# Boundary controllability of the Korteweg-de Vries equation: The Neumann case

Roberto Capistrano–Filhoa and Jandeilson Santos da Silvaa Corresponding author
email<EMAIL_ADDRESS><EMAIL_ADDRESS>aDepartamento de Matemática, Universidade Federal de Pernambuco (UFPE),
50740-545, Recife (PE), Brazil

###### Abstract

This note gives a necessary first step to understanding the critical set
phenomenon for the Korteweg-de Vries equation posed on the bounded domain.
Precisely, considering the Neumann boundary conditions with only one control
input, we showed that small perturbation of the equilibrium (steady state
solution $u\equiv c$) ensures that the resulting linear KdV equation is
exactly controllable in the critical case $L=2k\pi$, for $k\in\mathbb{N}^{*}$.
Additionally, considering $c\neq-1$ and

$\mathcal{R}_{c}:=\left\\{\frac{2\pi}{\sqrt{3(c+1)}}\sqrt{m^{2}+ml+m^{2}};\
m,l\in\mathbb{N}^{*}\right\\}\cup\left\\{\frac{m\pi}{\sqrt{c+1}};\
m\in\mathbb{N}^{*}\right\\},$

one can find $\epsilon_{c}>0$ such that when
$d\in(c-\epsilon_{c},c)\cup(c,c+\epsilon_{c})$, $d\neq-1$, we have
$L\notin\mathcal{R}_{d}$ and, consequently the nonlinear KdV system is exactly
controllable in $L^{2}(0,L)$ around of the steady state $u\equiv d$.

###### keywords:

Korteweg–de Vries equation, exact boundary controllability, Neumann boundary
conditions, critical set

##  1 Introduction

We had known when we formulated the waves as a free boundary problem of the
incompressible, irrotational Euler equation in an appropriate non-dimensional
form, there exist two non-dimensional parameters $\delta:=\frac{h}{\lambda}$
and $\varepsilon:=\frac{a}{h}$, where the water depth, the wavelength and the
amplitude of the free surface are parameterized as $h,\lambda$ and $a$,
respectively. See, for instance, Bona et al. (2002, 2005); Alvarez-Samaniego &
Lannes (2008); Bona et al. (2005); Lannes (2013); Saut & Scheurer (1987) and
references therein for a rigorous justification. Moreover, another non-
dimensional parameter $\mu$ appears, the Bond number, to measure the
importance of gravitational forces compared to surface tension forces also
appears. Considering the physical condition $\delta\ll 1$ we can characterize
the waves, called long waves or shallow water waves. In particular,
considering the relations between $\varepsilon$ and $\delta$, we can have the
following regime:

  * •

Korteweg-de Vries (KdV): $\varepsilon=\delta^{2}\ll 1$ and
$\mu\neq\frac{1}{3}$. Under this regime, Korteweg & de Vries (1985)111This
equation was first introduced by Boussinesq (1871), and Korteweg and de Vries
rediscovered it twenty years later. derived the following well-known equation
as a central equation among other dispersive or shallow water wave models
called the KdV equation from the equations for capillary-gravity waves:

$\pm 2\eta_{t}+3\eta\eta_{x}+\left(\frac{1}{3}-\mu\right)\eta_{xxx}=0.$

Today, it is well known that this equation has important phenomena that
directly affect the control problem related to them, so-called critical length
phenomenon. Let us briefly present the control problem, which makes the
phenomena of critical lengths emerge. The control problem was presented in a
pioneering work of Rosier Rosier (1997) that studied the following system

$\left\\{\begin{array}[c]{lll}u_{t}+u_{x}+uu_{x}+u_{xxx}=0&&\text{ in
}(0,L)\times(0,T),\\\ u(0,t)=0,\text{ }u(L,t)=0,\text{
}u_{x}(L,t)=g(t)&&\text{ in }(0,T),\\\ u(x,0)=u_{0}(x)&&\text{ in
}(0,L),\end{array}\right.$ (1)

where the boundary value function $g(t)$ is considered as a control input.
Precisely, the author answered the following control problem for the system
(1), giving the origin of the critical length phenomenon for the KdV equation.

Question $\mathcal{A}$: Given $T>0$ and $u_{0},u_{T}\in L^{2}(0,L)$, can one
find an appropriate control input $g(t)\in L^{2}(0,T)$ such that the
corresponding solution $u(x,t)$ of the system (1) satisfies

$u(x,0)=u_{0}(x)\quad\text{and}\quad u(x,T)=u_{T}(x)?$ (2)

Rosier (1997) answered the previous question. He showed that considering
$L\notin\mathcal{N}$, where

$\mathcal{N}:=\left\\{\frac{2\pi}{\sqrt{3}}\sqrt{k^{2}+kl+l^{2}}\,:k,\,l\,\in\mathbb{N}^{\ast}\right\\},$

the associated linear system (1)

$\left\\{\begin{array}[c]{lll}u_{t}+u_{x}+u_{xxx}=0&&\text{ in
}(0,L)\times(0,T),\\\ u(0,t)=0,\text{ }u(L,t)=0,\text{
}u_{x}(L,t)=g(t)&&\text{ in }(0,T),\\\ u(x,0)=u_{0}(x)&&\text{ in
}(0,L)\end{array}\right.$ (3)

is controllable; roughly speaking, if $L\in\mathcal{N}$ system (3) is not
controllable, that is, there exists a finite-dimensional subspace of
$L^{2}(0,L)$, denoted by $\mathcal{M}=\mathcal{M}(L)$, which is unreachable
from $0$ for the linear system. More precisely, for every nonzero state
$\psi\in\mathcal{M}$, $g\in L^{2}(0,T)$ and $u\in C([0,T];L^{2}(0,L))\cap
L^{2}(0,T;H^{1}(0,L))$ satisfying (3) and $u(\cdot,0)=0$, one has
$u(\cdot,T)\neq\psi$.

######  Definition 1.1.

A spatial domain $(0,L)$ is called critical for the system (3) if its domain
length $L$ belongs to $\mathcal{N}$.

Following the work of Rosier (1997), the boundary control system of the KdV
equation posed on the finite interval $(0,L)$ with various control inputs has
been intensively studied (cf. Caicedo et al. (2017); Cerpa (2007); Cerpa &
Crépeau (2009); Coron & Crépeau (2005); Crṕeau (2001); Glass & Guerrero (2008,
2010); Guilleron (2014) and see Cerpa (2014); Rosier & Zhang (2009) for more
complete reviews). Thus, this work gives a necessary first step to
understanding the critical set phenomenon for the KdV equation with Neumann
boundary conditions, giving a complementation of the results shown in Caicedo
et al. (2017).

###  1.1 Problem set

In this article, we study a class of distributed parameter control systems
described by the Korteweg–de Vries (KdV) equation posed on a bounded domain
$(0,L)$ with the Neumann boundary conditions

$\left\\{\begin{array}[c]{lll}u_{t}+u_{x}+u_{xxx}+uu_{x}=0&&\text{ in
}(0,L)\times(0,T),\\\ u_{xx}(0,t)=u_{xx}(L,t)=0&&\text{ in }(0,T),\\\
u_{x}(L,t)=h(t)&&\text{ in }(0,T),\\\ u(x,0)=u_{0}(x)&&\text{ in
}(0,L).\end{array}\right.$ (4)

Here, $h(t)$ will be considered as a control input. Recently, the authors in
Caicedo et al. (2017) dealt with the control problem related to the system
(4). Precisely, was proved that their solutions are exactly controllable in a
neighborhood of $c$ if the length $L$ of the spatial domain $(0,L)$ does not
belong to the set

$\mathcal{R}_{c}:=\left\\{\frac{2\pi}{\sqrt{3(c+1)}}\sqrt{m^{2}+ml+m^{2}};\
m,l\in\mathbb{N}^{*}\right\\}\cup\left\\{\frac{m\pi}{\sqrt{c+1}};\
m\in\mathbb{N}^{*}\right\\},$

that is, the relation (2) holds for the solution of the system (4). The result
can be read as follows.

######  Theorem 1.2 (Caicedo et al. (2017)).

Let $T>0$, $c\neq-1$ and $L\notin\mathcal{R}_{c}$. There exists $\delta>0$
such that for any $u_{0},u_{T}\in L^{2}(0,L)$ with

$\|u_{0}-c\|_{L^{2}(0,L)}<\delta\text{ \ and \
}\|u_{T}-c\|_{L^{2}(0,L)}<\delta$

one can find $h\in L^{2}(0,T)$ such that the system (4) admits a unique
solution

$u\in\mathcal{Z}_{T}=C\left([0,T];L^{2}(0,L)\right)\cap
L^{2}\left(0,T;H^{1}(0,L)\right)$

satisfying (2).

As in Rosier (1997), the first step is to obtain a control result for the
linear system, namely,

$\left\\{\begin{array}[c]{lll}v_{t}+(1+c)v_{x}+v_{xxx}=0&&\text{ \ in \
}(0,L)\times(0,T),\\\ v_{xx}(0,t)=v_{xx}(L,t)&\text{ \ in \ }(0,T),\\\
v_{x}(L,t)=h(t)&\text{ \ in \ }(0,T),\\\ v(x,0)=v_{0}(x)&\text{ \ in \
}(0,L).\end{array}\right.$ (5)

So, we have, for our case, the following definition.

######  Definition 1.3.

A spatial domain $(0,L)$ is called critical for the system (5) if its domain
length $L$ belongs to $\mathcal{R}_{c}$.

With this, the following result holds.

######  Theorem 1.4 (Caicedo et al. (2017)).

For $c\neq-1$, the linear system (5) is exactly controllable in the space
$L^{2}(0,L)$ if and only if $L\notin\mathcal{R}_{c}$. Otherwise, that is, if
$c=-1$, the system (5) is not exactly controllable in the space $L^{2}(0,L)$
for any $L>0$.

With this in hand, they extend the result for the nonlinear system (4) using
fixed point argument achieving Theorem 1.2 whenever $L\notin\mathcal{R}_{c}$.
It is important to point out that fixing $k\in\mathbb{N}^{*}$ and considering
$m=l=k$ we have $L=2k\pi$, when $c=0$, and, from Theorem 1.4, it follows that
(5) are not exactly controllable. Additionally, in this regime, before
mentioned, we do not know if the system (4) is exactly controllable. So, in
this context, the natural question appears:

Question $\mathcal{B}$: Given $T>0$, $L=2k\pi$, and $u_{0},u_{T}\in
L^{2}(0,L)$, can one find an appropriate control input $h(t)\in L^{2}(0,T)$
such that the corresponding solution $u(x,t)$ of the system (4) satisfies (2)?

###  1.2 Main result and paper’s outline

In this spirit, this note presents two results that give answers for Question
$\mathcal{B}$. The first result ensures that for $c$ near enough to $0$ (small
perturbations of $0$), the system (4) is exactly controllable in a
neighborhood of $c$ in $L^{2}(0,2k\pi)$, the result is presented as follows:

######  Theorem 1.5.

Let $L=2k\pi$. Then, there exists $\epsilon>0$ such that for every
$c\in(0,\epsilon]$ the system (5) is exactly controllable in $L^{2}(0,L)$ and,
consequently, system (4) is exactly controllable in a neighborhood of $c$ in
$L^{2}(0,L)$.

The second main result of the article is a generalization of the previous one.
Precisely, given $L\in\mathcal{R}_{c}$, for $d$ close enough to $c$, but not
equal, $L$ does not belong to the set $\mathcal{R}_{d}$. This generalization
can be read below.

######  Theorem 1.6.

Let $T>0$, $c\neq-1$ and $L\in\mathcal{R}_{c}$. There exists $\epsilon_{c}>0$
such that, for every $d\in(c-\epsilon_{c},c)\cup(c,c+\epsilon_{c})$,
$d\neq-1$, we have $L\notin\mathcal{R}_{d}$. Consequently, system (5) $($with
$c=d)$ is exactly controllable in $L^{2}(0,L)$ and the system (4) is exactly
controllable around of the steady state $u=d$ in $L^{2}(0,L)$.

The proof of these theorems is given in a very simple way and is based on the
topological properties of real numbers together with the Theorems 1.2 and 1.4,
completing in some sense the previous results given in Caicedo et al. (2017).

We finish this introduction with an outline of this work, which consists of
four parts including the introduction. Section 2 gives the proof of Theorem
1.5. Section 3 is devoted to proving carefully an extension of the Theorem
1.5, that is, we present the proof of Theorem 1.6. Finally, we present in
Section 4 a conclusion and some future perspectives concerning the critical
set phenomenon for the system (4).

##  2 Controllability in $L=2k\pi$: A particular case

Let us prove, in this section, Theorem 1.5. This result is a consequence of
Theorems 1.2, 1.4 and the result below which asserts that for every $c>0$
small enough, we have $L\notin\mathcal{R}_{c}$. So, with the previous theorems
presented in the introduction, showed in Caicedo et al. (2017), system (4) is
exactly controllable around $c$ for $h(t)\in L^{2}(0,T)$ and $L=2k\pi$.

######  Proposition 2.1.

There exists $\epsilon>0$ such that, for every $c\in(0,\epsilon]$ we have
$L=2k\pi\notin\mathcal{R}_{c}$.

###### Proof.

We will analyze the values of $c$ for which $2k\pi$ is a critical length. If
$2k\pi\in\mathcal{R}_{c}$ then

$2k\pi=\frac{2\pi\sqrt{m^{2}+ml+l^{2}}}{\sqrt{3(c+1)}},\
m,\l\in\mathbb{N}^{*}$ (6)

or

$2k\pi=\frac{m\pi}{\sqrt{c+1}};\ m\in\mathbb{N}^{*}.$ (7)

If (6) holds, then

$k=\sqrt{\frac{m^{2}+ml+l^{2}}{3(c+1)}},$

for some $m,l\in\mathbb{N}^{*}$. Hence,

$k^{2}=\frac{m^{2}+ml+l^{2}}{3(c+1)}$

which implies

$c+1=\frac{m^{2}+ml+l^{2}}{3k^{2}}.$

Thus, we have

$c=\frac{m^{2}+ml+l^{2}}{3k^{2}}-1,$

with $m,l\in\mathbb{N}^{*}$.

Now if (7) is verified, we get that

$2k=\frac{m}{\sqrt{c+1}},$

with $m,l\in\mathbb{N}^{*}$. Taking the square on both sides yields that

$4k^{2}=\frac{m^{2}}{c+1}$

implying that

$c+1=\frac{m^{2}}{4k^{2}}.$

Therefore,

$c=\frac{m^{2}}{4k^{2}}-1,$

where $m\in\mathbb{N}^{*}$. Thus, consequently, if $2k\pi\in\mathcal{R}_{c}$
then we must have $c\in\mathcal{U}\cup\mathcal{J}$ with

$\mathcal{U}:=\left\\{\frac{m^{2}+ml+l^{2}}{3k^{2}}-1;\
m,l\in\mathbb{N}^{*}\right\\}$

and

$\mathcal{J}=\left\\{\frac{m^{2}}{4k^{2}}-1;\ m\in\mathbb{N}^{*}\right\\}.$

The rest of the proof will show that all elements of
$\mathcal{U}\cup\mathcal{J}$ are isolated points. To proof it, let
$x,y\in\mathcal{U}$ with $x\neq y$ defined by

$x=\frac{m_{1}^{2}+m_{1}l_{1}+l_{1}^{2}}{3k^{2}}-1,\ \
y=\frac{m_{2}^{2}+m_{2}l_{2}+l_{2}^{2}}{3k^{2}}-1,$

with $m_{1},l_{1},m_{2},l_{2}\in\mathbb{N}^{*}$. Note that

$d(x,y)=|x-y|=\frac{1}{3k^{2}}\left|(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})-(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})\right|.$

If

$m_{1}^{2}+m_{1}l_{1}+l_{1}^{2}=m_{2}^{2}+m_{2}l_{2}+l_{2}^{2}$

then $x=y$ which is a contradiction. Thus $(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})$
and $(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})$ are distinct natural numbers so that

$\left|(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})-(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})\right|\geq
1,$

and consequently

$d(x,y)\geq\frac{1}{3k^{2}},\ \forall\ x,y\in\mathcal{U},\ x\neq y.$

Analogously, we get that

$\displaystyle d(x,y)\geq\frac{1}{4k^{2}},\ \forall x,y\in\mathcal{J},\ x\neq
y.$

Finally, suppose $x\in\mathcal{U}$ and $y\in\mathcal{J}$ with $x\neq y$. We
have

$x=\frac{m^{2}+ml+l^{2}}{3k^{2}}-1$

and

$y=\frac{j^{2}}{4k^{2}}-1,$

where $m,l,j\in\mathbb{N}^{*}$. Then

$\begin{split}x-y=&\frac{m^{2}+ml+l^{2}}{3k^{2}}-\frac{j^{2}}{4k^{2}}\\\
=&\frac{4(m^{2}+ml+l^{2})}{12k^{2}}-\frac{3j^{2}}{12k^{2}}\\\
=&\frac{1}{12k^{2}}\left[4(m^{2}+ml+l^{2})-3j^{2}\right].\end{split}$

If $4(m^{2}+ml+l^{2})=3j^{2}$ then $x=y$, which is a contradiction. Therefore
$4(m^{2}+ml+l^{2})$ and $3j^{2}$ are distinct natural numbers so that

$\left|4(m^{2}+ml+l^{2})-3j^{2}\right|\geq 1.$

Consequently,

$d(x,y)=|x-y|=\frac{1}{12k^{2}}\left|4(m^{2}+ml+l^{2})-3j^{2}\right|\geq\frac{1}{12k^{2}}.$

From all the above, we conclude that

$d(x,y)\geq\frac{1}{12k^{2}},\ \forall x,y\in\mathcal{U}\cup\mathcal{J},\
x\neq y.$ (8)

With this in hand, we claim that $\mathcal{U}\cup\mathcal{J}$ is a discrete
set. Indeed, given $x\in\mathcal{U}\cup\mathcal{J}$ consider
$\delta\in\left(0,1/12k^{2}\right)$. Then

$(x-\delta,x+\delta)\cap\left(\mathcal{U}\cup\mathcal{J}\right)=\\{x\\},$

due to (8). Finally, observe that $0\in\mathcal{U}\cup\mathcal{J}$. Since for
$m=l=k$ we have

$0=\frac{m^{2}+ml+l^{2}}{3k^{2}}-1.$

As $\mathcal{U}\cup\mathcal{J}$ is discrete, there exists $\delta>0$ such that

$(-\delta,\delta)\cap\left(\mathcal{U}\cup\mathcal{J}\right)=\\{0\\}.$

Therefore, for $c\in(-\delta,\delta)\backslash\\{0\\}$ we have
$c\notin\mathcal{U}\cup\mathcal{J}$ so that $2k\pi\notin\mathcal{R}_{c}$. ∎

##  3 Controllability in $L\in\mathcal{R}_{c}$: A generalization

Here, we can generalize the previous section, that is, showing Theorem 1.6.

###### Proof of Theorem 1.6.

We will split the proof into two cases.

First case: $L=2\pi\frac{\sqrt{j^{2}+jk+k^{2}}}{\sqrt{3(c+1)}}$ for some
$(j,k)\in\mathbb{N}^{*}\times\mathbb{N}^{*}$.

Let $d\neq-1$ and suppose that $L\in\mathcal{R}_{d}$. Then

$L=2\pi\frac{\sqrt{m^{2}+ml+l^{2}}}{\sqrt{3(d+1)}};\ m,l\in\mathbb{N}^{*}$ (9)

or

$L=\frac{m\pi}{\sqrt{d+1}};\ m\in\mathbb{N}^{*}.$ (10)

If (9) is the case, then

$2\pi\frac{\sqrt{j^{2}+jk+k^{2}}}{\sqrt{3(c+1)}}=2\pi\frac{\sqrt{m^{2}+ml+l^{2}}}{\sqrt{3(d+1)}},$

which implies

$d=\frac{(c+1)(m^{2}+ml+l^{2})}{j^{2}+jk+k^{2}}-1$

with $m,l\in\mathbb{N}^{*}$.

Otherwise, if (10) holds, so

$2\pi\frac{\sqrt{j^{2}+jk+k^{2}}}{\sqrt{3(c+1)}}=\frac{m\pi}{\sqrt{d+1}}$

which implies that

$d=\frac{3m^{2}(c+1)}{4(j^{2}+jk+k^{2})}-1$

where $m\in\mathbb{N}^{*}$. Therefore, if $L\in\mathcal{R}_{d}$ then we
necessarily have $d\in\mathcal{A}_{1}\cup\mathcal{B}_{1}$. Here,

$\mathcal{A}_{1}:=\left\\{\frac{(c+1)(m^{2}+ml+l^{2})}{j^{2}+jk+k^{2}}-1,\
m,l\in\mathbb{N}^{*}\right\\}$

and

$\mathcal{B}_{1}:=\left\\{\frac{3m^{2}(c+1)}{4(j^{2}+jk+k^{2})}-1;\
m\in\mathbb{N}^{*}\right\\}.$

We are now in a position to prove that $\mathcal{A}_{1}\cup\mathcal{B}_{1}$ is
discrete. To do that, consider $x,y\in\mathcal{A}_{1}$ such that $x\neq y$ in
the form

$x=\frac{(c+1)(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})}{j^{2}+jk+k^{2}}-1$

and

$y=\frac{(c+1)(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})}{j^{2}+jk+k^{2}}-1,$

with $m_{1},l_{1},m_{2},l_{2}\in\mathbb{N}^{*}$. Note that

$x-y=\frac{c+1}{j^{2}+jk+k^{2}}\left[(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})-(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})\right].$

Since $x\neq y$ we have

$(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})\neq(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2}).$

Thus

$\left|(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})-(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})\right|\geq
1$

so that

$d(x,y)\geq\frac{c+1}{j^{2}+jk+k^{2}},\ \forall x,y\in\mathcal{A}_{1},\ x\neq
y.$

Analogously, we get

$d(x,y)\geq\frac{3(c+1)}{4(j^{2}+jk+k^{2})},\ \forall\
x,y\in\mathcal{B}_{1},x\neq y.$

Now, let $x\in\mathcal{A}_{1}$ and $y\in\mathcal{B}_{1}$ with $x\neq y$, as
follow:

$x=\frac{(c+1)(m^{2}+ml+l^{2})}{j^{2}+jk+k^{2}}-1$

and

$y=\frac{3p^{2}(c+1)}{4(j^{2}+jk+k^{2})}-1,$

where $m,l,p\in\mathbb{N}^{*}$. Observe that

$\begin{split}x-y=&\frac{(c+1)(m^{2}+ml+l^{2})}{j^{2}+jk+k^{2}}-\frac{3p^{2}(c+1)}{4(j^{2}+jk+k^{2})}\\\
=&\frac{4(c+1)(m^{2}+ml+l^{2})}{4(j^{2}+jk+k^{2})}-\frac{3p^{2}(c+1)}{4(j^{2}+jk+k^{2})}\\\
=&\frac{c+1}{4(j^{2}+jk+k^{2})}\left[4(m^{2}+ml+l^{2})-3p^{2}\right].\end{split}$

Since $x\neq y$ we have that $4(m^{2}+ml+l^{2})$ and $3p^{2}$ are distinct
natural numbers so

$\left|4(m^{2}+ml+l^{2})-3p^{2}\right|\geq 1$

and, consequently, we have

$d(x,y)\geq\frac{c+1}{4(j^{2}+jk+k^{2})}.$

From all the above we conclude that

$d(x,y)\geq\frac{c+1}{4(j^{2}+jk+k^{2})},\ \forall\
x,y\in\mathcal{A}_{1}\cup\mathcal{B}_{1},\ x\neq y$

which implies that $\mathcal{A}_{1}\cup\mathcal{B}_{1}$ is discrete.

Note that $c\in\mathcal{A}_{1}\cup\mathcal{B}_{1}$ since

$c=\left[\frac{(c+1)(j^{2}+jk+k^{2})}{j^{2}+jk+k^{2}}-1\right]\in\mathcal{A}_{1}.$

As any point in $\mathcal{A}_{1}\cup\mathcal{B}_{1}$ is isolated, there exists
$\epsilon_{1}>0$ such that

$(c-\epsilon_{1},c+\epsilon_{1})\cap\left[\mathcal{A}_{1}\cup\mathcal{B}_{1}\right]=\\{c\\}.$

Therefore for $d\in(c-\epsilon_{1},c+\epsilon_{1})\backslash\\{c\\}$ we have
that $d\notin\mathcal{A}_{1}\cup\mathcal{B}_{1}$ and, therefore,
$L\notin\mathcal{R}_{d}$.

Second case: $L=\frac{k\pi}{\sqrt{c+1}}$ for some $k\in\mathbb{N}^{*}$.

Let $d\neq-1$ and suppose that $L\in\mathcal{R}_{d}$, that is, (9) or (10)
holds. If (9) holds, then

$\frac{k\pi}{\sqrt{c+1}}=2\pi\frac{\sqrt{m^{2}+ml+l^{2}}}{\sqrt{3(d+1)}}$

giving that

$d=\frac{(c+1)(m^{2}+ml+l^{2})}{3k^{2}}-1,$

with $m,l\in\mathbb{N}^{*}$. If (10) is the case, thus

$\frac{k\pi}{\sqrt{c+1}}=\frac{m\pi}{\sqrt{d+1}}$

so that

$d=\frac{(c+1)m^{2}}{k^{2}}-1.$

Therefore, if $L\in\mathcal{R}_{d}$ then we necessarily have
$d\in\mathcal{A}_{2}\cup\mathcal{B}_{2}$ where

$\mathcal{A}_{2}:=\left\\{\frac{(c+1)(m^{2}+ml+l^{2})}{3k^{2}}-1;\
m,l\in\mathbb{N}^{*}\right\\}$

and

$\mathcal{B}_{2}:=\left\\{\frac{(c+1)m^{2}}{k^{2}}-1;\
m\in\mathbb{N}^{*}\right\\}.$

As done before, taking $x,y\in\mathcal{A}_{2}$ with $x\neq y$, such that

$x=\frac{(c+1)(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})}{3k^{2}}-1$

and

$y=\frac{(c+1)(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})}{3k^{2}}-1,$

where $m_{1},l_{1},m_{2},l_{2}\in\mathbb{N}^{*}$,yields that

$x-y=\frac{c+1}{3k^{2}}\left[(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})-(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})\right]$

and, as $x\neq y$ we have

$\left|(m_{1}^{2}+m_{1}l_{1}+l_{1}^{2})-(m_{2}^{2}+m_{2}l_{2}+l_{2}^{2})\right|\geq
1.$

Consequently,

$d(x,y)\geq\frac{c+1}{3k^{2}},\forall x,y\in\mathcal{A}_{2},\ x\neq y.$

In an analogous way,

$d(x,y)\geq\frac{c+1}{k^{2}},\ \forall x,y\in\mathcal{B}_{2},\ x\neq y$

and

$d(x,y)\geq\frac{c+1}{3k^{2}},\ \forall x\in\mathcal{A}_{2},\
y\in\mathcal{B}_{2},\ x\neq y.$

Therefore,

$d(x,y)\geq\frac{c+1}{3k^{2}},\ \forall
x,y\in\mathcal{A}_{2}\cup\mathcal{B}_{2},\ x\neq y.$

Proceeding as in the first case, we conclude that there exists
$\epsilon_{2}>0$ such that, for every
$d\in(c-\epsilon_{2},c+\epsilon_{2})\backslash\\{c\\}$ we have
$d\notin\mathcal{A}_{2}\cup\mathcal{B}_{2}$ so that $L\notin\mathcal{R}_{d}$.
Considering $\epsilon_{c}:=\min\\{\epsilon_{1},\epsilon_{2}\\}$, thanks to the
Theorems 1.2 and 1.4, the proof of Theorem 1.6 is completed. ∎

##  4 Further comments and perspectives

The exact boundary controllability of the non-linear Korteweg–de Vries
equation with Neumann boundary conditions is studied. We consider only the
first spatial derivative at the right endpoint being controlled. In this case,
the exact controllability has been shown in Caicedo et al. (2017) when the
length $L$ does not belong to the set $\mathcal{R}_{c}$. In our work, we
complete this previous article studying the controllability in the critical
case $L=2k\pi$, with $k\in\mathbb{N}^{*}$. More precisely, we notice that the
critical sets $\mathcal{R}_{c}$ and, consequently, the controllability of the
systems (4) and (5) are sensitive to small perturbations in the steady state
solution $u\equiv c$, where we want to study the critical set phenomenon.

As mentioned before considering $c=0$, we have that $L=2k\pi$ is a critical
length to the system (4), and the analysis of the controllability is more
delicate. In a forthcoming paper, we will infer information about the local
controllability of (4) around the origin $c=0$ from its local controllability
around $c>0$ small enough in the critical case.

## Acknowledgment

Capistrano–Filho was supported by CAPES grant 88881.311964/2018-01 and
88881.520205/2020-01, CNPq grant 307808/2021-1 and 401003/2022-1, MATHAMSUD
grant 21-MATH-03 and Propesqi (UFPE). Da Silva acknowledges support from CNPq.
This work is part of the Ph.D. thesis of Santos at the Department of
Mathematics of the Universidade Federal de Pernambuco.

### Authors contributions

Capistrano-Filho and Santos work equality in Conceptualization; formal
analysis; investigation; writing–original draft; writing–review and editing.

### Conflict of interest statement

This work does not have any conflicts of interest.

### Data availability statement

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

## References

  * Alvarez-Samaniego & Lannes (2008) Alvarez-Samaniego, B., & Lannes, D. (2008). Large time existence for 3D water-waves and asymptotics, _Invent. Math._ , 171, 485–541.
  * Bona et al. (2002) Bona, J. L., Chen, M., & Saut, J.-C. (2002). Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. _J. Nonlinear. Sci._ , 12, 283–318.
  * Bona et al. (2005) Bona, J. L., Colin, T., & Lannes, D. (2005). Long wave approximations for water waves, _Arch. Ration. Mech. Anal._ , 178, 373–410.
  * Bona et al. (2005) Bona, J. L., Lannes, D., & Saut, J.-C. (2008). Asymptotic models for internal waves, _J. Math. Pures Appl._ , 9, 538–566.
  * Boussinesq (1871) Boussinesq, J. M. (1871). Thórie de l’intumescence liquide, applelée onde solitaire ou de, translation, se propageant dans un canal rectangulaire, _C. R. Acad. Sci. Paris._ , 72, 755–759.
  * Caicedo et al. (2017) Caicedo, M. A., Capistrano–Filho, R. A., & Zhang, B.-Y. (2017). Neumann Boundary Controllability of the Korteweg–de Vries Equation on a Bounded Domain. _SIAM Journal on Control and Optim._ , 55, 3503–3532.
  * Cerpa (2007) Cerpa, E. (2007). Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, _SIAM J. Control Optim._ , 46, 877–899.
  * Cerpa (2014) Cerpa, E. (2014). Control of a Korteweg-de Vries equation: a tutorial, _Math. Control Relat. Fields._ , 4, 45–99.
  * Cerpa & Crépeau (2009) Cerpa, E., & Crépeau, E. (2009). Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, _Ann. I.H. Poincaré_ , 26, 457–475.
  * Coron & Crépeau (2005) Coron, J.-M., & Crépeau, E. (2004) Exact boundary controllability of a nonlinear KdV equation with critical lengths, _Journal of the European Mathematical Society_ , 6, 367–398.
  * Crṕeau (2001) Crépeau, E. (2001). Exact boundary controllability of the Korteweg–de Vries equation around a non-trivial stationary solution, _International Journal of Control_ , 74, 1096–1106.
  * Glass & Guerrero (2008) Glass, O., & Guerrero, S. (2008). Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, _Asymptot. Anal._ , 60, 61–100.
  * Glass & Guerrero (2010) Glass, O., & Guerrero, S. (2010). Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, _Systems Control Lett._ , 59, 390–395.
  * Guilleron (2014) Guilleron, J.-P. (2014). Null controllability of a linear KdV equation on an interval with special boundary conditions, _Math. Control Signals Syst._ , 26, 375–401.
  * Korteweg & de Vries (1985) Korteweg, D. J., & de Vries, G. (1985). On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, _Phil. Mag._ , 39, 422–443.
  * Lannes (2013) Lannes, D. (2013). _The water waves problem. Mathematical analysis and asymptotics._ Mathematical Surveys and Monographs, 188. American Mathematical Society, Providence, RI, xx+32.
  * Rosier (1997) Rosier, L. (1997). Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, _ESAIM Control Optim. Cal. Var._ , 2, 33–55.
  * Rosier & Zhang (2009) Rosier, L., & Zhang, B.-Y. (2009). Control and stabilization of the Korteweg-de Vries equation: Recent progress, _J. Syst Sci $\&$ Complexity_, 22, 647–682.
  * Saut & Scheurer (1987) Saut, J.-C., & Scheurer, B. (1987). Unique continuation for some evolution equations. _J. Diff. Eqs._ , 66, 118–139.