  Theorem 1.2.

We consider the solutions of the equation (1.2). Let $N\geq 2$ be an integer,
$y_{1},\ldots,y_{N}\in\mathbb{T}$, and $I_{1},\ldots,I_{N}$ bounded intervals
of positive lengths.

  1. (i)

If $N=2$ and $(y_{1}-y_{2})/\pi$ is irrational, then the linear map

$u_{0}\mapsto\left(u_{\mathbb{T}\times\left\\{y_{1}\right\\}\times
I_{1}},u_{\mathbb{T}\times\left\\{y_{2}\right\\}\times I_{2}}\right)$

is one-to-one on $L^{2}(\mathbb{T}\times\mathbb{T})$.

  2. (ii)

If $y_{1},\ldots,y_{N+1},\pi$ are linearly independent over the field of
rational numbers and they belong to a real algebraic field of degree $N+2$,
then there exists a constant $C_{1}>0$ such that the solutions of (1.1) with
initial data $u_{0}\in L^{2}_{x}(\mathbb{T}\times\mathbb{T})$ satisfy the
following estimate:

$C_{1}\left\lVert
u_{0}\right\rVert_{H^{0,-1/N}(\mathbb{T}\times\mathbb{T})}^{2}\leq\sum_{j=1}^{N+1}\int_{I_{j}}\int_{\mathbb{T}}\left\lVert
u(x,y_{j},t)\right\rVert^{2}\,dx\,dt.$

######  Remark 1.

In case $N=1$, Theorem 1.2 (ii) states that if $(y_{1}-y_{2})/\pi$ is a
quadratic irrational number, then

$C_{1}\left\lVert
u_{0}\right\rVert_{H^{0,-1}(\mathbb{T}\times\mathbb{T})}^{2}\leq\sum_{j=1}^{2}\int_{I_{j}}\int_{\mathbb{T}}\left\lVert
u(x,y_{j},t)\right\rVert^{2}\,dx\,dt$

with some constant $C_{1}>0.$ Our proof will show that a slightly weaker
estimate holds under the much weaker assumption that $(y_{1}-y_{2})/\pi$ is an
arbitrary irrational algebraic number: for each real number $s>1$ there exists
a constant $C_{s}>0$ such that

$C_{s}\left\lVert
u_{0}\right\rVert_{H^{0,-s}(\mathbb{T}\times\mathbb{T})}^{2}\leq\sum_{j=1}^{2}\int_{I_{j}}\int_{\mathbb{T}}\left\lVert
u(x,y_{j},t)\right\rVert^{2}\,dx\,dt.$

The next observability theorem states that the initial data may also be
uniquely determined by the observation of the solutions on an arbitrarily
small disc in $\mathbb{T}\times\mathbb{T}$ during an arbitrarily small time
interval.

######  Theorem 1.3.

For every fixed nonempty open set
$U\subset\mathbb{T}\times\mathbb{T}\times\mathbb{R}$, there exist two
constants $C_{1},C_{2}>0$ such that the solutions of (1.2) with $u_{0}\in
L^{2}_{x}(\mathbb{T}\times\mathbb{T})$ satisfy the following estimates:

$C_{1}\left\lVert
u_{0}\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}^{2}\leq\int_{U}\left\lvert
u(x,y,t)\right\rvert^{2}\ dx\ dy\ dt\leq C_{2}\left\lVert
u_{0}\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}^{2}.$

For a better understanding of the observation regions considered in the
previous theorems, we present three figures below:

Theorems 1.1, 1.2 and 1.3

######  Remark 2.

Theorems 1.1, 1.2, and 1.3, improve the results obtained in [28] for the
system (1.1). The improvements are given to the observation regions. Indeed,
as remarked above, the results in [28] were proved by assuming that the
controls act on the union of a narrow vertical and a narrow horizontal strip.

Next, we consider the nonhomogeneous problem (1.3). We will prove the
following exact controllability and rapid stabilization theorems:

######  Theorem 1.4.

Fix $x_{0}\in\mathbb{T}$ and $T>0$ arbitrarily. Given any $z_{0},z_{T}\in
L^{2}(\mathbb{T}\times\mathbb{T})$, there exists a function $v\in
L^{2}(\left\\{x_{0}\right\\}\times\mathbb{T}\times(0,T))$ such that the
solution of the control problem (1.3) satisfies the final condition

$z(T)=z_{T}\quad\text{in}\quad\mathbb{T}\times\mathbb{T}.$

######  Theorem 1.5.

Fix $x_{0}\in\mathbb{T}$ and a number $\omega>0$ arbitrarily. There exists a
linear map

$F_{\omega}:L^{2}(\mathbb{T}\times\mathbb{T})\to
L^{2}(\left\\{x_{0}\right\\}\times\mathbb{T})$

and a constant $C>0$ such that the closed-loop problem

(1.5)
$\begin{cases}&z_{t}+z_{x}+z_{xxx}+z_{xyy}=F_{\omega}z\quad\text{in}\quad\mathbb{T}\times\mathbb{T}\times(0,+\infty),\\\
&z(0)=z_{0}\quad\text{in}\quad\mathbb{T}\times\mathbb{T},\end{cases}$

is well-posed in $L^{2}(\mathbb{T}\times\mathbb{T})$, and its solutions
satisfy the estimate

$\left\lVert z(t)\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}\leq
C\left\lVert z_{0}\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}e^{-\omega
t},$

for all $z_{0}\in L^{2}(\mathbb{T}\times\mathbb{T})$ and $t\geq 0$.

######  Theorem 1.6.

Fix a nonempty open set $\Omega\subset\mathbb{T}\times\mathbb{T}$ and $T>0$
arbitrarily. Given any
$z_{0},z_{T}\in\left(L^{2}_{x}(\mathbb{T}\times\mathbb{T})\right)^{\prime}$,
there exists a function $v\in L^{2}(\Omega\times(0,T))$ such that the solution
of the control problem (1.3) satisfies the final condition

$z(T)=z_{T}\quad\text{in}\quad\mathbb{T}\times\mathbb{T}.$

######  Theorem 1.7.

Fix a nonempty open set $\Omega\subset\mathbb{T}\times\mathbb{T}$ and a number
$\omega>0$ arbitrarily. There exists a linear map

$F_{\omega}:\left(L^{2}_{x}(\mathbb{T}\times\mathbb{T})\right)^{\prime}\to
L^{2}(\Omega)$

and a constant $C>0$ such that the closed-loop problem (1.5) is well-posed in
$\left(L^{2}_{x}(\mathbb{T}\times\mathbb{T})\right)^{\prime}$, and its
solutions satisfy the estimate

$\left\lVert z(t)\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}\leq
C\left\lVert z_{0}\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}e^{-\omega
t},$

for all $z_{0}\in\left(L^{2}_{x}(\mathbb{T}\times\mathbb{T})\right)^{\prime}$
and $t\geq 0$.

We finish this subsection by remarking that we could also obtain exact
controllability and rapid stabilization theorems corresponding to Theorem 1.2,
but the appropriate Hilbert spaces are not Sobolev spaces. The treatment is
similar to that of the wave equation with the Neumann boundary condition as in
[21] and [14], respectively.

###  1.4. Structure of the article

Our work is composed of nine sections, including the Introduction. In Section
2 we briefly discuss the well-posedness of the homogeneous system (1.3), and
the eigenfunction expansion of its solutions. The special form of its spectrum
will play a crucial role in all subsequent results. In Section 3 we give a
short review of some classical theorems of Ingham, Beurling, and Kahane. We
introduce the notion of a _sparse set_ ; this will simplify our applications
to observability. The most important mathematical proofs of the present paper
are given in Sections 4 and 5, where we prove Theorems 1.1, 1.2, and 1.3.
Section 6 is devoted to recalling two theorems of the kind

$\text{observability}\Longleftrightarrow\text{controllability}\quad\text{and}\quad\text{observability}\Longrightarrow\text{rapid
stabilizability}$

in an abstract framework. In particular, we emphasize that the well-posedness
of a linear control system may easily be derived from the observability of the
dual system. Applying the abstract results of Section 6, we deduce Theorems
1.4–1.7 from the results of Sections 4 and 5, in the Section 7. In our final
section 8, we indicate further possible applications of the methods developed
in this manuscript, and we formulate some related open problems.

##  2\. Well-posedness and eigenfunction expansions

We establish the well-posedness of the homogeneous system (1.2). We have the
following result:

######  Proposition 2.1.

Let $s\in\mathbb{R}$. Given any initial datum $u_{0}\in
H^{s}(\mathbb{T}\times\mathbb{T})$, the problem (1.3) has a unique solution

$u\in C(\mathbb{R};H^{s}(\mathbb{T}\times\mathbb{T}))\cap
C^{1}(\mathbb{R};H^{s-3}(\mathbb{T}\times\mathbb{T})).$

Furthermore, it is represented by the series

(2.1)
$u(x,y,t)=\sum_{m,n\in\mathbb{Z}}c_{m,n}e^{i(mx+ny+\omega_{m,n}t)},\quad\omega_{m,n}:=m^{3}+mn^{2}-m$

with suitable complex coefficients $c_{m,n}$, depending on $u_{0}$.

Moreover, the map $u_{S}\mapsto u_{0}$ is an isometric automorphism of
$L^{2}(\mathbb{T}\times\mathbb{T})$ for every $S\in\mathbb{R}$, and

$\left\lVert
u(S)\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}^{2}=4\pi^{2}\sum_{m,n\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}.$

###### Proof.

The proof follows a standard application of linear semigroup theory. The
present problem is even simpler because there is an orthonormal base formed of
eigenfunctions of the underlying operator

$\mathcal{A}u:=-(u_{x}+u_{xxx}+u_{xyy}),\quad u\in
H^{s+3}(\mathbb{T}\times\mathbb{T}).$

The first part of the theorem readily follows for example by applying [15,
Theorem 3.1] for this operator in the Hilbert space
$\mathcal{H}=H^{s}(\mathbb{T}\times\mathbb{T})$.

The isometry property is a consequence of Parseval’s formula:

$\displaystyle\left\lVert
u(S)\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}^{2}$
$\displaystyle=\int_{\mathbb{T}\times\mathbb{T}}\left\lvert\sum_{m,n\in\mathbb{Z}}c_{m,n}e^{i(mx+ny+\omega_{m,n}S)}\right\rvert^{2}\,dx\,dy$
$\displaystyle=4\pi^{2}\sum_{m,n\in\mathbb{Z}}\left\lvert
c_{m,n}e^{i\omega_{m,n}S}\right\rvert^{2}=4\pi^{2}\sum_{m,n\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}$

for every $S\in\mathbb{R}$.

Finally, each map $u_{S}\mapsto u_{0}$ in onto because if
$(c_{m,n})_{(m,n)\in\mathbb{Z}^{2}}$ runs over $\ell^{2}(\mathbb{Z}^{2})$,
then $(c_{m,n}e^{i\omega_{m,n}S})_{(m,n)\in\mathbb{Z}^{2}}$ also runs over
$\ell^{2}(\mathbb{Z}^{2})$. ∎

##  3\. A short review of Ingham type theorems

First, we recall a theorem of Ingham [8] and its strengthened form given by
Beurling [2] and Kahane [11].

######  Theorem 3.1.

Let $(\lambda_{k})_{k\in K}$ be a uniformly separated family of real numbers,
i.e., satisfying the following _gap condition_ :

$\gamma:=\inf_{\begin{subarray}{c}m,n\in K\\\ k\neq
n\end{subarray}}\left\lvert\lambda_{k}-\lambda_{n}\right\rvert>0.$

Then for each bounded interval $I$ of length $\left\lvert
I\right\rvert>\frac{2\pi}{\gamma}$ there exist two constants $C_{1},C_{2}>0$
such that every square summable family $(c_{k})_{k\in K}$ of complex numbers
satisfies the following inequalities:

(3.1) $C_{1}\sum_{k\in K}\left\lvert
x_{k}\right\rvert^{2}\leq\int_{I}\left\lvert\sum_{k\in
K}c_{k}e^{i\lambda_{k}t}\right\rvert^{2}\,dt\leq C_{2}\sum_{k\in K}\left\lvert
x_{k}\right\rvert^{2}.$

Moreover, the conclusion holds (with different constants) under the weaker
condition

$\left\lvert
I\right\rvert>\frac{2\pi}{\gamma_{L}}\quad\text{with}\quad\gamma_{L}:=\inf_{\begin{subarray}{c}m,n\in
K\setminus L\\\ k\neq
n\end{subarray}}\left\lvert\lambda_{k}-\lambda_{n}\right\rvert,$

for every finite subset $L$ of $K$.

######  Example 1.

The family $(\lambda_{k}:=k^{3})_{k\in\mathbb{Z}}$ satisfies the assumptions
of the theorem with $\gamma=1$. Furthermore, if $L_{m}=\left\\{k\in\mathbb{Z}\
:\ |k|<m\right\\}$ for some integer $m\geq 1$, then $\gamma(L_{m})=2m$ by a
simple verification. Since $\gamma_{L_{m}}=+\infty$ as $m\to+\infty$, the
inequalities (3.1) (with $K=\mathbb{Z}$) hold for bounded intervals of
arbitrary positive length.

Theorem 3.1 was extended to higher dimensions by Kahane [11]. We only state
here a special case of his results that will be sufficient for our
applications. To do that, let us introduce some notations.

######  Definition.

Consider a set $\Lambda=\left\\{\lambda_{k}\ :\ k\in K\right\\}$ in a normed
space.

$\Lambda$ is _uniformly separated_ if

$\gamma(\Lambda):=\inf_{\begin{subarray}{c}m,n\in K\\\ k\neq
n\end{subarray}}\left\lVert\lambda_{k}-\lambda_{n}\right\rVert>0.$

$\Lambda$ is _sparse_ if for every $\varepsilon>0$ there exists a finite cover
$\Lambda=\Lambda_{1}\cup\cdots\cup\Lambda_{m}$ of $K$ such that

$\frac{1}{\gamma(\Lambda_{1})}+\cdots+\frac{1}{\gamma(\Lambda_{m})}<\varepsilon.$

######  Remarks 1.

  1. (i)

It follows from the definition that both properties are invariant for
similarities, and they are preserved for taking subsets.

  2. (ii)

The empty set is sparse. The singletons $\left\\{\lambda\right\\}$ are sparse
because $\gamma(\left\\{\lambda\right\\})=+\infty$ by definition.

  3. (iii)

The union of finitely many sparse sets $\Lambda_{1}$, …, $\Lambda_{m}$ is
still sparse. Indeed, given $\varepsilon>0$, it suffices to apply the
definition to each $\Lambda_{j}$ with $\varepsilon/m$ instead of $\varepsilon$
and then to take the union of all these covers.

  4. (iv)

Every finite set is sparse by (ii) and (iii).

Let us give a typical nontrivial example:

######  Example 2.

The set $\Lambda=\left\\{k^{3}\ :\ k\in\mathbb{Z}\right\\}$ is sparse in
$\mathbb{R}$. Indeed, for any fixed $\varepsilon>0$ choose an integer
$m>1/3\varepsilon$, and consider the finite cover
$\Lambda_{1},\ldots,\Lambda_{2m+1}$ of $\Lambda$ by the sets

$\left\\{k^{3}\ :\ k\geq m\right\\},\quad\left\\{k^{3}\ :\ k\leq-m\right\\}$

and the singletons $\left\\{j^{3}\right\\}$ for $j=-m+1,\ldots,m-1$. The first
set has a uniform gap $>6m$ because if $k_{1}>k_{2}\geq m$, then

$k_{1}^{3}-k_{2}^{3}\geq(k_{2}+1)^{3}-k_{2}^{3}=3k_{2}^{2}+3k_{2}+1\geq 6m+1.$

The second set is a reflection of $\Lambda_{1}$, hence it has the same uniform
gap. Since the singletons have infinite uniform gaps, we conclude that

$\sum_{j=1}^{2m+1}\frac{1}{\gamma(\Lambda_{j})}<\frac{1}{6m}+\frac{1}{6m}+\frac{1}{+\infty}+\cdots+\frac{1}{+\infty}=\frac{1}{3m}<\varepsilon.$

The following theorem shows the importance of sparse sets; is an easy
corollary of a more general theorem of Kahane [11] (see also [1] and [15,
Theorem 8.1 and Proposition 8.4] for some improvements).

######  Theorem 3.2.

Let $(\lambda_{k})_{k\in K}$ be a uniformly separated and sparse family of
vectors in $\mathbb{R}^{N}$. Then for each nonempty bounded open set
$G\subset\mathbb{R}^{N}$ there exist two constants $C_{1},C_{2}>0$ such that

$C_{1}\sum_{k\in K}\left\lvert
x_{k}\right\rvert^{2}\leq\int_{G}\left\lvert\sum_{k\in
K}c_{k}e^{i\lambda_{k}\cdot z}\right\rvert^{2}\,dz\leq C_{2}\sum_{k\in
K}\left\lvert x_{k}\right\rvert^{2},$

for every square summable family $(x_{k})_{k\in K}$ of complex numbers.

##  4\. Observability on segments

This section is dedicated to giving the proof of Theorems 1.1 and 1.2. Let us
start with the proof of Theorem 1.1. We will apply Ingham’s inequality
infinitely many times, by ensuring the existence of the same constants for
each application. This kind of proof was first applied in [7].

###### Proof of Theorem 1.1.

Fix an integer $N\geq 1$ such that $\frac{2\pi}{N^{2}}<|I|$. For each fixed
$n$ with $\left\lvert n\right\rvert\geq N$ the sequence

$(\omega_{m,n})_{m\in\mathbb{Z}}=(m^{3}+mn^{2}-m)_{m\in\mathbb{Z}}$

has a uniform gap $\geq N^{2}$ because

$(m+1)^{3}+(m+1)n^{2}-(m+1)-(m^{3}+mn^{2}-m)=3m(m+1)+n^{2}\geq n^{2}\geq
N^{2},$

for every $m\in\mathbb{Z}$.

It follows that we may apply Ingham’s theorem for each of these sequences with
common uniform constants $C_{1}^{\prime},C_{2}^{\prime}>0$, depending only on
$|I|$:

(4.1) $C_{1}^{\prime}\sum_{m\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}\leq\int_{I}\left\lvert\sum_{m\in\mathbb{Z}}c_{m,n}e^{i(mx_{0}+\omega_{m,n}t)}\right\rvert^{2}\,dt\leq
C_{2}^{\prime}\sum_{m\in\mathbb{Z}}\left\lvert c_{m,n}\right\rvert^{2},$

for each fixed $n\in\mathbb{Z}\setminus\left\\{-N+1,\ldots,N-1\right\\}$, in
the previous inequality we have used the following trivial equality
$\left\lvert c_{m,n}e^{imx_{0}}\right\rvert=\left\lvert c_{m,n}\right\rvert$.

For each of the remaining $2N-1$ values of $n$, the truncated sequence

$(m^{3}+mn^{2}-m)_{\left\lvert m\right\rvert\geq N+1}$

also has a uniform gap $\geq N^{2}$ because

$3m(m+1)+n^{2}=3|m|\cdot|m+1|+n^{2}\geq N^{2},$

whenever $\left\lvert m\right\rvert\geq N+1$ and thus $\left\lvert
m+1\right\rvert\geq N$. Applying the last part of Theorem 3.1, there exist
suitable constants $C_{1,n}^{\prime},C_{2,n}^{\prime}>0$, depending only on
$|I|$, such that

(4.2) $C_{1,n}^{\prime}\sum_{m\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}\leq\int_{I}\left\lvert\sum_{m\in\mathbb{Z}}c_{m,n}e^{i(mx_{0}+\omega_{m,n}t)}\right\rvert^{2}\,dt\leq
C_{2,n}^{\prime}\sum_{m\in\mathbb{Z}}\left\lvert c_{m,n}\right\rvert^{2},$

for each fixed $n\in\left\\{-N+1,\ldots,N-1\right\\}$.

Setting

$C_{1}^{\prime\prime}:=\min\left\\{C_{1}^{\prime},C_{1,1-N}^{\prime},\ldots,C_{1,N-1}^{\prime}\right\\}\quad\text{and}\quad
C_{2}^{\prime\prime}:=\max\left\\{C_{2}^{\prime},C_{2,1-N}^{\prime},\ldots,C_{2,N-1}^{\prime}\right\\},$

and, adding the equations (4.1) and (4.2), we obtain that

$C_{1}^{\prime\prime}\sum_{m\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}\leq\int_{I}\left\lvert\sum_{m\in\mathbb{Z}}c_{m,n}e^{i(mx_{0}+\omega_{m,n}t)}\right\rvert^{2}\,dt\leq
C_{2}^{\prime\prime}\sum_{m\in\mathbb{Z}}\left\lvert c_{m,n}\right\rvert^{2},$

for each $n\in\mathbb{Z}$. Summing them we obtain the following estimate:

(4.3) $C_{1}^{\prime\prime}\sum_{m,n\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}\leq\sum_{n\in\mathbb{Z}}\int_{I}\left\lvert\sum_{m\in\mathbb{Z}}c_{m,n}e^{i(mx_{0}+\omega_{m,n}t)}\right\rvert^{2}\,dt\leq
C_{2}^{\prime\prime}\sum_{m,n\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}.$

Since

$\displaystyle\int_{I}\int_{\mathbb{T}}\left\lvert
u(x_{0},y,t)\right\rvert^{2}\ dy\ dt$
$\displaystyle=\int_{I}\int_{\mathbb{T}}\left\lvert\sum_{n\in\mathbb{Z}}e^{iny}\sum_{m\in\mathbb{Z}}c_{m,n}e^{i(mx_{0}+\omega_{m,n}t)}\right\rvert^{2}\
dy\ dt$
$\displaystyle=2\pi\sum_{n\in\mathbb{Z}}\int_{I}\left\lvert\sum_{m\in\mathbb{Z}}c_{m,n}e^{i(mx_{0}+\omega_{m,n}t)}\right\rvert^{2}\
dt$

by Parseval’s formula, and

$\left\lVert
u_{0}\right\rVert_{L^{2}(\mathbb{T}\times\mathbb{T})}^{2}=4\pi^{2}\sum_{m,n\in\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}$

by Proposition 2.1, the estimates (1.4) of Theorem 1.1 follow from (4.3) with
$C_{1}=C_{1}^{\prime\prime}/2\pi$ and $C_{2}=C_{2}^{\prime\prime}/2\pi$. ∎

Now we turn to the proof of Theorem 1.2. First, we observe that observation on
only one horizontal segment is not sufficient. Indeed, since
$\omega_{m,n}=\omega_{m,-n}$, we have for any fixed $y_{0}\in\mathbb{T}$ the
following identity by a similar computation as in the preceding proof:

$\displaystyle\int_{I}\int_{\mathbb{T}}\left\lvert
u(x,y_{0},t)\right\rvert^{2}\,dx\,dt$
$\displaystyle=\int_{I}\int_{\mathbb{T}}\left\lvert\sum_{m\in\mathbb{Z}}e^{imx}\sum_{n\in\mathbb{Z}}c_{m,n}e^{i(ny_{0}+\omega_{m,n}t)}\right\rvert^{2}\,dx\,dt$
$\displaystyle=2\pi\sum_{m\in\mathbb{Z}}\int_{I}\left\lvert\sum_{n\in\mathbb{Z}}c_{m,n}e^{i(ny_{0}+\omega_{m,n}t)}\right\rvert^{2}\,dt$
$\displaystyle=2\pi\sum_{m\in\mathbb{Z}}\int_{I}\left\lvert
c_{m,0}+\sum_{n=1}^{\infty}\left(c_{m,n}e^{iny_{0}}+c_{m,-n}e^{-iny_{0}}\right)e^{i\omega_{m,n}t}\right\rvert^{2}\,dt.$

The last expression is equivalent for every non-degenerate bounded open
interval $I$ to

$\left\lvert\sum_{n\in\mathbb{Z}}c_{0,n}e^{iny_{0}}\right\rvert^{2}+\sum_{m\in\mathbb{Z}^{*}}\left(\left\lvert
c_{m,0}\right\rvert^{2}+\sum_{n=1}^{\infty}\left\lvert
c_{m,n}e^{iny_{0}}+c_{m,-n}e^{-iny_{0}}\right\rvert^{2}\right),$

so that these observations do not allow us to distinguish the coefficients
$c_{0,n}$, and for $m\neq 0$ the coefficients $c_{m,n}$ and $c_{m,-n}$.

If we choose initial data that are orthogonal to the set of functions
depending only on $x$, then $c_{0,n}=0$ for all $n$, and we obtain the
equivalence

$\int_{I}\int_{\mathbb{T}}\left\lvert
u(x,y_{0},t)\right\rvert^{2}\,dx\,dt\asymp\sum_{m\in\mathbb{Z}^{*}}\left(\left\lvert
c_{m,0}\right\rvert^{2}+\sum_{n=1}^{\infty}\left\lvert
c_{m,n}e^{iny_{0}}+c_{m,-n}e^{-iny_{0}}\right\rvert^{2}\right).$

This shows that the observation on one horizontal segment does not allow us to
determine separately $c_{m,n}$ and $c_{m,-n}$, but only a linear combination
of them.

If we observe the solution on two horizontal segments on identical or
different time intervals $I_{1}$ and $I_{2}$, then we obtain that

(4.4) $\int_{I_{1}}\int_{\mathbb{T}}\left\lvert
u(x,y_{1},t)\right\rvert^{2}\,dx\,dt+\int_{I_{2}}\int_{\mathbb{T}}\left\lvert
u(x,y_{2},t)\right\rvert^{2}\,dx\,dt$

is equivalent to

$\sum_{m\in\mathbb{Z}^{*}}\left(\left\lvert
c_{m,0}\right\rvert^{2}+\sum_{n=1}^{\infty}\left(\left\lvert
c_{m,n}e^{iny_{1}}+c_{m,-n}e^{-iny_{1}}\right\rvert^{2}+\left\lvert
c_{m,n}e^{iny_{2}}+c_{m,-n}e^{-iny_{2}}\right\rvert^{2}\right)\right),$

or to

(4.5) $\sum_{m\in\mathbb{Z}^{*}}\left(\left\lvert
c_{m,0}\right\rvert^{2}+\sum_{n=1}^{\infty}\left(\left\lvert
c_{m,n}e^{2iny_{1}}+c_{m,-n}\right\rvert^{2}+\left\lvert
c_{m,n}e^{2iny_{2}}+c_{m,-n}\right\rvert^{2}\right)\right).$

With this in hand, we are now in a position to prove the second main result of
this article.

###### Proof of Theorem 1.2.

(i) In view of the equivalence between (4.4) and (4.5) it suffices to show the
implications

$c_{m,n}e^{2iny_{1}}+c_{m,-n}=c_{m,n}e^{2iny_{2}}+c_{m,-n}=0\Longrightarrow
c_{m,n}=c_{m,-n}=0.$

This holds if and only if $e^{2iny_{1}}\neq e^{2iny_{2}}$, that is, if
$(y_{1}-y_{2})/\pi$ is irrational, showing item (i).

(ii) Let us first assume that $N=1$, i.e., we observe the solution on two
segments. Then we have to prove the following estimate with some constant
$\gamma>0$:

(4.6) $\int_{I_{1}}\int_{\mathbb{T}}\left\lvert
u(x,y_{1},t)\right\rvert^{2}\,dx\,dt+\int_{I_{2}}\int_{\mathbb{T}}\left\lvert
u(x,y_{2},t)\right\rvert^{2}\,dx\,dt\geq\gamma\sum_{m\in\mathbb{Z}^{*}}\sum_{n\in\mathbb{Z}}\frac{\left\lvert
c_{m,n}\right\rvert^{2}}{n^{2}}.$

Since $(y_{1}-y_{2})/\pi$ is irrational, we infer from the equalities

$c_{m,n}=\frac{c_{m,n}e^{2iny_{1}}+c_{m,-n}}{e^{2iny_{1}}-e^{2iny_{2}}}\quad\text{and}\quad
c_{m,-n}=\frac{c_{m,-n}e^{-2iny_{1}}+c_{m,n}}{e^{-2iny_{1}}-e^{-2iny_{2}}}$

that

$\left\lvert c_{m,n}e^{2iny_{1}}+c_{m,-n}\right\rvert^{2}+\left\lvert
c_{m,n}e^{2iny_{2}}+c_{m,-n}\right\rvert^{2}\geq\left\lvert
e^{2in(y_{1}-y_{2})}-1\right\rvert^{2}\left(\left\lvert
c_{m,n}\right\rvert^{2}+\left\lvert c_{m,-n}\right\rvert^{2}\right).$

Thanks to the fact that

$\left\lvert e^{2\pi i\theta}-1\right\rvert\geq
c\operatorname{dist}(\theta,\mathbb{Z}),$

for all real numbers $z$ with some constant $\gamma_{1}>0$, yields that

$\left\lvert c_{m,n}e^{2iny_{1}}+c_{m,-n}\right\rvert^{2}+\left\lvert
c_{m,n}e^{2iny_{2}}+c_{m,-n}\right\rvert^{2}\\\
\geq\gamma_{1}^{2}\left(\operatorname{dist}(n(y_{1}-y_{2})/\pi,\mathbb{Z})\right)^{2}\left(\left\lvert
c_{m,n}\right\rvert^{2}+\left\lvert c_{m,-n}\right\rvert^{2}\right).$

Now we recall, e.g., from [3, Case $n=1$ of Theorem III on p. 79] that if
$\theta$ is an irrational quadratic algebraic number, then there exists a
constant $\gamma_{2}>0$ such that

$\operatorname{dist}(n\theta,\mathbb{Z})\geq\frac{\gamma_{2}}{n},$

for every integer $n\geq 1$. Applying this with $\theta=(y_{1}-y_{2})/\pi$ we
obtain the following estimate:

$\left\lvert c_{m,n}e^{2iny_{1}}+c_{m,-n}\right\rvert^{2}+\left\lvert
c_{m,n}e^{2iny_{2}}+c_{m,-n}\right\rvert^{2}\geq\frac{\gamma_{1}^{2}\gamma_{2}^{2}}{n^{2}}\left(\left\lvert
c_{m,n}\right\rvert^{2}+\left\lvert c_{m,-n}\right\rvert^{2}\right).$

Using this inequality the required estimate (4.6) follows from the equivalence
of the expressions (4.4) and (4.5).

Under the weaker assumption that $(y_{1}-y_{2})/\pi$ is an irrational
algebraic number, we may repeat the preceding proof, by using at the end
Roth’s theorem (see. e.g., [3, Theorem I on p. 104]):

$\operatorname{dist}(n\theta,\mathbb{Z})\geq\frac{\gamma_{s}}{n^{s}},$

for every integer $n\geq 1$. This yields the theorem mentioned in Remark 1.

Now let us turn to the case $N\geq 2$. Applying the first part of the proof
for $N=1$ with $y_{1}-y_{j}$ for $j=2,\ldots,N+1$ we obtain the inequality

$\displaystyle\sum_{j=1}^{N+1}\int_{I_{j}}\int_{\mathbb{T}}\left\lVert
u(x,y_{j},t)\right\rVert^{2}\,dx\,dt$
$\displaystyle\geq\gamma_{1}^{2}\sum_{j=2}^{N+1}\left(\operatorname{dist}(n(y_{1}-y_{j})/\pi,\mathbb{Z})\right)^{2}\left(\left\lvert
c_{m,n}\right\rvert^{2}+\left\lvert c_{m,-n}\right\rvert^{2}\right)$
$\displaystyle\geq\gamma_{1}^{2}\max_{j=2}^{N+1}\left\\{\operatorname{dist}(n(y_{1}-y_{j})/\pi,\mathbb{Z})\right\\}^{2}\left(\left\lvert
c_{m,n}\right\rvert^{2}+\left\lvert c_{m,-n}\right\rvert^{2}\right).$

Thanks to our assumptions on $y_{1},\ldots,y_{N+1}$ we may apply [3, Theorem
III on p. 79] to obtain the inequality

$\max_{j=2}^{N+1}\left\\{\operatorname{dist}(n(y_{1}-y_{j})/\pi,\mathbb{Z})\right\\}\geq\gamma_{2}n^{-1/N},$

for all $n$. Therefore we have

$\sum_{j=1}^{N+1}\int_{I_{j}}\int_{\mathbb{T}}\left\lVert
u(x,y_{j},t)\right\rVert^{2}\,dx\,dt\geq\gamma_{1}^{2}\gamma_{2}^{2}n^{-2/N},$

for all integers $n\geq 1$, so the proof of item (ii) is achieved, and Theorem
1.2 is proved. ∎

##  5\. Observability on small balls

This section deals with the presentation of the proof of Theorem 1.3. In this
theorem, we consider only initial data belonging to
$L^{2}_{x}(\mathbb{T}\times\mathbb{T}):=H^{0,0}_{x}(\mathbb{T}\times\mathbb{T})$.
Then the terms with $m=0$ are missing in formula (2.1), so that the solution
of (1.2) is given by

$u(x,y,t)=\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}c_{m,n}e^{i(mx+ny+\omega_{m,n}t)},\quad\omega_{m,n}:=m^{3}+mn^{2}-m,$

with square summable complex coefficients $c_{m,n}$.

In view of Theorem 3.2, Theorem 1.3 will be proved if we establish the
following result:

######  Proposition 5.1.

The set

$\Lambda=\left\\{\lambda_{m,n}:=(m,n,m^{3}+mn^{2}-m)\in\mathbb{Z}^{3}\ :\
(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}\right\\}$

is uniformly separated and sparse.

For simplicity, we will consider the usual Euclidean norm of $\mathbb{R}^{N}$;
this is not essential because all other norms are equivalent to this one. Then
$\Lambda$ is obviously uniformly separated with $\gamma(\Lambda)\geq 1$. We
now prove that $\Lambda$ is sparse. To do that, we need four lemmas.

######  Lemma 5.2.

A set $\Lambda$ in a normed space is sparse if it has the following property:
for every $R>0$ there exists a sparse set $\Lambda^{\prime}$ such that
$\gamma(\Lambda\setminus\Lambda^{\prime})\geq R$.

###### Proof.

For any fixed $\varepsilon>0$ choose $R>2/\varepsilon$, and let $\Lambda_{1}$,
…, $\Lambda_{m}$ be a finite cover of the corresponding sparse set
$\Lambda^{\prime}$ such that

$\frac{1}{\gamma(\Lambda_{1})}+\cdots+\frac{1}{\gamma(\Lambda_{m})}<\frac{\varepsilon}{2}.$

Then the finitely many sets $\Lambda_{1},\ldots,\Lambda_{m},\Lambda^{\prime}$
cover of $\Lambda$, and

$\frac{1}{\gamma(\Lambda_{1})}+\cdots+\frac{1}{\gamma(\Lambda_{m})}+\frac{1}{\gamma(\Lambda\setminus\Lambda^{\prime})}<\frac{\varepsilon}{2}+\frac{1}{R}\leq\varepsilon.\qed$

The following is a generalization of Example 2:

######  Lemma 5.3.

Consider a set of the form

$\Lambda=\left\\{(g(m),f(m))\ :\
m\in\mathbb{Z}\right\\}\subset\mathbb{R}^{N},$

where $g:\mathbb{Z}\to\mathbb{Z}^{N-1}$ is an arbitrary function, and
$f:\mathbb{Z}\to\mathbb{Z}$ is a function satisfying the limit relation

(5.1) $\lim_{|m|\to-\infty}\left(f(m+1)-f(m)\right)=+\infty.$

Then $\Lambda$ is sparse. The same conclusion holds if the limit is $-\infty$.

###### Proof.

As mentioned before, we consider the Euclidean norm of $\mathbb{R}^{N}$, so
that

$\left\lVert(x_{1},\ldots,x_{N})\right\rVert\geq\max\left\\{\left\lvert
x_{1}\right\rvert,\ldots,\left\lvert x_{N}\right\rvert\right\\}.$

It follows from our assumptions that

$\lim_{m\to-\infty}f(m)=-\infty\quad\text{and}\quad\lim_{m\to+\infty}f(m)=+\infty.$

Therefore, for any fixed $R>0$ we may choose an integer $n\geq 1$ such that

$\displaystyle f(n)-f(-n)\geq R,$ $\displaystyle f(m+1)-f(m)\geq
R\quad\text{for all}\quad m\leq-n,\quad\text{and}$ $\displaystyle
f(m+1)-f(m)\geq R\quad\text{for all}\quad m\geq n.$

In particular, $f$ is increasing in the intervals $(-\infty,-n]$ and
$[n,+\infty)$. We complete the proof of the first part by showing that the set

$\Lambda^{\prime}:=\left\\{(g(m),f(m))\ :\ |m|<n\right\\}$

satisfies the hypotheses of Lemma 5.2.

Indeed, first of all, $\Lambda\setminus\Lambda^{\prime}$ is a finite set,
hence it is sparse. It remains to show that if $|m_{1}|\geq n$, $|m_{2}|\geq
n$ and $m_{1}\neq m_{2}$, then

$\left\lVert(g(m_{1}),f(m_{1}))-(g(m_{2}),f(m_{2}))\right\rVert\geq R.$

We may assume by symmetry that $m_{1}>m_{2}$, so that $m_{1}\geq m_{2}+1$.

We are going to prove the stronger inequality $f(m_{1})-f(m_{2})\geq R$. It is
satisfied if $m_{1}$ and $m_{2}$ have different signs, because then

$f(m_{1})-f(m_{2})\geq f(n)-f(-n)\geq R.$

If $m_{1}$ and $m_{2}$ have equal signs, then

$f(m_{1})-f(m_{2})\geq f(m_{2}+1)-f(m_{2})\geq R.$

If the limit is $-\infty$ is the assumption of the lemma, then $-\Lambda$ is
sparse by the first part, and then its reflection $\Lambda$ is also sparse. ∎

Based on Lemmas 5.2, 5.3 and on the fact that a finite union of sparse sets is
still sparse, our strategy for the proof of Proposition 5.1 is the following:
for any given $R>0$ we will construct a finite number of sparse sets
$\Lambda_{k,\ell,R}$ of the type studied in Lemma 5.3 such that their union
$\Lambda^{\prime}$ satisfies the uniform gap condition
$\gamma(\Lambda\setminus\Lambda^{\prime})\geq R$.

The construction of the sets $\Lambda_{k,\ell,R}$ is inspired by a method
developed by Jaffard [9] for the proof of the interior observability of
rectangular plates.

To do that, we will need some preliminary technical results. If

$\displaystyle\lambda_{m,n}=(m,n,m^{3}+mn^{2}-m)$ and
$\displaystyle\lambda_{m+k,n+\ell}=\left((m+k),n+\ell,(m+k)^{3}+(m+k)(n+\ell)^{2}-(m+k)\right),$

are two distinct elements of

$\Lambda=\left\\{\lambda_{m,n}:=(m,n,m^{3}+mn^{2}-m)\in\mathbb{Z}^{3}\ :\
(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}\right\\},$

then $(k,\ell)\in\mathbb{Z}^{2}\setminus\left\\{(0,0)\right\\}$, and

(5.2) $\lambda_{m+k,n+\ell}-\lambda_{m,n}=(k,\ell,Q(k,\ell,m,n))$

with

$Q(k,\ell,m,n):=k(3m^{2}+n^{2})+2\ell mn+(3k^{2}+\ell^{2})m+2k\ell
n+k^{3}+k\ell^{2}-k.$

######  Lemma 5.4.

The polynomial $Q(k,\ell,m,n)$ satisfies the following relation:

(5.3) $Q(k,\ell,m,n)=k(3m^{2}+n^{2})+2\ell
mn+o\left(m^{2}+n^{2}\right)\quad\text{as}\quad\left\lVert(m,n)\right\rVert\to+\infty.$

Furthermore, we have

(5.4) $Q(k,\ell,m,n)=m\ell(2n+\ell)\quad\text{if}\quad k=0,$

and

(5.5)
$\begin{split}3kQ(k,\ell,m,n)=&\left(3k\left(m+\frac{k}{2}\right)+\ell\left(n+\frac{\ell}{2}\right)\right)^{2}\\\
&-(\ell^{2}-3k^{2})\left(n+\frac{\ell}{2}\right)^{2}+\frac{3k^{2}}{4}\left(k^{2}+\ell^{2}-4\right)\end{split}$

in the general case.

###### Proof.

The relations (5.3) and (5.4) can be obtained according to the classical
procedure to obtain the canonical form a quadratic form. Note that we may
write $3kQ(k,\ell,m,n)$ in the form

$(3k+\ell n+c_{1})^{2}+c_{2}(n+c_{3})^{2}+c_{4},$

with suitable constants $c_{1},\ldots,c_{4}$. An elementary computation allows
us to determine these constants, leading to the identity (5.5). ∎

In order to estimate the norms $\lambda_{m+k,n+\ell}-\lambda_{m,n}$ by (5.2),
first we investigate the sets

$B_{k,\ell,R}:=\left\\{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}\ :\ \left\lvert
Q(k,\ell,m,n)\right\rvert<R\right\\},$

for $(k,\ell)\in\mathbb{Z}^{2}\setminus\left\\{(0,0)\right\\}$ and $R>0$.

######  Lemma 5.5.

Let $(k,\ell)\in\mathbb{Z}^{2}\setminus\left\\{(0,0)\right\\}$ and $R>0$.

  1. (i)

If $3k^{2}>\ell^{2}$, then the set $B_{k,\ell,R}$ is finite.

  2. (ii)

If $k=0$ and $\ell$ is even, then $B_{k,\ell,R}$ is covered by a finite set
and the sequence $\left(m,-\frac{\ell}{2}\right)$, with $m\in\mathbb{Z}^{*}$.

  3. (iii)

If $k=0$ and $\ell$ is odd, then $B_{k,\ell,R}$ is finite.

  4. (iv)

If $0<3k^{2}<\ell^{2}$, then $B_{k,\ell,R}$ is covered by a finite set and two
sequences $(m_{1}(n),n)_{n\in\mathbb{Z}}$ and $(m_{2}(n),n)_{n\in\mathbb{Z}}$
satisfying the relations

(5.6) $m_{j}(n)=\alpha_{j}n+\beta_{j}+o(1),\quad|n|\to+\infty,\quad j=1,2$

with suitable constants $\alpha_{j}\neq 0$ and $\beta_{j}$ (see Figure 1 for
an example of this case).

Figure 1. (iv) The region containing $B_{2,4,7}$

###### Proof.

(i) If $3k^{2}>\ell^{2}$, then the quadratic form $k(3m^{2}+n^{2})+2\ell mn$
of $(m,n)$ is (positive or negative) definite. Then we infer from (5.3) the
relation

$\left\lvert
Q(k,\ell,m,n)\right\rvert\to+\infty\quad\text{as}\quad\left\lVert(m,n)\right\rVert\to+\infty,$

so that $B_{k,\ell,R}$ is a finite set.

(ii) If $k=0$, then $\ell\neq 0$ and

$Q(k,\ell,m,n)=m\ell(2n+\ell)$

by (5.4). This expression vanishes if $\ell$ is even and $n=-\ell/2$, so that
$\mathbb{Z}^{*}\times\left\\{-\ell/2\right\\}\subset B_{k,\ell,R}$ in this
case.

Since $m\neq 0$ by our assumption, if $(m,n)$ does not belong to
$\mathbb{Z}^{*}\times\left\\{-\ell/2\right\\}$, then

$\left\lvert
Q(k,\ell,m,n)\right\rvert\geq\min\left\\{|m|,|\ell(2n+\ell)|\right\\},$

and the last expression becomes larger than $R$ when
$\left\lVert(m,n)\right\rVert\to+\infty$ is sufficiently large. Therefore
$B_{k,\ell,R}$ is covered by $\mathbb{Z}^{*}\times\left\\{-\ell/2\right\\}$
and a finite set.

(iii) We may repeat the proof of (ii). Since now $\ell$ is odd,
$\mathbb{Z}^{*}\times\left\\{-\ell/2\right\\}$ is disjoint from
$B_{k,\ell,R}\subset\mathbb{Z}^{2}$, and hence
$\mathbb{Z}^{*}\times\left\\{-\ell/2\right\\}$ is not necessary for the
covering of $B_{k,\ell,R}$.

(iv) Setting

$y=3k\left(m+\frac{k}{2}\right)+\ell\left(n+\frac{\ell}{2}\right)\quad\text{and}\quad
x=\sqrt{\ell^{2}-3k^{2}}\left(n+\frac{\ell}{2}\right)$

for brevity, (5.5) may be written in the form

$3kQ(k,\ell,m,n)=y^{2}-x^{2}+\frac{3k^{2}}{4}\left(k^{2}+\ell^{2}-4\right).$

Using this equality, the condition $\left\lvert Q(k,\ell,m,n)\right\rvert<R$
in the definition of $B_{k,\ell,R}$ implies the relation

$\left\lvert y^{2}-x^{2}\right\rvert<3|k|R+\frac{3k^{2}}{4}\left\lvert
k^{2}+\ell^{2}-4\right\rvert=:C,$

or equivalently

$-C<y^{2}-x^{2}<C.$

Observe that $|x|\to+\infty\Longleftrightarrow|y|\to+\infty$ in this region
(see Figure 2).

Figure 2. The region $-9<y^{2}-x^{2}<9$

Hence either

$\displaystyle|x|<\sqrt{C}\quad\text{and}\quad|y|<\sqrt{2C},$ or
$\displaystyle|x|\geq\sqrt{C}\quad\text{and}\quad\sqrt{x^{2}-C}<|y|<\sqrt{x^{2}+C}.$

Observe that the set of these vectors $(x,y)$ shrinks to asymptotes $y=\pm x$
of the corresponding hyperbola because, as $|x|\to+\infty$,

$\Bigl{\lvert}|y|-|x|\Bigr{\rvert}<\sqrt{x^{2}+C}-\sqrt{x^{2}-C}=\frac{2C}{\sqrt{x^{2}+C}+\sqrt{x^{2}-C}}\to
0.$

We may also write it in the following form:

$\min\left\\{\left\lvert y-x\right\rvert,\left\lvert
y+x\right\rvert\right\\}=o(1)\quad\text{as}\quad|x|\to+\infty.$

Returning to the variables $m,n$, dividing by $3k$, and introducing the
sequences

$\widetilde{m_{1}}(n):=\frac{\sqrt{\ell^{2}-3k^{2}}-\ell}{3k}\left(n+\frac{\ell}{2}\right)-\frac{k}{2}\quad\text{and}\quad\widetilde{m_{2}}(n):=\frac{-\sqrt{\ell^{2}-3k^{2}}-\ell}{3k}\left(n+\frac{\ell}{2}\right)-\frac{k}{2},$

the last relation becomes

(5.7) $\min\left\\{\left\lvert m-\widetilde{m_{1}}(n)\right\rvert,\left\lvert
m-\widetilde{m_{2}}(n)\right\rvert\right\\}=o(1)\quad\text{as}\quad|n|\to+\infty;$

because the relation $|n|\to+\infty$ implies $|x|\to+\infty$. If $|n|$ is
sufficiently large, say $|n|>N$, then

$\min\left\\{\left\lvert m-\widetilde{m_{1}}(n)\right\rvert,\left\lvert
m-\widetilde{m_{2}}(n)\right\rvert\right\\}<\frac{1}{2}.$

Observe that at most two integers $m$ may satisfy this inequality: the closest
integers to $\widetilde{m_{1}}(n)$ and $\widetilde{m_{2}}(n)$. Denoting these
closest integers by $m_{1}(n)$ and $m_{2}(n)$ if $|n|>N$, and defining
$m_{j}(n):=\widetilde{m_{j}}(n)$ otherwise, the set

$\left\\{(m,n)\in B_{k,\ell,R}\ :\ |n|>N\right\\}$

is covered by the two sequences $(m_{1}(n),n)_{n\in\mathbb{Z}}$ and
$(m_{2}(n),n)_{n\in\mathbb{Z}}$. Furthermore, in view of (5.7) and the
definition of the sequences $(\widetilde{m_{j}}(n))$ the relations (5.6) are
satisfied with

$\alpha_{j}=\frac{\pm\sqrt{\ell^{2}-3k^{2}}-\ell}{3k}\quad\text{and}\quad\beta_{j}=\frac{\alpha_{j}\ell-k}{2},\quad
j=1,2.$

Note that $\alpha_{j}\neq 0$.

To complete the proof it remains to show that the set

$\left\\{(m,n)\in B_{k,\ell,R}\ :\ |n|\leq N\right\\}$

is finite. This follows by observing that for any fixed $n$, by (5.5) at most
finitely many integers $m$ may satisfy the inequality $\left\lvert
Q(k,\ell,m,n)\right\rvert<R$, showing the item (iv). ∎

As a consequence of the previous lemma, we have the following:

######  Corollary 5.6.

The sets

$\Lambda_{k,\ell,R}:=\left\\{\lambda_{m,n}\ :\ (m,n)\in B_{k,\ell,R}\right\\}$

are sparse for all $(k,\ell)\in\mathbb{Z}^{2}\setminus\left\\{(0,0)\right\\}$
and $R>0$.

###### Proof.

We distinguish the four cases of Lemma 5.5.

(i) If $3k^{2}>\ell^{2}$, then the set $B_{k,\ell,R}$ is finite, so that
$\Lambda_{k,\ell,R}$ is finite, too, and hence it is sparse.

(ii) If $k=0$, then $B_{k,\ell,R}$ is covered by a finite set and the sequence

$\left(m,-\frac{\ell}{2}\right),\quad m\in\mathbb{Z}^{*}.$

Hence it is sufficient to prove that the set

$\left\\{\left(m,-\frac{\ell}{2},m^{3}+\frac{m\ell^{2}}{4}-1\right)\ :\
m\in\mathbb{Z}\right\\}$

is sparse. This follows from Lemma 5.3 with $N=3$ and
$f(m)=m^{3}+\frac{m\ell^{2}}{4}-1$. The assumption (5.1) of the lemma is
satisfied because

$f(m+1)-f(m)=3m(m+1)+1+\frac{\ell^{2}}{4}\to+\infty\quad\text{as}\quad|m|\to+\infty.$

(iii) As in case (i), $B_{k,\ell,R}$ is finite, so that $\Lambda_{k,\ell,R}$
is finite, and hence sparse.

(iv) Similarly to the case (ii), it suffices to show that the two sets

$S_{j}:=\left\\{(m_{j}(n),n,m_{j}(n)^{3}+m_{j}(n)n^{2}-m_{j}(n))\ :\
n\in\mathbb{Z}\right\\},\quad j=1,2$

are sparse. We are going to apply Lemma 5.3, so let us rewrite these sets in
the form

$S_{j}=\left\\{(g_{j}(n),f_{j}(n))\ :\ n\in\mathbb{Z}\right\\}$

with

$g_{j}(n)=(m_{j}(n),n)\quad\text{and}\quad
f_{j}(n)=m_{j}(n)^{3}+m_{j}(n)n^{2}-m_{j}(n).$

Since the following treatment is identical for both values of $j$, we omit the
subscript $j$ for simplicity. By Lemma 5.5 (iv) there exist two constants
$\alpha\neq 0$ and $\beta$ such that

$m(n)=\alpha n+\beta+o(1)\quad\text{as}\quad|n|\to+\infty.$

Using this we obtain the relation (henceforth all relations are understood for
$|n|\to+\infty$)

$f(n)=m(n)^{3}+m(n)n^{2}-m(n)=(\alpha^{3}+\alpha)n^{3}+(3\alpha^{2}\beta+\beta)n^{2}+o(n^{2}).$

It follows that

$f(n+1)=(\alpha^{3}+\alpha)(n^{3}+3n^{2})+(3\alpha^{2}\beta+\beta)n^{2}+o(n^{2}),$

and therefore

$f(n+1)-f(n)=3(\alpha^{3}+\alpha)n^{2}+o(n^{2}).$

Since $3(\alpha^{3}+\alpha)\neq 0$ by our condition $\alpha\neq 0$, the
assumption (5.1) of Lemma 5.3 is satisfied, and we conclude that $S_{1}$ and
$S_{2}$ are sparse sets. ∎

Now we are ready to prove the main proposition of this section which ensures
that Theorem 1.3 holds.

###### Proof of Proposition 5.1.

Fix a real number $R>0$, and consider the sets $\Lambda_{k,\ell,R}$, where
$(k,\ell)\in\mathbb{Z}^{2}\setminus\left\\{(0,0)\right\\}$, $|k|<R$ and
$|\ell|<R$. They are sparse by Corollary 5.6, therefore their union
$\Lambda^{\prime}$ is also sparse. By Lemma 5.2 it remains to show that
$\gamma(\Lambda\setminus\Lambda^{\prime})\geq R$.

If $\lambda_{m+k,n+\ell}$ and $\lambda_{m,n}$ are two distinct elements of
$\Lambda\setminus\Lambda^{\prime}$, then (see (5.2))

$\left\lVert\lambda_{m+k,n+\ell}-\lambda_{m,n}\right\rVert=\left\lVert(k,\ell,Q(k,\ell,m,n))\right\rVert\geq\max\left\\{|k|,|\ell|,|Q(k,\ell,m,n)|\right\\}.$

This obviously yields
$\left\lVert\lambda_{m+k,n+\ell}-\lambda_{m,n}\right\rVert\geq R$ if $|k|\geq
R$ or $|\ell|\geq R$. Otherwise, we infer from the definition of
$\Lambda^{\prime}$ that

$\left\lVert\lambda_{m+k,n+\ell}-\lambda_{m,n}\right\rVert\geq|Q(k,\ell,m,n)|\geq
R.$

This proves the relation $\gamma(\Lambda\setminus\Lambda^{\prime})\geq R$. ∎

##  6\. A review on observation, control and stabilization

Consider a linear evolutionary problem

(6.1) $z^{\prime}=\mathcal{A}z+\mathcal{B}v,\quad z(0)=z_{0},$

where $\mathcal{A}$ is a densely defined, closed linear operator in some
Hilbert space $H$, and $\mathcal{B}$ is a bounded linear operator of another
Hilbert space $G$ into $D(\mathcal{A}^{*})^{\prime}$. Let us also consider the
dual problem

(6.2) $u^{\prime}=-\mathcal{A}^{*}u,\quad
u(0)=u_{0},\quad\psi=\mathcal{B}^{*}u.$

Here $\mathcal{A}^{*},\mathcal{B}^{*}$ denote the adjoints of $\mathcal{A}$
and $\mathcal{B}$.

In control-theoretical terminology, $\mathcal{B}$ is a control operator, $u$
is a control, and $\mathcal{B}^{*}$ is an observability operator. Assume that
the following three hypotheses are satisfied (we denote by
$G^{\prime},H^{\prime}$ the (anti)dual spaces of $G$ and $H$, respectively):

  * (H1)

The operator $\mathcal{A}^{*}$ generates a group $e^{s\mathcal{A}^{*}}$ in
$H^{\prime}$;

  * (H2)

We have $D(\mathcal{A}^{*})=D(\mathcal{B}^{*})$, and there exist two numbers
$\lambda\in\mathbb{C}$ and $c\in\mathbb{R}$ such that

$\left\lVert\mathcal{B}^{*}u\right\rVert\leq c\left\lVert(\mathcal{A}+\lambda
I)^{*}u\right\rVert\quad\text{for all}\quad u_{0}\in D(\mathcal{A}^{*}).$

  * (H3)

There exist two numbers $T^{\prime},c^{\prime}>0$ such that

$\left\lVert\mathcal{B}^{*}u\right\rVert_{L^{2}(0,T^{\prime};G^{\prime})}\leq
c^{\prime}\left\lVert u_{0}\right\rVert_{H^{\prime}}\quad\text{for
all}\quad\varphi_{0}\in D(\mathcal{A}^{*}).$

It follows from (H1) that for every $u_{0}\in H^{\prime}$ the equation in
(6.2) has a unique solution $u\in C(\mathbb{R},H^{\prime})$, and that $u\in
C(\mathbb{R},D(\mathcal{A}^{*}))\cap C^{1}(\mathbb{R},H^{\prime})$ if
$u_{0}\in D(\mathcal{A}^{*})$. Moreover, note that hypothesis (H2) is a
weakening of the boundedness of the operator $\mathcal{B}$; it is satisfied in
many problems where the observation is made on the boundary or some non-open
set of the underlying domain like in the case of Theorem 1.1.

It follows from hypotheses (H1) and (H2) that hypothesis (H3) is meaningful.
It is a (hidden) regularity property, allowing us to extend the observation to
$\psi\in L^{2}_{\text{loc}}(\mathbb{R},G^{\prime})$ for every $u_{0}\in
H^{\prime}$. Moreover, (H1)–(H3) imply the following strengthened version of
(H3): for every $T>0$ there exists a constant $c_{T}$ such that the solutions
of (6.2) satisfy the estimates

(6.3) $\left\lVert\psi\right\rVert_{L^{2}(-T,T;G^{\prime})}\leq
c_{T}\left\lVert u_{0}\right\rVert_{H^{\prime}}\quad\text{for all}\quad
u_{0}\in H^{\prime}.$

This result also shows that the value of $T^{\prime}$ in (H3) has no
importance: if it is satisfied for some $T^{\prime}>0$, then, in fact, it is
satisfied for _every_ $T^{\prime}>0$.

Next, we define the solutions of (6.1) by transposition. Fix $z_{0}\in H$ and
$v\in L^{2}_{\text{loc}}(\mathbb{R},G)$ arbitrarily. Multiply the equation in
(6.1) by the solution $u$ of the equation in (6.2). Integrating by parts
formally between $0$ and $T\in\mathbb{R}$, we easily obtain the identity

(6.4) $\left\langle z(T),u(T)\right\rangle_{H,H^{\prime}}=\left\langle
z_{0},u_{0}\right\rangle_{H,H^{\prime}}+\int_{0}^{T}\left\langle
z(S),\psi(S)\right\rangle_{G,G^{\prime}}\,ds.$

######  Definition.

We define a solution of (6.1) as a function $z:\mathbb{R}\to H$ satisfying the
identity (6.4) for all $u_{0}\in H^{\prime}$ and $T\in\mathbb{R}$.

The previous definition is justified by the following proposition:

######  Proposition 6.1.

Given $z_{0}\in H$ and $v\in L^{2}_{\text{loc}}(\mathbb{R},G)$ arbitrarily,
the problem (6.1) has a unique solution. Moreover, $z:\mathbb{R}\to H$ is
continuous, and satisfies for every $T>0$ the following estimate:

$\left\lVert z\right\rVert_{L^{\infty}(-T,T;H)}\leq c_{T}\left(\left\lVert
z_{0}\right\rVert_{H}+\left\lVert v\right\rVert_{L^{2}(-T,T;G)}\right).$

######  Definitions.

Now we introduce two definitions:

  * •

The system (6.1) is _exactly controllable_ in time $T>0$ if for every pair of
data $z_{0},z_{1}\in H$ there exists a function $v\in
L^{2}_{\text{loc}}(\mathbb{R},G)$ such that

$\left\lVert v\right\rVert_{L^{2}(0,T;G)}\leq c^{\prime}\left(\left\lVert
z_{0}\right\rVert_{H}+\left\lVert z_{1}\right\rVert_{H}\right),$

and that the solution of (6.1) satisfies the final condition $z(T)=z_{1}$.

  * •

The system (6.2) is _observable_ in time $T>0$ if its solutions satisfy the
estimates

$\left\lVert u_{0}\right\rVert_{H^{\prime}}\leq
c^{\prime\prime}\left\lVert\psi\right\rVert_{L^{2}(0,T;G^{\prime})}\quad\text{for
all}\quad u_{0}\in H^{\prime}.$

We note that, similarly to (6.3), by a density argument the observability is
equivalent to the following, apparently weaker condition:

(H4) $\left\lVert u_{0}\right\rVert_{H^{\prime}}\leq
c^{\prime\prime}\left\lVert\psi\right\rVert_{L^{2}(0,T;G^{\prime})}\quad\text{for
all}\quad u_{0}\in D(A^{*}).$

The following important duality result extends a classical finite-dimensional
theorem; it was essentially proved in [5]:

######  Theorem 6.2.

Assume (H1)–(H3). For any given $T>0$, the system (6.1) is exactly
controllable if and only if the system (6.2) is observable.

Next, we state from [14] a rapid stabilization theorem that extended a former
result of Slemrod [31] on bounded control operators:

######  Theorem 6.3.

Assume (H1)–(H4) for some $T>0$, and fix $\omega>0$ arbitrarily. There exists
a linear map $F_{\omega}:H\to G$ and a constant $C>0$ such that the closed-
loop problem

$z^{\prime}=\mathcal{A}z+\mathcal{B}F_{\omega}z,\quad z(0)=z_{0},$

is well-posed in $H$, and its solutions satisfy the estimate

$\left\lVert z(t)\right\rVert_{H}\leq C\left\lVert
z_{0}\right\rVert_{H}e^{-\omega t},$

for all $z_{0}\in H$ and $t\geq 0$.

##  7\. Controllability and stabilization results

With the presentation of the previous review, we are in a position to prove
Theorems 1.4–1.7 by applying Theorems 6.2 and 6.3.

###### Proof of Theorems 1.6 and 1.7.

These theorems will follow with the following choices:

$\displaystyle H^{\prime}:=L^{2}_{x}(\mathbb{T}\times\mathbb{T}),\quad
G^{\prime}:=L^{2}(\Omega);$
$\displaystyle\mathcal{A}^{*}u:=u_{x}+u_{xxx}+u_{xyy},\quad\mathcal{B}^{*}u:=u|_{\Omega};$

and

$\displaystyle D(\mathcal{A}^{*})=D(\mathcal{B}^{*})=\left\\{u\in H^{\prime}\
:\ u_{x}+u_{xxx}+u_{xyy}\in H^{\prime}\right\\}.$

Using the Fourier expansion

(7.1)
$u(x,y)=\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}c_{m,n}e^{i(mx+ny)}$

of the elements of $H^{\prime}$ and Parseval’s formula

$\left\lVert
u\right\rVert_{H^{\prime}}^{2}=4\pi^{2}\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2},$

we may show that an element $u\in H^{\prime}$ belongs to
$D(\mathcal{A}^{*})=D(\mathcal{B}^{*})$ if and only if

(7.2) $\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}\left\lvert
m(m^{2}+n^{2}-m)\right\rvert^{2}\left\lvert c_{m,n}\right\rvert^{2}<+\infty.$

Indeed, using formula (7.1) we have

(7.3)
$(u_{x}+u_{xxx}+u_{xyy})(x,y)=\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}m(m^{2}+n^{2}-m)c_{m,n}e^{i(mx+ny)},$

whence $u_{x}+u_{xxx}+u_{xyy}\in L^{2}(\mathbb{T}\times\mathbb{T})$ if and
only if the condition (7.2) is satisfied. Furthermore, since the index $m=0$
is missing in (7.3) by our assumption $u\in H^{\prime}$,
$u_{x}+u_{xxx}+u_{xyy}$ also belongs to $H^{\prime}$ if and only if it belongs
to $L^{2}(\mathbb{T}\times\mathbb{T})$.

Hypothesis (H1) follows at once from the representation (7.1). The following
computation shows that hypothesis (H2) is satisfied with $\lambda=1$:

$\displaystyle\left\lVert(I+\mathcal{A}^{*})u\right\rVert_{H^{\prime}}^{2}$
$\displaystyle=\int_{\mathbb{T}\times\mathbb{T}}\left\lvert
u+u_{x}+u_{xxx}+u_{xyy}\right\rvert^{2}\,dx\,dy$
$\displaystyle=\int_{\mathbb{T}\times\mathbb{T}}\left\lvert\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}\left(1+im(m^{2}+n^{2}-1)\right)c_{m,n}e^{i(mx+ny)}\right\rvert^{2}\,dx\,dy$
$\displaystyle=4\pi^{2}\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}\left\lvert
1+im(m^{2}+n^{2}-1)\right\rvert^{2}\left\lvert c_{m,n}\right\rvert^{2}$
$\displaystyle\geq
4\pi^{2}\sum_{(m,n)\in\mathbb{Z}^{*}\times\mathbb{Z}}\left\lvert
c_{m,n}\right\rvert^{2}=\left\lVert
u\right\rVert_{H^{\prime}}^{2}=\int_{\mathbb{T}\times\mathbb{T}}\left\lvert
u\right\rvert^{2}\,dx\,dy$ $\displaystyle\geq\int_{\Omega}\left\lvert
u\right\rvert^{2}\,dx\,dy=\left\lVert\mathcal{B}^{*}u\right\rVert_{G^{\prime}}^{2}.$

Hypotheses (H3) and (H4) are equivalent to the two-sided estimates of Theorem
1.4 with $U:=\Omega\times(0,T)$. We may thus applying Theorems 6.2 and 6.3 to
obtain Theorems 1.6 and 1.7, respectively. ∎

###### Proof of Theorems 1.4 and 1.5.

Now, we make the following choices:

$\displaystyle H^{\prime}:=L^{2}(\mathbb{T}\times\mathbb{T}),\quad
G^{\prime}:=L^{2}(\left\\{x_{0}\right\\}\times\mathbb{T});$
$\displaystyle\mathcal{A}^{*}u:=u_{x}+u_{xxx}+u_{xyy},\quad\mathcal{B}^{*}u:=u|_{\left\\{x_{0}\right\\}\times\mathbb{T}};$

and

$\displaystyle D(\mathcal{A}^{*})=D(\mathcal{B}^{*})=\left\\{u\in H^{\prime}\
:\ u_{x}+u_{xxx}+u_{xyy}\in H^{\prime}\right\\}.$

Note that $L^{2}(\left\\{x_{0}\right\\}\times\mathbb{T})$ is isometrically
isomorphic to (and hence may be identified with) $L^{2}(\mathbb{T})$.

Hypothesis (H1) now follows from the representation

$u(x,y)=\sum_{(m,n)\in\mathbb{Z}^{2}}c_{m,n}e^{i(mx+ny)}$

of the elements of $H^{\prime}$ and Parseval’s formula

$\left\lVert
u\right\rVert_{H^{\prime}}^{2}=4\pi^{2}\sum_{(m,n)\in\mathbb{Z}^{2}}\left\lvert
c_{m,n}\right\rvert^{2}.$

For the verification of hypothesis (H2) first, we observe that

(7.4)
$\left\lVert(I+\mathcal{A}^{*})u\right\rVert_{H^{\prime}}^{2}=4\pi^{2}\sum_{(m,n)\in\mathbb{Z}^{2}}\left\lvert
1+im(m^{2}+n^{2}-1)\right\rvert^{2}\left\lvert c_{m,n}\right\rvert^{2}$

because this part of the preceding similar computation remains valid without
the assumption $m\neq 0$.

Next, we apply the Cauchy–Schwarz inequality to obtain the following
relations, where we write

$\mu_{m,n}:=\left\lvert 1+im(m^{2}+n^{2}-1)\right\rvert$

for brevity:

$\displaystyle\left\lVert\mathcal{B}^{*}u\right\rVert_{G^{\prime}}^{2}$
$\displaystyle=\int_{\mathbb{T}}\left\lvert\sum_{(m,n)\in\mathbb{Z}^{2}}c_{m,n}e^{i(mx_{0}+ny)}\right\rvert^{2}\,dy=2\pi\sum_{n\in\mathbb{Z}}\left\lvert\sum_{m\in\mathbb{Z}}c_{m,n}e^{imx_{0}}\right\rvert^{2}$
$\displaystyle\leq
2\pi\sum_{n\in\mathbb{Z}}\left(\left(\sum_{m\in\mathbb{Z}}\left\lvert
c_{m,n}e^{imx_{0}}\mu_{m,n}\right\rvert^{2}\right)\cdot\left(\sum_{m\in\mathbb{Z}}\frac{1}{\mu_{m,n}^{2}}\right)\right)$
$\displaystyle=2\pi\sum_{n\in\mathbb{Z}}\left(\left(\sum_{m\in\mathbb{Z}}\left\lvert
c_{m,n}\mu_{m,n}\right\rvert^{2}\right)\cdot\left(\sum_{m\in\mathbb{Z}}\frac{1}{\mu_{m,n}^{2}}\right)\right).$

Since $\mu_{0,n}=1$, and $\mu_{m,n}\geq\left\lvert m\right\rvert$ if
$m\in\mathbb{Z}^{*}$, we have

$\sum_{m\in\mathbb{Z}}\frac{1}{\mu_{m,n}^{2}}\leq
1+2\sum_{k=1}^{\infty}\frac{1}{k^{2}}<5,$

for every $n\in\mathbb{Z}$, and therefore we infer from the preceding estimate
that

$\left\lVert\mathcal{B}^{*}u\right\rVert_{G^{\prime}}^{2}\leq
10\pi\sum_{(m,n)\in\mathbb{Z}^{2}}\left\lvert
1+im(m^{2}+n^{2}-1)\right\rvert^{2}\left\lvert c_{m,n}\right\rvert^{2}.$

Combining with (7.4) and noticing that $10\pi<4\pi^{2}$, we obtain that

$\left\lVert\mathcal{B}^{*}u\right\rVert_{G^{\prime}}\leq\left\lVert(I+\mathcal{A}^{*})u\right\rVert_{H^{\prime}},$

i.e., hypothesis (H2) is satisfied.

Finally, hypotheses (H3) and (H4) are equivalent to the two-sided estimates of
Theorem 1.3. Applying Theorems 6.2 and 6.3, Theorems 1.6 and 1.7 follow. ∎

##  8\. Further comments

In this paper, we considered only internal observability and controllability
properties of the system (1.1), however boundary observability and
controllability results may also be obtained by adapting our approach; see
e.g., [17] and [16] for analogous results on membrane and plate models.

Concerning Theorems 1.1, 1.2 and 1.3 it is natural to ask whether they remain
valid in more general situations, for example, if the segments are oblique or
they are moving in time (moving observation). For linear plate models, these
questions were studied in [10], and they probably may be adapted to the
Zakharov–Kuznetsov equation (1.1), however, these issues are still open to
study.

Note that the method developed in this work was based on the notion of sparse
sets, it may be adapted to more general equations, containing further higher-
order terms, in the two-dimensional cases.

## Acknowledgements

The first author was supported by CAPES grant numbers 88881.311964/2018-01 and
88881.520205/2020-01, CNPq grant numbers 307808/2021-1 and 401003/2022-1,
MATHAMSUD grant 21-MATH-03 and Propesqi (UFPE). The second author was
supported by the following grants: FACEPE APV-0002-1.01/23, Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) – finance Code 001 and
NSFC no. 11871348. The third author was partially supported by CNPq (Brasil)
and CAPES (Brasil) – finance Code 001.

Part of this work was done during the stay of the second author at the
Department of Mathematics of the Federal University of Rio de Janeiro (UFRJ)
and the Department of Mathematics of the Federal University of Pernambuco
(Recife) in September 2023. He thanks the departments for their hospitality.

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