#  On the small-time bilinear control of a nonlinear heat equation: global
approximate controllability and exact controllability to trajectories

Alessandro Duca,111Université de Lorraine, CNRS, INRIA, IECL, F-54000 Nancy,
France; e-mail<EMAIL_ADDRESS>Eugenio Pozzoli 222Université de
Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France; e-mail:
<EMAIL_ADDRESS>Cristina Urbani333Dipartimento di Scienze
Tecnologiche e dell’Innovazione, Universitas Mercatorum, Piazza Mattei 10,
00186, Roma, Italy; e-mail<EMAIL_ADDRESS>
###### Abstract

In this work we analyse the small-time reachability properties of a nonlinear
parabolic equation, by means of a bilinear control, posed on a torus of
arbitrary dimension $d$. Under a saturation hypothesis on the control
operators, we show the small-time approximate controllability between states
sharing the same sign. Moreover, in the one-dimensional case $d=1$, we combine
this property with a local exact controllability result, and prove the small-
time exact controllability of any positive states towards the ground state of
the evolution operator.

Keywords: bilinear control, heat equation, approximate controllability, exact
controllability, moment problem, biorthogonal family

2010 MSC: 93B05, 35Q93, 35K05.

##  1 Introduction

In this paper we study controllability properties of the following Nonlinear
Heat Equation on the $d$-dimensional torus ${\mathbb{T}}^{d}$

$\begin{split}\begin{cases}\partial_{t}\psi(t,x)=\Big{(}\Delta-\kappa\psi(t,x)^{p}+\langle
u(t),Q(x)\rangle\Big{)}\psi(t,x),\ \ \ \ \ \ \ \ x\in{\mathbb{T}}^{d},\
t>0,\\\ \psi(t=0,\cdot)=\psi_{0}(\cdot),\end{cases}\end{split}$ (NHE)

where $d,p\in{\mathbb{N}}^{*}$, $\kappa\in{\mathbb{R}}$, the operator
$\Delta=\sum_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}}$ is the
Laplacian and
$Q=(Q_{1},...,Q_{q},\mu_{1},\mu_{2}):{\mathbb{T}}\to{\mathbb{R}}^{q+2}$ is a
smooth fixed function. The ${\mathbb{R}}^{q+2}$-valued function $u\in
L^{2}_{loc}({\mathbb{R}}^{+},{\mathbb{R}}^{q+2})$ plays the role of a control.
This means that $u$ is the function that can be chosen to steer the solution
of the problem towards a desired state. Observe that our control depends only
on time.

In numerous practical problems from chemistry, neurobiology and life science,
the evolution of a specific system is subjected to a control which is not
external but rather a modification of the principal parameter of the
evolution. In these cases, it can be appropriate to consider evolution
equations in the presence of multiplicative controls. Such controls are called
bilinear and take the form of $\langle u(t),Q(x)\rangle$ (as in the equation
(NHE)) when only the time-depending intensities can be adjusted, while the
spatial parts are fixed.

An example of a parabolic model with a dynamics governed by a multiplicative
control is the distributed parameter control model studied by Lenhart and Bhat
in [13]. In this work, the authors consider the wildlife damage management for
controlling population of diffusive small mammal species such as beavers,
raccoons and muskrats. These small mammal species often damage human interests
and it is important to study the possibility to control their dispersal
behaviour. The migratory habit of such animals presents obviously an
additional complication. For instance, their removal from any habitat may
cause the attraction of other individuals from nearby lands and a consequent
increase of the trapping cost. In [13], the authors study the dynamics of the
population density of one of this species given by a control model
incorporating dispersive dynamics and a multiplicative control which
represents the trapping. In such a work, the evolution is modelled by an
equation of the form

$\displaystyle\partial_{t}\varphi(t,x)=(\alpha\Delta+a-b\varphi(t,x)+p(t,x))\varphi(t,x),\
\ \ \ \ \ \ t>0,$ (1.1)

where $\varphi$ is the population density, $\alpha$ is constant, $a$ and $b$
are growth parameters and $p$ is the rate of trapping which is used as a
control. When one can control the time-dependent intensity of the trapping but
not its spatial distribution, it is possible to separate the variables of
function $p$ and write

$p(t,x)=u(t)\mu(x).$

This choice leads to a new formulation for the evolution model where the
actual control is the function $u$ and the equation, for $a=0$, takes the form
of (NHE).

### Global approximate controllability on the $d$-dimensional torus

Let us consider the following vector space:

$\mathcal{H}_{0}:={\rm span}_{{\mathbb{R}}}\\{Q\in\widehat{\mathcal{H}}\\},\ \
\ \ \text{with}\ \ \ \ \ \widehat{\mathcal{H}}:=\\{Q_{1},\dots,Q_{q}\\}.$

We also introduced the vectors

$\mathcal{K}=\Big{\\{}\big{(}1,0,\dots,0\big{)},\big{(}0,1,\dots,0\big{)},\dots,\big{(}0,\dots,1,0\big{)},\big{(}1,\dots,1\big{)}\Big{\\}}.$

######  Assumptions I.

The potential $Q=(Q_{1},...,Q_{q},\mu_{1},\mu_{2})$ is such that
$\widehat{\mathcal{H}}\subset C^{\infty}({\mathbb{T}}^{d},{\mathbb{R}})$ and

$\begin{split}&\big{\\{}1,\ \cos\langle k,x\rangle,\ \sin\langle
k,x\rangle\big{\\}}_{k\in\mathcal{K}}\subset\mathcal{H}_{0}.\end{split}$ (1.2)

We now present the first main result of the paper, that is, a small-time
global approximate controllability property of (NHE) between states sharing
the same sign.

In what follows, we denote by

$\psi(t;\psi_{0},u)$

the solution of (NHE) at time $t$, associated with initial condition
$\psi_{0}$ and control $u$. Of course, it is intended that we will look at
such solutions only when existence is ensured (see Proposition (2.1).

######  Main Theorem A.

Let $s\in{\mathbb{N}}^{*}$ be such that $s>d/2$. Assume Assumptions I be
verified. Then, (NHE) verifies the following small-time approximate
controllability properties.

  * •

Let $\psi_{0},\psi_{1}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ be such that
${\rm sign}(\psi_{0})={\rm sign}(\psi_{1})$. For any $\epsilon>0$ and $T>0$,
there exist $\tau\in[0,T]$ and $(u_{1},...u_{q})\in
L^{2}((0,\tau),{\mathbb{R}}^{q})$ such that

$\big{\|}\psi(\tau;\psi_{0},u)-\psi_{1}\big{\|}_{L^{2}}<\epsilon,$

with $u=(u_{1},...,u_{q},0,0)$.

  * •

Let $\psi_{0},\psi_{1}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ be such that
$\psi_{0},\psi_{1}>0$ (or $\psi_{0},\psi_{1}<0$). For any $\epsilon>0$ and
$T>0$, there exists $(u_{1},...u_{q})\in L^{2}((0,T),{\mathbb{R}}^{q})$ such
that

$\big{\|}\psi(T;\psi_{0},u)-\psi_{1}\big{\|}_{H^{s}}<\epsilon,$

with $u=(u_{1},...,u_{q},0,0)$.

Main Theorem A is a specific case of Theorem 3.3 where approximate
controllability is ensured with general control potentials $Q_{j}$ verifying
specific saturating assumptions.

To the best of our knowledge, the most recent works on approximate
controllability for nonlinear parabolic equations via multiplicative controls
are [18, 28]. In such works, the authors studied $1$-dimensional problems with
globally Lipschitz continuous nonlinearities. They proved approximate
controllability between states with the same number of sign changes when time
is sufficiently large. They considered space-time dependent controls
$\alpha(t,x)$ built according to the initial and the final state. The main
novelties of Main Theorem A are the following.

  * •

The approximate controllability is achieved in arbitrarily small times.

  * •

The approximate controllability holds on ${\mathbb{T}}^{d}$ for any
$d\in{\mathbb{N}}$, that is, in arbitrary spatial dimensions.

  * •

The spatial profile $Q(x)$ of the term $\langle u(t),Q(x)\rangle$ is fixed and
only $u$ plays the role of control.

  * •

Equation (NHE) exhibits a polynomial nonlinearity.

### Global exact controllability to the ground state solution for the
$1$-dimensional problem

The second main result of our work is the exact controllability of (NHE) to
the ground state solution in the $1$-dimensional flat torus ${\mathbb{T}}$.
Let us consider the ordered eigenvalues $\\{\lambda_{k}\\}_{k\in{\mathbb{N}}}$
of the Laplacian $-\Delta$ (not counted with their multiplicity)

$\lambda_{k}=k^{2},\qquad\forall\,k\in{\mathbb{N}}$ (1.3)

Note that, except for the first one $\lambda_{0}=0$, all the eigenvalues are
double. We denote by $\\{c_{0},c_{k},s_{k}\\}_{k\in{\mathbb{N}}}$ the
corresponding orthonormal eigenfunctions of $-\Delta$

$c_{0}=\frac{1}{\sqrt{2\pi}},\qquad
c_{k}(x)=\frac{1}{\sqrt{\pi}}\cos(kx),\qquad
s_{k}(x)=\frac{1}{\sqrt{\pi}}\sin(kx),\qquad\forall\,k\in{\mathbb{N}}^{*},$
(1.4)

which form a Hilbert basis of $L^{2}({\mathbb{T}})$. Notice that $c_{0}$
represents the free evolution ($u=0$) of the linear ($\kappa=0$) heat equation
(NHE) with initial condition $\psi_{0}=c_{0}$. Such a solution is usually
called the _ground state solution_. Henceforth, we will denote it by $\Phi$.
To study the exact controllability of (NHE), we introduce the following
additional assumption.

######  Assumptions II.

The potential $Q=(Q_{1},...,Q_{q},\mu_{1},\mu_{2})$ is such that $Q_{1}=1$ and
$\mu_{1},\mu_{2}\in H^{3}({\mathbb{T}},{\mathbb{R}})$ verify

$\begin{split}\langle\mu_{1},c_{0}\rangle_{L^{2}}\neq 0,\ \ \ \ \ &\ \ \
\langle\mu_{2},c_{0}\rangle_{L^{2}}=0,\\\
\exists\,b_{1},q_{1}>0\,:\>\lambda_{k}^{q_{1}}\left|\langle\mu_{1},c_{k}\rangle_{L^{2}}\right|\geq
b_{1},\,\,&\text{ and
}\,\,\langle\mu_{1},s_{k}\rangle_{L^{2}}=0,\quad\forall\,k\in{\mathbb{N}}^{*},\\\
\exists\,b_{2},q_{2}>0\,:\>\lambda_{k}^{q_{2}}\left|\langle\mu_{2},s_{k}\rangle_{L^{2}}\right|\geq
b_{2},\,\,&\text{ and
}\,\,\langle\mu_{2},c_{k}\rangle_{L^{2}}=0,\quad\forall\,k\in{\mathbb{N}}^{*}.\end{split}$
(1.5)

Observe that, for any admissible potential $Q$ in the sense of Assumption II,
$\Phi$ is also a trajectory of (NHE). Indeed, denoting by
$u_{\kappa}=\frac{\kappa}{(2\pi)^{p/2}}$ and setting
$\hat{u}=(u_{\kappa},0,\dots,0)$, one has that
$\Phi\equiv\psi(t;c_{0},\hat{u})$.

We now state our second main result which ensures global small-time exact
controllability of the 1-dimensional (NHE) to the ground state solution,
starting from a positive state.

######  Main Theorem B.

Let $d=1$, $\kappa>0$ and $p\in 2{\mathbb{N}}^{*}$. Assume Assumptions I and
Assumptions II be verified. Then, (NHE) is exactly controllable in
$H^{3}({\mathbb{T}})$ to the ground state solution $\Phi$ in any positive time
from any positive state. In details, for any $T>0$ and $\psi_{0}\in\\{\psi\in
H^{3}({\mathbb{T}},{\mathbb{R}}):\ sgn(\psi)>0\\},$ there exists $u\in
L^{2}((0,T),{\mathbb{R}}^{q+2})$ such that

$\psi(T;\psi_{0},u)=\Phi.$

Analogously, for any $T>0$ and $\psi_{0}\in\\{\psi\in
H^{3}({\mathbb{T}},{\mathbb{R}}):\ sgn(\psi)<0\\},$ there exists $u\in
L^{2}((0,T),{\mathbb{R}}^{q+2})$ such that

$\psi(T;\psi_{0},u)=-\Phi.$

Main Theorem B yields small-time exact controllability to the ground state
solution in $H^{3}$ when $\kappa>0$ and $p\in 2{\mathbb{N}}^{*}$. Such a
property is obtained by using the global approximate controllability result
between positive states (Main Theorem A) together with a local exact
controllability result to the ground state solution $\Phi$ in any positive
time (see Theorem 4.3). The specific choice of the parameters $\kappa$ and $p$
ensures that equation (NHE) is globally well-posed in $H^{3}$ which is crucial
for the methodology of the proof of the local exact controllability.

Note that the result of Main Theorem B can be achieved by means of $5$
controls. Indeed, we consider a potential of the form

$Q=(1,\cos(x),\sin(x),\mu_{1}(x),\mu_{2}(x))$

satisfying Assumptions I and II. An example of suitable functions $\mu_{1}$
and $\mu_{2}$ are

$\mu_{1}=x^{3}(2\pi-x)^{3},\ \ \ \ \ \ \ \ \ \ \ \ \ \
\mu_{2}=x^{3}(x-\pi)^{3}(x-2\pi)^{3}$

(see Example 4.2 for further details). Hence, it is sufficient to consider a
potential of the form

$Q=(1,\ \cos(x),\ \sin(x),\ x^{3}(2\pi-x)^{3},\
x^{3}(x-\pi)^{3}(x-2\pi)^{3}),$

to apply Main Theorem B and deduce the controllability of the nonlinear
equation (NHE) to the ground state solution $\Phi$ in any positive time $T>0$.

An interesting aspect of Main Theorem B is the validity of an exact
controllability result on the torus where the Laplacian exhibits double
eigenvalues. Indeed, the first step for proving local exact controllability of
(NHE) is based on the solvability of a suitable moment problem. The method for
showing local controllability for bilinear control problem of parabolic type,
introduced in [5, 6], is not directly applicable in our framework. Indeed, we
face two additional difficulties

  * •

The presence of doubles eigenvalues is, a priori, an obstacle to the
solvability of the moment problem at hand. We address the problem by
“filtering” the spectrum of the Laplacian via the two potentials $\mu_{1}$ and
$\mu_{2}$. We exploit hypotheses (1.5) so that $\mu_{1}$ acts only on the
frequencies associated to the eigenfunctions $\\{c_{k}\\}_{k\in{\mathbb{N}}}$
and $\mu_{2}$ only on those associated to
$\\{s_{k}\\}_{k\in{\mathbb{N}}^{*}}$. Hence, we decompose the moment problem
into two subproblems, each characterized by simple eigenvalues, making both
solvable.

  * •

In order to adapt the technique proposed in [5, 6] the solutions of equation
(NHE) need to be globally defined and unique. These properties may be trivial
in the linear case, whereas a more careful analysis should be developed for a
nonlinear equation like (NHE). We propose a method which require more
regularity on the initial condition and on the control.

### Some references

The classical controllability problem of parabolic type equations as (NHE),
with bilinear control, is a delicate matter even in the linear case
($\kappa=0$). The main reason is a structural obstacle, described in [9],
which forbid from proving controllability results in
$L^{2}({\mathbb{T}},{\mathbb{R}})$. In details, the authors of [9] showed that
the reachable set of the (NHE), with $\kappa=0$, starting from any
$\psi_{0}\in L^{2}({\mathbb{T}}^{d},{\mathbb{R}})$, is contained in a
countable union of compact subsets of $L^{2}({\mathbb{T}}^{d},{\mathbb{R}})$
and then it has dense complement. Therefore, this property prevents obtaining
any classical exact controllability result in $L^{2}$. Hence, we shall explore
different notions of controllability such as the approximate controllability
or the exact controllability to trajectories.

Approximate controllability results via multiplicative controls have been
obtained in [30] for $1$-D linear parabolic problems in sufficiently large
time $T>0$ and between non-negative states. In [30] the control term is not
scalar input type, but depends on space and time (see also [31]). Similar
results were proved in [16, 17] for linear degenerate parabolic problems.
Finally, approximate controllability for nonlinear problem was established in
[18, 28].

The techniques leading to approximate controllability of Main Theorem A are
inspired by a saturating geometric control approach introduced by Agrachev and
Sarychev in [1, 2] in the case of additive controls. In such works the authors
proved global approximate controllability of the 2D Navier–Stokes and Euler
systems. Their approach has been extended to study different equations with
additive controls (see, for instance, [3, 33, 36, 37]). The saturating control
methodology has been implemented for bilinear control problems in the recent
work [25]. In this paper the authors studied the non-linear Schrödinger
equation, proving small-time approximate controllability between eigenmodes.
Main Theorem A is inspired by the approach developed in [25]. Nevertheless,
the saturating techniques for parabolic equation as (NHE) leads to different
controllability properties like small-time approximate controllability between
states with the same sign. For additional results on small-time
controllability of PDEs through bilinear control, achieved using similar
methods, see [22, 24, 34, 23].

Another type of controllability not ruled out by the negative findings of [9]
is the exact controllability to trajectories. This property was firstly
studied by Alabau-Boussouira, Cannarsa and Urbani in [5, 6] in an abstract
setting for parabolic PDEs by means of scalar input bilinear controls. An
example of application of such results is the local and semi-global
controllability of a heat equation as (NHE) (for $\kappa=0$) on the interval
$(0,1)$. The methodology has been later extended in [15] to network-type
domains to cope with condensation phenomena of the eigenvalues of the
diffusion operator. The approach of [5, 6] used for studying local exact
controllability to eigensolutions relies on the solvability of a suitable
moment problem related to the null controllability of a linearized version of
the problem at hand. The resolution of moment problems has been intensively
studied in literature over a long period, starting from the classical works
[26, 27], to the more recent ones [7, 8, 11, 12, 15, 19, 20, 29, 10].

### Scheme of the work

The paper is organized as follows. In Section 2 we present some preliminary
results, mamely, local and global well-posedness of the non-linear heat
equation (NHE) and a crucial limit adopted in the proof of Main Theorem A. In
Section 3 we prove the small-time global approximate controllability and Main
Theorem A. Section 4 is devoted to the exact controllability to the ground
state solution and contains the proof of Main Theorem B. In the appendices A
and B, we present the proofs of the local and global well-posedness results
for equation (NHE), respectively. Appendix C contains a technical estimate
that we use in the proof of Main Theorem B.

### Acknowledgments

The authors would like to thank the colleagues Vahagn Nersesyan and Takéo
Takahashi for the fruitful discussions on the saturation control method
adopted in this work. The third author acknowledges support from INdAM
National Group for Mathematical Analysis, Probability and their Applications
(GNAMPA) and from the MIUR Excellence Department Project awarded to the
Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006
(postdoc position 2020-2022). This project has received financial support from
the CNRS through the MITI interdisciplinary programs.

##  2 Preliminaries

The aim of this section is to present some preliminary results. We start by
ensuring existence and uniqueness of solutions of equation (NHE). In the
second part of the section we prove a limit of conjugated dynamics, which is a
key point in the proof of Main Theorem A.

###  2.1 Local and global well-posedness

We start by stating the following local well-posedness result for the Cauchy
problem (NHE).

######  Proposition 2.1.

Let $s>d/2$ and $Q\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}}^{q+2})$. For any
$\psi_{0}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ and $u\in
L^{2}_{loc}(\mathbb{R}^{+},\mathbb{R}^{q+2})$, there exists a maximal time
$\mathcal{T}=\mathcal{T}(\psi_{0},u)>0$ and a unique mild solution $\psi$ of
(NHE). Namely, for any $T<\mathcal{T}$, $\psi\in
C^{0}([0,T],H^{s}({\mathbb{T}}^{d},{\mathbb{R}}))$ and is represented by the
formula

$\psi(t;\psi_{0},u)=e^{t\Delta}\psi_{0}+\int_{0}^{t}e^{(t-s)\Delta}\left(\langle
u(s),Q(x)\rangle\psi(s,x)-\kappa\psi(s,x)^{p+1}\right)ds.$

Moreover, if ${\cal T}<+\infty$, then
$\left\lVert\psi(t)\right\rVert_{H^{s}}\to+\infty$ as $t\to{\cal T}^{-}$. In
addition, we have the following properties.

  * i.

Assume that $\psi_{0},\phi_{0}\in
B_{H^{s}({\mathbb{T}}^{d},{\mathbb{R}})}(0,R)$ for some $R>0$ and $u,v\in
L^{2}_{loc}({\mathbb{R}}^{+},{\mathbb{R}}^{q+2})$. Then, for any $0\leq
T\leq\min\\{{\cal T}(\psi_{0},u),{\cal T}(\phi_{0},v)\\}$, there exists
$C=C(u,v)$ such that

$\sup_{0\leq t\leq
T}\left\lVert\psi(t;\psi_{0},u)-\psi(t;\phi_{0},v)\right\rVert_{H^{s}}\leq
C\left(\left\lVert\psi_{0}-\phi_{0}\right\rVert_{H^{s}}+\left\lVert
u-v\right\rVert_{L^{2}}\right).$ (2.1)

  * ii.

Set $K=\|\psi\|_{C([0,T],H^{s})}+\|\psi_{0}\|_{H^{s}}+\|u\|_{L^{2}}$. There
exists $\delta=\delta({\cal T}(\psi_{0},u),K)>0$ such that, for any
$\hat{\psi}_{0}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ and $\hat{u}\in
L^{2}((0,T),{\mathbb{R}}^{q+2})$ satisfying

$\|\hat{\psi}_{0}-\psi_{0}\|_{H^{s}}+\|\hat{u}-u\|_{L^{2}}<\delta,$ (2.2)

problem (NHE) admits a unique mild solution $\hat{\psi}\in
C\big{(}[0,T],H^{s}({\mathbb{T}}^{d},{\mathbb{R}})\big{)}$ with initial
condition $\hat{\psi}_{0}$ and control $\hat{u}$.

The proof of Proposition 2.1 follows from a fix point argument and from
Sobolev embeddings for $s>d/2$. It can be found in Appendix A.

We further present a global well-posedness result for equation (NHE) in the
case $d=1$, $\kappa\geq 0$ and $p\in 2{\mathbb{N}}^{*}$.

######  Proposition 2.2.

Let $d=1$, $p\in 2{\mathbb{N}}^{*}$, $\psi_{0}\in
H^{3}({\mathbb{T}},{\mathbb{R}})$, $Q\in
H^{3}({\mathbb{T}},{\mathbb{R}}^{q+2})$, $u\in
H^{1}_{loc}((0,+\infty),{\mathbb{R}}^{q+2})$ and $\kappa\geq 0$. Then, for any
$T>0$ there exists a unique mild solution $\psi\in
C^{0}([0,T],H^{3}({\mathbb{T}},{\mathbb{R}}))$ of (NHE).

The proof of Proposition 2.2 follows from some energy estimates and can be
found in Appendix B.

###  2.2 Small-time limit of conjugated dynamics

Let us introduce the nonlinear operator

${\mathbb{B}}(\varphi)(x)=\sum_{j=1}^{d}\left(\partial_{x_{j}}\varphi(x)\right)^{2},\qquad\forall\,\varphi\in
C^{1}({\mathbb{T}}^{d},{\mathbb{R}}).$ (2.3)

Then, the following result holds true.

######  Proposition 2.3.

Let $s\in{\mathbb{N}}^{*}$, $s>d/2$ and $(Q_{1},...,Q_{q})\in
H^{2s+1}({\mathbb{T}}^{d},{\mathbb{R}}^{q})$. Assume $\psi_{0}\in
H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$, $(u_{1},...,u_{q})\in{\mathbb{R}}^{q}$,
and $\varphi\in H^{2s+1}({\mathbb{T}}^{d},{\mathbb{R}})$ be non-negative.
Then, there exists a constant $\delta_{0}>0$ such that, for any
$\delta\in(0,\delta_{0})$, the solution
$\psi(t;e^{-\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)$ of (NHE) with
$u=(u_{1},...,u_{q},0,0)$ is well-defined in $[0,\delta]$. Furthermore, the
following limit holds

$e^{\delta^{-1/2}\varphi}\psi(\delta;e^{-\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)\to
e^{{\mathbb{B}}(\varphi)+\langle u,Q\rangle}\psi_{0}\quad\text{in $H^{s}$, as
$\delta\to 0^{+}$}.$

###### Proof.

Preliminaries. For any $\delta>0$ we set
$\phi(t):=e^{\delta^{-1/2}\varphi}\psi(t;e^{-\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)$.
According to Proposition 2.1, $\phi(t)$ exists up to some maximal time ${\cal
T}^{\delta}={\cal T}(e^{-\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)$ and it
has the same regularity of $\psi$. Furthermore, if ${\cal T}^{\delta}<\infty$

$\|e^{-\delta^{-1/2}\varphi}\phi(t)\|_{H^{s}}\to+\infty\quad\text{as~{}$t\to{\cal
T}^{\delta-}$.}$

We introduce the following functions

$\displaystyle w(t)=e^{\left({\mathbb{B}}(\varphi)+\langle
u,Q\rangle\right)t}\psi_{0}^{\delta},\qquad v(t)=\phi(\delta t)-w(t),$ (2.4)

where $\psi_{0}^{\delta}=e^{\delta^{1/4}\Delta}\psi_{0}\in
H^{r}({\mathbb{T}}^{d},{\mathbb{R}})$, with $r=s+2$, is such that

$\displaystyle\|\psi_{0}-\psi_{0}^{\delta}\|_{H^{s}}\to 0,\qquad\text{ as
$\delta\to 0^{+}$,}$ (2.5)

and there exists $C>0$ independent of $\delta>0$ such that

$\displaystyle\|\psi_{0}^{\delta}\|_{H^{s}}\leq
C,\qquad\|\psi_{0}^{\delta}\|_{H^{r}}\leq C\delta^{-1/4},\qquad\text{for
}\delta\leq 1.$ (2.6)

Our aim is to show that $\phi(\delta)\xrightarrow{\delta\rightarrow
0^{+}}e^{\left({\mathbb{B}}(\varphi)+\langle u,Q\rangle\right)}\psi_{0}$ in
$H^{s}$. Thanks to (2.5) it is sufficient to prove that

$\displaystyle\|v(t=1)\|_{H^{s}}\xrightarrow{\delta\rightarrow 0^{+}}0.$ (2.7)

However, before proving (2.7), we need to ensure the existence of
$\delta_{0}>0$ small enough such that, for every $0<\delta<\delta_{0}$, $v(t)$
is well-defined in $[0,1]$ and moreover that

$\displaystyle\delta^{-1}{\cal T}^{\delta}\geq 1.$ (2.8)

An intermediate inequality. In view of (2.6), there exists $C>0$ such that,
for every $t\in[0,2]$,

$\displaystyle\|w(t)\|_{H^{s}}\leq C,\quad\,\,\quad\quad\|w(t)\|_{H^{r}}\leq
C{\delta^{-1/4}}.$ (2.9)

We observe that $v$ is a solution of the following equation

$\begin{split}\partial_{t}v&=\delta\Delta(v+w)-\delta\kappa(e^{-\delta^{-\frac{1}{2}}\varphi}(v+w))^{p}(v+w)-\delta^{1/2}(v+w)\Delta\varphi\\\
&\quad-2\delta^{\frac{1}{2}}\nabla(v+w)\cdot\nabla\varphi+{\mathbb{B}}(\varphi)v+\langle
u,Q\rangle v,\end{split}$ (2.10)

with initial condition

$v(0)=\psi_{0}-\psi_{0}^{\delta}.$ (2.11)

Let us start by assuming that $\psi_{0}\in
H^{2s+2}({\mathbb{T}}^{d},{\mathbb{R}})$ which implies $\psi(t)\in
H^{2s+2}({\mathbb{T}}^{d},{\mathbb{R}})$ and then $v(t)\in
H^{2s+2}({\mathbb{T}}^{d},{\mathbb{R}})$ for every $t\in(0,{\cal
T}^{\delta})$. Let $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in{\mathbb{N}}^{d}$
be such that $|\alpha|=|\alpha_{1}|+\ldots+|\alpha_{d}|\leq s$. Thanks to the
accreativity of the operator $-\Delta$, we get

$\displaystyle\partial_{t}\|\partial^{\alpha}v\|_{L^{2}}^{2}$
$\displaystyle=-2\delta\langle(-\Delta\partial^{\alpha}v),\partial^{\alpha}v\rangle_{L^{2}}+2\langle\delta\Delta
w+\delta\kappa(e^{-\delta^{-\frac{1}{2}}\varphi}(v+w))^{p}(v+w)$
$\displaystyle\quad-\delta^{\frac{1}{2}}(v+w)\Delta\varphi+2\delta^{\frac{1}{2}}\nabla(v+w)\cdot\nabla\varphi+{\mathbb{B}}(\varphi)v+\langle
u,Q\rangle v,\partial^{2\alpha}v\rangle_{L^{2}}$ $\displaystyle\leq
2\langle\delta\Delta w+\delta\kappa
e^{-p\delta^{-\frac{1}{2}}\varphi}(v+w)^{p+1}-\delta^{\frac{1}{2}}(v+w)\Delta\varphi$
$\displaystyle\quad+2\delta^{\frac{1}{2}}\nabla(v+w)\cdot\nabla\varphi+{\mathbb{B}}(\varphi)v+\langle
u,Q\rangle v,\partial^{2\alpha}v\rangle_{L^{2}}$ $\displaystyle\leq
2\Big{(}\delta|\langle\partial^{\alpha}\Delta
w,\partial^{\alpha}v\rangle_{L^{2}}|+\delta|\kappa|\big{|}\big{\langle}\partial^{\alpha}\big{(}e^{-p\delta^{-\frac{1}{2}}\varphi}(v+w)^{p+1}\big{)},\partial^{\alpha}v\big{\rangle}_{L^{2}}\big{|}$
$\displaystyle\quad+\delta^{\frac{1}{2}}|\langle\partial^{\alpha}\big{(}(v+w)\Delta\varphi\big{)},\partial^{\alpha}v\rangle_{L^{2}}|+\delta^{1/2}|\langle\partial^{\alpha}\big{(}\nabla(v+w)\cdot\nabla\varphi\big{)},\partial^{\alpha}v\rangle_{L^{2}}|$
(2.12)
$\displaystyle\quad+|\langle\partial^{\alpha}\big{(}{\mathbb{B}}(\varphi)v\big{)},\partial^{\alpha}v\rangle_{L^{2}}|+|\big{\langle}\partial^{\alpha}\big{(}\langle
u,Q\rangle v\big{)},\partial^{2\alpha}v\big{\rangle}_{L^{2}}|\Big{)}.$

We observe that since all the functions involved in the above estimate are in
$H^{2s+2}({\mathbb{T}}^{d},{\mathbb{R}})$, no boundary terms appear when
integrating by parts.

Notice that there exist constants $C>0$, independent of $\delta$, such that
$|\langle\nabla v\cdot\nabla\varphi,\partial^{2\alpha}v\rangle_{L^{2}}|\leq
C\|\varphi\|_{H^{2s+1}}\|v\|_{H^{s}}^{2}$ and

$\displaystyle\big{|}\big{\langle}\partial^{\alpha}\big{(}e^{-p\delta^{-\frac{1}{2}}\varphi}(v+w)^{p+1}\big{)},\partial^{\alpha}v\big{\rangle}_{L^{2}}\big{|}$
$\displaystyle\leq
C\|e^{-p\delta^{-\frac{1}{2}}\varphi}\|_{L^{\infty}}\|\varphi\|_{H^{s+1}}^{s}\Big{(}p^{s}\delta^{-\frac{s}{2}}+1\Big{)}\|v+w\|_{H^{s}}^{p+1}\|v\|_{H^{s}}.$

Hence, we deduce the existence of a constant $C>0$, independent of $\delta$,
such that

$\displaystyle\partial_{t}\|\partial^{\alpha}v\|_{L^{2}}^{2}$
$\displaystyle\leq
C\delta\|w\|_{H^{r}}\|v\|_{H^{s}}+C\delta\|e^{-p\delta^{-\frac{1}{2}}\varphi}\|_{L^{\infty}}(\delta^{-\frac{s}{2}}+1)\|v+w\|_{H^{s}}^{p+1}\|v\|_{H^{s}}+C\delta^{\frac{1}{2}}\|v+w\|_{H^{s}}\|v\|_{H^{s}}$
$\displaystyle\quad+C\delta^{1/2}\|v\|_{H^{s}}^{2}+C\delta^{1/2}\|w\|_{H^{r}}\|v\|_{H^{s}}+C\|v\|_{H^{s}}^{2}+C\|v\|_{H^{s}}^{2}.$

Now, we have that
$\|e^{-\delta^{-\frac{1}{2}}\varphi}\|_{L^{\infty}}(\delta^{-\frac{s}{2}}+1)\xrightarrow[]{\delta\rightarrow
0}0$ thanks to the positivity of $\varphi$. Using (2.9) and the Young’s
inequality we deduce that there exists $C>0$, independent of $\delta$, such
that

$\displaystyle\partial_{t}\|\partial^{\alpha}v\|_{L^{2}}^{2}$
$\displaystyle\leq
C\delta^{3/4}\|v\|_{H^{s}}+C\delta\|v\|_{H^{s}}^{p+2}+C\delta\|v\|_{H^{s}}+C\delta^{\frac{1}{2}}\|v\|_{H^{s}}^{2}+C\delta^{\frac{1}{2}}\|v\|_{H^{s}}$
$\displaystyle\quad+C\delta^{1/2}\|v\|_{H^{s}}^{2}+C\delta^{1/4}\|v\|_{H^{s}}+C\|v\|_{H^{s}}^{2}+C\|v\|_{H^{s}}^{2}$
$\displaystyle\leq
C\delta^{1/2}+C(1+\delta^{1/2})\|v\|_{H^{s}}^{2}+C\delta\|v\|_{H^{s}}^{p+2}.$

The last relation holds for $t\leq\delta^{-1}{\cal T}^{\delta}$. We recall
that $\|\cdot\|_{H^{s}}^{2}=\sum_{\underset{|\alpha|\leq
s}{\alpha\in{\mathbb{N}}^{d}}}\|\partial^{\alpha}\cdot\|_{L^{2}}^{2}$ and
therefore there exists $C>0$ such that

$\displaystyle\partial_{t}\|v\|_{H^{s}}^{2}$ $\displaystyle\leq
C\delta^{1/2}+C(1+\delta^{1/2})\|v\|_{H^{s}}^{2}+C\delta\|v\|_{H^{s}}^{p+2}.$

By the Grönwall Lemma and recalling (2.11) we obtain that

$\|v(t)\|_{H^{s}}^{2}\leq
e^{C(1+\delta^{1/2})t}\left(C\delta^{1/2}t+\|\psi_{0}-\psi_{0}^{\delta}\|_{H^{s}}^{2}+C\delta\int_{0}^{t}\|v(y)\|_{H^{s}}^{p+2}{\textup{d}}y\right)$
(2.13)

for $t\leq\delta^{-1}{\cal T}^{\delta}$ and for every $\psi_{0}\in
H^{2s+2}({\mathbb{T}}^{d},{\mathbb{R}})$. Finally, we can extend the validity
of (2.13) for every $\psi_{0}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ thanks
to item _ii._ of Proposition 2.1 and to the density of
$H^{2s+2}({\mathbb{T}}^{d},{\mathbb{R}})$ into
$H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ with respect to the $H^{s}$-norm.

Properties of the maximal time. It remains to prove (2.8). We consider
$\delta_{0}>0$ sufficiently small so that, for $0<\delta<\delta_{0}$, we have
$\|\psi_{0}-\psi_{0}^{\delta}\|_{H^{s}}^{2}<1/2$ and then

$\|v(0)\|_{H^{s}}^{2}<1/2.$

Denote $\tau^{\delta}:=\sup\left\\{t<\delta^{-1}{\cal
T}^{\delta}:\|v(t)\|_{H^{s}}<1\right\\}$. The above inequality yields
$\tau^{\delta}>0$. If $\tau^{\delta}=+\infty$ then (2.8) is obviously
verified. Thus, let us assume that

$\tau^{\delta}<+\infty.$

To prove (2.8) we show that for $\delta_{0}>0$ sufficiently small and
$0<\delta<\delta_{0}$ we have $\tau^{\delta}\geq 1$. Assume by contradiction
that, for every $\delta_{0}>0$ small, there exists $0<\delta<\delta_{0}$ such
that $\tau^{\delta}<1$. Thanks to (2.13), we get

$\displaystyle
1=\|v(\tau^{\delta})\|_{H^{s}}^{2}<e^{C(1+\delta^{1/2})\tau^{\delta}}\left(C\delta^{1/2}\tau^{\delta}+\|\psi_{0}-\psi_{0}^{\delta}\|_{H^{s}}^{2}+C\delta\int_{0}^{\tau^{\delta}}\|v(y)\|_{H^{s}}^{p+2}{\textup{d}}y\right).$
(2.14)

We recall that, by definition, $\|v(t)\|_{H^{s}}<1$ in $[0,\tau^{\delta})$.
For $\delta_{0}$ sufficiently small we have

$\displaystyle
e^{C(1+\delta^{1/2})\tau^{\delta}}\left(C\delta^{1/2}\tau^{\delta}+\|\psi_{0}-\psi_{0}^{\delta}\|_{H^{s}}^{2}\right)<\frac{1}{2},$
(2.15)

since $0<\delta<\delta_{0}$. Moreover,

$\displaystyle
e^{C(1+\delta^{1/2})\tau^{\delta}}\left(C\delta^{1/2}\tau^{\delta}+\|\psi_{0}-\psi_{0}^{\delta}\|_{H^{s}}^{2}+C\delta\int_{0}^{\tau^{\delta}}\|v(y)\|_{H^{s}}^{p+2}{\textup{d}}y\right)<1,$

which contradicts (2.14). Hence, we conclude that there exists $\delta_{0}>0$
sufficiently small such that $\tau^{\delta}>1$ for every
$0<\delta<\delta_{0}$. Thus, (2.8) holds true.

Conclusion. Finally, $1\in[0,\tau^{\delta})\subset[0,\delta^{-1}{\cal
T}^{\delta})$ and thanks to (2.13) we have established the validity of (2.7)
since

$\|v(1)\|_{H^{s}}^{2}\leq
e^{C(1+\delta^{1/2})}\left(C\delta^{1/2}+\|\psi_{0}-\psi_{0}^{\delta}\|_{H^{s}}^{2}+C\delta\right)\to
0\quad\text{as $\delta\to 0^{+}$}.$

∎

A first simple consequence of Proposition 2.3, concerning the control of
(NHE), is the following small-time global approximate null-controllability.

######  Corollary 2.4.

Let $s\in{\mathbb{N}}^{*}$ be such that $s>d/2$ and $1\in{\rm
span}\\{Q_{1},...,Q_{q}\\}$. Assume $\psi_{0}\in
H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$. For any $\epsilon,T>0$, there exists
$(u_{1},...,u_{q})\in L^{2}((0,T),{\mathbb{R}}^{q})$ such that the solution
$\psi(t;\psi_{0},u)$ of the (NHE) in $[0,T]$ with control
$u=(u_{1},...,u_{q},0,0)$ is well-defined and

$\|\psi(T;\psi_{0},u)\|_{H^{s}}<\epsilon.$

###### Proof.

For any $\tilde{\epsilon}>0$, consider $c>0$ sufficiently large so that

$|e^{-c}|<\frac{\tilde{\epsilon}}{2\|\psi_{0}\|_{H^{s}}}\ \ \ \
\Longrightarrow\ \ \ \ \
\|e^{-c}\psi_{0}\|_{H^{s}}<\frac{\tilde{\epsilon}}{2}.$

Let $-c=\sum_{j=1}^{q}u_{j}Q_{j}$ for some
$(u_{1},\dots,u_{q})\in{\mathbb{R}}^{q}$. Thanks to Proposition 2.3, there
exists $\delta>0$ such that the constant control
$u=(u_{1},\dots,u_{q})/\delta:[0,\delta]\to{\mathbb{R}}^{q}$ is such that the
solution of (NHE) is well-defined in $[0,\delta]$ and

$\|\psi(\delta;\psi_{0},u/\delta)-e^{-c}\psi_{0}\|_{H^{s}}<\tilde{\epsilon}/2.$

The triangular inequality ensures the result since, when we consider the
control $v=\frac{u}{\delta}$, we have

$\|\psi(\delta;\psi_{0},v)\|_{H^{s}}\leq\|\psi(\delta;\psi_{0},v)-e^{-c}\psi_{0}\|_{H^{s}}+\|e^{-c}\psi_{0}\|_{H^{s}}<\tilde{\epsilon}.$

The dynamics can be obtained exactly in a time $T$, since $0$ is stationary
solution of (NHE) with control $u=0$. Indeed, thanks to the second point of
Proposition 2.1, when $\tilde{\epsilon}$ is sufficiently small the two
solutions $\psi(T-\delta;\psi(\delta;\psi_{0},v),0),\psi(T-\delta;0,0)\equiv
0,$ are defined in the same time interval $[0,T]$. Now, the first point of
Proposition 2.1 yields the existence of $C>0$, independent of
$\psi(\delta;\psi_{0},v)$, such that

$\|\psi(T-\delta;\psi(\delta;\psi_{0},v),0)-\psi(T-\delta;0,0)\|_{H^{s}}\leq
C\|\psi(\delta;\psi_{0},v)\|_{H^{s}}<\tilde{\epsilon}C.$

The proof is therefore concluded since, for every $\epsilon>0$, it is
sufficient to chose $\tilde{\epsilon}=\frac{\epsilon}{C}.$ ∎

##  3 Small-time approximate controllability

The aim of this section is to prove the small-time approximate controllability
results stated in Main Theorem A.

###  3.1 An intermediate controllability result

Let us recall the definition of $\widehat{\mathcal{H}}$ and $\mathcal{H}_{0}$
provided in Assumption I

$\begin{split}\mathcal{H}_{0}={\rm
span}_{{\mathbb{R}}}\\{Q\in\widehat{\mathcal{H}}\\},\qquad\widehat{\mathcal{H}}=\\{Q_{1},\dots,Q_{q}\\}.\end{split}$

We define $\mathcal{H}_{j}$, for every $j\in{\mathbb{N}}^{*}$, iteratively as
the largest vector space containing elements $\psi$ of the form

$\begin{split}\psi=\varphi_{0}+\sum_{k=1}^{p}{\mathbb{B}}(\varphi_{k}),\qquad\varphi_{0},\dots,\varphi_{p}\in\mathcal{H}_{j-1},\quad
p\in{\mathbb{N}},\end{split}$ (3.1)

where the non-linear operator ${\mathbb{B}}$ is defined in (2.3). We also
denote

$\mathcal{H}_{\infty}=\bigcup_{j=0}^{\infty}\mathcal{H}_{j}.$

Let us recall the following result from the work [25].

######  Proposition 3.1.

[25, Proposition 2.6] If Assumptions I are verified, then
$\mathcal{H}_{\infty}$ is dense in $H^{s}({\mathbb{T}}^{d},{\mathbb{R}}),s\geq
0$.

We start by ensuring the following property of small-time approximate
controllability for (NHE).

######  Proposition 3.2.

Let $s\in{\mathbb{N}}^{*}$ be such that $s>d/2$ and $(Q_{1},...,Q_{q})\in
C^{\infty}({\mathbb{T}}^{d},{\mathbb{R}}^{q})$ be such that
$1\in\mathcal{H}_{0}$. Assume that $\mathcal{H}_{\infty}$ is dense in
$H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$. Let $\psi_{0}\in
H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ and $\varphi\in
H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$. For any $\epsilon,T>0$, there exist
$\tau\in[0,T)$ and $(u_{1},...,u_{q})\in L^{2}((0,\tau),{\mathbb{R}}^{q})$
such that the solution $\psi(t;\psi_{0},u)$ of (NHE) with control
$u=(u_{1},...,u_{q},0,0)$ is well-defined in $[0,\tau]$ and

$\|\psi(\tau;\psi_{0},u)-e^{\varphi}\psi_{0}\|_{H^{s}}<\epsilon.$

###### Proof.

Let us recall that the concatenation $v*u$ of two scalar control laws
$u:[0,T_{1}]\to\mathbb{R}^{q+2},\ v:[0,T_{2}]\to\mathbb{R}^{q+2}$ is the
control law defined on $[0,T_{1}+T_{2}]$ as follows

$(v*u)(t)=\begin{cases}u(t),&t\in[0,T_{1}]\\\
v(t-T_{1}),&t\in(T_{1},T_{1}+T_{2}].\end{cases}$

Such definition extends to controls with values in $\mathbb{R}^{q}$,
componentwise. We will often use the fact that

$\psi(T_{1}+t;\psi_{0},v*u)=\psi(t;\psi(T_{1},\psi_{0},u),v),\quad t>0.$

Let us start by assuming that the following property holds for any
$n\in\mathbb{N}$:

  * ($P_{n}$)

for any $\psi_{0}\in H^{s}(\mathbb{T}^{d},{\mathbb{R}})$,
$\phi\in\mathcal{H}_{n}$, and any $\varepsilon,T>0$, there exist
$\tau\in[0,T)$ and $(u_{1},...,u_{q}):[0,\tau]\to\mathbb{R}^{q}$ piecewise
constant such that the solution of the (NHE) associated with the initial
condition $\psi_{0}$ and control $u=(u_{1},...,u_{q},0,0)$ satisfies

$\left\|\psi(\tau;\psi_{0},u)-e^{\phi}\psi_{0}\right\|_{H^{s}(\mathbb{T}^{d})}<\varepsilon.$

Property $(P_{n})$ combined with the density feature given in Proposition 3.1
implies at once the statement.

We are thus left to prove $(P_{n})$. An analogous property appeared in [25,
Theorem 2.2] in the study of nonlinear Schrödinger equations with bilinear
control. We provide the proof for completeness. Let us proceed by induction on
the index $n$.

Inductive basis: $n=0$
If $\phi\in\mathcal{H}_{0}$, there exists $(u_{1},...,u_{q})\in\mathbb{R}^{q}$
such that $\phi(x)=\langle u,Q(x)\rangle$ with $u=(u_{1},...,u_{q},0,0)$.
Consider then the solution of (NHE) associated with the constant control
$u^{\tau}:=u/\tau\in\mathbb{R}^{q}$ and with the initial condition $\psi_{0}$.
Applying Proposition 2.3 with $\varphi=0$, we find $\tau\in[0,T)$ such that

$\left\|\psi(\tau;\psi_{0},u^{\tau})-e^{\phi}\psi_{0}\right\|_{H^{s}(\mathbb{T}^{d})}<\varepsilon,$

which proves the desired property.

Inductive step: $n\Rightarrow n+1$
By assuming that $(P_{n})$ holds, we prove $(P_{n+1})$. If
$\phi\in\mathcal{H}_{n+1}$, there exist $N\in\mathbb{N}$ and
$\phi_{0},\dots,\phi_{N}\in\mathcal{H}_{n}$ such that

$\phi=\phi_{0}+\sum_{j=1}^{N}{\mathbb{B}}(\phi_{j}).$

Consider $\phi_{1}$ and $c>0$ such that $\tilde{\phi}_{1}=\phi_{1}+c\geq 0$
and note that ${\mathbb{B}}(\tilde{\phi}_{1})={\mathbb{B}}(\phi_{1})$. Thanks
to Proposition 2.3, we can fix $\gamma\in[0,T/3)$ small enough such that

$\left\|e^{\gamma^{-1/2}\tilde{\phi}_{1}}\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0)-e^{{\mathbb{B}}(\phi_{1})}\psi_{0}\right\|_{H^{s}}<\varepsilon/2.$

Since $c\in\mathcal{H}_{0}$, $\phi_{1}\in\mathcal{H}_{n}$, then
$\tilde{\phi}_{1}\in\mathcal{H}_{n}$. Thanks to the inductive hypothesis, for
any $\sigma,T,\gamma>0$ there exist $\delta\in[0,T/3)$ and a piecewise
constant control
$u^{\delta,\gamma}=(u^{\delta,\gamma}_{1},...,u^{\delta,\gamma}_{q},0,0):[0,\delta]\to\mathbb{R}^{q+2}$
such that

$\left\|\psi(\delta;\psi_{0},u^{\delta,\gamma})-e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0}\right\|_{H^{s}}<\sigma.$
(3.2)

Let now the dynamics evolve freely in a time interval of leght $\gamma$, that
is, we consider the control
$0|_{[0,\gamma]}=(0,\dots,0):[0,\gamma]\to\mathbb{R}^{q+2}$. From (2.1) we
deduce that there exists $C=C(\gamma)$ such that

$\left\|\psi(\delta+\gamma;\psi_{0},0|_{[0,\gamma]}*u^{\delta,\gamma})-\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0)\right\|_{H^{s}}=\left\|\psi(\gamma;\psi(\delta,\psi_{0},u^{\delta,\gamma}),0)-\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0)\right\|_{H^{s}}<C\sigma.$

We use again the inductive hypothesis to deduce that there exist
$\delta^{\prime}\in[0,T/3)$ and a piecewise constant control
$u^{\delta^{\prime},\gamma}=(u^{\delta^{\prime},\gamma}_{1},...,u^{\delta^{\prime},\gamma}_{q},0,0):[0,\delta^{\prime}]\to\mathbb{R}^{q+2}$
such that

$\left\|\psi(\delta^{\prime};\psi(\gamma,e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0),u^{\delta^{\prime},\gamma})-e^{\gamma^{-1/2}\tilde{\phi}_{1}}\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0)\right\|_{H^{s}}<\sigma.$

Then, thanks to (2.1), there exists
$C^{\prime}=C^{\prime}(\|u^{\delta^{\prime},\gamma}\|_{L^{2}},\delta^{\prime})$
such that

$\begin{split}&\left\|\psi(\delta+\gamma+\delta^{\prime};\psi_{0},u^{\delta^{\prime},\gamma}*0|_{[0,\gamma]}*u^{\delta,\gamma})-e^{{\mathbb{B}}(\phi_{1})}\psi_{0}\right\|_{H^{s}}\\\
&\qquad\qquad\qquad\qquad\qquad\leq\left\|\psi(\delta^{\prime};\psi(\delta+\gamma;\psi_{0},0|_{[0,\gamma]}*u^{\delta,\gamma}),u^{\delta^{\prime},\gamma})-\psi(\delta^{\prime};\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0),u^{\delta^{\prime},\gamma})\right\|_{H^{s}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\left\|\psi(\delta^{\prime};\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0),u^{\delta^{\prime},\gamma})-e^{\gamma^{-1/2}\tilde{\phi}_{1}}\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0)\right\|_{H^{s}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\left\|e^{\gamma^{-1/2}\tilde{\phi}_{1}}\psi(\gamma;e^{-\gamma^{-1/2}\tilde{\phi}_{1}}\psi_{0},0)-e^{{\mathbb{B}}(\phi_{1})}\psi_{0}\right\|_{H^{s}}\\\
&\qquad\qquad\qquad\qquad\qquad\leq
C^{\prime}C\sigma+\sigma+\varepsilon/2.\end{split}$

Choosing $\sigma>0$ small enough such that
$C^{\prime}C\sigma+\sigma<\varepsilon/2$, we have then proved that the
piecewise constant control
$u^{\delta^{\prime},\gamma}*0|_{[0,\gamma]}*u^{\delta,\gamma}$ steers the
initial state $\psi_{0}$ $\varepsilon$-close to the state
$e^{{\mathbb{B}}(\phi_{1})}\psi_{0}$ in time
$\tau:=\delta^{\prime}+\gamma+\delta<T$. We can now repeat the same argument
for $\phi_{2}$, reasoning as if we were starting from the initial state
$e^{{\mathbb{B}}(\phi_{1})}\psi_{0}$, and prove that the system can be driven
arbitrarily close to the state
$e^{{\mathbb{B}}(\phi_{1})+{\mathbb{B}}(\phi_{2})}\psi_{0}$ in arbitrarily
small time, and hence, by iteration, to
$e^{\sum_{i=1}^{N}{\mathbb{B}}(\phi_{i})}\psi_{0}$. By inductive hypothesis we
conclude that there exists a piecewise constant control $u$ leading the state
$e^{\sum_{i=1}^{N}{\mathbb{B}}(\phi_{i})}\psi_{0}$ arbitrarily close to
$e^{\phi_{0}+\sum_{i=1}^{N}{\mathbb{B}}(\phi_{i})}\psi_{0}$. in arbitrarily
small time. This completes the proof of property $(P_{n})$. ∎

###  3.2 Small-time global approximate controllability

In this section we prove a more general results which in fact has inspired
Main Theorem A.

######  Theorem 3.3.

Let $s\in{\mathbb{N}}^{*}$, $s>d/2$ and let $(Q_{1},...,Q_{q})\in
C^{\infty}({\mathbb{T}}^{d},{\mathbb{R}}^{q})$ be such that
$1\in\mathcal{H}_{0}$ and $\mathcal{H}_{\infty}$ is dense in
$H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$.

  * i.

Let $\psi_{0},\psi_{1}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ be such that
${\rm sign}(\psi_{0})={\rm sign}(\psi_{1})$. For any $\epsilon>0$ and $T>0$,
there exist $\tau\in(0,T]$ and $(u_{1},...,u_{q})\in
L^{2}((0,\tau),{\mathbb{R}}^{q})$ for which the solution $\psi(t;\psi_{0},u)$
of (NHE) with control $u=(u_{1},...,u_{q},0,0)$ is well-defined in $[0,\tau]$
and satisfies

$\|\psi(\tau;\psi_{0},u)-\psi_{1}\|_{L^{2}}<\epsilon.$

  * ii.

Let $\psi_{0},\psi_{1}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ be such that
$\psi_{0},\psi_{1}>0$ (or $\psi_{0},\psi_{1}<0$). For any $\epsilon>0$ and
$T>0$, there exists $(u_{1},...,u_{q})\in L^{2}((0,T),{\mathbb{R}}^{q})$ such
that the solution $\psi(t;\psi_{0},u)$ of (NHE) with control
$u=(u_{1},...,u_{q},0,0)$ is well-defined in $[0,T]$ and verifies

$\|\psi(T;\psi_{0},u)-\psi_{1}\|_{H^{s}}<\epsilon.$

###### Proof.

Let us start by proving _i._. Denote by $Z$ the set of zeroes of $\psi_{j}$,
for $j=0,1$. Consider for $\eta>0$ the set

$Z_{\eta}:=\\{x\in{\mathbb{T}}^{d}\mid\mathop{\rm
dist}\nolimits(x,Z)<\eta\\},$

and its complement in ${\mathbb{T}}^{d}$, $Z^{c}_{\eta}$. For $\eta>0$, we
define

$\phi_{\eta}=\rho_{\eta}\log(\psi_{1}/\psi_{0}),$

where $\rho_{\eta}$ is a smooth function compactly supported inside
${\mathbb{Z}}^{c}_{\eta}$. $\phi_{\eta}$ is well-defined because
$\psi_{1}/\psi_{0}>0$ on $Z^{c}_{\eta}$. Furthermore, $\rho_{\eta}$ belongs to
$H^{s}({\mathbb{T}}^{d})$. Notice that

$\|e^{\phi_{\eta}}\psi_{0}-\psi_{1}\|_{L^{2}({\mathbb{T}}^{d})}\leq\|e^{\phi_{\eta}}\psi_{0}-\psi_{1}\|_{L^{2}(Z^{c}_{\eta})}+\|\psi_{0}+\psi_{1}\|_{L^{2}(Z_{\eta})}$

Fixed any $\epsilon,T>0$, observe that we can choose $\eta>0$ small enough
such that

$\|e^{\phi_{\eta}}\psi_{0}-\psi_{1}\|_{L^{2}({\mathbb{T}}^{d})}<\epsilon/2.$

We then apply Proposition 3.2 with $\varphi=\phi_{\eta}$ and we deduce that
there exist a time $\tau\in[0,T)$ and a control $u=(u_{1},...,u_{q},0,0)\in
L^{2}((0,\tau),{\mathbb{R}}^{q+2})$ such that the solution
$\psi(t;\psi_{0},u)$ of (NHE) is well-defined in $[0,\tau]$ and

$\|\psi(\tau;\psi_{0},u)-e^{\varphi}\psi_{0}\|_{L^{2}}<\epsilon/2.$

By the triangular inequality we conclude that

$\|\psi(\tau;\psi_{0},u)-\psi_{1}\|_{L^{2}}\leq\|\psi(\tau;\psi_{0},u)-e^{\varphi}\psi_{0}\|_{L^{2}}+\|e^{\varphi}\psi_{0}-\psi_{1}\|_{L^{2}}<\epsilon,$

and the first item of the Proposition is then proved.

The proof of _ii._ follows from the same strategy used for proving _i._.
However, we now consider the $H^{s}$ instead of the $L^{2}$-norm. Observe that
Proposition 3.2 is valid for the $H^{s}$-norm. In this case, we substitute
$\phi_{\eta}$ with two functions $\phi_{1}=\log(1/\psi_{0})$ and
$\phi_{2}=\log(\psi_{1})$ which are well-defined everywhere in
${\mathbb{T}}^{d}$. Note that $\phi_{1},\phi_{2}\in
H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$ since $\psi_{1},\psi_{0}\in
H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$. Proposition 3.2 with $\varphi=\phi_{1}$
yields the existence of a control
$u^{1}=(u^{1}_{1},...,u^{1}_{q},0,0):[0,\tau_{1})\rightarrow{\mathbb{R}}^{q+2}$
which steers $\psi_{0}$ close to the constant state $1$ in time $\tau_{1}\leq
T/2$. We apply again Proposition 3.2 with $\varphi=\phi_{2}$. Thus, we can
find a control
$u^{2}=(u^{2}_{1},...,u^{2}_{q},0,0):[0,\tau_{2})\rightarrow{\mathbb{R}}^{q+2}$
such that the solution of (NHE) starting from a state close to $1$ at time
$\tau_{2}\leq T/2$ reaches a neighbourhood of the final target $\psi_{1}$.
Hence, the approximate controllability in $H^{s}$ can be achieved in time $T$
by exploiting that $1$ is a stationary solution of (NHE) associated to the
control
$u^{stat}=(u^{stat}_{1},...,u^{stat}_{q},0,0):[0,T-\tau_{1}-\tau_{2}]\rightarrow{\mathbb{R}}^{q+2}$,
thanks to the assumption that $1\in\mathcal{H}_{0}$. The control will
therefore be defined as

$(u^{2}*u^{stat}*u^{1})(t)={\mathbf{1}}_{[0,\tau_{1})}u^{1}(t)+{\mathbf{1}}_{[\tau_{1},T-\tau_{2}]}u^{stat}(t-\tau_{1})+{\mathbf{1}}_{(T-\tau_{2},T]}u^{2}(t-T+\tau_{2}).$

∎

######  Remark 3.4.

The reason why the first part of Theorem 3.3 (and, consequently, of Main
Theorem A) is stated only for the $L^{2}$-norm, while the second part is valid
with the stronger $H^{s}$-norm, is due to our approximation technique. More
precisely, the term
$\|e^{\phi_{\eta}}\psi_{0}-\psi_{1}\|_{L^{2}(Z^{c}_{\eta})}$ cannot be small
for $\eta\to 0$ with respect to the $H^{s}$-norm, $s>0$, as soon as
$Z\neq\emptyset$.

###### Proof of Main Theorem A.

Main Theorem A follows from Theorem 3.3. Indeed, thanks to Proposition 3.1,
the space $\mathcal{H}_{\infty}$ is dense in
$H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$. Furthermore, the potentials $Q_{j}$
belong to $H^{r}({\mathbb{T}}^{d},{\mathbb{R}})$ for every $r>0$. Therefore,
the proof is completed. ∎

##  4 Exact controllability to the ground state solution

We now move to the proof of Main Theorem B. Henceforth, we suppose $d=s=1$.
Let the solution of (NHE) exist for any time $T>0$. For instance, it is enough
to require $p\in 2{\mathbb{N}}^{*}$, $\kappa\geq 0$ and $\psi_{0}\in
H^{3}({\mathbb{T}})$, as stated in Proposition 2.2. Main Theorem B will be a
direct consequence of the following more general result.

######  Theorem 4.1.

Let $\kappa\geq 0$ and $p\in 2{\mathbb{N}}^{*}$. Assume Assumptions II be
verified and $\mathcal{H}_{\infty}$ be dense in
$H^{3}({\mathbb{T}},{\mathbb{R}})$. Then, (NHE) is exactly controllable to the
ground state solution $\Phi$ in any positive time from any positive state. In
other words, for any $T>0$ and

$\psi_{0}\in\\{\psi\in H^{3}({\mathbb{T}},{\mathbb{R}}):\ sgn(\psi)>0\\},$

there exists $u\in L^{2}((0,T),{\mathbb{R}}^{q+2})$ such that

$\psi(T;\psi_{0},u)=\Phi.$

Analogously, for any $T>0$ and

$\psi_{0}\in\\{\psi\in H^{3}({\mathbb{T}},{\mathbb{R}}):\ sgn(\psi)<0\\},$

there exists $u\in L^{2}((0,T),{\mathbb{R}}^{q+2})$ such that

$\psi(T;\psi_{0},u)=-\Phi.$

######  Example 4.2.

Examples of suitable functions $\mu_{1}$ and $\mu_{2}$ verifying hypotheses
(1.5) in Assumptions II are

$\mu_{1}=x^{3}(2\pi-x)^{3},\qquad\mu_{2}=x^{3}(x-\pi)^{3}(x-2\pi)^{3}.$

Indeed, both functions belong to $H^{3}({\mathbb{T}},{\mathbb{R}})$, $\mu_{1}$
is symmetric with respect to $x=\pi$ and $\mu_{2}$ is antisymmetric. Thus,
$\langle\mu_{1},c_{0}\rangle_{L^{2}}\neq 0$ and
$\langle\mu_{2},c_{0}\rangle_{L^{2}}=0$. Moreover,

$\begin{split}\langle\mu_{1},s_{k}\rangle_{L^{2}}=0,\qquad\langle\mu_{2},c_{k}\rangle_{L^{2}}=0.\\\
\end{split}$

By further computations, the remaining hypotheses in (1.5) are verified since

$\begin{split}\langle\mu_{1},c_{k}\rangle_{L^{2}}=\frac{96\pi(k^{2}\pi^{2}-15)}{k^{6}},\qquad\langle\mu_{2},s_{k}\rangle_{L^{2}}=\frac{-864\pi(840-105k^{2}\pi^{2}+2k^{4}\pi^{4})}{k^{9}}.\\\
\end{split}$

Before proving Theorem 4.1, and consecutively Main Theorem B, we need to
ensure the following local exact controllability result to the ground state
solution $\Phi=\frac{1}{\sqrt{2\pi}}$.

######  Theorem 4.3.

Let $\kappa\geq 0$ and $p\in 2{\mathbb{N}}^{*}$. Assume Assumptions II be
verified. Then, (NHE) is locally exactly controllable to the ground state
solution $\Phi$ in any positive time. In other words, for any $T>0$ there
exists $R_{T}>0$ such that, for any

$\psi_{0}\in\\{\psi\in
H^{3}({\mathbb{T}},{\mathbb{R}}):\|\psi-\Phi\|_{H^{1}}<R_{T}\\},$

there exists $(u_{1},u_{2})\in H^{1}((0,T),{\mathbb{R}}^{2})$ such that
$\psi(T;\psi_{0},u)=\Phi$, where
$u=(\frac{\kappa}{\Phi^{p}},0,...,0,u_{1},u_{2})$. Furthermore,

$\left\lVert
u\right\rVert_{H^{1}(0,T)}\leq\frac{e^{-\pi^{2}\Gamma_{0}/T}}{e^{2\pi^{2}\Gamma_{0}/(3T)}-1},$
(4.1)

where

$\Gamma_{0}:=2\nu+(\max\\{\ln\gamma_{1},0\\}+\max\\{\ln
C^{2}_{Q},0\\}+\gamma_{2}+\ln 8)/2$ (4.2)

with

$\gamma_{1}:=2\kappa(p+1)^{2}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j},\qquad\gamma_{2}:=2\kappa(p+1)\Phi^{p}+\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}+1,$

and

$R_{T}=e^{-6\Gamma_{0}/T_{1}},$ (4.3)

with

$T_{1}:=\min\Big{\\{}\frac{6}{\pi^{2}}T,1,T_{0}\Big{\\}}.$

The constant $T_{0}$ is defined in (4.5) and $C_{Q}$ in (C.4).

###  4.1 Control of the linearized system

The result of Theorem 4.3 follows from the null-controllability of an
associated linear system and the iteration of a control procedure on a clever
choice of time steps, as proposed in [6].

We first observe that the ground state $\Phi=1/\sqrt{2\pi}$ is solution of
(NHE) in any time interval $[0,T]$ for $u=(u_{\kappa},0,0,\dots,0)$, with
$u_{\kappa}=\frac{\kappa}{(2\pi)^{p/2}}$ and $\psi_{0}=\Phi$. Indeed, we
recall that $Q_{1}=1$, thanks to Assumptions II.

Let $s_{1},s_{0}>0$. Consider the following linear system

$\begin{cases}\partial_{t}\xi(t)-\partial_{x}^{2}\xi(t)+\kappa p\xi(t)=\langle
v(t),Q\rangle\Phi,&t\in(s_{0},s_{1}).\\\ \xi(s_{0})=\xi_{0},\end{cases}$ (4.4)

with $Q=(Q_{1},...,Q_{q},\mu_{1},\mu_{2})\in
H^{3}({\mathbb{T}},{\mathbb{R}}^{q+2})$ and $q\in{\mathbb{N}}^{*}$. We denote
by $\xi(\cdot;s_{0},\xi_{0},v)$ the solution of (4.4) with initial condition
$\xi_{0}$ at time $s_{0}$ and control $v$.

######  Definition 4.4.

The pair $(-\partial^{2}_{x},Q)$ is said to be $1$-_null controllable_ in time
$T>s_{0}$ if there exists a constant $N(T)>0$ such that for any $\xi_{0}\in
L^{2}({\mathbb{T}},{\mathbb{R}})$, there exists a control $v\in
L^{2}((s_{0},T),{\mathbb{R}}^{q+2})$ such that $\xi(T;s_{0},\xi_{0},v)=0$ and
moreover $\left\lVert v\right\rVert_{L^{2}(0,T)}\leq
N(T)\left\lVert\xi_{0}\right\rVert_{L^{2}}$. The best constant, that is,

$N(T):=\sup_{\left\lVert\xi_{0}\right\rVert_{L^{2}}=1}\inf\left\\{\left\lVert
v\right\rVert_{L^{2}}\,:\,\xi(T;s_{0},\xi_{0},v)=0\right\\}$

is called the _control cost_. If $v\in
H^{1}_{0}((s_{1},T),{\mathbb{R}}^{q+2}),$ the pair $(-\partial^{2}_{x},Q)$ is
called _smoothly_ $1$-null controllable and

$N(T):=\sup_{\left\lVert\xi_{0}\right\rVert_{L^{2}}=1}\inf\left\\{\left\lVert
v\right\rVert_{H^{1}}\,:\,\xi(T;s_{0},\xi_{0},v)=0\right\\}$

######  Remark 4.5.

Let us observe that, fixed $v\in L^{2}((s_{0},s_{1}),{\mathbb{R}}^{q+2})$ and
$Q\in H^{1}({\mathbb{T}},{\mathbb{R}}^{q+2})$, for any $\xi_{0}\in
L^{2}({\mathbb{T}},{\mathbb{R}})$, there exists a unique mild solution

$\xi\in C([s_{0},s_{1}],L^{2}({\mathbb{T}}))$

of (4.4) (see [6, Proposition 2.1] adapted to the current case). Furthermore,
given a smoother initial condition of (4.4), for instance $\xi_{0}\in
H^{1}({\mathbb{T}})$, the solution $\xi$ belongs to the space

$C([s_{0},s_{1}],H^{1}({\mathbb{T}}))\cap
H^{1}((s_{0},s_{1}),L^{2}({\mathbb{T}}))\cap
L^{2}((s_{0},s_{2}),H^{2}({\mathbb{T}})).$

Such regularity is called _maximal regularity_ , and it is due to the
analiticity of the semigroup generated by the operator $\partial^{2}_{x}$
(see, for instance, [6, Proposition 4] and [4, Corollary 3] that can be
adapted to (4.4)).

The result that follows shows that we use the last two components of the
control to drive the linear system (4.4) to rest. Henceforth, where constants
$C$ come with no specific index, they may vary from line to line.

######  Proposition 4.6.

Let Assumptions II be verified. Then, $(-\partial^{2}_{x},Q)$ is smoothly null
controllable in any time $T>0$. Namely, for any $\xi_{0}\in
L^{2}({\mathbb{T}},{\mathbb{R}})$, there exists $(v_{1},v_{2})\in
H^{1}_{0}((0,T),{\mathbb{R}}^{2})$ such that the solution of (4.4) with
$(s_{0},s_{1})=(0,T)$ and $v=(0,...,0,v_{1},v_{2})$ satisfies

$\xi(T;\xi_{0},v)=0.$

Furthermore, there exist $\nu,T_{0}>0$ such that

$N(\tau)\leq e^{\nu/\tau},\quad\forall\,0<\tau\leq T_{0}.$ (4.5)

###### Proof.

For all $T>0$, we want to prove the existence of $N(T)>0$ such that for any
initial condition $\xi_{0}\in L^{2}({\mathbb{T}},{\mathbb{R}})$ there exists a
control $v=(0,...,0,v_{1},v_{2})$ so that $\xi(T;\xi_{0},v)=0$. To this
purpose, we first note that the operator $-\partial_{x}^{2}+p\kappa$ exhibits
double eigenvalues $\tilde{\lambda}_{j}=\lambda_{j}+p\kappa$ with
$j\in{\mathbb{N}}$ (see definition (1.3) of $\lambda_{j}$), with the exception
of the first one, associated to the eigenfunctions
$\\{c_{0},c_{j},s_{j}\\}_{j\in{\mathbb{N}}^{*}}$, defined in (1.4).

Let us consider the reduced problem (4.4) with $\hat{Q}=(\mu_{1},\mu_{2})$ and
$(s_{0},s_{1})=(0,T)$. We look for a control of the form
$\hat{v}=(v_{1},v_{2})$. We decompose the solution with respect to the Hilbert
basis $\\{c_{0},c_{k},s_{k}\\}_{k\in{\mathbb{N}}^{*}}$. Recalling that
$\langle\mu_{1},s_{k}\rangle_{L^{2}}=\langle\mu_{2},c_{k}\rangle_{L^{2}}=0$,
the smooth-null controllability property is equivalent to finding $\hat{v}\in
H^{1}_{0}((0,T);{\mathbb{R}}^{2})$ such that

$0=\xi(T;\xi_{0},\hat{v})=\sum_{k\in{\mathbb{N}}}e^{-\tilde{\lambda}_{k}T}\langle\xi_{0},c_{k}\rangle_{L^{2}}c_{k}+\sum_{k\in{\mathbb{N}}^{*}}e^{-\tilde{\lambda}_{k}T}\langle\xi_{0},s_{k}\rangle_{L^{2}}s_{k}\\\
-\int_{0}^{T}v_{1}(s)\sum_{k\in{\mathbb{N}}}e^{-\tilde{\lambda}_{k}(T-s)}\langle\mu_{1}\Phi,c_{k}\rangle_{L^{2}}c_{k}\,ds-\int_{0}^{T}v_{2}(s)\sum_{k\in{\mathbb{N}}^{*}}e^{-\tilde{\lambda}_{k}(T-s)}\langle\mu_{2}\Phi,s_{k}\rangle_{L^{2}}s_{k}\,ds.$

Notice that $\Phi=c_{0}$ is a constant. The above relation is verified when
the following infinite number of identities are satisfied for a control
function $\hat{v}=(v_{1},v_{2})$

$\begin{split}\langle\xi_{0},c_{k}\rangle_{L^{2}}=c_{0}\int_{0}^{T}e^{\tilde{\lambda}_{k}s}v_{1}(s)\langle\mu_{1},c_{k}\rangle_{L^{2}}\,ds,\quad\forall\,k\in{\mathbb{N}},\\\
\langle\xi_{0},s_{k}\rangle_{L^{2}}=c_{0}\int_{0}^{T}e^{\tilde{\lambda}_{k}s}v_{2}(s)\langle\mu_{2},s_{k}\rangle_{L^{2}}\,ds,\quad\forall\,k\in{\mathbb{N}}^{*},\end{split}$

that can be rewritten in compact form as

$\displaystyle\int_{0}^{T}e^{\lambda_{k}s}\tilde{v}_{1}(s)\,ds=d_{k}^{1},\quad\forall\,k\in{\mathbb{N}},$
(4.6)
$\displaystyle\int_{0}^{T}e^{\lambda_{k}s}\tilde{v}_{2}(s)\,ds=d_{k}^{2},\quad\forall\,k\in{\mathbb{N}}^{*},$
(4.7)

where

$d_{k}^{1}:=\frac{\langle\xi_{0},c_{k}\rangle_{L^{2}}}{c_{0}\langle\mu_{1},c_{k}\rangle_{L^{2}}},\quad\quad\quad
d_{k}^{2}:=\frac{\langle\xi_{0},s_{k}\rangle_{L^{2}}}{c_{0}\langle\mu_{2},s_{k}\rangle_{L^{2}}}\quad\forall\,k\in{\mathbb{N}}^{*},$

are well-defined thanks to Assumptions II and

$\tilde{v}_{1}(s)=e^{p\kappa s}v_{1}(s),\qquad\tilde{v}_{2}(s)=e^{p\kappa
s}v_{2}(s).$

We treat (4.6) an (4.7) as two separate moment problems. Let us start solving
(4.6). We seek for a function $\tilde{v}_{1}$ such that

$\tilde{v}_{1}^{\prime}(t)=r_{1}\left(\frac{T}{2}-t\right)e^{-(\frac{T}{2}-t)},\qquad
r_{1}\in L^{2}\left(-\frac{T}{2},\frac{T}{2}\right).$

Therefore, it should hold

$\tilde{v}_{1}(t)-\tilde{v}_{1}(0)=\int_{0}^{t}\tilde{v}_{1}^{\prime}(s)ds=\int_{0}^{t}r_{1}\left(\frac{T}{2}-s\right)e^{-(\frac{T}{2}-s)}ds.$

We require that $\tilde{v}_{1}(0)=\tilde{v}_{1}(T)=0$, which implies that
$r_{1}$ must satisfy

$\int_{-\frac{T}{2}}^{\frac{T}{2}}r_{1}(s)e^{-s}ds=0.$ (4.8)

Integrating by parts (4.6) and taking into account that
$\tilde{v}_{1}(0)=\tilde{v}_{1}(T)=0$, we get for any $k\in{\mathbb{N}}^{*}$

$\begin{split}d^{1}_{k}&=\int_{0}^{T}e^{\lambda_{k}s}\tilde{v}_{1}(s)ds=\frac{1}{\lambda}_{k}\left(\left.e^{\lambda_{k}t}\tilde{v}_{1}(t)\right|^{T}_{0}-\int_{0}^{T}e^{-(\frac{T}{2}-t)+\lambda_{k}t}r_{1}\left(\frac{T}{2}-t\right)dt\right)\\\
&=\frac{1}{\lambda_{k}}\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{-s+\lambda_{k}(\frac{T}{2}-s)}r_{1}(s)ds.\end{split}$

The above identities can be rewritten as follows

$\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{-(1+\lambda_{k})s}r_{1}(s)ds=\lambda_{k}d^{1}_{k}e^{-\frac{T}{2}\lambda_{k}},\qquad
k\in{\mathbb{N}}^{*}.$

For $k=0$ we have that

$\begin{split}d^{1}_{0}&=\int_{0}^{T}\tilde{v}_{1}(s)ds=\left.s\tilde{v}_{1}(s)\right|^{T}_{0}-\int_{0}^{T}sr_{1}\left(\frac{T}{2}-s\right)e^{-(\frac{T}{2}-s)}ds=\int_{-\frac{T}{2}}^{\frac{T}{2}}\left(\frac{T}{2}-s\right)r_{1}(s)e^{-s}ds\\\
&=-\int_{-\frac{T}{2}}^{\frac{T}{2}}se^{-s}r_{1}(s)ds,\end{split}$

where we have used (4.8). Define the family

$\omega_{k}:=1+\lambda_{k},\qquad k\in{\mathbb{N}}^{*},$

and the sequence $\tilde{d}^{1}_{kj}$

$\tilde{d}^{1}_{01}=-d_{0},\qquad\tilde{d}^{1}_{k1}=0,\qquad
k\in{\mathbb{N}}^{*},$
$\tilde{d}^{1}_{00}=0,\qquad\tilde{d}^{1}_{k0}=\lambda_{k}d^{1}_{k}e^{-\frac{T}{2}\lambda_{k}},\qquad
k\in{\mathbb{N}}^{*}.$

Thus, we look for $r_{1}\in L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)$ such
that

$\int_{-\frac{T}{2}}^{\frac{T}{2}}s^{j}e^{-\omega_{k}s}r_{1}(s)ds=\tilde{d}^{1}_{kj},\qquad
j=0,1,\quad k\in{\mathbb{N}}.$ (4.9)

As proved in [11, Theorem 1.5], there exists $T_{0}>0$ such that for every
$0<T<T_{0}$ there exists a family

$\\{\sigma_{k,j}\\}_{k,j\in{\mathbb{N}}}\subset
L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)$

which is biorthogonal to

$e_{kj}(s)=s^{j}e^{-\omega_{k}s},\qquad k,j\in{\mathbb{N}},$

and moreover

$\left\lVert\sigma_{k,j}\right\rVert_{L^{2}(-T/2,T/2)}\leq
Ce^{C\sqrt{\omega_{k}}+\frac{C}{T}},\qquad j=0,1,\quad k\in{\mathbb{N}}^{*}.$

Therefore, by defining

$r_{1}(s):=\tilde{d}^{1}_{01}\sigma_{01}+\sum_{k=1}^{\infty}\tilde{d}^{1}_{k0}\sigma_{k0}(s),$

we deduce that $r_{1}$ solves the moment problem (4.9). Let us finally show
that $r_{1}\in L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)$:

$\begin{split}\left\lVert
r_{1}\right\rVert_{L^{2}(-T/2,T/2)}&\leq|\tilde{d}^{1}_{00}|\left\lVert\sigma_{01}\right\rVert_{L^{2}(-T/2,T/2)}+\sum_{k=1}^{\infty}|\tilde{d}^{1}_{k0}|\left\lVert\sigma_{k0}\right\rVert_{L^{2}(-T/2,T/2)}\\\
&\leq
Ce^{C/T}|\tilde{d}^{1}_{00}|+C\sum_{k=1}^{\infty}\lambda_{k}|d^{1}_{k}|e^{-\frac{T}{2}\lambda_{k}+C\sqrt{1+\lambda_{k}}+\frac{C}{T}}\\\
&\leq
Ce^{C/T}\left(\frac{|\langle\xi_{0},c_{0}\rangle_{L^{2}}|}{c_{0}|\langle\mu_{1},c_{0}\rangle_{L^{2}}|}+\left(\sum_{k=1}^{\infty}\frac{\lambda_{k}^{2}e^{-T\lambda_{k}+C\sqrt{\lambda_{k}+1}}}{c_{0}^{2}|\langle\mu_{1},c_{k}\rangle_{L^{2}}|^{2}}\right)^{1/2}\left\lVert\xi_{0}\right\rVert_{L^{2}}\right)\\\
&\leq
Ce^{C/T}\left(1+\left(\sum_{k=1}^{\infty}\lambda_{k}^{2(q_{1}+1)}e^{-T\lambda_{k}+C\sqrt{\lambda_{k}+1}}\right)^{1/2}\right)\left\lVert\xi_{0}\right\rVert_{L^{2}},\end{split}$

where $q_{1}$ is the parameter introduced in Assumptions II. Let us analyse
the behaviour with respect to $T$ of the following series:

$S(T):=\sum_{k=1}^{\infty}\lambda_{k}^{2(q_{1}+1)}e^{-T\lambda_{k}+C\sqrt{\lambda_{k}+1}}=\sum_{k=1}^{\infty}\left(\lambda^{2(q_{1}+1)}_{k}e^{-\frac{T}{2}\lambda_{k}}\right)e^{-\frac{T}{2}\lambda_{k}+C\sqrt{\lambda_{k}+1}}.$
(4.10)

For any $\lambda\geq 0$, we introduce the function
$f(\lambda):=e^{-\frac{T}{2}\lambda+C\sqrt{\lambda+1}}$. Its derivative is
given by

$f^{\prime}(\lambda)=e^{-\frac{T}{2}\lambda+C\sqrt{\lambda}}\left(-\frac{T}{2}+\frac{C}{2\sqrt{\lambda+1}}\right),$
(4.11)

and its maximum is attended at $\lambda=\left(\frac{C}{T}\right)^{2}-1$.
Hence, for every $0<T\leq 1$ we have

$S(T)\leq
e^{\frac{T}{2}+\frac{C}{T}}\sum_{k=1}^{\infty}\left(\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}\right)\leq
e^{\frac{C}{T}}\sum_{k=1}^{\infty}\left(\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}\right).$
(4.12)

Now, for any $\lambda\geq 0$, we consider the function
$g(\lambda):=\lambda^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda}$. From its derivative
$g^{\prime}(\lambda)=\lambda^{2q_{1}+1}e^{-\frac{T}{2}\lambda}\left(2(q_{1}+1)-\frac{T}{2}\lambda\right)$,
we deduce that

$g(\lambda)\text{ is }\left\\{\begin{array}[]{ll}\text{increasing }&\text{if
}0\leq\lambda<\frac{4(q_{1}+1)}{T}\\\ \\\ \text{decreasing }&\text{if
}\lambda\geq\frac{4(q_{1}+1)}{T}\end{array}\right.$ (4.13)

and $g$ has a maximum for $\lambda=\frac{4(q_{1}+1)}{T}$. We define the index

$k_{1}:=k_{1}(T)=\sup\left\\{k\in{\mathbb{N}}^{*}\,:\,\lambda_{k}\leq\frac{4(q_{1}+1)}{T}\right\\}.$
(4.14)

We can rewrite the sum in (4.12) as

$\sum_{k=1}^{\infty}\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}=\sum_{1\leq
k\leq
k_{1}-1}\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}+\sum_{k_{1}\leq
k\leq k_{1}+1}\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}+\sum_{k\geq
k_{1}+2}\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}.$ (4.15)

For any $1\leq k\leq k_{1}-1$, we have

$\int_{\lambda_{k}}^{\lambda_{k+1}}\lambda^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda}d\lambda\geq(\lambda_{k+1}-\lambda_{k})\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}\geq\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}},$
(4.16)

and for any $k\geq k_{1}+2$, it holds that

$\int_{\lambda_{k-1}}^{\lambda_{k}}\lambda^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda}d\lambda\geq(\lambda_{k}-\lambda_{k-1})\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}\geq\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}.$
(4.17)

Therefore, we obtain that

$\sum_{k=1}^{\infty}\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}\leq
2\int_{0}^{\infty}\lambda^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda}d\lambda+\sum_{k_{1}\leq
k\leq k_{1}+1}\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}.$ (4.18)

Recalling that $g$ has a maximum at $\lambda=\frac{4(q_{1}+1)}{T}$, we have
that

$\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}\leq\left(\frac{4(q_{1}+1)}{T}\right)^{2(q_{1}+1)}e^{-2(q_{1}+1)},\quad\text{for
}k=k_{1},k_{1}+1.$ (4.19)

Moreover, we can rewrite the integral term in (4.18) as

$\begin{split}\int_{0}^{\infty}\lambda^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda}d\lambda&=\frac{2}{T}\int_{0}^{\infty}\left(\frac{2s}{T}\right)^{2(q_{1}+1)}e^{-s}ds\\\
&=\left(\frac{2}{T}\right)^{2q_{1}+3}\int_{0}^{\infty}s^{2(q_{1}+1)}e^{-s}ds=\Gamma(2q_{1}+3)\left(\frac{2}{T}\right)^{2q_{1}+3},\end{split}$
(4.20)

where by $\Gamma(\cdot)$ we indicate the Euler integral of the second kind.
Therefore, thanks to (4.19) and (4.20) we conclude that there exist two
positive constants $C^{1}_{q_{1}},C^{2}_{q_{1}}$ such that

$\sum_{k=1}^{\infty}\lambda_{k}^{2(q_{1}+1)}e^{-\frac{T}{2}\lambda_{k}}\leq\frac{C^{1}_{q_{1}}}{T^{2q_{1}+2}}+\frac{C^{2}_{q_{1}}}{T^{2q_{1}+3}}.$
(4.21)

From the above estimate we can prove that there exist positive constants $C$
for which

$\left\lVert r_{1}\right\rVert_{L^{2}(-T/2,T/2)}\leq
Ce^{C/T}\left\lVert\xi_{0}\right\rVert_{L^{2}},\quad\forall\,0<T<\min\\{T_{0},1\\},$
(4.22)

and thus

$\left\lVert(\tilde{v}_{1})^{\prime}\right\rVert_{L^{2}(0,T)}\leq
e^{T/2}\left\lVert r_{1}\right\rVert_{L^{2}(-T/2,T/2)}\leq Ce^{C/T}\left\lVert
y_{0}\right\rVert,\quad\forall\,0<T<\min\\{T_{0},1\\}.$ (4.23)

From the Poincaré inequality, there exists a constant $C>0$ such that for any
$\tilde{v}\in H^{1}_{0}(0,T)$ it holds that
$\left\lVert\tilde{v}_{1}\right\rVert_{H^{1}(0,T)}\leq
C\left\lVert(\tilde{v}_{1})^{\prime}\right\rVert_{L^{2}(0,T)}$, and we deduce
that

$\left\lVert\tilde{v}_{1}\right\rVert_{H^{1}(0,T)}\leq
C\left\lVert(\tilde{v}_{1})^{\prime}\right\rVert_{L^{2}(0,T)}\leq
Ce^{C/T}\left\lVert\xi_{0}\right\rVert_{L^{2}},\quad\forall\,0<T<\min\\{T_{0},1\\}.$
(4.24)

Finally, recalling that $v_{1}(t)=e^{-p\kappa t}\tilde{v}_{1}(t)$, we conclude
that

$\begin{split}\left\lVert
v_{1}\right\rVert^{2}_{H^{1}(0,T)}&=\int_{0}^{T}e^{-2p\kappa
t}|\tilde{v}_{1}(t)|^{2}dt+\int_{0}^{T}\left(\left(e^{-2pkt}|\tilde{v}_{1}(t)|\right)^{\prime}\right)^{2}dt\\\
&\leq C\left\lVert\tilde{v}_{1}\right\rVert^{2}_{H^{1}(0,T)}\leq
Ce^{C/T}\left\lVert\xi_{0}\right\rVert_{L^{2}}^{2},\end{split}$

for all $0<T\leq\min\\{T_{0},1\\}$. The same computations are also valid for
the second component of the control $v_{2}$ and the proof is then completed. ∎

###  4.2 Proof of Main Theorem B

We now prove Theorem 4.3.

###### Proof of Theorem 4.3.

$\mathbf{(}1)$ Time decomposition and preliminaries.  Let $T>0$ and define

$T_{f}:=\min\\{T,\frac{\pi^{2}}{6},\frac{\pi^{2}}{6}T_{0}\\},$

where $T_{0}$ is the constant in (4.5). Let $T_{1}$ be defined as follows

$T_{1}:=\frac{6}{\pi^{2}}T_{f}.$

Observe that, with this choice $0<T_{1}\leq 1$. We now defined the sequences

$T_{j}:=\frac{T_{1}}{j^{2}},\quad j\geq
1\qquad\text{and}\qquad\tau_{n}:=\sum_{j=1}^{n}T_{j},\quad n\geq 0,$

with the convention $\sum_{j=1}^{0}T_{j}=0$. It is easy to prove that

$\sum_{j=1}^{\infty}T_{j}=T_{f}$

and thus we will perform an iterative control procedure on the consecutive
time intervals $[\tau_{n},\tau_{n+1}]$, $n\geq 0$ so that at the limit
$n\to\infty$ we will prove exact controllability of (NHE) to the ground state
solution in time $T_{f}$.

$\mathbf{(}2)$ Estimates in the first time step: inductive basis. Let us set
$y=\psi-\Phi$, where $\psi$ and $\Phi$ are solutions of (NHE) associated with
the initial conditions $\psi_{0},\Phi\in H^{3}({\mathbb{T}})$ and controls
$u\in H^{1}_{loc}({\mathbb{R}}^{+},{\mathbb{R}}^{q+2})$ and
$\hat{u}:=(u_{\kappa},0,0,\dots,0)$ with $u_{\kappa}=\frac{\kappa}{\Phi^{p}}$,
respectively. Consider the equation satisfied by $y$ in a general time
interval $(s_{0},s_{1})$:

$\begin{cases}\partial_{t}y(t)-\partial^{2}_{x}y(t)+\kappa(y(t)+\Phi)^{p+1}=\langle
u(t),Q\rangle y(t)+\langle u(t),Q\rangle\Phi,&t\in(s_{0},s_{1})\\\
y(s_{0})=y_{s_{0}}=\psi(s_{0})-\Phi.\end{cases}$ (4.25)

Our aim is to prove that system (4.25) is null controllable in time $T_{f}$ by
means of a bilinear control $u$. This would imply that
$\psi(T_{f};\psi_{0},0,u)=\Phi$. For this purpose, we shall first consider
problem (4.25) and its linearization (4.4), with initial condition
$\xi_{0}=y_{0}:=\psi_{0}-\Phi$, in the time interval
$[s_{0},s_{1}]=[\tau_{0},\tau_{1}]=[0,T_{1}]$. Observe that, thanks to the
regularity of $\psi_{0}$ and $u$, the solution of (NHE) is globally in time
well-defined (see Proposition 2.2), as well as the solution of the associated
linear system

$\begin{cases}\partial_{t}\xi(t)-\partial_{x}^{2}\xi(t)+\kappa p\xi(t)=\langle
v^{1}(t),Q\rangle\Phi,&t\in(\tau_{0},\tau_{1})\\\
\xi(\tau_{0})=\xi_{0},\end{cases}$ (4.26)

(see Remark 4.5).

From Proposition 4.6 we deduce that there exists a control, defined as
$v^{1}=(0,\dots,0,v_{1}^{1},v_{2}^{1})$, with $(v_{1}^{1},v_{2}^{1})\in
H^{1}_{0}((\tau_{0},\tau_{1}),{\mathbb{R}}^{2})$ such that

$\xi(\tau_{1};\tau_{0},\xi_{0},v^{1})=0,$

and

$\left\lVert v^{1}\right\rVert_{H^{1}(\tau_{0},\tau_{1})}\leq
N(\tau_{1})\left\lVert\xi_{0}\right\rVert_{L^{2}}=N(\tau_{1})\left\lVert
y_{0}\right\rVert_{L^{2}}\leq N(\tau_{1})\left\lVert
y_{0}\right\rVert_{H^{1}},$ (4.27)

with $N(\tau_{1})$ that satisfies (4.5) (because $\tau_{1}\leq T_{0}$). We now
define

$u^{1}=(u_{\kappa},0,\dots,0,v_{1},v_{2})\in
H^{1}((\tau_{0},\tau_{1}),{\mathbb{R}}^{q+2}).$

Using such control in equation (NHE) (and so in (4.25) in the time interval
$[\tau_{0},\tau_{1}]$), one easily finds that (4.25) reads as

$\begin{cases}\partial_{t}y(t)-\partial^{2}_{x}y(t)+\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}y^{j}(t)\Phi^{p+1-j}=\langle
v^{1}(t),Q\rangle y(t)+\langle
v^{1}(t),Q\rangle\Phi,&t\in(\tau_{0},\tau_{1})\\\
y(\tau_{0})=y_{0}.\end{cases}$ (4.28)

We recall that $y_{0}\in H^{3}({\mathbb{T}},{\mathbb{R}})$ and that
$\mu_{1},\mu_{2}\in H^{3}({\mathbb{T}},{\mathbb{R}})$, thanks to Assumptions
II. Hence, from the definition of $y=\psi-\Phi$ and since $\psi$ and $\Phi$
are both solutions of (NHE), we deduce from _i._ of Proposition 2.1 that

$\sup_{\tau_{0}\leq t\leq\tau_{1}}\left\lVert
y(t)\right\rVert_{H^{1}}=\sup_{\tau_{0}\leq
t\leq\tau_{1}}\left\lVert\psi(t)-\Phi\right\rVert_{H^{1}}\leq
C\left(\left\lVert\psi_{0}-\Phi\right\rVert_{H^{1}}+\left\lVert
u^{1}-\hat{u}\right\rVert_{L^{2}(\tau_{1},\tau_{1})}\right)\leq
C\left(\left\lVert y_{0}\right\rVert_{H^{1}}+\left\lVert
v^{1}\right\rVert_{H^{1}(\tau_{1},\tau_{1})}\right).$ (4.29)

Thanks to (4.27), we conclude that

$\sup_{\tau_{0}\leq t\leq\tau_{1}}\left\lVert y(t)\right\rVert_{H^{1}}\leq
C\left(1+N(\tau_{1})\right)\left\lVert y_{0}\right\rVert_{H^{1}}.$

Now, we introduce $w:=y-\xi$ (we use the control $u^{1}$ for $y$ and $v^{1}$
for $\xi$ on the time interval $[\tau_{0},\tau_{1}]$) and we observe that $w$
solves the following problem

$\begin{cases}\partial_{t}w(t)-\partial^{2}_{x}w(t)-\kappa(p+1)\Phi^{p}w(t)+\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}y^{j}(t)=\langle
v^{1}(t),Q\rangle y(t),&t\in(\tau_{0},\tau_{1})\\\ w(\tau_{0})=0.\end{cases}$
(4.30)

Thanks to estimate (C.3) of Proposition C.1, we have that

$\sup_{t\in[\tau_{0},\tau_{1}]}\left\lVert w(t)\right\rVert_{H^{1}}\leq
A_{4}(\tau_{1},\left\lVert y_{0}\right\rVert_{H^{1}})\left\lVert
y_{0}\right\rVert^{2}_{H^{1}}.$ (4.31)

Observe that our initial condition $\psi_{0}$ satisfies $\psi_{0}\in\\{\psi\in
H^{3}({\mathbb{T}},{\mathbb{R}}):\|\psi-\Phi\|_{H^{1}}<R_{T}\\}$, with $R_{T}$
defined in (4.3). Therefore, since $R_{T}<1$, we have that $\left\lVert
y_{0}\right\rVert_{H^{1}}<1$ and moreover

$N(\tau_{1})\left\lVert y_{0}\right\rVert_{H^{1}}\leq
e^{\nu/\tau_{1}}e^{-6\Gamma_{0}/\tau_{1}}<1$

since $\Gamma_{0}>\nu$ (see definition (4.2)). Hence, we obtain that

$A_{4}(\tau_{1},\left\lVert y_{0}\right\rVert_{H^{1}})\leq K(\tau_{1})$ (4.32)

where

$K^{2}(\tau):=2\left(2\kappa\tau(p+1)^{2}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N(\tau)^{4}\right)+C_{Q}^{2}N(\tau)^{2}\left(1+N(\tau)^{2}\right)\right)\cdot\\\
e^{\tau\left(2\kappa(p+1)\Phi^{p}+\kappa(p+1)\sum_{j=2}^{p+1}\Phi^{p+1-j}+1\right)}.$
(4.33)

######  Remark 4.7.

Notice that for any $\tau\leq 1$ it is possible to prove that

$K(\tau)\leq e^{\Gamma_{0}/\tau}.$ (4.34)

Since $\tau_{1}\leq 1$, we deduce that (4.34) holds for $\tau=\tau_{1}$. Using
(4.32) and (4.34) (for $\tau=\tau_{1}$) in (4.31) and recalling that
$\xi(\tau_{1};\tau_{0},\xi_{0},v^{1})=0,$ we conclude that

$\left\lVert y(\tau_{1};\tau_{0},y_{0},u^{1})\right\rVert_{H^{1}}\leq
e^{\Gamma_{0}/\tau_{1}}e^{-12\Gamma_{0}/\tau_{1}}=e^{-11\Gamma_{0}/\tau_{1}}<1.$
(4.35)

$\mathbf{(}3)$ Induction argument. Inequality (4.36) enables us to apply an
iterative argument. In fact, we have just proved the first step of an
induction procedure which consists in building in consecutive time intervals
of the form $[\tau_{n-1},\tau_{n}]$, $n\geq 1$, a control
$u^{n}=(u_{\kappa},0,...,0,u_{1}^{n},u_{2}^{n})\in
H^{1}((\tau_{n-1},\tau_{n}),{\mathbb{R}}^{q+2})$ such that

$\begin{array}[]{ll}1.&\left\lVert
u^{n}\right\rVert_{H^{1}(\tau_{n-1},\tau_{n})}\leq N(T_{n})\left\lVert
y_{n-1}\right\rVert_{H^{1}},\\\
2.&\xi(\tau_{n};\tau_{n-1},y_{n-1},v^{n})=0,\\\ 3.&\left\lVert
y(\tau_{n};\tau_{n-1},y_{n-1},u^{n})\right\rVert_{H^{1}}\leq
e^{\left(\sum_{j=1}^{n}2^{n-j}j^{2}-2^{n}6\right)\Gamma_{0}/T_{1}},\\\
4.&\left\lVert
y(\tau_{n};\tau_{n-1},y_{n-1},u^{n})\right\rVert_{H^{1}}\leq\prod_{j=1}^{n}K(T_{j})^{2^{n-j}}\left\lVert
y_{0}\right\rVert^{2^{n}}_{H^{1}},\end{array}$ (4.36)

where $v^{n}:=u^{n}-\hat{u}$ and, thanks to Proposition 2.2,

$y_{n-1}:=y(\tau_{n-1},0,y_{0},q^{n-1})\in H^{3}({\mathbb{T}},{\mathbb{R}}),$
(4.37)
$q^{n-1}(t)=\sum_{j=1}^{n-1}u^{j}(t)\chi_{[\tau_{j-1},\tau_{j}]}(t)\qquad\text{(component-
wise).}$ (4.38)

Observe that, by construction

$y_{n}=y(\tau_{n};\tau_{n-1},y_{n-1},u^{n})\in
H^{3}({\mathbb{T}},{\mathbb{R}}),\qquad\forall\,n\geq 1.$

We underline that, thanks to the global well-posedness of the solution of
(NHE) (and so of (4.25)), we do not meet any problem of existence when
changing at each step of the proof the initial condition, the control and the
time interval.

Let us now prove the iterative step of the induction argument. We suppose
properties 1.–4. of (4.36) hold for each $j=1,\dots,n-1$. Hence, suppose we
have built controls $u^{j}=(u_{\kappa},0,...,u_{1}^{j},u_{2}^{j})\in
H^{1}((\tau_{j-1},\tau_{j}),{\mathbb{R}}^{q+2})$ such that 1.–4. of (4.36) are
satisfied. In particular, for $j=n-1$, assume that

$\begin{array}[]{ll}1.&\left\lVert
u^{n-1}\right\rVert_{H^{1}(\tau_{n-2},\tau_{n-1})}\leq N(T_{n-1})\left\lVert
y_{n-2}\right\rVert_{H^{1}}\\\
2.&\xi(\tau_{n-1};\tau_{n-2},y_{n-2},v^{n-1})=0\\\ 3.&\left\lVert
y(\tau_{n-1};\tau_{n-2},y_{n-2},u^{n-1})\right\rVert_{H^{1}}\leq
e^{\left(\sum_{j=1}^{n-1}2^{n-1-j}j^{2}-2^{n-1}6\right)\Gamma_{0}/T_{1}}\\\
4.&\left\lVert
y(\tau_{n-1};\tau_{n-2},y_{n-2},u^{n-1})\right\rVert_{H^{1}}\leq\prod_{j=1}^{n-1}K(T_{j})^{2^{n-1-j}}\left\lVert
y_{0}\right\rVert^{2^{n-1}}_{H^{1}}.\end{array}$ (4.39)

We now prove the existence of
$u^{n}=(u_{\kappa},0,...,0,u_{1}^{n},u_{2}^{n})\in
H^{1}((\tau_{n-1},\tau_{n}),{\mathbb{R}}^{q+2})$ such that 1.–4. of (4.36) are
fulfilled. Define $q^{n-1}$ and $y_{n-1}$ as in (4.38) and (4.37). We apply
Proposition 4.6 to the linear system (4.4) with $(s_{0},s_{1})=(0,T_{n})$ and
$\xi(0)=y_{n-1}$. The result ensures the existence of a control
$\tilde{v}^{n}=(0,...,0,\tilde{u}_{1}^{n},\tilde{u}_{2}^{n})\in
H^{1}_{0}((0,T_{n}),{\mathbb{R}}^{q+2})$ such verifies

$\left\lVert\tilde{v}^{n}\right\rVert_{H^{1}(0,T_{n})}\leq N(T_{n})\left\lVert
y_{n-1}\right\rVert_{H^{1}},$

with $N(T_{n})\leq e^{\nu/T_{n}}$ and

$\xi(T_{n};0,y_{n-1},\tilde{v}^{n})=0.$

We set
$\tilde{u}^{n}(s):=\hat{u}+\tilde{v}^{n}(s)=(u_{\kappa},0,...,0,\tilde{u}_{1}^{n},\tilde{u}_{2}^{n})$
with $s\in[0,T_{n}].$ Consider problem (4.25) for $(s_{0},s_{1})=(0,T_{n})$
with $y(0)=y_{n-1}$. By using the control $\tilde{u}^{n}$ in (4.25), we get
(4.28) in $[0,T_{n}]$ with control $\tilde{v}^{n}$ instead of $v^{1}$. We
denote by $y(s;0,y_{n-1},\tilde{u}^{n})$ with $s\in[0,T_{n}]$ its solution.
Observe that, since

$y_{n-1}=y(\tau_{n-1};0,y_{0},q_{n-1})=y(\tau_{n-1};\tau_{n-2},y_{n-2},u^{n-1})\in
H^{3}({\mathbb{T}},{\mathbb{R}}),$

from 3. of (4.39) we deduce that

$N(T_{n})\left\lVert y_{n-1}\right\rVert_{H^{1}}\leq e^{\nu
n^{2}/T_{1}}e^{\left(\sum_{j=1}^{n-1}2^{n-1-j}j^{2}-2^{n-1}6\right)\Gamma_{0}/T_{1}}\leq
e^{-(2n+3)\Gamma_{0}/T_{1}}<1,$ (4.40)

where we have used that $\nu<\Gamma_{0}$ and the identity

$\sum_{j=0}^{n}\frac{j^{2}}{2^{j}}=2^{-n}(-n^{2}-4n+6(2^{n}-1)),\qquad\forall\,n\geq
0.$

Now, for $s\in[0,T_{n}]$, we define

$w(s):=y(s;0,y_{n-1},\tilde{u}^{n})-\xi(s;0,y_{n-1},\tilde{v}^{n}).$

Thus, we apply Proposition C.1, and we deduce that

$\left\lVert y(T_{n};0,y_{n-1},\tilde{u}^{n})\right\rVert_{H^{1}}=\left\lVert
w(T_{n};0,\tilde{v}^{n})\right\rVert_{H^{1}}\leq A_{4}(T_{n};\left\lVert
y_{n-1}\right\rVert_{H^{1}})\left\lVert y_{n-1}\right\rVert_{H^{1}}^{2}.$

We shift forward time interval into $[\tau_{n-1},\tau_{n}]$ and we define

$u^{n}(t):=\tilde{u}^{n}(t-\tau_{n-1}),\qquad
v^{n}(t):=\tilde{v}^{n}(t-\tau_{n-1}),\qquad t\in(\tau_{n-1},\tau_{n}),$

and we obtain

$\left\lVert u^{n}\right\rVert_{H^{1}(\tau_{n-1},\tau_{n})}\leq
N(T_{n})\left\lVert y_{n-1}\right\rVert_{H^{1}},$

and

$\xi(\tau_{n};\tau_{n-1},y_{n-1},v^{n})=\xi(T_{n};0,y_{n-1},\tilde{v}^{n})=0.$

Thus, 1. and 2. of (4.36) are fulfilled.

Recalling definition (4.33), property (4.40) and $\nu<\Gamma_{0}$, we deduce
that

$\left\lVert
y(\tau_{n};\tau_{n-1},y_{n-1},u^{n})\right\rVert_{H^{1}}=\left\lVert
y(T_{n};0,y_{n-1},\tilde{u}^{n})\right\rVert_{H^{1}}\leq K(T_{n})\left\lVert
y_{n-1}\right\rVert^{2}_{H^{1}}.$ (4.41)

Thus, using 3. of (4.39) and the estimate above, we obtain

$\left\lVert y(\tau_{n};\tau_{n-1},y_{n-1},u^{n})\right\rVert_{H^{1}}\leq
e^{n^{2}\Gamma_{0}/T_{1}}\left[e^{\left(\sum_{j=1}^{n-1}2^{n-1-j}j^{2}-2^{n-1}6\right)\Gamma_{0}/T_{1}}\right]^{2}=e^{\left(\sum_{j=1}^{n}2^{n-1-j}j^{2}-2^{n-1}6\right)\Gamma_{0}/T_{1}},$

that is, 3. of (4.36) is satisfied. Finally, using again (4.41) and thanks to
4. of (4.39) we conclude that

$\left\lVert y(\tau_{n};\tau_{n-1},y_{n-1},u^{n})\right\rVert_{H^{1}}\leq
K(T_{n})\left[\prod_{j=1}^{n-1}K(T_{j})^{2^{n-1-j}}\left\lVert
y_{0}\right\rVert^{2^{n-1}}_{H^{1}}\right]^{2}=\prod_{j=1}^{n}K(T_{j})^{2^{n-j}}\left\lVert
y_{0}\right\rVert^{2^{n}}_{H^{1}},$

which is exactly 4. of (4.36). The induction argument is therefore concluded.

$\mathbf{(}4)$ Conclusion. Let us now observe that, from 4. of (4.36), for
every $n\in{\mathbb{N}}$ it holds that

$\begin{split}\left\lVert
y(\tau_{n};\tau_{n-1},y_{n-1},u^{n})\right\rVert_{H^{1}}&\leq\prod_{j=1}^{n}K(T_{j})^{2^{n-j}}\left\lVert
y_{0}\right\rVert^{2^{n}}_{H^{1}}\leq\prod_{j=1}^{n}\left(e^{\Gamma_{0}j^{2}/T_{1}}\right)^{2^{n-j}}\left\lVert
y_{0}\right\rVert^{2^{n}}_{H^{1}}\\\
&=e^{(\Gamma_{0}2^{n}/T_{1})\sum_{j=1}^{n}j^{2}/2^{j}}\left\lVert
y_{0}\right\rVert^{2^{n}}_{H^{1}}\leq
e^{(\Gamma_{0}2^{n}/T_{1})\sum_{j=1}^{\infty}j^{2}/2^{j}}\left\lVert
y_{0}\right\rVert^{2^{n}}_{H^{1}}\\\
&\leq\left(e^{6\Gamma_{0}/T_{1}}\left\lVert
y_{0}\right\rVert_{H^{1}}\right)^{2n},\end{split}$

where in the last inequality we have used that
$\sum_{j=1}^{\infty}\frac{j^{2}}{2^{j}}=6.$ By definition, the above estimate
reads as

$\left\lVert
y(\tau_{n};0,y_{0},q^{n})\right\rVert_{H^{1}}\leq\left(e^{6\Gamma_{0}/T_{1}}\left\lVert
y_{0}\right\rVert_{H^{1}}\right)^{2n},$

with

$q^{n}(t)=\sum_{j=1}^{n}u^{j}(t)\chi_{(\tau_{j-1},\tau_{j})}(t)\qquad\text{(component-
wise)}.$

Taking the limit as $n\to+\infty$, we deduce that

$\left\lVert\psi(\pi^{2}T_{1}/6;0,\psi_{0},q^{\infty})-\Phi\right\rVert_{H^{1}}=\left\lVert
y(\pi^{2}T_{1}/6;0,\psi_{0},q^{\infty})\right\rVert_{H^{1}}=\left\lVert
y(T_{f};0,\psi_{0},q^{\infty})\right\rVert_{H^{1}}\leq 0$

thanks to the definition (4.3) of $R_{T}$. Thus, we construct a control $u\in
H^{1}_{loc}({\mathbb{R}}^{+},{\mathbb{R}}^{q+2})$

$u(t):=\begin{cases}\sum_{j=1}^{\infty}u^{j}(t)\chi_{(\tau_{j-1},\tau_{j})}(t),&0<t\leq
T_{f},\\\ 0&t>T_{f},\end{cases}\qquad\text{(component-wise)},$

such that, at time $T_{f}\leq T$, the solution of (NHE) reaches exactly the
ground state, that is,

$\psi(T_{f};0,\psi^{0},u)=\Phi.$

Furthermore, we can derive a bound for the $H^{1}$-norm of the control

$\left\lVert
u\right\rVert^{2}_{H^{1}(0,T_{f})}\leq\sum_{n=1}^{\infty}\left\lVert
u^{n}\right\rVert^{2}_{H^{1}(0,T_{f})}\leq\sum_{n=1}^{\infty}\left(N(T_{n})\left\lVert
y_{n-1}\right\rVert_{H^{1}}\right)^{2}\leq\sum_{n=1}^{\infty}e^{-2(2n+3)\Gamma_{0}/T_{1}}=\frac{e^{-\pi^{2}\Gamma_{0}/T_{f}}}{e^{2\pi^{1}\Gamma_{0}/(3T_{f})}-1}.$

where we have used (4.40). Since $T_{f}\leq T$, we easily obtain (4.1). ∎

We are finally ready to prove Theorem 4.1 and Main Theorem B.

###### Proof of Theorem 4.1.

The result is obtained by combining Theorem 3.3 with $s=3$ and Theorem 4.3. In
details, for every $\psi_{0}\in H^{3}({\mathbb{T}},{\mathbb{R}})$ strictly
positive, Theorem 3.3 allows defining a dynamics steering $\psi_{0}$ close to
$\Phi$ in the $H^{3}$ metric in an arbitrarily small time. If we are close
enough to $\Phi$, we can use the local controllability result of Theorem 4.3.
Thus, the solution of (NHE) reaches the target $\Phi$ in any finite time. ∎

###### Proof of Main Theorem B.

Main Theorem B is a direct consequence of Theorem 4.1 and the density property
of $\mathcal{H}_{\infty}$ in $H^{1}({\mathbb{T}},{\mathbb{R}})$ ensured by
Proposition 3.1. ∎

## Appendices

##  Appendix A Proof of Proposition 2.1

In this section, we shall prove the existence and uniqueness of solutions of
(NHE).

###### Proof.

$\mathbf{(}1)$ Existence and uniqueness of solutions. Let $s>d/2$ and
$\psi_{0}\in H^{s}({\mathbb{T}}^{d},{\mathbb{R}})$. For the sake of shortness,
we consider $\kappa=1$. However, the proof remains valid in the general case.
We are going to show that there exists $t_{1}>0$ such that the Cauchy problem
(NHE) admits a unique solution $\psi\in
C^{0}([0,t_{1}],H^{s}({\mathbb{T}}^{d},{\mathbb{R}}))$. We define the
following quantities

$M:=\sup\\{\left\lVert
e^{t\Delta}-I\right\rVert_{\mathcal{L}(H^{s}({\mathbb{T}}^{d},{\mathbb{R}}))},\,0\leq
t\leq 1\\},\ \ \ \ \ \ \
r(\psi_{0}):=2M\left\lVert\psi_{0}\right\rVert_{H^{s}}$ $C(Q):=\max_{1\leq
i\leq d}\left\lVert Q_{i}\right\rVert_{H^{s}},\qquad
C(d,u,Q):=\sqrt{2d}\left\lVert u\right\rVert_{L^{2}(0,1)}C(Q).$

Observe that, since $s>d/2$, we deduce that the embedding
$H^{s}({\mathbb{T}}^{d},{\mathbb{R}})\hookrightarrow
C^{0}({\mathbb{T}}^{d},{\mathbb{R}})$ is continuous, that is, there exists a
constant $C({\mathbb{T}}^{d})$ such that

$\sup_{x\in{\mathbb{T}}^{d}}|y(x)|\leq C({\mathbb{T}}^{d})\left\lVert
y\right\rVert_{H^{s}},\quad\forall\,y\in
H^{s}({\mathbb{T}}^{d},{\mathbb{R}}).$ (A.1)

We now define

$\delta:=\min\left\\{1,\frac{(r(\psi_{0}))^{2}}{4C({\mathbb{T}}^{d})^{2}\left(C(d,u,Q)(r(\psi_{0})+\left\lVert\psi_{0}\right\rVert_{H^{s}})+2^{p}((r(\psi_{0}))^{p+1}+\left\lVert\psi_{0}\right\rVert^{p+1}_{H^{s}})\right)^{2}}\right\\},$
(A.2)

and set $t_{1}=\delta$.

We denote $B:=B_{C^{0}([0,t_{1}],H^{s}))}(\psi_{0},r(\psi_{0}))$ the ball in
the space $C^{0}([0,t_{1}],H^{s}({\mathbb{T}}^{d},{\mathbb{R}}))$ of center
$\psi_{0}$ and radius $r(\psi_{0})$. For every $\psi\in B$ we define the
following function

$\Phi(\psi)(t):=e^{t\Delta}\psi_{0}+\int_{0}^{t}e^{(t-s)\Delta}\left(\langle
u(s),Q(x)\rangle\psi(s,x)+\psi(s,x)^{p+1}\right)ds.$ (A.3)

Let us show that $\Phi$ maps $B$ into itself:

$\begin{split}\left\lVert\Phi(\psi)(t)-\psi_{0}\right\rVert_{H^{s}}&\leq\left\lVert
e^{t\Delta}\psi_{0}-\psi_{0}\right\rVert_{H^{s}}\quad+\left\lVert\int_{0}^{t}e^{(t-s)\Delta}\left(\langle
u(s),Q\rangle\psi(s)+\psi(s)^{p+1}\right)ds\right\rVert_{H^{s}}\\\ &\leq
M\left\lVert\psi_{0}\right\rVert_{H^{s}}+\int_{0}^{t}\left\lVert\langle
u(s),Q\rangle\psi(s)\right\rVert_{H^{s}}+\left\lVert\psi(s)^{p+1}\right\rVert_{H^{s}}ds\\\
&\leq
M\left\lVert\psi_{0}\right\rVert_{H^{s}}+C({\mathbb{T}}^{d})\left(C(Q)\int_{0}^{t}\sum_{i=1}^{q}|u_{i}(s)|\left\lVert\psi(s)\right\rVert_{H^{s}}ds+\int_{0}^{t}\left\lVert\psi(s)\right\rVert_{H^{s}}^{p+1}ds\right)\\\
&\leq
M\left\lVert\psi_{0}\right\rVert_{H^{s}}+C({\mathbb{T}}^{d})\sqrt{d}C(Q)\left(\int_{0}^{t}\sum_{i=1}^{q}|u_{i}(s)|^{2}ds\right)^{1/2}\left(2\int_{0}^{t}\left\lVert\psi(s)-\psi_{0}\right\rVert^{2}_{H^{s}}+\left\lVert\psi_{0}\right\rVert^{2}_{H^{s}}ds\right)^{1/2}\\\
&\quad\quad+C({\mathbb{T}}^{d})2^{p}\int_{0}^{t}\left\lVert\psi(s)-\psi_{0}\right\rVert_{H^{s}}^{p+1}+\left\lVert\psi_{0}\right\rVert^{p+1}_{H^{s}}ds.\end{split}$

Thus, we deduce that $\Phi(\psi)\in B$ since

$\sup_{t\in[0,t_{1}]}\left\lVert\Phi(\psi)(t)-\psi_{0}\right\rVert_{H^{s}}\leq\frac{r(\psi_{0})}{2}+C({\mathbb{T}}^{d})\sqrt{2d}C(Q)\left\lVert
u\right\rVert_{L^{2}(0,1)}\left(\sup_{t\in[0,t_{1}]}\left\lVert\psi(t)-\psi_{0}\right\rVert_{H^{s}}+\left\lVert\psi_{0}\right\rVert_{H^{s}}\right)\sqrt{t_{1}}\\\
+C({\mathbb{T}}^{d})2^{p}\left(\sup_{t\in[0,t_{1}]}\left\lVert\psi(t)-\psi_{0}\right\rVert_{H^{s}}^{p+1}+\left\lVert\psi_{0}\right\rVert^{p+1}_{H^{s}}\right)\sqrt{t_{1}}\leq\frac{r(\psi_{0})}{2}+\frac{r(\psi_{0})}{2}=r(\psi_{0}).$

Now, we show that $\Phi$ is a contraction over $B$. Let $\psi,\phi\in B$, then

$\begin{split}\left\lVert\Phi(\psi)(t)-\Phi(\phi)(t)\right\rVert_{H^{s}}&=\left\lVert\int_{0}^{t}\langle
u(s),Q(x)\rangle(\psi(s)-\phi(s))+\psi(s)^{p+1}-\phi(s)^{p+1}ds\right\rVert_{H^{s}}\\\
&\leq\int_{0}^{t}\left\lVert\langle
u(s),Q(x)\rangle(\psi(s)-\phi(s))\right\rVert_{H^{s}}ds\\\
&\quad+C({\mathbb{T}}^{d})\int_{0}^{t}\left\lVert\psi(s)-\phi(s)\right\rVert_{H^{s}}\sum_{j=0}^{p}\left\lVert\psi(s)\right\rVert^{j}_{H^{s}}\left\lVert\phi(s)\right\rVert^{p-j}_{H^{s}}ds\\\
&\leq C(Q)\sqrt{2d}\left\lVert
u\right\rVert_{L^{2}((0,1),{\mathbb{R}}^{q+2})}\left(\int_{0}^{t}\left\lVert\psi(s)-\phi(s)\right\rVert^{2}_{H^{s}}ds\right)^{1/2}\\\
&\quad+C({\mathbb{T}}^{d})2^{p-2}\int_{0}^{t}\left\lVert\psi(s)-\phi(s)\right\rVert_{H^{s}}L(\phi,\phi)ds\end{split}$

where

$L(\psi,\phi):=\sum_{j=0}^{p}\left(\left\lVert\psi(s)-\psi_{0}\right\rVert^{j}_{H^{s}}+\left\lVert\psi_{0}\right\rVert^{j}_{H^{s}}\right)\left(\left\lVert\phi(s)-\psi_{0}\right\rVert^{p-j}_{H^{s}}+\left\lVert\psi_{0}\right\rVert^{p-j}_{H^{s}}\right)$

Therefore, we get that

$\begin{split}\sup_{0\leq t\leq
t_{1}}\left\lVert\Phi(\psi)(t)-\Phi(\phi)(t)\right\rVert_{H^{s}}&\leq
C(Q)\sqrt{2d}\left\lVert u\right\rVert_{L^{2}((0,1)}\sqrt{t_{1}}\sup_{0\leq
t\leq t_{1}}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}\\\
&\quad+C({\mathbb{T}}^{d})2^{p-2}\sqrt{t_{1}}\sup_{0\leq t\leq
t_{1}}L(\psi,\phi)\sup_{0\leq t\leq
t_{1}}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}\\\
&\leq\left(C(d,u,Q)+C({\mathbb{T}}^{d})2^{p-2}\tilde{L}(\psi_{0})\right)\sqrt{t_{1}}\sup_{0\leq
t\leq t_{1}}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}},\end{split}$

where

$\tilde{L}(\psi_{0}):=\sum_{j=0}^{p}\left(r(\psi_{0})^{j}+\left\lVert\psi_{0}\right\rVert^{j}_{H^{s}}\right)\left(r(\psi_{0})^{p-j}+\left\lVert\psi_{0}\right\rVert^{p-j}_{H^{s}}\right).$

With the same kind of computation one can prove that

$\sup_{0\leq t\leq
t_{1}}\left\lVert\Phi^{2}(\psi)(t)-\Phi^{2}(\phi)(t)\right\rVert_{H^{s}}\leq\left(C(d,u,Q)+C({\mathbb{T}}^{d})2^{p-2}\tilde{L}(\psi_{0})\right)^{2}\frac{(\sqrt{t_{1}})^{2}}{\sqrt{2}}\sup_{0\leq
t\leq t_{1}}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}},$

and, iterating the procedure, one shows that

$\sup_{0\leq t\leq
t_{1}}\left\lVert\Phi^{n}(\psi)(t)-\Phi^{n}(\phi)(t)\right\rVert_{H^{s}}\leq\left(C(d,u,Q)+C({\mathbb{T}}^{d})2^{p-2}\tilde{L}(\psi_{0})\right)^{n}\frac{(\sqrt{t_{1}})^{n}}{\sqrt{n!}}\sup_{0\leq
t\leq t_{1}}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}.$

For $n$ large enough, it holds that

$\left(C(d,u,Q)+C({\mathbb{T}}^{d})2^{p-2}\tilde{L}(\psi_{0})\right)^{n}\frac{(\sqrt{t_{1}})^{n}}{\sqrt{n!}}<1.$

Hence, form a well-known corollary of the Banach fixed point Theorem, we
deduce that $\Phi$ is a contraction over $B$ and so there exists a unique
fixed point $\psi\in B$ which is the solution of (NHE). Furthermore, it holds
that

$\sup_{t\in[0,t_{1}]}\left\lVert\psi(t)\right\rVert_{H^{s}}\leq(2M+1)\left\lVert\psi_{0}\right\rVert_{H^{s}}.$

We have just proved that if $\psi$ is solution of (NHE) in $[0,\tau]$, then we
can extend it into $[0,\tau+\delta(\tau)]$. Indeed, by defining the quantities

$M(\tau):=\sup_{\tau\leq t\leq\tau+1}\left\\{\left\lVert
e^{t\Delta}-I\right\rVert_{\mathcal{L}(H^{s}({\mathbb{T}}^{d},{\mathbb{R}}))}\right\\},\quad
r(\tau,\psi(\tau)):=2M(\tau)\left\lVert\psi(\tau)\right\rVert_{H^{s}},\quad
C(d,u,Q):=\sqrt{2d}\left\lVert u\right\rVert_{L^{2}(\tau,\tau+1)}C(Q),$

and

$\delta(\tau):=\min\left\\{1,\frac{(r(\psi(\tau))^{2}}{4C({\mathbb{T}}^{d})^{2}\left(C(d,u,Q)(r(\psi(\tau))+\left\lVert\psi(\tau)\right\rVert_{H^{s}})+2^{p}((r(\psi(\tau)))^{p+1}+\left\lVert\psi(\tau)\right\rVert^{p+1}_{H^{s}})\right)^{2}}\right\\},$

then one just sets $\psi(t)=\zeta(t)$ for $t\in[\tau,\tau+\delta(\tau)]$,
where

$\zeta(t)=e^{(t-\tau)\Delta}\psi(\tau)+\int_{\tau}^{t}e^{(t-s)\Delta}\left(\langle
u(s),Q(x)\rangle\psi(s,x)+\psi(s,x)^{p+1}\right)ds.$

Let $[0,{\cal T})$ be the maximal interval of existence of $\psi$, solution of
the (NHE). If ${\cal T}<+\infty$, then
$\left\lVert\psi(t)\right\rVert_{H^{s}}\to+\infty$ as $t\to{\cal T}^{-}$,
otherwise $\psi$ could be extended, which contradicts the maximality of ${\cal
T}$. Observe that, for any $0<T<{\cal T}$,

$\sup_{t\in[0,T]}\left\lVert\psi(t)\right\rVert_{H^{s}}\leq
C\left\lVert\psi_{0}\right\rVert_{H^{s}}.$ (A.4)

$\mathbf{(}2)$ Proof of the continuity (2.1) and the stability (2.2). Let
$\psi,\phi\in C^{0}([0,T],H^{s}({\mathbb{T}}^{d},{\mathbb{R}}))$, with $0\leq
T\leq\min\\{{\cal T}(\psi_{0}),{\cal T}(\phi_{0})\\}$, be the solutions of the
(NHE) corresponding to the initial conditions $\psi_{0}$ and $\phi_{0}$ and
controls $u$ and $v$, respectively. Then,

$\begin{split}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}&\leq\left\lVert\psi_{0}-\phi_{0}\right\rVert_{H^{s}}\\\
&\quad+\int_{0}^{t}(\left\lVert\langle u(s),Q\rangle\psi(s)-\langle
v(t),Q\rangle\phi(s)\right\rVert_{H^{s}}+\left\lVert\psi(s)^{p+1}-\phi(s)^{p+1}\right\rVert_{H^{s}})ds\\\
&\leq\left\lVert\psi_{0}-\phi_{0}\right\rVert_{H^{s}}+\int_{0}^{t}\left\lVert\langle
u(s),Q\rangle(\psi(s)-\phi(s))\right\rVert_{H^{s}}ds+\\\
&\quad+\int_{0}^{t}\left\lVert(\langle u(s),Q\rangle-\langle
v(s),Q\rangle)\phi(s)\right\rVert_{H^{s}}ds\\\
&\quad+C({\mathbb{T}}^{d})\int_{0}^{t}\left\lVert\psi(s)-\phi(s)\right\rVert_{H^{s}}\sum_{j=0}^{p}\left\lVert\psi(s)\right\rVert^{j}_{H^{s}}\left\lVert\phi(s)\right\rVert^{p-j}_{H^{s}}ds\\\
&\leq\left\lVert\psi_{0}-\phi_{0}\right\rVert_{H^{s}}+C(Q)\sqrt{2d}\left\lVert
u\right\rVert_{L^{2}(0,t)}\sqrt{t}\sup_{0\leq s\leq
t}\left\lVert\psi(s)-\phi(s)\right\rVert_{H^{s}}\\\
&\quad+C(Q)\sqrt{2d}\left\lVert
u-v\right\rVert_{L^{2}(0,t)}\sqrt{t}\sup_{0\leq s\leq
t}\left\lVert\phi(s)\right\rVert_{H^{s}}\\\
&\quad+C({\mathbb{T}}^{d})\sqrt{t}\left(\sum_{j=0}^{p}\sup_{0\leq s\leq
t}\left\lVert\psi(s)\right\rVert^{j}_{H^{s}}\sup_{0\leq s\leq
t}\left\lVert\phi(s)\right\rVert^{p-j}_{H^{s}}\right)\sup_{0\leq s\leq
t}\left\lVert\psi(s)-\phi(s)\right\rVert_{H^{s}}\end{split}$

Therefore, we get the existence of $C_{1},C_{2},C_{3}>0$, only depending on
the parameters of the problem, such that

$\begin{split}\sup_{0\leq t\leq
T}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}&\leq\left\lVert\psi_{0}-\phi_{0}\right\rVert_{H^{s}}+\left(C(Q)\sqrt{2d}\sqrt{T}\sup_{0\leq
t\leq T}\left\lVert\phi(t)\right\rVert_{H^{s}}\right)\left\lVert
u-v\right\rVert_{L^{2}(0,T)}\\\
&\quad+\left(C(d,u,Q)+C({\mathbb{T}}^{d})\sum_{j=0}^{p}\sup_{0\leq t\leq
T}\left\lVert\psi(t)\right\rVert^{j}_{H^{s}}\sup_{0\leq t\leq
T}\left\lVert\phi(t)\right\rVert^{p-j}_{H^{s}}\right)\sqrt{T}\sup_{0\leq t\leq
T}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}\\\
&\leq\left\lVert\psi_{0}-\phi_{0}\right\rVert_{H^{s}}+C_{1}R\sqrt{T}\left\lVert
u-v\right\rVert_{L^{2}(0,T)}+(C_{2}+C_{3}R^{p})\sqrt{T}\sup_{0\leq t\leq
T}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}\end{split}$

where we have used (A.4) and that $\psi_{0},\phi_{0}\in B_{H^{s}}(0,R)$. If
$(C_{2}+C_{3}R^{p})\sqrt{T}<1$, then we obtain the validity of (2.1) and then

$\sup_{0\leq t\leq T}\left\lVert\psi(t)-\phi(t)\right\rVert_{H^{s}}\leq
C\left(\left\lVert\psi_{0}-\phi_{0}\right\rVert_{H^{s}}+\left\lVert
u-v\right\rVert_{L^{2}(0,T)}\right).$

Otherwise, we can subdivide the interval $[0,T]$ into subintervals where
inequality $(C_{2}+C_{3}R^{p})\sqrt{T}<1$ holds and obtain the result.
Finally, the same techniques also imply the validity of (2.2). ∎

##  Appendix B Global well-posedness and proof of Proposition 2.2

This appendix aims to prove Proposition 2.2. We show that for more regular
initial conditions, $\psi_{0}\in H^{3}({\mathbb{T}},{\mathbb{R}})$, the
solution of the unidimensional problem (NHE) is global in time, that is, the
maximum time of existence is ${\cal T}(\psi_{0})=+\infty$. First, we observe
that if $\psi_{0}\in H^{1}({\mathbb{T}},{\mathbb{R}})$ then, from Proposition
2.1, there exists ${\cal T}(\psi_{0})$ such that for every $0<T<{\cal
T}(\psi_{0})$ there exists a unique solution $\psi\in
C^{0}([0,T],H^{1}({\mathbb{T}},{\mathbb{R}}))$. Second, if
$\mathcal{T}(\psi_{0})=+\infty$ then the solution is global in time, otherwise

$\left\lVert\psi(t)\right\rVert_{H^{1}}\to+\infty\qquad\text{as
}t\to\mathcal{T}(\psi_{0}).$

It can be proved, thanks to the analiticity of the semigroup generated by
$\Delta$ and applying [14, Theorem 3.1, p. 143] to the fixed point argument
used in the proof of Proposition 2.1, that

$\psi\in H^{1}((0,T),L^{2}({\mathbb{T}},{\mathbb{R}}))\cap
L^{2}((0,T),H^{2}({\mathbb{T}},{\mathbb{R}})).$

Let us consider $\Psi:=\psi_{t}$ that formally solves the problem

$\begin{cases}\partial_{t}\Psi-\Delta\Psi+\kappa(p+1)\psi^{p}\Psi=\langle
u^{\prime}(t),Q(x)\rangle\psi+\langle u(t),Q(x)\rangle\Psi\\\
\Psi(0)=\Psi_{0}:=\Delta\psi_{0}-\kappa\,\psi_{0}^{p+1}+\langle
u(0),Q(x)\rangle\psi_{0}.\end{cases}$ (B.1)

Let us now proved the following result which is necessary for the proof of
Proposition 2.2.

######  Proposition B.1.

Let $\psi_{0}\in H^{3}({\mathbb{T}},{\mathbb{R}})$, $Q\in
H^{3}({\mathbb{T}},{\mathbb{R}}^{q+2})$, $u\in
H^{1}_{\text{loc}}((0,+\infty),{\mathbb{R}}^{q+2})$, $0<T<{\cal
T}(\psi_{0},u)$ and $\kappa\geq 0$. Then, there exists a unique solution
$\Psi\in L^{2}((0,T),H^{1}({\mathbb{T}},{\mathbb{R}}))$ of (B.1) given by

$\Psi(t)=e^{t\Delta}\Psi_{0}+\int_{0}^{t}e^{(t-s)\Delta}\left(\langle
u^{\prime}(s),Q(x)\rangle\psi(s)+\langle
u(s),Q(x)\rangle\Psi(s)-\kappa(p+1)\psi(s)^{p}\Psi(s)\right)ds,$

where $\psi$ is the unique solution of (NHE) with initial condition
$\psi_{0}$. Furthermore, it holds that $\Psi=\psi_{t}$.

###### Proof.

$\mathbf{(}1)$ Existence and uniqueness of solutions. We already know from the
local well-posedness of (NHE) that $\Psi=\psi_{t}\in
L^{2}((0,T),L^{2}({\mathbb{T}},{\mathbb{R}}))$. We now consider equation
(B.1). In order to apply a fix point argument, for every $\xi\in
C([0,T],H^{1}({\mathbb{T}},{\mathbb{R}}))$, we consider the following map

$\Phi(\xi)(t):=e^{t\Delta}\Psi_{0}+\int_{0}^{t}e^{(t-s)\Delta}\Big{(}\langle
u^{\prime}(s),Q(x)\rangle\psi(s)+\langle
u(s),Q(x)\rangle\xi(s)-\kappa(p+1)\psi(s)^{p}\xi(s)\Big{)}ds,$

where $\psi\in H^{1}((0,T),L^{2}({\mathbb{T}},{\mathbb{R}}))\cap
L^{2}((0,T),H^{2}({\mathbb{T}},{\mathbb{R}}))\cap
C([0,T],H^{3}({\mathbb{T}},{\mathbb{R}}))$ is the solution of (NHE) with
initial condition $\psi_{0}\in H^{3}({\mathbb{T}},{\mathbb{R}})$. We first
observe that $\Psi_{0}=\Delta\psi_{0}-\kappa\psi_{0}^{p}\psi_{0}+\langle
u(0),Q(x)\rangle\psi_{0}$ is well-defined and is in
$H^{1}({\mathbb{T}},{\mathbb{R}})$. Let us first prove that $\Phi$ maps
$C([0,T],H^{1}({\mathbb{T}},{\mathbb{R}}))$ into itself. Since

$f(\cdot):=\langle u^{\prime}(\cdot),Q\rangle\psi+\langle
u(\cdot),Q\rangle\xi(\cdot)-\kappa(p+1)\psi(\cdot)^{p}\xi(\cdot)\in
L^{2}((0,T),L^{2}({\mathbb{T}},{\mathbb{R}})),$

from [14, Theorem 3.1, p. 143] we deduce that $\Phi(\xi)\in
H^{1}([0,T],L^{2}({\mathbb{T}},{\mathbb{R}}))\cap
L^{2}((0,T),H^{2}({\mathbb{T}},{\mathbb{R}}))$ for every $\xi\in
C([0,T],H^{1}({\mathbb{T}},{\mathbb{R}}))$. Thus, we deduce from [32,
Proposition 2.1, p. 22 and Theorem 3.1, p. 23] that $\Phi$ maps
$C([0,T],H^{1}({\mathbb{T}},{\mathbb{R}}))$ into itself. We now prove that
$\Phi$ is a contraction. Let $\xi,\tilde{\xi}\in
C([0,T],H^{1}({\mathbb{T}},{\mathbb{R}}))$. Then,

$\begin{split}&\sup_{t\in[0,T]}\left\lVert\Phi(\xi)(t)-\Phi(\tilde{\xi})\right\rVert_{H^{1}}=\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}e^{(t-s)\Delta}\left(\langle
u(s),Q(x)\rangle(\xi(s)-\tilde{\xi}(s))-\kappa(p+1)\psi(s)^{p}(\xi(s)-\tilde{\xi}(s))\right)ds\right\rVert_{H^{1}}\\\
&\quad\leq C\left\lVert\left(\langle
u(\cdot),Q\rangle-\kappa(p+1)|\psi(s)|^{p}\right)(\xi(\cdot)-\tilde{\xi}(\cdot))\right\rVert_{L^{2}((0,T),L^{2}({\mathbb{T}}))}\leq
C\left(\int_{0}^{t}\left\lVert\xi(s)-\tilde{\xi}(s)\right\rVert_{L^{2}}^{2}ds\right)^{1/2}\\\
&\quad\leq
C\sqrt{T}\sup_{t\in[0,T]}\left\lVert\xi(t)-\tilde{\xi}(t)\right\rVert_{L^{2}}\leq
C\sqrt{T}\sup_{t\in[0,T]}\left\lVert\xi(t)-\tilde{\xi}(t)\right\rVert_{H^{1}}\end{split}$

If $C\sqrt{T}<1$ then $\Phi$ is a contraction. Otherwise, one can divide the
interval $[0,T]$ in a finite number of sub-intervals where $\Phi$ is a
contraction and conclude the argument. So, we deduce that $\Phi$ admits a
unique fix point $\Psi$ in the space
$C([0,T],H^{1}({\mathbb{T}},{\mathbb{R}}))$ which is the solution of (B.1).

$\mathbf{(}1)$ Proof of the identity $\Psi=\psi_{t}$. We now prove that the
unique solution of (B.1) is indeed $\psi_{t}$. Let $t\in[0,T]$, $\tau>0$ such
that $0<t+\tau<T$, and consider the difference

$\frac{\psi(t+\tau)-\psi(t)}{\tau}-\Psi(t).$

Our aim is to prove that the above quantity converges to $0$ as $\tau\to 0$.
Thanks to the expression of $\psi$ and $\Psi$ as mild solutions, we get

$\begin{split}&\frac{\psi(t+\tau)-\psi(t)}{\tau}-\Psi(t)=\frac{1}{\tau}\left\\{e^{(t+\tau)\Delta}\psi_{0}-e^{t\Delta}\psi_{0}\right\\}+\frac{1}{\tau}\int_{0}^{\tau}e^{(t+\tau-s)\Delta}\left[-\kappa\,\psi(s)^{p+1}+\langle
u(s),Q\rangle\psi(s)\right]ds\\\
&\qquad+\frac{1}{\tau}\int_{0}^{t}e^{(t-s)\Delta}\left[-\kappa\,\psi(s+\tau)^{p+1}+\langle
u(s+\tau),Q\rangle\psi(s+\tau)+\kappa\,\psi(s)^{p+1}-\langle
u(s),Q\rangle\psi(s)\right]ds\\\
&\qquad-e^{t\Delta}(\Delta\psi_{0}-\kappa\,\psi_{0}^{p+1}+\langle
u(0),Q\rangle\psi_{0})-\int_{0}^{t}e^{(t-s)\Delta}\left(\langle
u^{\prime}(s),Q(x)\rangle\psi(s)+\langle
u(s),Q(x)\rangle\xi(s)-\kappa(p+1)\psi(s)^{p}\xi(s)\right)ds\\\
&\quad=\frac{1}{\tau}\left\\{e^{(t+\tau)\Delta}\psi_{0}-e^{t\Delta}\psi_{0}\right\\}-e^{t\Delta}\psi_{0}-e^{t\Delta}(-\kappa\,\psi_{0}^{p+1}+\langle
u(0),Q\rangle\psi_{0}))\\\
&\qquad+\frac{1}{\tau}\int_{0}^{\tau}e^{(t+\tau-s)\Delta}\left[-\kappa\,\psi(s)^{p+1}+\langle
u(s),Q\rangle\psi(s)\right]ds+\int_{0}^{t}e^{(t-s)\Delta}\left[-\kappa\,\psi(s+\tau)^{p}\left(\frac{\psi(s+\tau)-\psi(s)}{\tau}-\Psi(s)\right)\right]ds\\\
&\qquad+\int_{0}^{t}e^{(t-s)\Delta}\left[-\kappa\left[p\,\psi(s)^{p}\left(\frac{\psi(s+\tau)-\psi(s)}{\tau}-\Psi(s)\right)+o\left(\frac{\psi(s+\tau)-\psi(s)}{\tau}\right)\right]\right]ds\\\
&\qquad+\int_{0}^{t}e^{(t-s)\Delta}\left[\langle\frac{u(s+\tau)-u(s)}{\tau},Q\rangle\psi(s+\tau)+\langle
u(s+\tau),Q\rangle\left(\frac{\psi(s+\tau)-\psi(s)}{\tau}-\Psi(s)\right)\right]ds\\\
&\qquad+\int_{0}^{t}e^{(t-s)\Delta}\left[-\kappa\,\psi(s+\tau)^{p}\Psi(s)-p\kappa\,\psi(s)^{p}\Psi(s)+\langle
u(s+\tau),Q\rangle\Psi(s)\right]ds\\\
&\qquad-\int_{0}^{t}e^{(t-s)\Delta}\left(\langle
u^{\prime}(s),Q(x)\rangle\psi(s)+\langle
u(s),Q(x)\rangle\Psi(s)-\kappa(p+1)\psi(s)^{p}\Psi(s)\right)ds.\end{split}$

Now, by taking the absolute value of the above identity, we obtain

$\begin{split}&\left|\frac{\psi(t+\tau)-\psi(t)}{\tau}-\Psi(t)\right|\leq\left|\frac{1}{\tau}\left\\{e^{(t+\tau)\Delta}\psi_{0}-e^{t\Delta}\psi_{0}\right\\}-e^{t\Delta}\psi_{0}\right|\\\
&\qquad+\left|\frac{1}{\tau}\int_{0}^{\tau}e^{(t+\tau-s)\Delta}\left[-\kappa\psi(s)^{p+1}+\langle
u(s),Q\rangle\psi(s)\right]ds-e^{t\Delta}(-\kappa\,\psi_{0}^{p+1}+\langle
u(0),Q\rangle\psi_{0}))\right|\\\
&\qquad+\int_{0}^{t}\Big{|}e^{(t-s)\Delta}\Big{[}\langle\frac{u(s+\tau)-u(s)}{\tau},Q\rangle\psi(s+\tau)-\langle
u^{\prime}(s),Q(x)\rangle\psi(s)-\kappa\,\psi(s+\tau)^{p}\Psi(s)\\\
&\qquad\qquad-p\kappa\,\psi(s)^{p}\Psi(s)+\langle
u(s+\tau),Q\rangle\Psi(s)-\langle
u(s),Q(x)\rangle\Psi(s)+\kappa(p+1)\psi(s)^{p}\Psi(s)\Big{]}\Big{|}ds\\\
&\qquad+\int_{0}^{t}\left|-\kappa\,\psi(s+\tau)^{p}-\kappa
p\,\psi(s)^{p}+\langle
u(s+\tau),Q\rangle\right|\left|\frac{\psi(s+\tau)-\psi(s)}{\tau}-\Psi(s)\right|ds\\\
&\qquad+\int_{0}^{t}\left|e^{(t-s)\Delta}o\left(\frac{\psi(s+\tau)-\psi(s)}{\tau}\right)\right|ds,\\\
\end{split}$

which is of the type $f(t)\leq\alpha(t)+\int_{0}^{t}\beta(s)f(s)ds.$ We apply
the Grönwall’s Lemma, and we deduce that
$f(t)\leq\alpha(t)+\int_{0}^{t}\alpha(s)\beta(s)e^{\int_{s}^{t}\beta(r)dr}ds,$
and then, by taking the limit as $\tau\to 0$, we finally obtain that

$\left|\frac{\psi(t+\tau)-\psi(t)}{\tau}-\Psi(t)\right|\to 0.$

Hence, we have proved that the unique solution $\Psi$ of (B.1) coincides with
$\psi_{t}$. ∎

We are now ready to prove the global well-posedness of (NHE) with $d=1$ stated
in Proposition 2.2.

###### Proof of Proposition 2.2.

From the local well-posedness result, we know that for any $\psi_{0}\in
H^{1}({\mathbb{T}},{\mathbb{R}})$ there exists ${\cal T}(\psi_{0},u)>0$ such
that for any $0<t<{\cal T}(\psi_{0},u)$ there exists a unique solution
$\psi\in H^{1}((0,t),L^{2}({\mathbb{T}},{\mathbb{R}}))\cap
L^{2}((0,t),H^{2}({\mathbb{T}},{\mathbb{R}}))\cap
C([0,t],H^{1}({\mathbb{T}},{\mathbb{R}}))$ of problem (NHE). Furthermore, if
${\cal T}(\psi_{0},u)<+\infty$, then
$\left\lVert\psi(t)\right\rVert_{H^{1}}\to+\infty$ as $t\to{\cal
T}(\psi_{0},u).$ We shall prove that

$\left\lVert\psi(t)\right\rVert_{H^{1}}\leq C\qquad\text{as }t\to{\cal
T}(\psi_{0},u),$

and we would deduce that ${\cal T}(\psi_{0},u)=+\infty$, that is, the solution
is globally well-defined. We recall that
$\left\lVert\psi\right\rVert^{2}_{H^{1}}:=\left\lVert\psi\right\rVert_{L^{2}}^{2}+\left\lVert\partial_{x}\psi\right\rVert_{L^{2}}^{2}.$
For almost every $0<t<\mathcal{T}(\psi_{0},u)$, we multiply the equation in
(NHE) by $\psi$ and we obtain

$\langle\partial_{t}\psi,\psi\rangle_{L^{2}}-\langle\Delta\psi,\psi\rangle_{L^{2}}+\kappa\langle\psi^{p+1},\psi\rangle_{L^{2}}=\big{\langle}\langle
u(t),Q(x)\rangle\psi,\psi\big{\rangle}_{L^{2}}.$

Recalling that $\kappa\geq 0$ and $p\in 2{\mathbb{N}}^{*}$, we get
$\kappa\langle\psi^{p+1},\psi\rangle_{L^{2}}>0$ and then
$\frac{1}{2}\frac{d}{dt}\left\lVert\psi(t)\right\rVert_{L^{2}}^{2}\leq
C(t)\left\lVert\psi(t)\right\rVert^{2}_{L^{2}}$ thanks to the accreativity of
$-\Delta$. Therefore, for a.e. $t\in(0,{\cal T}(\psi_{0},u))$, we have the
inequality

$\left\lVert\psi(t)\right\rVert^{2}_{L^{2}}\leq\left\lVert\psi_{0}\right\rVert^{2}_{L^{2}}e^{2\int_{0}^{t}C(s)ds}\leq\left\lVert\psi_{0}\right\rVert^{2}_{L^{2}}e^{2C\sqrt{{\cal
T}(\psi_{0},u)}\left\lVert u\right\rVert_{L^{2}(0,{\cal T}(\psi_{0},u))}}.$
(B.2)

Now we multiply equation (NHE) by $-\Delta\psi$ and we get

$-\langle\partial_{t}\psi,\Delta\psi\rangle_{L^{2}}+\langle\Delta\psi,\Delta\psi\rangle_{L^{2}}-\kappa\langle\psi^{p+1},\Delta\psi\rangle_{L^{2}}=-\big{\langle}\langle
u(t),Q(x)\rangle\psi,\Delta\psi\big{\rangle}_{L^{2}}.$ (B.3)

Let us observe that thanks to the properties of $\Delta$, for every $\xi\in
H^{2}({\mathbb{T}})$, it holds that

$t\mapsto\langle\Delta\xi(t),\xi(t)\rangle_{L^{2}}$

is absolutely continuous and

$\frac{d}{dt}\langle\Delta\xi(t),\xi(t)\rangle_{L^{2}}=2\langle\partial_{t}\xi(t),\Delta\xi(t)\rangle_{L^{2}}.$

We rewrite (B.3) in the equivalent form

$-\int_{0}^{2\pi}\partial_{t}\psi\Delta\psi
dx+\int_{0}^{2\pi}(\Delta\psi)^{2}dx-\kappa\int_{0}^{2\pi}\psi^{p+1}\Delta\psi
dx=-\int_{0}^{2\pi}\langle u(t),Q(x)\rangle\psi\Delta\psi dx$

and, integrating by parts, we obtain

$-\left.\partial_{t}\psi\partial_{x}\psi\right|_{0}^{2\pi}+\int_{0}^{2\pi}\partial_{x}(\partial_{t}\psi)\partial_{x}\psi
dx+\int_{0}^{2\pi}(\Delta\psi)^{2}dx-\left.\kappa\psi^{p+1}\partial_{x}\psi\right|_{0}^{2\pi}+\kappa(p+1)\int_{0}^{2\pi}\psi^{p}(\partial_{x}\psi)^{2}dx\\\
\leq
C(t)\left|\left.\psi\partial_{x}\psi\right|_{0}^{2\pi}-\int_{0}^{2\pi}(\partial_{x}\psi)^{2}dx\right|,$

and as above we get an inequality of the form
$\int_{0}^{2\pi}\partial_{x}(\partial_{t}\psi)\partial_{x}\psi dx\leq
C(t)\int_{0}^{2\pi}(\partial_{x}\psi)^{2}dx,$ where we have used that the
first integral on the left-hand side is well-defined since $\psi_{t}\in
L^{2}((0,T),H^{1}({\mathbb{T}},{\mathbb{R}}))$ thanks to the fact that
$\psi_{0}\in H^{3}({\mathbb{T}},{\mathbb{R}})$ and to Proposition B.1. Hence,
we deduce that

$\frac{1}{2}\partial_{t}\left\lVert\partial_{x}\psi(t)\right\rVert^{2}_{L^{2}}\leq
C(t)\left\lVert\partial_{x}\psi_{0}\right\rVert^{2}_{L^{2}},$

and by the Grönwall’s inequality we get

$\left\lVert\partial_{x}\psi(t)\right\rVert^{2}_{L^{2}}\leq\left\lVert\partial_{x}\psi_{0}\right\rVert^{2}_{L^{2}}e^{2\int_{0}^{t}C(s)ds}\leq\left\lVert\partial_{x}\psi_{0}\right\rVert^{2}_{L^{2}}e^{2C\sqrt{{\cal
T}(\psi_{0},u)}\left\lVert u\right\rVert_{L^{2}(0,{\cal T}(\psi_{0},u))}}.$
(B.4)

Therefore, for almost every $t\in(0,{\cal T}(\psi_{0},u))$ we have proved that

$\left\lVert\psi(t)\right\rVert^{2}_{H^{1}}=\left\lVert\psi(t)\right\rVert^{2}_{L^{2}}+\left\lVert\partial_{x}\psi(t)\right\rVert^{2}_{L^{2}}\leq(\left\lVert\psi_{0}\right\rVert^{2}_{L^{2}}+\left\lVert\partial_{x}\psi_{0}\right\rVert^{2}_{L^{2}})e^{2C\sqrt{{\cal
T}(\psi_{0},u)}\left\lVert u\right\rVert_{L^{2}(0,{\cal T}(\psi_{0},u))}},$

where we have used (B.2) and (B.4). Since the right-hand side does not depend
on $t$, we can take the limit as $t\to{\cal T}(\psi_{0},u)$ and conclude that
$\left\lVert\psi(t)\right\rVert^{2}_{H^{1}}\leq C$ as $t\to{\cal
T}(\psi_{0},u).$ ∎

##  Appendix C An estimate for the exact controllability to the ground state
solution

In this section we derive an estimate for the solution $w$ of (4.30) which is
useful for the proof of Theorem 4.3. Recall that $\Phi=1/\sqrt{2\pi}$. We
shall take advantage of estimate (4.29) that we have obtained for the solution
of

$\begin{cases}\partial_{t}y(t)-\partial^{2}_{x}y(t)+\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}y^{j}(t)\Phi^{p+1-j}=\langle
v^{1}(t),Q\rangle y(t)+\langle v^{1}(t),Q\rangle\Phi,&t\in(s_{0},s_{1}),\\\
y(s_{0})=y_{s_{0}}:=\psi(s_{0})-\Phi.\end{cases}$ (C.1)

######  Proposition C.1.

Let $\psi(s_{0})\in H^{3}({\mathbb{T}},{\mathbb{R}})$ and $v^{1}\in
H^{1}((s_{0},s_{1}),{\mathbb{R}}^{q+2})$ with $0\leq s_{0}<s_{1}$. Consider
$Q\in H^{3}({\mathbb{T}},{\mathbb{R}}^{q+2})$ and $y\in
C([s_{0},s_{1}],H^{3}({\mathbb{T}},{\mathbb{R}}))\cap
C^{1}([s_{0},s_{1}],H^{1}({\mathbb{T}},{\mathbb{R}}))$ be the solution of
(C.1). Let $v^{1}$ satisfy (4.27). Then, the solution $w\in
C([s_{0},s_{1}],H^{1}({\mathbb{T}},{\mathbb{R}}))\cap
H^{1}([s_{0},s_{1}],L^{2}({\mathbb{T}},{\mathbb{R}}))$ of

$\begin{cases}\partial_{t}w(t)-\partial^{2}_{x}w(t)-\kappa(p+1)\Phi^{p}w(t)+\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}y^{j}(t)=\langle
v^{1}(t),Q\rangle y(t),&t\in(s_{0},s_{1})\\\ w(s_{0})=0\end{cases}$ (C.2)

satisfies the following inequality with $\sigma=s_{1}-s_{0}$ and
$A_{4}(\sigma,\left\lVert y_{s_{0}}\right\rVert_{H^{1}})$ defined in (C.7):

$\sup_{t\in[s_{0},s_{1}]}\left\lVert w(t)\right\rVert_{H^{1}}\leq
A_{4}(\sigma,\left\lVert y_{s_{0}}\right\rVert_{H^{1}})\left\lVert
y_{s_{0}}\right\rVert^{2}_{H^{1}}.$ (C.3)

###### Proof.

We recall that $w$ has been defined as $w:=y-\xi$ with $\xi\in
C([s_{0},s_{1}],H^{1}({\mathbb{T}},{\mathbb{R}}))\cap
H^{1}(s_{0},s_{1},L^{2}({\mathbb{T}},{\mathbb{R}}))\cap
L^{2}(s_{0},s_{1},H^{2}({\mathbb{T}},{\mathbb{R}}))$ solution of the linear
system (4.4) with control $v^{1}\in
H^{1}_{0}((s_{0},s_{1}),{\mathbb{R}}^{q+2})$ such that

$\xi(s_{1};s_{0},\xi_{0},v^{1})=0,\quad\text{with}\quad\xi(s_{0})=y_{s_{0}}=\psi(s_{0})-\Phi$

and $\left\lVert v^{1}\right\rVert_{H^{1}(s_{0},s_{1})}\leq
N(\sigma)\left\lVert y_{s_{0}}\right\rVert_{H^{1}},$ with
$\sigma=s_{1}-s_{0}.$ The existence of such control has been established in
Proposition 4.6. Let us estimate the norm of $w$ at time $s_{1}$. We first
multiply the equation in (C.2) by $w$ and we obtain

$\langle\partial_{t}w(t),w(t)\rangle_{L^{2}}-\langle\partial^{2}_{x}w(t),w(t)\rangle_{L^{2}}-\kappa(p+1)\Phi^{p}\langle
w(t),w(t)\rangle_{L^{2}}+\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\langle
y^{j}(t),w(t)\rangle_{L^{2}}\\\ =\big{\langle}\langle v^{1}(t),Q\rangle
y(t),w(t)\big{\rangle}_{L^{2}}.$

Using the accreativity of $-\partial_{x}^{2}$ and that
$H^{1}({\mathbb{T}})\hookrightarrow C({\mathbb{T}})$, we get

$\frac{1}{2}\frac{d}{dt}\left\lVert
w(t)\right\rVert^{2}_{L^{2}}\leq\kappa(p+1)\Phi^{p}\left\lVert
w(t)\right\rVert^{2}_{L^{2}}+\\\
+\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(\frac{\left\lVert
y(t)\right\rVert^{2j}_{H^{1}}}{2}+\frac{\left\lVert
w(t)\right\rVert_{L^{2}}^{2}}{2}\right)+C_{Q}^{2}\sum_{j=1}^{q+2}|v^{1}_{j}(t)|^{2}\frac{\left\lVert
y(t)\right\rVert_{L^{2}}^{2}}{2}+\frac{\left\lVert
w(t)\right\rVert_{L^{2}}^{2}}{2},$

where

$C_{Q}:=\sup_{i=1,\dots,{q+2}}\left\lVert Q^{i}\right\rVert_{C^{0}}$ (C.4)

By the Grönwall’s Lemma, for any $t\in(s_{0},s_{1})$ it holds that

$\left\lVert
w(t)\right\rVert_{L^{2}}^{2}\leq\left(\int_{s_{0}}^{t}\kappa\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left\lVert
y(s)\right\rVert^{2j}_{H^{1}}+C_{Q}^{2}\sum_{j=1}^{q+2}|v^{1}_{j}(s)|^{2}\left\lVert
y(s)\right\rVert_{H^{1}}^{2}ds\right)\cdot\\\
e^{\int_{s_{0}}^{t}\left(2\kappa(p+1)\Phi^{p}+\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}+1\right)ds}.$

Therefore, by taking the supremum over $[s_{0},s_{1}]$ we obtain

$\begin{split}&\sup_{t\in[s_{0},s_{1}]}\left\lVert
w(t)\right\rVert^{2}_{L^{2}}\leq\left(\kappa\sigma\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\sup_{t\in[s_{0},s_{1}]}\left\lVert
y(t)\right\rVert^{2j}_{H^{1}}+C_{Q}^{2}\left\lVert
v^{1}\right\rVert^{2}_{L^{2}(s_{0},s_{1})}\sup_{t\in[\tau_{0},\tau_{1}]}\left\lVert
y(t)\right\rVert_{H^{1}}^{2}\right)e^{A_{1}(\sigma)}\\\
&\leq\left(2\kappa\sigma(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(\left\lVert
y_{s_{0}}\right\rVert^{2j}_{H^{1}}+\left\lVert
v^{1}\right\rVert^{2j}_{L^{2}(s_{0},s_{1})}\right)+C_{Q}^{2}\left\lVert
v^{1}\right\rVert^{2}_{L^{2}(s_{0},s_{1})}\left(\left\lVert
y_{s_{0}}\right\rVert^{2}_{H^{1}}+\left\lVert
v^{1}\right\rVert^{2}_{L^{2}(s_{0},s_{1})}\right)\right)e^{A_{1}(\sigma)}\\\
&\leq\left(2\kappa\sigma(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N(\sigma)^{2j}\right)\left\lVert
y_{s_{0}}\right\rVert^{2j}_{H^{1}}+C_{Q}^{2}N(\sigma)^{2}\left(1+N(\sigma)^{2}\right)\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}}\right)e^{A_{1}(\sigma)}\\\
&\leq\left(2\kappa\sigma(p+1)\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N(\sigma)^{2j}\right)\left\lVert
y_{s_{0}}\right\rVert^{2(j-2)}_{H^{1}}+C_{Q}^{2}N(\sigma)^{2}\left(1+N(\sigma)^{2}\right)\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}}\right)e^{A_{1}(\sigma)}\\\ &\leq
A_{2}(\sigma,\left\lVert y_{s_{0}}\right\rVert_{H^{1}})^{2}\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}}\end{split}$ (C.5)

where

$A_{1}(\sigma):=\sigma\left(2\kappa(p+1)\Phi^{p}+\kappa\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}+1\right)$
$A_{2}(\sigma,\left\lVert y_{s_{0}}\right\rVert_{H^{1}}):=\\\
\left(2\kappa\sigma(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N(\sigma)^{2j}\right)\left\lVert
y_{s_{0}}\right\rVert^{2(j-2)}_{H^{1}}+C_{Q}^{2}N(\sigma)^{2}\left(1+N(\sigma)^{2}\right)\right)^{1/2}e^{A_{1}(\sigma)/2}.$

Let us now multiply the equation in (4.30) by $-\partial^{2}_{x}w(t)$

$-\langle\partial_{t}w(t),\partial^{2}_{x}w(t)\rangle_{L^{2}}+\langle\partial^{2}_{x}w(t),\partial^{2}_{x}w(t)\rangle_{L^{2}}+\kappa(p+1)\Phi^{p}\langle
w(t),\partial^{2}_{x}w(t)\rangle_{L^{2}}\\\
-\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\langle
y^{j}(t),\partial^{2}_{x}w(t)\rangle_{L^{2}}=-\big{\langle}\langle
v^{1}(t),Q\rangle y(t),\partial^{2}_{x}w(t)\big{\rangle}_{L^{2}}.$

We now perform integrations by parts, and we get

$\langle\partial_{t}(\partial_{x}w(t)),\partial_{x}w(t)\rangle_{L^{2}}+\left\lVert\partial^{2}_{x}w(t)\right\rVert^{2}_{L^{2}}-\kappa(p+1)\Phi^{p}\langle\partial_{x}w(t),\partial_{x}w(t)\rangle_{L^{2}}\\\
+\kappa\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}j\langle
y^{j-1}(t)\partial_{x}y(t),\partial_{x}w(t)\rangle_{L^{2}}=-\langle\langle
v^{1}(t),Q\rangle y(t),\partial^{2}_{x}w(t)\rangle_{L^{2}}$

and therefore

$\frac{1}{2}\frac{d}{dt}\left\lVert\partial_{x}w(t)\right\rVert^{2}_{L^{2}}+\left\lVert\partial^{2}_{x}w(t)\right\rVert^{2}_{L^{2}}\leq\kappa(p+1)\Phi^{p}\left\lVert\partial_{x}w(t)\right\rVert_{L^{2}}^{2}\\\
+\kappa(p+1)\displaystyle\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(\frac{\left\lVert
y(t)\right\rVert^{2j}_{H^{1}}}{2}+\frac{\left\lVert\partial_{x}w(t)\right\rVert_{L^{2}}^{2}}{2}\right)+C_{Q}^{2}\sum_{j=1}^{q+2}|v^{1,j}(t)|^{2}\left\lVert
y(t)\right\rVert^{2}_{H^{1}}+\frac{\left\lVert\partial^{2}_{x}w(t)\right\rVert_{L^{2}}^{2}}{2},$

where we have used that $\partial_{t}w=\partial_{t}y-\partial_{t}\xi\in
L^{2}((s_{0},s_{1}),H^{1}({\mathbb{T}},{\mathbb{R}}))$ (see Proposition B.1
for $y$ and Remark 4.5 for $\xi$). We apply the Grönwall’s Lemma, and we
obtain that, for any $t\in[s_{0},s_{1}]$,

$\left\lVert\partial_{x}w(t)\right\rVert_{L^{2}}^{2}\leq\left(\int_{s_{0}}^{t}\kappa(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left\lVert
y(s)\right\rVert^{2j}_{H^{1}}+C^{2}_{Q}\sum_{j=1}^{q+2}|v^{1,j}(s)|^{2}\left\lVert
y(s)\right\rVert^{2}_{H^{1}}ds\right)e^{A_{3}(s_{0},t)}$

with

$A_{3}(s_{0},t):=\int_{s_{0}}^{t}\Big{(}2\kappa(p+1)\Phi^{p}+\kappa(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\Big{)}ds.$

Taking the supremum over the interval $[s_{0},s_{1}]$, we obtain

$\begin{split}&\sup_{t\in[s_{0},s_{1}]}\left\lVert\partial_{x}w(t)\right\rVert^{2}_{L^{2}}\leq\left(\int_{s_{0}}^{s_{1}}\kappa(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left\lVert
y(s)\right\rVert^{2j}_{H^{1}}+C^{2}_{Q}\sum_{j=1}^{q+2}|v^{1,j}(s)|^{2}\left\lVert
y(s)\right\rVert^{2}_{H^{1}}ds\right)e^{A_{3}(s_{0},s_{1})}\\\
&\quad\leq\left(\kappa\sigma(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\sup_{t\in[s_{0},s_{1}]}\left\lVert
y(t)\right\rVert^{2j}_{H^{1}}+C^{2}_{Q}\left\lVert
v^{1}\right\rVert^{2}_{L^{2}(s_{0},s_{1})}\sup_{t\in[s_{0},s_{1}]}\left\lVert
y(t)\right\rVert^{2}_{H^{1}}\right)e^{A_{3}(s_{0},s_{1})}\\\
&\quad\leq\left(2\kappa\sigma(p+1)^{2}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(\left\lVert
y_{s_{0}}\right\rVert^{2j}_{H^{1}}+\left\lVert
v^{1}\right\rVert^{2j}_{L^{2}(s_{0},s_{1})}\right)+C^{2}_{Q}\left\lVert
v^{1}\right\rVert^{2}_{L^{2}(s_{0},s_{1})}\left(\left\lVert
y_{s_{0}}\right\rVert^{2}_{H^{1}}+\left\lVert
v^{1}\right\rVert^{2}_{L^{2}(s_{0},s_{1})}\right)\right)e^{A_{3}(s_{0},s_{1})}.\end{split}$

Thanks to the assumption on $y_{0}$ we deduce

$\begin{split}\sup_{t\in[s_{0},s_{1}]}&\left\lVert\partial_{x}w(t)\right\rVert^{2}_{L^{2}}\\\
&\leq\left(2\kappa\sigma(p+1)^{2}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N^{2j}(\sigma)\right)\left\lVert
y_{s_{0}}\right\rVert^{2j}_{H^{1}}+C_{Q}^{2}N^{2}(\sigma)\left(1+N^{2}(\sigma)\right)\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}}\right)e^{A_{3}(s_{0},s_{1})}\\\
&=\left(2\kappa\sigma(p+1)^{2}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N^{2j}(\sigma)\right)\left\lVert
y_{s_{0}}\right\rVert^{2(j-2)}_{H^{1}}+C^{2}_{Q}N^{2}(\sigma)\left(1+N^{2}(\sigma)\right)\right)e^{A_{3}(s_{0},s_{1})}\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}}\\\ &=A_{3}(\sigma,\left\lVert
y_{s_{0}}\right\rVert_{H^{1}})^{2}\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}},\end{split}$ (C.6)

where

$A_{3}(\sigma,\left\lVert y_{s_{0}}\right\rVert_{H^{1}}):=\\\
\left(2\kappa\sigma(p+1)^{2}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N^{2j}(\sigma)\right)\left\lVert
y_{s_{0}}\right\rVert^{2(j-2)}_{H^{1}}+C^{2}_{Q}N^{2}(\sigma)\left(1+N^{2}(\sigma)\right)\right)^{1/2}e^{A_{3}(s_{0},s_{1})/2}.$

Finally, from (C.5) and (C.6), we conclude that

$\begin{split}\sup_{t\in[s_{0},s_{1}]}\left\lVert
w(t)\right\rVert^{2}_{H^{1}}&\leq 2\left(\sup_{t\in[s_{0},s_{1}]}\left\lVert
w(t)\right\rVert^{2}_{L^{2}}+\sup_{t\in[s_{0},s_{1}]}\left\lVert\partial_{x}w(t)\right\rVert^{2}_{L^{2}}\right)\\\
&\leq 2\left(A_{2}(\sigma,\left\lVert
y_{s_{0}}\right\rVert_{H^{1}})^{2}+A_{3}(\sigma,\left\lVert
y_{s_{0}}\right\rVert_{H^{1}})^{2}\right)\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}}\leq A_{4}^{2}(\sigma,\left\lVert
y_{s_{0}}\right\rVert_{H^{1}})^{2}\left\lVert
y_{s_{0}}\right\rVert^{4}_{H^{1}},\end{split}$

with

$A_{4}(\sigma,\left\lVert
y_{s_{0}}\right\rVert_{H^{1}}):=\sqrt{2}\left(2\kappa\sigma(p+1)^{2}\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}\left(1+N(\sigma)^{2j}\right)\left\lVert
y_{s_{0}}\right\rVert^{2(j-2)}_{H^{1}}+C^{2}_{Q}N^{2}(\sigma)\left(1+N(\sigma)^{2}\right)\right)^{1/2}\cdot\\\
e^{\sigma(2\kappa(p+1)\Phi^{p}+\kappa(p+1)\sum_{j=2}^{p+1}\binom{p+1}{j}\Phi^{p+1-j}+1))/2}.$
(C.7)

The proof is therefore concluded. ∎

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