# A Virtual Reality Teleoperation Interface for
Industrial Robot Manipulators

Eric Rosen1 and Devesh K. Jha2 1Eric Rosen is with Department of Computer
Science, Brown University, Providence, RI<EMAIL_ADDRESS>K. Jha
is with Mitsubishi Electric Research Labs, Cambridge, MA<EMAIL_ADDRESS>
###### Abstract

We address the problem of teleoperating an industrial robot manipulator via a
commercially available Virtual Reality (VR) interface. Previous works on VR
teleoperation for robot manipulators focus primarily on collaborative or
research robot platforms (whose dynamics and constraints differ from
industrial robot arms), or only address tasks where the robot’s dynamics are
not as important (e.g: pick and place tasks). We investigate the usage of
commercially available VR interfaces for effectively teleoeprating industrial
robot manipulators in a variety of contact-rich manipulation tasks. We find
that applying standard practices for VR control of robot arms is challenging
for industrial platforms because torque and velocity control is not exposed,
and position control is mediated through a black-box controller. To mitigate
these problems, we propose a simplified filtering approach to process command
signals to enable operators to effectively teleoperate industrial robot arms
with VR interfaces in dexterous manipulation tasks. We hope our findings will
help robot practitioners implement and setup effective VR teleoperation
interfaces for robot manipulators. The proposed method is demonstrated on a
variety of contact-rich manipulation tasks which can also involve very precise
movement of the robot during execution (videos can be found at
https://www.youtube.com/watch?v=OhkCB9mOaBc).

##  I INTRODUCTION

Robots have been steadily increasing their presence in our daily lives where
they can be used to create and provide more assistance for various
collaborative tasks. It is envisioned that the next-generation industrial
robotic systems will have humans and robots working next to each where robots
can easily learn complex tasks from human experts as part of the collaborative
system. However, creating robotic systems which can easily collaborate while
being versatile is extremely challenging. One of the key challenges is to
design interfaces which be used intuitively for a large variety of tasks for
collaborative human-robot systems. Such interfaces are very important for
designing collaborative systems so that a human can easily demonstrate complex
manipulation tasks to the robot. In this paper, we consider teleoperation of
industrial robots using virtual reality interfaces and demonstrate the
efficiency of the system over a wide variety of manipulation tasks. Figure 2
shows a rendering of the manipulator arm in the virtual game environment as
well as an actual image of the robot used in this study.

Figure 1: Example images from two of the manipulation tasks we succesfully
performed with our VR teleoperation interface. Top: stacking Bandu pieces.
Bottom: Flipping a bottle upwards using extrinsic contacts.

Teleoperation is the problem and set of solutions associated with remotely
controlling a semi-autonomous robot platform. Traditionally, teleoperation is
performed by system experts because standard interfaces like keyboards, mice,
and teach pendents require advanced training to effectively use. More
recently, commercially available Virtual Reality (VR) hardware have gained
tractions as viable interfaces for robot teleopertaion because the spatially-
tracked hand controllers make it easier for non-experts to command high DoF
arms [1]. Most research into VR teleoperation for robots has done work on
research robot platforms. We address the problem of applying these techniques
to industrial robot arm platforms.

(a) Virtual reality rendering of the MELFA Assista robot

(b) Actual Image of the MELFA Assista robot used in the study

Figure 2: VR rendering and the actual robot image used in this study.

In general, the industrial robotic arms with pure position control have
several characteristics which make VR teleoperation very challenging for these
manipulators when compared to research robotic platforms. Firstly, industrial
robot arms focus primarily on precision, and therefore often only have high-
precision position control scheme exposed, whereas many research robot
platforms have access to velocity or torque control. Secondly, industrial
robot arms tend to be heavier due to the mechanical construction compared to
cobots or research platforms, and so the higher amounts of inertia introduce a
higher amount of latency between control signals from the operator to
commanded movements on the robot platform. We find that these two important
differences make it extremely challenging to take existing practices for VR
teleoperation developed in research laboratories and effectively deploy them
on industrial robot arms to perform dexterous manipulation tasks (Figure 1).

To overcome these issues, we propose some key system implementations that
improve teleoperation efficacy. We propose a signal-processing algorithm for
interpolating user-specified position controls for a robot arm that prevents
the robot from faulting even when the robot’s controller dynamics are not
known or adjustable. We also use a hybrid force-position control scheme where
operators command both the robot’s end-effector position (via spatially-
tracked hand controllers) while using the robot in impedance control mode to
help compensate for discrepancies due to latency as well as provide some
compliance during manipulation tasks.

Our paper is structured as follows: we first describe existing practices and
approaches for VR teleoperation of robot manipulators, and describe in detail
the challenges we found when applying them to an industrial robot arm. Then,
we describe in detail our proposed set of system practices for improving the
efficacy of VR teleoperation for industrial robot arms. Finally, we conclude
with an experimental demonstration of how our system changes enable operators
to teleoperate an industrial robot arm with VR to perform dexterous
manipulation tasks like stacking, peg-in-hole insertion and various other
challenging manipulation tasks.

We believe our findings and contributions will be useful to practitioners who
aim to deploy existing VR teleoperation research to industrial robot
platforms.

##  II RELATED WORK

Virtual, augmented, and mixed reality devices have recently garnered interest
within the robotics community as powerful visual interfaces for facilitating
human-robot interaction [2]. Virtual Reality interfaces for teleoperating
robots have empirically improved the performance of novice operators in
metrics such as increased task success and lower cognitive workload [1, 3, 4].
In this section, we discuss different VR systems and interfaces that have been
developed for robot teleoperation in both research and industrial contexts.

[5] proposed a VR system to teleoperate a Baxter robot using a homonuculus
view, which embeds the user’s view in a control room that is located inside
the “head” of the robot. This decouples the view of the user from the view of
the robot, which makes it easy to control the robot arm based on the distance
between the user’s hand and the robot’s virtual arm markers. [6] created a
virtual reality operator station for the Military mobile robot TAROS which
enabled users to command a virtual arm of the robot while accounting for
obstacle avoidance.

Most similar to our work are those that investigate developing VR
teleoperation interfaces for industrial robot arms. [7] developed an Augmented
Reality (AR) interface for visualizing the state of an industrial robot arm
and used a gamepad to control the robot arm. [8] developed a VR interface for
increasing the flexibility of existing industrial robot workspaces by enabling
end-users to specify Cartesian waypoints when programming motion plans. [9]
developed a robust adapative control scheme for teleoperating robots amongst
delays, which is especially important when latency is high between the end-
user and robot platform.

##  III VIRTUAL REALITY TELEOPERATION FOR INDUSTRIAL MANIPULATORS

In this section, we first motivate why we think using VR teleoperation is
difficult for typical industrial manipulators. This discussion is followed by
our proposed method for designing the VR teleoperation system.

###  III-A Why is VR Teleoperation difficult for Industrial Robots?

We investigate developing an VR interface for controlling an industrial robot
manipulator, specifically the MELFA RV-5AS-D Assita robot platform (see Figure
2). The Assista manipulator is a 6 DoF manipulator equipped with a single DoF
parallel-jaw gripper. The robot is equipped with Mitsubishi Electric F/T
sensor $1$F-FS$001$-W$200$ at the wrist (see Figure 2). This sensor is used to
implement an indirect force controller using the stock stiffness controller of
the robot [10, 11]. The Assista is an industrial collaborative manipulator arm
with pose repeatability of $\pm 0.03$mm.

Algorithm 1 Filter for VR Teleoperation

Delta end-effector pose $\Delta$p, control frequency $f$’, max position speed
$s_{p}$, max orientation speed $s_{o}$, noise position threshold $n_{p}$,
noise orientation threshold $n_{o}$ $\Delta$$p^{p}$, $\Delta$$p^{o}$ =
$\Delta$p $\triangleright$ Separate delta position and orientation

if norm($\Delta$$p^{p}$) $>n_{p}$ then $\triangleright$ Delta position above
noise threshold

$\Delta$p${}_{filtered}^{p}$ = $\Delta$$p^{p}$/ norm($\Delta$$p^{p}$)

$\Delta$p${}_{filtered}^{p}$ *= $s_{p}$

else$\triangleright$ Delta position below noise threshold

$\Delta$p${}_{filtered}^{p}$ = [0,0,0]

end if

if norm($\Delta$$p^{o}$) $>n_{o}$ then $\triangleright$ Delta orientation
above noise threshold

$\Delta$p${}_{filtered}^{o}$ = $\Delta$$p^{o}$/ norm($\Delta$$p^{o}$)

$\Delta$p${}_{filtered}^{o}$ *= $s_{o}$

else$\triangleright$ Delta orientation below noise threshold

$\Delta$p${}_{filtered}^{o}$ = [0,0,0]

end ifreturn
linspace([$\Delta$p${}_{filtered}^{p}$,$\Delta$p${}_{filtered}^{o}$],
$\frac{1}{f^{\prime}}$)

In general, industrial manipulator arms focus on very high precision during
operation. Consequently, depending on their pay-load capacity, they tend to be
bulky as they use very powerful servo motors during movement. To allow high
precision, the robots have a well-tuned control method for the movement of the
joints which is not exposed to an end-user. Furthermore, these robots, in
general, do not provide access to velocity control methods. Instead, there is
a high-frequency, continuous trajectory control method exposed for smooth and
continuous movement of the robot. Note that this controller can be used to
design a psuedo-velocity controller using a pre-defined position trajectory.
An important point to note here is that the high inertia of the industrial
manipulators introduce significant latency in following the teleoperation
commands through a virtual reality setup.

Another important aspect of these robots is that to allow safe operation,
controllers of these robots have a fault system which stops the operation of
the robot if a certain command sent by an end-user results in saturation of
motor commands. To ensure safe operation, this system is not exposed to the
end-user and thus, it acts as a black-box system. For the high-frequency
position control, this implies that we can only send very small incremental
commands such that the robot operates in the feasible range of the motor
controllers. This feasible incremental pose command depends on the on the
current pose of the robot. To ensure operation of the robot without incurring
faults, we need to ensure that we operate within the unknown, feasible set of
commands for robot movement (when using the high-frequency position control).
This requirement imposes several challenges regarding how the VR commands
should be sent to the robot controller so that we achieve a smooth, error-free
movement of the robot while following the human teleoperation commands with
sufficient accuracy.

(a) Block stacking (irregular)

(b) Peg in hole

(c) Block stacking

(d) Peg in hole unplugging

Figure 3: Sample images of four of the manipulation tasks performed in our
experiment using our VR teleoperation interface.

The above characteristics of a high-precision, position-controlled industrial
manipulator arms make it difficult to use a teleoperation system designed for
a research robotic system. Consequently, these characteristics require that we
filter the signals received from tracking the VR controllers appropriately so
that we can achieve low latency while ensuring feasible movement of the
manipulator arm. Furthermore, to allow safety during contact interactions, we
use the default impedance controller for the robot to allow compliance during
contact interactions. One can design a complex compliance controller as shown
in our previous work [10] but this is out of scope of the current paper.

###  III-B Proposed System

For controlling the robot arm with a VR headset, we use the HTC Vive Pro. To
communicate between the robot and the VR headset, we use ROS# to generate a
virtual environment in Unity3D and create a websockets server to facilitate
networking between the ROS and Unity computing nodes. We generate a 3D model
of the robot arm in Unity based on the robot’s URDF, and update the pose of
the model based on the joint states of the real robot. For sending position
commands to the robot, we use a similar scheme to [12], where a duplicate
virtual model of the robot’s end effector is instantiated into the scene, and
the operator is able to move their 3D spatially-tracked hand-controller to
apply relative pose transformations to the copy. Based on the difference
between the copy and the robot’s end effector transform, a delta end-effector
pose $\Delta$p is computed and send as a control command to the robot.

(a) Bandu

(b) Block destacking

(c) Object flipping

(d) Handover

Figure 4: Sample images of four of the manipulation tasks performed in our
experiment using our VR teleoperation interface.

Given a desired delta end-effector pose $\Delta$p, we can directly use a
position-based controller to command the robot arm to the desired
configuration. Directly controlling the robot arm via raw position control has
been done with relative success in a variety of works using non-industrial
robot arms [12, 3]. For these robot platforms, a fixed max speed specification
can be set and used to enable the robot to eventually attain any specified
$\Delta$p after a finite amount of time. However, for industrial robot
platforms like the Assista manipulator, a black-box controller is used to
compute sufficient velocities to attain the $\Delta$p at a fixed frequency
within a certain accuracy threshold. Due to the robot’s built-in fault system,
if the commanded $\Delta$p requires a sufficiently high torque to be executed,
the robot will fault out, which is problematic for effective teleoperation.
This problem is exacerbated since the requisite torques to minimally satisfy
the command are a function of the robot’s current joint configuration. This
means that if a single maximal speed is chosen as a cap to limit how much the
user can control the robot’s position using the teleoperation system, there
will be configurations where the robot motion is limited significantly more
than what is necessary.

(a) Bandu

(b) Block destacking (irregular)

(c) Block destacking

(d) Block stacking

(e) Flipping

(f) Handover

(g) Peg-in-hole (Unplugging)

(h) Peg-in-hole

Figure 5: Visualizations of commanded and executed 3D position trajectories
from the different manipulation tasks. The triangular points indicate where
the trajectory started.

While this problem may occur in any robot platform, it is especially prevalent
in industrial robot platforms that have black-box controllers and built-in
fault systems. We propose addressing this problem with a simplified filtering
approach designed to accommodate the lack of certainty over the robot’s
dynamics and built-in fault system. More formally, we assume a commanded delta
end-effector pose $\Delta$p is commanded at a given frequency $f$, and our
goal is to produce a suitable filtered command signal $\Delta$p’ for
satisfying the command without faulting. We propose a two-stage approach,
where we first transform $\Delta$p based on a speed cap and noise filtering
threshold, and then use a linear transformation to interpolate from the
current position to the desired end-effector pose at a higher frequency $f$’
$>$ $f$. The first step ensures that the commanded end-effector pose does not
activate the faults and that overshooting for the desired pose is mitigated,
while the second step ensures that the robot is operating in a reactively to
new commanded inputs from the user. Our algorithmic implementation for the
two-stage filtering approach can be found in Algorithm 1.

Task | Time (s) | RMS (mm) | Comm. Jerk | Exe. Jerk
---|---|---|---|---
Bandu | 69.675 | 28.87 | 0.000611 | 0.000147
Block destacking | 83.07 | 28.21 | 0.000617 | 0.000125
Block stacking | 71.92 | 21.5 | 0.000525 | 0.000124
Block stacking (irreg.) | 80.35 | 25.63 | 0.000632 | 0.000124
Flipping | 61.02 | 15.34 | 0.000432 | 0.000134
Handover | 41.27 | 19.2 | 0.000602 | 0.000152
Peg-in-hole unplugging | 85.62 | 38.382 | 0.000635 | 0.000134
Peg-in-hole | 49.67 | 38.38 | 0.000542 | 0.000122
Figure 6: A table of the task competition times, RMS between commanded and
executed trajectories and jerk between the commanded and the executed
trajectories.

##  IV EXPERIMENTS

To evaluate our proposed filtering approach for improving the efficacy of VR
teleoperation for industrial robot arms, we tested our ability to perform $8$
different complex manipulation tasks when using our proposed scheme on the
Assista robot manipulator. Furthermore, we operate the robot in the stiffness
control mode with a suitable choice of stiffness matrix for all the task to
avoid excessive contact forces during manipulation.

For each task, we had an experienced operator (one of the author’s of this
paper) use the VR teleoperation system with our filtering approach implemented
to perform each manipulation task. For every task, the robot was initially
homed to the same starting configuration before recording. Each task was
performed 2 times, and we recorded both the command trajectory from the user
and the trajectory executed by the robot. Sample images of each of the tasks
can be found in Figure 3 and 4.

###  IV-A Manipulation Tasks

We evaluate the proposed VR teleoperation task on a variety of contact-rich
manipulation tasks. Some of these tasks require precise placement and
interaction during execution, and are challenging to solve using planning or
Reinforcement Learning (RL) [13]. So, we try to understand how precise
manipulation and interaction can be performed using the proposed system. In
particular, we perform the following tasks using the proposed system:

  1. 1.

Bandu - Stack four Bandu blocks on top of each other in two columns (Figure
4(a)).

  2. 2.

Block stacking - Stack three colored cubes on top of each other into a column
(Figure 3(c)).

  3. 3.

Block destacking - Destack three cubes and place them into a new stacked
column (Figure 4(b)).

  4. 4.

Block stacking irregular - Stack three irregular-shaped blocks on top of each
other into a column (Figure 3(a)).

  5. 5.

Object flipping - Flip a bottle from a horizontal position to a vertical
position using non-prehensile manipulation (Figure 4(c)).

  6. 6.

Handover - Pick up a screw driver from a slot and hand it over to a co-located
human (Figure 4(d)).

  7. 7.

Peg in hole - Lift a peg from a peg holder and place it into a fitted hole
(Figure 3(b)).

  8. 8.

Peg in hole unplugging - Unplug a peg from a tight hole and put it back in a
holder. (Figure 3(d)).

##  V Discussion

Sampled visualizations of the commanded and executed trajectories for the
different manipulation tasks can be found in Figure 5. We also report the
average task completion time for each of the manipulation tasks, the RMS
between the commanded and executed trajectories, and the average jerk of the
trajectories, which are included in Table 6. Qualitatively, we can see that
there are a wide-range of different motion trajectories across the different
tasks. Tasks like Block stacking and the handover task involve many linear
motions associated with space motion, whereas tasks like Bandu and block
destacking (irregular) involved a lot of subtle, small reposition movements
due to the contact-rich nature involved in successfully completing the tasks.
Whereas the commanded trajectories from the VR involved are relatively not
smooth (due in part to operator error and position tracking error), our
filtering approach results in a smoother trajectory that still keeps
relatively low tracking error with the commanded signal. This is
quantitatively evaluated based on the average jerk norm of the trajectories,
which, as can be seen in Table 6, are much lower for the executed trajectories
compared to the commanded trajectories. We found that this was crucial for
tasks where motions with high jerk easily resulted in task failure, such as
the flipping task.

##  VI CONCLUSION

We address the problem of using commercially-available VR hardware to
effectively teleoperate an industrial robot manipulator. We propose a
simplistic filtering scheme for processing position-commanded signals that
reduces the number of faults and tracking error when teleoperating an
industrial robot arm with a black-box controller. We demonstrate the efficacy
of our approach by teleoperating a MELFA Assista robot to perform $8$
dexterous manipulation tasks. Future work will investigate leveraging our
system and interface to enable industrial end-users to teach robot motor
skills via imitation learning.

## References

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# Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger
equation

Alessandro Duca 111Université Paris-Saclay, UVSQ, CNRS, Laboratoire de
Mathématiques de Versailles, 78000, Versailles, France; e-mail:
<EMAIL_ADDRESS>Vahagn Nersesyan 222Université Paris-Saclay, UVSQ,
CNRS, Laboratoire de Mathématiques de Versailles, 78000, Versailles, France;
e-mail<EMAIL_ADDRESS>
###### Abstract

We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary
dimension. The equation is studied in presence of an external potential field
whose time-dependent amplitude is taken as control. Assuming that the
potential satisfies a saturation property, we show that the NLS equation is
approximately controllable between any pair of eigenstates in arbitrarily
small time. The proof is obtained by developing a multiplicative version of a
geometric control approach introduced by Agrachev and Sarychev. We give an
application of this result to the study of the large time behavior of the NLS
equation with random potential. More precisely, we assume that the amplitude
of the potential is a random process whose law is $1$-periodic in time and
non-degenerate. Combining the controllability with a stopping time argument
and the Markov property, we show that the trajectories of the random equation
are almost surely unbounded in regular Sobolev spaces.

AMS subject classifications: 35Q55, 35R60, 37L55, 81Q93, 93B05

Keywords: Nonlinear Schrödinger equation, approximate controllability,
geometric control theory, growth of Sobolev norms, random perturbation

###### Contents

  1. 0 Introduction
  2. 1 Preliminaries
  3. 2 Approximate controllability
  4. 3 Proof of Proposition 1.2
  5. 4 Saturating subspaces
  6. 5 Growth of Sobolev norms

##  0 Introduction

In this paper, we study the controllability and the growth of Sobolev norms
for the following nonlinear Schrödinger (NLS) equation on the torus
${\mathbb{T}}^{d}={\mathbb{R}}^{d}/2\pi{\mathbb{Z}}^{d}$:

$i\partial_{t}\psi=-\Delta\psi+V(x)\psi+\kappa|\psi|^{2p}\psi+\langle
u(t),Q(x)\rangle\psi.$ (0.1)

We assume that $V:{\mathbb{T}}^{d}\to{\mathbb{R}}$ is an arbitrary smooth
potential, $Q:{\mathbb{T}}^{d}\to{\mathbb{R}}^{q}$ is a given smooth external
field subject to some geometric condition, $d,p\geq 1$ are arbitrary integers,
and $\kappa$ is an arbitrary real number. The role of the control (or the
random perturbation) is played by ${\mathbb{R}}^{q}$-valued function (or
random process) $u$ which is assumed to depend only on time. Eq. (0.1) is
equipped with the initial condition

$\psi(0,x)=\psi_{0}(x)$ (0.2)

belonging to a Sobolev space $H^{s}=H^{s}({\mathbb{T}}^{d};{\mathbb{C}})$ of
order $s>d/2$, so that the problem is locally well-posed.

The purpose of this paper is to study the NLS equation (0.1) when the driving
force $u$ acts multiplicatively through only few low Fourier modes. Referring
the reader to the subsequent sections for the general setting, let us
formulate in this Introduction particular cases of our main results. Let
${\cal K}\subset{\mathbb{Z}}^{d}_{*}$ be the set of $d$ vectors defined by

${\cal
K}=\\{(1,0,\ldots,0),\,(0,1,\ldots,0),\,\ldots,\,(0,0,\ldots,1,0),\,(1,\ldots,1)\\},$
(0.3)

and assume that the field $Q=(Q_{1},\ldots,Q_{q})$ is such that

$\left\\{{\bf 1},\,\sin\langle x,k\rangle,\,\cos\langle x,k\rangle:k\in{\cal
K}\right\\}\subset\mathop{\rm span}\nolimits\\{Q_{j}:j=1,\ldots,q\\}.$ (0.4)

Let $s_{d}$ be the least integer strictly greater than $d/2$.

######  Theorem A.

The problem (0.1), (0.2) is approximately controllable in the following sense:
for any $s\geq s_{d}$, $\varepsilon>0$, $\varkappa>0$, $\psi_{0}\in H^{s}$,
and $\theta\in C^{\infty}({\mathbb{T}}^{d};{\mathbb{R}})$, there is a time
$T\in(0,\varkappa)$, a control $u\in L^{2}([0,T];{\mathbb{R}}^{q})$, and a
unique solution $\psi\in C([0,T];H^{s})$ of (0.1), (0.2) such that

$\|\psi(T)-e^{i\theta}\psi_{0}\|_{H^{s}}<\varepsilon.$

More general formulation of this result is given in Theorem 2.2, where the
controllability is proved under an abstract saturation condition for the field
$Q$ (see Condition (H1)). Note that the time $T$ may depend on the initial
condition $\psi_{0}$, the target $e^{i\theta}\psi_{0}$, and the parameters in
the equation. In the second result, we show that, when $V=0$ and $\psi_{0}$ is
an eigenstate $\phi_{l}(x)=(2\pi)^{-d/2}e^{i\langle x,l\rangle},$
$l\in{\mathbb{Z}}^{d}$ of the Laplacian, the system can be approximately
controlled in any fixed time $T>0$ to any target of the form
$e^{i\theta}\phi_{m}$ with $m\in{\mathbb{Z}}^{d}$.

######  Theorem B.

For any $s\geq s_{d},$ $\varepsilon>0$, $l,m\in{\mathbb{Z}}^{d}$, $\theta\in
C^{\infty}({\mathbb{T}}^{d};{\mathbb{R}})$, and $T>0$, there is a control
$u\in L^{2}([0,T];{\mathbb{R}}^{q})$ and a unique solution $\psi\in
C([0,T];H^{s})$ of (0.1), (0.2) with $V=0$ and $\psi_{0}=\phi_{l}$ such that

$\|\psi(T)-e^{i\theta}\phi_{m}\|_{L^{2}}<\varepsilon.$

The controllability of the Schrödinger equation with time-dependent bilinear
(multiplicative) control has attracted a lot of attention during the last
fifteen years. In the one-dimensional case, local exact controllability
results are established by Beauchard, Coron, and Laurent [Bea05, BC06, BL10].
There is a vast literature on the approximate controllability in the
multidimensional case. For the first achievements, we refer the reader to the
papers by Boscain et al. [CMSB09, BCCS12], Mirrahimi [Mir09], and the second
author [Ner10]. Except the paper [BL10], all the other works consider the
linear Schrödinger equation, i.e., the one obtained by taking $\kappa=0$ in
Eq. (0.1); note that in that case the control problem is still nonlinear in
$u$.

Theorems A and B are the first to deal with the problem of bilinear
approximate controllability of the NLS equation. Let us emphasise that the
controllability between any pair of eigenstates in arbitrarily small time is
new even in the linear case $\kappa=0$. It is interesting to note that Theorem
B complements a result by Beauchard et al. [BCT18], which proves that, for
some choices of the field $Q$, there is a minimal time for the approximate
controllability to some particular states in the phase space.

The approach adopted in the proofs of Theorems A and B is quite different from
those used in the literature for bilinear control systems. We proceed by
developing Agrachev–Sarychev type arguments which were previously employed in
the case of additive controls. Let us recall that Agrachev and Sarychev [AS05,
AS06] considered the global approximate controllability of the 2D
Navier–Stokes and Euler systems. Their approach has been further extended by
many authors to different equations. Let us mention, for example, the papers
[Shi06, Shi07] by Shirikyan who considered the approximate controllability of
the 3D Navier–Stokes system and Sarychev [Sar12] who considered the case of
the 2D defocusing cubic Schrödinger equation. The configuration we use in the
present paper is closer to the one elaborated in the recent paper [Ner21],
where parabolic PDEs are studied with polynomial nonlinearities. We refer the
reader to the reviews [AS08, Shi18] and the paper [Ner21] for more references
and discussions.

The present paper is the first to deal with Agrachev–Sarychev type arguments
in a bilinear setting. To explain the scheme of the proof of Theorem A, let us
denote by ${\cal R}_{t}(\psi_{0},u)$ the solution of problem (0.1), (0.2)
defined up to some maximal time. A central role in the proof is played by the
limit

$e^{-i\delta^{-1/2}\varphi}{\cal
R}_{\delta}(e^{i\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)\to
e^{-i\left({\mathbb{B}}(\varphi)+\langle
u,Q\rangle\right)}\psi_{0}\quad\text{in $H^{s}$ as $\delta\to 0^{+}$}$ (0.5)

which holds for any $\psi_{0}\in H^{s}$, $\varphi\in
C^{\infty}({\mathbb{T}}^{d};{\mathbb{R}})$, and constant
$u\in{\mathbb{R}}^{q}$. Here we denote
${\mathbb{B}}(\varphi)(x)=\sum_{j=1}^{d}\left(\partial_{x_{j}}\varphi(x)\right)^{2}$.
Applying this limit with $\varphi=0$ and using the assumption (0.4), we see
that the equation can be controlled in small time from initial point
$\psi_{0}$ arbitrarily close to $e^{i\theta}\psi_{0}$ for any $\theta$ in the
vector space

${\cal H}_{0}=\mathop{\rm span}\nolimits\left\\{{\bf 1},\,\sin\langle
x,k\rangle,\,\cos\langle x,k\rangle:k\in{\cal K}\right\\}.$

By applying again the limit (0.5) with functions $\varphi=\theta_{j}\in{\cal
H}_{0}$, $j=1,\ldots,n$, we add more directions in $\theta$. That is, we show
that the system can be steered from $\psi_{0}$ close to $e^{i\theta}\psi_{0}$,
where $\theta$ now belongs to a larger vector space ${\cal H}_{1}$ whose
elements are of the form

$\theta_{0}-\sum_{j=1}^{n}{\mathbb{B}}(\theta_{j}).$

We iterate this argument and construct an increasing sequence of subspaces
$\\{{\cal H}_{j}\\}$ such that the equation can be approximately controlled to
any target $e^{i\theta}\psi_{0}$ with any $\theta\in{\cal H}_{j}$ and $j\geq
1$. Using trigonometric computations, we show that the union $\cup_{j=1}{\cal
H}_{j}$ is dense in $C^{k}({\mathbb{T}}^{d},{\mathbb{R}})$ for any $k\geq 1$
(in other words, ${\cal H}_{0}$ is a saturating space for the NLS equation,
see Definition 2.1). This completes the proof of Theorem A.

Theorem B is derived from Theorem A by noticing that the eigenstate $\phi_{l}$
can be approximated in $L^{2}$ by functions of the form $e^{i\theta}\phi_{m}$
and that the eigenstates are constant solutions333This follows immediately
from the assumptions that $V=0$ and ${\mathbf{1}}\in{\cal H}_{0}$. of Eq.
(0.1) corresponding to some control. This allows to appropriately adjust the
controllability time and choose it the same for any initial condition and
target.

As an application of Theorem A, we study the large time behavior of the
trajectories of the random NLS equation. We show that if a random process
perturbes the same Fourier modes as in the above controllability results, then
the energy is almost surely transferred to higher modes resulting in the
unboundedness of the trajectories in regular Sobolev spaces. More precisely,
we replace the control $u$ by an ${\mathbb{R}}^{q}$-valued random process
$\eta$ of the form

$\eta(t)=\sum_{k=1}^{+\infty}{\mathbb{I}}_{[k-1,k)}(t)\eta_{k}(t-k+1),$ (0.6)

where ${\mathbb{I}}_{[k-1,k)}$ is the indicator function of the interval
$[k-1,k)$ and $\\{\eta_{k}\\}$ are independent identically distributed random
variables in $L^{2}([0,1];{\mathbb{R}}^{q})$ with non-degenerate law (see
Condition (H2)). The solution $\psi$ of the problem (0.1), (0.2), (0.6) will
be itself a random process in $H^{s}$. We prove the following result.

######  Theorem C.

For any $s>s_{d}$ and any non-zero $\psi_{0}\in H^{s}$, the trajectory of
(0.1), (0.2), (0.6) is almost surely unbounded in $H^{s}$.

The idea of constructing unbounded solutions by using random perturbations is
not new. Such results have been obtained by Bourgain [Bou99] and Erdogan et
al. [EgKS03] for linear one-dimensional Schrödinger equations. They also
provided polynomial lower bounds for the growth. Unboundedness of trajectories
for multidimensional linear Schrödinger equations is obtained in [Ner09]. In
that paper, the assumptions on the law of the random perturbation are rather
general and no estimates for the growth are given; Theorem C is a
generalisation of that result to the case of the NLS equations. There are also
examples of linear Schrödinger equations with various deterministic time-
dependent potentials which admit unbounded trajectories: e.g., see the papers
by Bambusi et al. [BGMR18], Delort [Del14], Haus and Maspero [HM20, Mas19],
and the references therein.

There are only few results in the case of unperturbed NLS equations. For cubic
defocusing Schrödinger equations on bounded domains or manifolds, the
existence of unbounded trajectories in regular Sobolev spaces is a challenging
open problem (see Bourgain [Bou00]). In different situations, existence of
trajectories with arbitrarily large finite growth has been shown by Kuksin
[Kuk97], Colliander et al. [CKS+10], Guardia and Kaloshin [GK15], and others.
Hani et al. [HPTV15] show the existence of unbounded trajectories in the case
of the cubic defocusing Schrödinger equation on the infinite cylinder
${\mathbb{R}}\times{\mathbb{T}}^{d}$. In the case of the cubic Szegő equation
on the circle, Gérard and Grellier [GG17] show that the trajectories are
generically unbounded in Sobolev spaces. Moreover, they exhibit the existence
of a family of solutions with superpolynomial growth.

Let us give a brief (and not entirely accurate) description of the main ideas
of the proof of Theorem C. By starting from any initial point $\psi_{0}\in
H^{s}$, Theorem A allows to increase the Sobolev norms by choosing
appropriately the control. This, together with a compactness argument and the
assumption that the law of the process $\eta$ is non-degenerate, leads to a
uniform estimate of the form

$c_{M}=\sup_{\psi_{0}\in
H^{s}}{\mathbb{P}}\left\\{\sup_{t\in[0,1]}\|\psi(t)\|_{H^{s}}>M\right\\}<1$

for any $M>0$. By combining the latter with the Markov property, we show that

${\mathbb{P}}\left\\{\sup_{t\in[0,n]}\|\psi(t)\|_{H^{s}}>M\right\\}\leq
c_{M}^{n}$

for any $\psi_{0}\in H^{s}$. Then, the Borel–Cantelli lemma implies that the
norm of any trajectory becomes almost surely larger than $M$ in some random
time that is almost surely finite. As $M$ is arbitrary, this proves the
required result.

The paper is organised as follows. In Section 1, we discuss the local well-
posedness and some stability properties of the NLS equation. In Section 2, we
formulate more general versions of Theorems A and B and give their proofs.
Section 3 is devoted to the derivation of limit (0.5). In Section 4, we
establish a general criterion for the validity of the saturation property.
Finally, in Section 5, we prove Theorem C.

### Acknowledgement

The authors thank Armen Shirikyan for his valuable comments. The research of
AD was supported by the ANR grant ISDEEC ANR-16-CE40-0013. The research of VN
was supported by the ANR grant NONSTOPS ANR-17-CE40-0006-02.

### Notation

In what follows, we use the following notation.

$\langle\cdot,\cdot\rangle$ is the Euclidian scalar product in
${\mathbb{R}}^{q}$ and $\|\cdot\|$ is the corresponding norm. We write $m\bot
l$ when the vectors $m,l\in{\mathbb{R}}^{q}$ are orthogonal and $m\not\perp l$
when they are not.

$H^{s}=H^{s}({\mathbb{T}}^{d};{\mathbb{C}}),~{}s\geq 0$ and
$L^{p}=L^{p}({\mathbb{T}}^{d};{\mathbb{C}}),~{}p\geq 1$ are the standard
Sobolev and Lebesgue spaces of functions $f:{\mathbb{T}}^{d}\to{\mathbb{C}}$
endowed with the norms $\|\cdot\|_{s}$ and $\|\cdot\|_{L^{p}}$. The space
$L^{2}$ is endowed with the scalar product

$\langle
f,g\rangle_{L^{2}}=\int_{{\mathbb{T}}^{d}}f(x)\overline{g(x)}{\textup{d}}x.$

$C^{s}=C^{s}({\mathbb{T}}^{d};{\mathbb{C}})$,
$s\in{\mathbb{N}}\cup\\{\infty\\}$ is the space of $s$-times continuously
differentiable functions $f:{\mathbb{T}}^{d}\to{\mathbb{C}}$.

Let $X$ be a Banach space. We denote by $B_{X}(a,r)$ the closed ball of radius
$r>0$ centred at $a\in X$.

We write $J_{T}$ instead of $[0,T]$ and $J$ instead of $[0,1]$.

$C(J_{T};X)$ is the space of continuous functions $f:J_{T}\to X$ with the norm

$\|f\|_{C(J_{T};X)}=\max_{t\in J_{T}}\|f(t)\|_{X}.$

$L^{p}(J_{T};X),1\leq p<\infty$ is the space of Borel-measurable functions
$f:J_{T}\to X$ with

$\|f\|_{L^{p}(J_{T};X)}=\left(\int_{0}^{T}\|f(t)\|_{X}^{p}{\textup{d}}t\right)^{1/p}<\infty.$

$\lceil x\rceil$ is the least integer greater than or equal to
$x\in{\mathbb{R}}$.

$s_{d}$ is the least integer strictly greater than $d/2$.

${\bf 1}$ is the function identically equal to $1$ on ${\mathbb{T}}^{d}$.

##  1 Preliminaries

In this section, we consider the NLS equation (0.1), where $u$ is a
deterministic ${\mathbb{R}}^{q}$-valued function and
$V:{\mathbb{T}}^{d}\to{\mathbb{R}}$ and
$Q:{\mathbb{T}}^{d}\to{\mathbb{R}}^{q}$ are arbitrary smooth functions. In
what follows, we shall always assume that the parameters $d\geq 1$, $p\geq 1$,
and $\kappa\in{\mathbb{R}}$ are arbitrary. Here we formulate two propositions
that will be used in the proofs of our main results. The first one gathers
some well-known facts about the local well-posedness and stability of the NLS
equation in regular Sobolev spaces.

######  Proposition 1.1.

For any $s>d/2$, $\hat{\psi}_{0}\in H^{s}$, and $\hat{u}\in
L^{2}_{loc}({\mathbb{R}}_{+};{\mathbb{R}}^{q})$, there is a maximal time
${\cal T}={\cal T}(\hat{\psi}_{0},\hat{u})>0$ and a unique solution
$\hat{\psi}$ of the problem (0.1), (0.2) with
$(\psi_{0},u)=(\hat{\psi}_{0},\hat{u})$ whose restriction to the interval
$J_{T}$ belongs to $C(J_{T};H^{s})$ for any $T<{\cal T}$. If ${\cal
T}<\infty$, then $\|\hat{\psi}(t)\|_{s}\to+\infty$ as $t\to{\cal T}^{-}$.
Furthermore, for any $T<{\cal T}$, there are constants
$\delta=\delta(T,\Lambda)>0$ and $C=C(T,\Lambda)>0$, where

$\Lambda=\|\hat{\psi}\|_{C(J_{T};H^{s})}+\|\hat{u}\|_{L^{2}(J_{T};{\mathbb{R}}^{q})},$

such that the following two properties hold.

  1. (i)

For any $\psi_{0}\in H^{s}$ and $u\in L^{2}(J_{T};{\mathbb{R}}^{q})$
satisfying

$\|\psi_{0}-\hat{\psi}_{0}\|_{s}+\|u-\hat{u}\|_{L^{2}(J_{T};{\mathbb{R}}^{q})}<\delta,$
(1.1)

the problem (0.1), (0.2) has a unique solution $\psi\in C(J_{T};H^{s}).$

  2. (ii)

Let ${\cal R}$ be the resolving operator for Eq. (0.1), i.e., the mapping
taking a couple $(\psi_{0},u)$ satisfying (1.1) to the solution $\psi$. Then

$\displaystyle\|{\cal R}(\psi_{0},u)-{\cal
R}(\hat{\psi}_{0},\hat{u})\|_{C(J_{T};H^{s})}$ $\displaystyle\leq
C\left(\|\psi_{0}-\hat{\psi}_{0}\|_{s}+\|u-\hat{u}\|_{L^{2}(J_{T};{\mathbb{R}}^{q})}\right).$

The proof of this proposition is rather standard, so we omit it (e.g., see
Section 3.3 in [Tao06] or Section 4.10 in [Caz03] for similar results). Let
${\cal S}$ be the unit sphere in $L^{2}$. As the functions $V,Q$, and $u$ are
real-valued, the solution $\psi$ belongs to ${\cal S}$ throughout its
lifespan, provided that $\psi_{0}\in{\cal S}\cap H^{s}$.

Before formulating the second proposition, let us introduce some notation. For
any $\psi_{0}\in H^{s}$ and $T>0$, let $\Theta(\psi_{0},T)$ be the set of
functions $u\in L^{2}(J_{T};{\mathbb{R}}^{q})$ such that the problem (0.1),
(0.2) has a solution in $C(J_{T};H^{s})$. By the previous proposition, the set
$\Theta(\psi_{0},T)$ is open in $L^{2}(J_{T};{\mathbb{R}}^{q})$. For any
$\varphi\in C^{1}({\mathbb{T}}^{d};{\mathbb{R}})$, let

${\mathbb{B}}(\varphi)(x)=\sum_{j=1}^{d}\left(\partial_{x_{j}}\varphi(x)\right)^{2}.$
(1.2)

We have the following asymptotic property in small time.

######  Proposition 1.2.

For any $s\geq s_{d}$, $\psi_{0}\in H^{s}$, $u\in{\mathbb{R}}^{q}$, and
$\varphi\in C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$, where $r={\lceil
s\rceil+2}$, there is a constant $\delta_{0}>0$ such that 444For any vector
$u\in{\mathbb{R}}^{q}$, with a slight abuse of notation, we denote by the same
letter the constant function equal to $u$.
$\delta^{-1}u\in\Theta(e^{i\delta^{-1/2}\varphi}\psi_{0},\delta)$ for any
$\delta\in(0,\delta_{0})$ and the following limit holds

$e^{-i\delta^{-1/2}\varphi}{\cal
R}_{\delta}(e^{i\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)\to
e^{-i\left({\mathbb{B}}(\varphi)+\langle
u,Q\rangle\right)}\psi_{0}\quad\text{in $H^{s}$ as $\delta\to 0^{+}$},$ (1.3)

where ${\cal R}_{\delta}$ is the restriction of the solution at time
$t=\delta$.

The proof of this proposition is postponed to Section 4. Limit (1.3) is a
multiplicative version of a limit established in Proposition 2 in [Ner21] in
the case of parabolic PDEs with additive controls.

##  2 Approximate controllability

In what follows, we assume that $s\geq s_{d}$ and denote $r={\lceil
s\rceil+2}$ as in Proposition 1.2. We start this section with a definition of
a saturation property inspired by the papers [AS06, Shi06]. Let ${\cal H}$ be
a finite-dimensional subspace of $C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$, and
let ${\cal F}({\cal H})$ be the largest subspace of
$C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$ whose elements can be represented in
the form

$\theta_{0}-\sum_{j=1}^{n}{\mathbb{B}}(\theta_{j})$

for some integer $n\geq 1$ and functions $\theta_{j}\in{\cal H}$,
$j=0,\ldots,n$, where ${\mathbb{B}}$ is given by (1.2). As ${\mathbb{B}}$ is
quadratic, ${\cal F}({\cal H})$ is well-defined and finite-dimensional. Let us
define a non-decreasing sequence $\\{{\cal H}_{j}\\}$ of finite-dimensional
subspaces by ${\cal H}_{0}={\cal H}$ and ${\cal H}_{j}={\cal F}({\cal
H}_{j-1})$, $j\geq 1$, and denote

${\cal H}_{\infty}=\bigcup_{j=1}^{+\infty}{\cal H}_{j}.$ (2.1)

######  Definition 2.1.

A finite-dimensional subspace ${\cal H}\subset
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$ is said to be saturating if ${\cal
H}_{\infty}$ is dense in $C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$.

We assume that the following condition is satisfied.

(H1) The field $Q=(Q_{1},\ldots,Q_{q})$ is saturating, i.e., the subspace

${\cal H}=\mathop{\rm span}\nolimits\\{Q_{j}:j=1,\ldots,q\\}$

is saturating in the sense of Definition 2.1

In this section, we prove the following result. As we will see below, it
implies Theorems A and B formulated in the Introduction.

######  Theorem 2.2.

Assume that Condition (H1) is satisfied. Then for any $\varepsilon>0$,
$\varkappa>0$, $\psi_{0}\in H^{s}$, and $\theta\in
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$, there is a time $T\in(0,\varkappa)$ and
a control $u\in\Theta(\psi_{0},T)$ such that

$\|{\cal R}_{T}(\psi_{0},u)-e^{i\theta}\psi_{0}\|_{s}<\varepsilon.$

###### Proof.

By using an induction argument in $N$, we show that the approximate
controllability property in this theorem is true for any $\theta\in{\cal
H}_{N}$ and $N\geq 0$. Combined with the saturation hypothesis, this will lead
to approximate controllability with any $\theta\in
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$.

Step 1. Case $N=0$. Let us show that, for any $\varepsilon>0$, $\varkappa>0$,
$\psi_{0}\in H^{s}$, and $\theta\in{\cal H}$, there is a time
$T\in(0,\varkappa)$ and a control $u\in\Theta(\psi_{0},T)$ such that

$\|{\cal R}_{T}(\psi_{0},u)-e^{i\theta}\psi_{0}\|_{s}<\varepsilon.$ (2.2)

By applying Proposition 1.2 with $\varphi=0$ and $u\in{\mathbb{R}}^{q}$ such
that $\theta=-\langle u,Q\rangle$, we obtain

${\cal R}_{\delta}(\psi_{0},\delta^{-1}u)\to e^{i\theta}\psi_{0}\quad\text{in
$H^{s}$ as $\delta\to 0^{+}$}.$

This implies (2.2) with sufficiently small time $T=\delta$ and control
$\delta^{-1}u$.

Step 2. Case $N\geq 1$. We assume that the result is true for any
$\theta\in{\cal H}_{N-1}$. Let $\tilde{\theta}\in{\cal H}_{N}$ be of the form

$\tilde{\theta}=\theta_{0}-\sum_{j=1}^{n}{\mathbb{B}}(\theta_{j}),$

where $n\geq 1$ and $\theta_{j}\in{\cal H}_{N-1}$, $j=0,\ldots,n$. By applying
Proposition 1.2 with $\varphi=\theta_{1}$ and $u=0$, we get

$e^{-i\delta^{-1/2}\theta_{1}}{\cal
R}_{\delta}(e^{i\delta^{-1/2}\theta_{1}}\psi_{0},0)\to
e^{-i{\mathbb{B}}(\theta_{1})}\psi_{0}\quad\text{in $H^{s}$ as $\delta\to
0^{+}$}.$

The induction hypothesis, the assumption that $\theta_{1}\in{\cal H}_{N-1}$,
and Proposition 1.1 imply that, for any $\varepsilon>0$ and $\varkappa>0$,
there is a time $T_{1}\in(0,\varkappa)$ and a control
$u_{1}\in\Theta(\psi_{0},T_{1})$ such that

$\|{\cal
R}_{T_{1}}(\psi_{0},u_{1})-e^{-i{\mathbb{B}}(\theta_{1})}\psi_{0}\|_{s}<\varepsilon.$

By iterating this argument with $\theta_{j}\in{\cal H}_{N-1}$, $j=0,\ldots,n$,
we obtain that for any $\varepsilon>0$ and $\varkappa>0$, there is
$T_{n}\in(0,\varkappa)$ and $u_{n}\in\Theta(\psi_{0},T_{n})$ such that

$\|{\cal
R}_{T_{n}}(\psi_{0},u_{n})-e^{i\left(\theta_{0}-\sum_{j=1}^{n}{\mathbb{B}}(\theta_{j})\right)}\psi_{0}\|_{s}=\|{\cal
R}_{T_{n}}(\psi_{0},u_{n})-e^{i\tilde{\theta}}\psi_{0}\|_{s}<\varepsilon.$

As $\tilde{\theta}\in{\cal H}_{N}$ is arbitrary, this proves the required
property for $N$.

Step 3. Conclusion. Finally, let $\theta\in
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$ be arbitrary. By the saturation
hypothesis, ${\cal H}_{\infty}$ is dense in
$C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$. Hence, we can find $N\geq 1$ and
$\tilde{\theta}\in H_{N}$ such that

$\|e^{i\theta}\psi_{0}-e^{i\tilde{\theta}}\psi_{0}\|_{s}<\varepsilon.$

Applying the controllability property proved in the previous steps for
$\tilde{\theta}\in H_{N}$, we complete the proof. ∎

As a consequence of this result, we have the following two theorems.

######  Theorem 2.3.

Under the conditions of Theorem 2.2, for any $M>0$, $\varkappa>0$, and non-
zero $\psi_{0}\in H^{s}$, there is a time $T\in(0,\varkappa)$ and a control
$u\in\Theta(\psi_{0},T)$ such that

$\|{\cal R}_{T}(\psi_{0},u)\|_{s}>M.$

###### Proof.

It suffices to apply Theorem 2.2 by choosing $\theta\in
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$ such that

$\|e^{i\theta}\psi_{0}\|_{s}>M.$

To find such $\theta$, we take any $\theta_{1}\in
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$ verifying
$\|e^{i\theta_{1}}\psi_{0}\|_{1}\neq 0$, put $\theta=\lambda\theta_{1}$ with
sufficiently large $\lambda>0$, and use the inequality
$\|\cdot\|_{1}\leq\|\cdot\|_{s}.$ ∎

######  Theorem 2.4.

Assume that the conditions of Theorem 2.2 are satisfied and

${\mathbf{1}}\in\mathop{\rm
span}\nolimits\\{Q_{j}:j=1,\ldots,q\\}\quad\quad\text{and}\quad\quad V=0.$
(2.3)

Then, for any $\varepsilon>0$, $l,m\in{\mathbb{Z}}^{d}$, $\theta\in
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$, and $T>0$, there is a control
$u\in\Theta(\phi_{l},T)$ such that

$\|{\cal R}_{T}(\phi_{l},u)-e^{i\theta}\phi_{m}\|_{L^{2}}<\varepsilon.$

###### Proof.

Let us take any $\theta_{1}\in C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$. Applying
Theorem 2.2, we find a time $T_{1}\in(0,T)$ and a control
$u_{1}\in\Theta(\phi_{l},T_{1})$ such that

$\|{\cal
R}_{T_{1}}(\phi_{l},u)-e^{i\theta_{1}}\phi_{l}\|_{s}<\frac{\varepsilon}{2}.$

Choosing $\theta_{1}\in C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$ such that

$\|e^{i\theta_{1}}\phi_{l}-e^{i\theta}\phi_{m}\|_{L^{2}}<\frac{\varepsilon}{2},$

we arrive at

$\|{\cal R}_{T_{1}}(\phi_{l},u)-e^{i\theta}\phi_{m}\|_{L^{2}}<\varepsilon.$

Now, notice that $\phi_{l}$ is a stationary solution of Eq. (0.1)
corresponding a control $u_{0}\in
L^{2}_{loc}({\mathbb{R}}_{+};{\mathbb{R}}^{q})$ satisfying the relation

$\langle u_{0}(t),Q(x)\rangle=-|l|^{2}-\kappa(2\pi)^{-dp}\quad\text{for any
$t\geq 0$ and $x\in{\mathbb{T}}^{d}$}.$

Such a choice of $u_{0}$ is possible in view of assumption (2.3). Thus,
$u_{0}\in\Theta(\phi_{l},t)$ and $\phi_{l}={\cal R}_{t}(\phi_{l},u_{0})$ for
any $t\geq 0$. Setting

$u(t)=\begin{cases}u_{0}(t)&\text{for }t\in[0,T-T_{1}],\\\
u_{1}(t-T+T_{1})&\text{for }t\in(T-T_{1},T],\end{cases}$

we complete the proof of the theorem. ∎

Let us close this section with an example of a saturating subspace. Let ${\cal
I}\subset{\mathbb{Z}}^{d}_{*}$ be a finite set and let

${\cal H}={\cal H}({\cal I})=\text{span}\left\\{{\bf 1},\,\sin\langle
x,k\rangle,\,\cos\langle x,k\rangle:k\in{\cal I}\right\\}.$ (2.4)

Recall that ${\cal I}$ is a generator if any vector of ${\mathbb{Z}}^{d}$ is a
linear combination of vectors of ${\cal I}$ with integer coefficients. The
following proposition is proved in Section 4.

######  Proposition 2.5.

The subspace ${\cal H}({\cal I})$ is saturating in the sense of Definition 2.1
if and only if ${\cal I}$ is a generator and for any $l,m\in{\cal I}$, there
are vectors $\\{n_{j}\\}_{j=1}^{k}\subset{\cal I}$ such that $l\not\perp
n_{1}$, $n_{j}\not\perp n_{j+1}$, $j=1,\ldots,k-1,$ and $n_{k}\not\perp m$.

Clearly, the set ${\cal K}\subset{\mathbb{Z}}^{d}_{*}$ defined by (0.3)
satisfies the condition in this proposition. Therefore, the subspace ${\cal
H}({\cal K})$ is saturating, and Theorems A and B follow from Theorems 2.2 and
2.4, respectively.

##  3 Proof of Proposition 1.2

We start by proving the result in the case when $s>d/2$ is an integer, so
$r=s+2$. Let us fix any $R>0$ and assume that $\psi_{0}\in H^{s}$, $\varphi\in
C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$, and $u\in{\mathbb{R}}^{q}$ are such
that

$\|\psi_{0}\|_{s}+\|\varphi\|_{C^{r}}+\|u\|_{{\mathbb{R}}^{q}}\leq R.$ (3.1)

For any $\delta>0$, we denote $\phi(t)=e^{-i\delta^{-1/2}\varphi}{\cal
R}_{t}(e^{i\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)$. According to
Proposition 1.1, $\phi(t)$ exists up to some maximal time ${\cal
T}^{\delta}={\cal T}(e^{i\delta^{-1/2}\varphi}\psi_{0},\delta^{-1}u)$, and

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

We need to show that

  * $(a)$

there is a constant $\delta_{0}>0$ such that ${\cal T}^{\delta}>\delta$ for
any $\delta<\delta_{0}$;

  * $(b)$

the following limit holds

$\phi(\delta)\to e^{-i\left({\mathbb{B}}(\varphi)+\langle
u,Q\rangle\right)}\psi_{0}\quad\text{in $H^{s}$ as $\delta\to 0^{+}$}.$

To prove these properties, we introduce the functions

$\displaystyle w(t)$ $\displaystyle=e^{-i\left({\mathbb{B}}(\varphi)+\langle
u,Q\rangle\right)t}\psi_{0}^{\delta},$ (3.2) $\displaystyle v(t)$
$\displaystyle=\phi(\delta t)-w(t),$

where $\psi_{0}^{\delta}\in H^{r}$ is such that 555In what follows, $C$
denotes positive constants which may change from line to line. These constants
depend on the parameters $R,V,Q,\kappa,p,d,s$, but not on $\delta$.

$\displaystyle\|\psi_{0}^{\delta}\|_{s}\leq C\quad\quad\quad\quad\text{for
}\delta\leq 1,$ (3.3) $\displaystyle\|\psi_{0}^{\delta}\|_{r}\leq
C\delta^{-1/4}\,\,\,\,\quad\text{for }\delta\leq 1,$ (3.4)
$\displaystyle\|\psi_{0}-\psi_{0}^{\delta}\|_{s}\to 0\quad\,\,\,\text{ as
$\delta\to 0^{+}$}.$

For example, we can define $\psi_{0}^{\delta}$ by using the heat semigroup:
$\psi_{0}^{\delta}=e^{\delta^{1/4}\Delta}\psi_{0}$. In view of (3.1)-(3.4), we
have

$\displaystyle\|w(t)\|_{s}$ $\displaystyle\leq C,\quad\,\,\quad\quad t\geq 0,$
(3.5) $\displaystyle\|w(t)\|_{r}$ $\displaystyle\leq C{\delta^{-1/4}},\quad
t\geq 0.$ (3.6)

Furthermore, $v(t)$ is well-defined for $t<\delta^{-1}{\cal T}^{\delta}$ and
satisfies the equation

$\displaystyle i\partial_{t}v$ $\displaystyle=-\delta\Delta(v+w)+\delta
V(v+w)+\delta\kappa|v+w|^{2p}(v+w)$
$\displaystyle\quad-i\delta^{\frac{1}{2}}{\mathbb{D}}(v+w,\varphi)+{\mathbb{B}}(\varphi)v+\langle
u,Q\rangle v,$ (3.7)

and the initial condition

$v(0)=\psi_{0}-\psi_{0}^{\delta},$ (3.8)

where

${\mathbb{D}}(v+w,\varphi)=(v+w)\Delta\varphi+2\sum_{j=1}^{d}\partial_{x_{j}}(v+w)\,\partial_{x_{j}}\varphi.$

Let $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in{\mathbb{N}}^{d}$ be such that
$|\alpha|=|\alpha_{1}|+\ldots+|\alpha_{d}|\leq s$. We take the scalar product
of Eq. (3.7) with $\partial^{2\alpha}v$ in $L^{2}$ and integrating by parts,
we obtain

$\displaystyle\partial_{t}\|\partial^{\alpha}v\|_{L^{2}}^{2}$
$\displaystyle\leq C\Big{(}\delta|\langle\Delta
w,\partial^{2\alpha}v\rangle_{L^{2}}|+\delta|\langle
V(v+w),\partial^{2\alpha}v\rangle_{L^{2}}|$
$\displaystyle\quad+\delta|\langle|v+w|^{2p}(v+w),\partial^{2\alpha}v\rangle_{L^{2}}|+\delta^{1/2}|\langle{\mathbb{D}}(v+w,\varphi),\partial^{2\alpha}v\rangle_{L^{2}}|$
$\displaystyle\quad+|\langle{\mathbb{B}}(\varphi)v+\langle u,Q\rangle
v,\partial^{2\alpha}v\rangle_{L^{2}}|\Big{)}=\sum_{j=1}^{5}I_{j}.$ (3.9)

We estimate the terms $I_{1},I_{2},I_{3},$ and $I_{5}$ by integrating by parts
and by using (3.1), (3.5), and (3.6):

$\displaystyle|I_{1}|$ $\displaystyle\leq C\delta\|w\|_{r}\|v\|_{s}\leq
C\delta^{3/4}\|v\|_{s},$ $\displaystyle|I_{2}|$ $\displaystyle\leq
C\delta\|v+w\|_{s}\|v\|_{s}\leq C\delta\|v\|_{s}^{2}+C\delta\|v\|_{s},$
$\displaystyle|I_{3}|$ $\displaystyle\leq
C\delta\|v+w\|_{s}^{2p+1}\|v\|_{s}\leq
C\delta\|v\|_{s}^{2(p+1)}+C\delta\|v\|_{s},$ $\displaystyle|I_{5}|$
$\displaystyle\leq C\|v\|_{s}^{2}.$

We estimate $I_{4}$ as follows

$|I_{4}|\leq C\delta^{1/2}\|v\|_{s}^{2}+C\delta^{1/2}\|w\|_{s+1}\|v\|_{s}\leq
C\delta^{1/2}\|v\|_{s}^{2}+C\delta^{1/4}\|v\|_{s},$

In the last relation, we used again the integration by parts, the identities
(3.1), (3.5) and (3.6), and the equality

$\displaystyle\langle\partial_{x_{j}}\varphi\,\partial_{x_{j}}\partial^{\alpha}v,\partial^{\alpha}v\rangle_{L^{2}}$
$\displaystyle=\frac{1}{2}\langle\partial_{x_{j}}\varphi,\partial_{x_{j}}|\partial^{\alpha}v|^{2}\rangle_{L^{2}}=-\langle\partial^{2}_{x_{j}}\varphi,|\partial^{\alpha}v|^{2}\rangle_{L^{2}}.$

Summing up inequalities (3.9) for all $\alpha\in{\mathbb{N}}^{d}$,
$|\alpha|\leq s$, combining the resulting inequality with the estimates for
$I_{j}$ and the Young inequality, and recalling that $\delta\leq 1$, we obtain

$\partial_{t}\|v\|^{2}_{s}\leq
C\delta^{1/2}+C(1+\delta^{1/2})\|v\|_{s}^{2}+C\delta\|v\|_{s}^{2(p+1)},\quad
t\leq\delta^{-1}{\cal T}^{\delta}.$

This inequality, together with (3.8) and the Gronwall inequality, implies that

$\|v(t)\|_{s}^{2}\leq
e^{C(1+\delta^{1/2})t}\left(C\delta^{1/2}t+\|\psi_{0}-\psi_{0}^{\delta}\|_{s}^{2}+C\delta\int_{0}^{t}\|v(y)\|_{s}^{2(p+1)}{\textup{d}}y\right)$
(3.10)

for $t\leq\delta^{-1}{\cal T}^{\delta}$. Let us take $\delta_{0}\in(0,1)$ so
small that, for $\delta<\delta_{0}$,

$\displaystyle\|\psi_{0}-\psi_{0}^{\delta}\|_{s}^{2}<1,$ (3.11) $\displaystyle
e^{C(1+\delta^{1/2})}\left(C\delta^{1/2}+\|\psi_{0}-\psi_{0}^{\delta}\|_{s}^{2}\right)<\frac{1}{2},$
(3.12)

and denote

$\tau^{\delta}=\sup\left\\{t<\delta^{-1}{\cal
T}^{\delta}:\|v(t)\|_{s}<1\right\\}.$

From (3.8) and (3.11) it follows that $\tau^{\delta}>0$ for
$\delta<\delta_{0}$. Let us show that $\tau^{\delta}>1$ provided that

$\delta_{0}<\left(2Ce^{2C}\right)^{-1}.$ (3.13)

Assume, by contradiction, that $\tau^{\delta}\leq 1$. Let $t=\tau^{\delta}$ in
(3.10). By using (3.12) and (3.13), we obtain

$1=\|v(\tau^{\delta})\|_{s}^{2}<\frac{1}{2}+\frac{1}{2}\int_{0}^{\tau^{\delta}}\|v(y)\|_{s}^{2(p+1)}{\textup{d}}y\leq
1.$

This contradiction shows that $\tau^{\delta}>1$ for $\delta<\delta_{0}$, hence
also $1<\delta^{-1}{\cal T}^{\delta}$. Thus, property (a) is proved. Taking
$t=1$ in (3.10), we arrive at

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

This implies (b) and completes the proof in the case when $s>d/2$ is an
integer.

To derive properties (a) and (b) in the general case, i.e., when $s\geq s_{d}$
is an arbitrary number, we use inequality (3.10) for integer values of $s$ and
an interpolation argument.

##  4 Saturating subspaces

###### Proof of Proposition 2.5.

The proof is divided into four steps.

Step 1. First, let us assume that ${\cal I}\subset{\mathbb{Z}}^{d}_{*}$ is an
arbitrary finite set, ${\cal H}_{0}({\cal I})={\cal H}({\cal I})$ is the
subspace defied by (2.4), ${\cal H}_{j}({\cal I})={\cal F}({\cal
H}_{j-1}({\cal I}))$ for $j\geq 1$, and ${\cal H}_{\infty}({\cal I})$ is
defined by (2.1).

Step 1.1. Let us show that, if

$\cos\langle x,m\rangle,\,\sin\langle x,m\rangle\in{\cal H}_{\infty}({\cal
I})\quad\text{for some $m\in{\mathbb{Z}}^{d}_{*}$},$

then

${\mathbb{B}}(\cos\langle x,m\rangle),\,{\mathbb{B}}(\sin\langle
x,m\rangle)\in{\cal H}_{\infty}({\cal I}).$

Indeed, assume that

$\cos\langle x,m\rangle,\,\sin\langle x,m\rangle\in{\cal H}_{N}({\cal
I})\quad\text{ for some $N\geq 0$}.$ (4.1)

The equalities

$\cos\langle x,2m\rangle=1-\frac{2}{|m|^{2}}{\mathbb{B}}(\cos\langle
x,m\rangle)=\frac{2}{|m|^{2}}{\mathbb{B}}(\sin\langle x,m\rangle)-1,$ (4.2)

the assumptions ${\bf 1}\in{\cal H}({\cal I})$ and (4.1), and the definition
of ${\cal F}$ imply that

$\cos\langle x,2m\rangle\in{\cal H}_{N+1}({\cal I}).$ (4.3)

As a consequence of (4.2) and (4.3), we have

$\displaystyle{\mathbb{B}}(\cos\langle x,m\rangle)$
$\displaystyle=\frac{|m|^{2}}{2}(1-\cos\langle x,2m\rangle)\in{\cal
H}_{N+1}({\cal I}),$ $\displaystyle{\mathbb{B}}(\sin\langle x,m\rangle)$
$\displaystyle=\frac{|m|^{2}}{2}(1+\cos\langle x,2m\rangle)\in{\cal
H}_{N+1}({\cal I}),$

which imply the required result.

Step 1.2. Let us show that, if

$\cos\langle x,m\rangle,\,\sin\langle x,m\rangle,\,\cos\langle
x,l\rangle,\,\sin\langle x,l\rangle\in{\cal H}_{\infty}({\cal I})$

for some $m,l\in{\mathbb{Z}}^{d}_{*}$ such that $m\not\perp l$, then

$\cos\langle x,m+l\rangle,\,\sin\langle x,m+l\rangle\in{\cal H}_{\infty}({\cal
I}).$

Indeed, this follows immediately from the equalities

$\displaystyle\cos\langle x,m+l\rangle=$ $\displaystyle\,\pm\frac{1}{\langle
m,l\rangle}\Big{(}{\mathbb{B}}(\sin\langle x,m\rangle\pm\sin\langle
x,l\rangle)+{\mathbb{B}}(\cos\langle x,m\rangle\mp\cos\langle x,l\rangle)$
$\displaystyle-{\mathbb{B}}(\sin\langle x,m\rangle)-{\mathbb{B}}(\sin\langle
x,l\rangle)-{\mathbb{B}}(\cos\langle x,m\rangle)-{\mathbb{B}}(\cos\langle
x,l\rangle)\Big{)},$ $\displaystyle\sin\langle x,m+l\rangle=$
$\displaystyle\,\pm\frac{1}{\langle m,l\rangle}\Big{(}{\mathbb{B}}(\sin\langle
x,m\rangle\mp\cos\langle x,l\rangle)+{\mathbb{B}}(\cos\langle
x,m\rangle\mp\sin\langle x,l\rangle)$ $\displaystyle-{\mathbb{B}}(\sin\langle
x,m\rangle)-{\mathbb{B}}(\sin\langle x,l\rangle)-{\mathbb{B}}(\cos\langle
x,m\rangle)-{\mathbb{B}}(\cos\langle x,l\rangle)\Big{)}$

and the result of step 1.1.

Step 2. Now, let us suppose that ${\cal I}\subset{\mathbb{Z}}^{d}_{*}$ is a
finite set such that, for any $l,m\in{\cal I}$, there are vectors
$\\{n_{j}\\}_{j=1}^{k}\subset{\cal I}$ satisfying $l\not\perp n_{1}$,
$n_{j}\not\perp n_{j+1}$, $j=1,\ldots,k-1,$ and $n_{k}\not\perp m$. Let
$N=\text{card}({\cal I})$ and ${\cal I}=\\{m_{1},\ldots,m_{N}\\}$. Arguing by
induction on $N$, we show in this step that

$\cos\langle x,a_{1}m_{1}+\ldots+a_{N}m_{N}\rangle,\,\sin\langle
x,a_{1}m_{1}+\ldots+a_{N}m_{N}\rangle\in{\cal H}_{\infty}({\cal I})$ (4.4)

for any $a_{1},\ldots,a_{N}\in{\mathbb{Z}}$.

Step 2.1. Let ${\cal I}=\\{m_{1},m_{2}\\}\subset{\mathbb{Z}}^{d}_{*}$ with
$m_{1}\not\perp m_{2}$. By the result of step 1.2, we have

$\cos\langle x,a_{1}m_{1}\rangle,\,\sin\langle
x,a_{1}m_{1}\rangle,\,\cos\langle x,a_{2}m_{2}\rangle,\,\sin\langle
x,a_{2}m_{2}\rangle\in{\cal H}_{\infty}({\cal I})$

for any $a_{1},a_{2}\in{\mathbb{Z}}$. Again, in view of step 1.2, this implies
that

$\cos\langle x,a_{1}m_{1}+a_{2}m_{2}\rangle,\,\sin\langle
x,a_{1}m_{1}+a_{2}m_{2}\rangle\in{\cal H}_{\infty}({\cal I})$

for any $a_{1},a_{2}\in{\mathbb{Z}}$.

Step 2.2. Assume that the required property is true if the cardinality of the
set ${\cal I}$ is less or equal to $N-1$. Let ${\cal
I}\subset{\mathbb{Z}}^{d}_{*}$ be such that $N=\text{card}({\cal I})$ and
${\cal I}=\\{m_{1},\ldots,m_{N}\\}$. Without loss of generality, we can assume
$m_{N-1}\not\perp m_{N}$ and the set $\\{m_{1},\ldots,m_{N-1}\\}$ satisfies
the condition formulated in the beginning of step 2. Let us take any
$a_{1},\ldots,a_{N}\in{\mathbb{Z}}$ and $k\geq 1$ and write

$\displaystyle a_{1}m_{1}+\ldots+a_{N}m_{N}$
$\displaystyle=\left(a_{1}m_{1}+\ldots+a_{N-2}m_{N-2}+(a_{N-1}-k)m_{N-1}\right)$
$\displaystyle\quad+\left(km_{N-1}+a_{N}m_{N}\right).$ (4.5)

Then,

$\displaystyle\langle
a_{1}m_{1}+\ldots+(a_{N-1}-k)m_{N-1},km_{N-1}+a_{N}m_{N}\rangle$
$\displaystyle=(a_{N-1}-k)k\|m_{N-1}\|^{2}$
$\displaystyle\quad+O(k)\quad\text{as $k\to+\infty$.}$

As $m_{N-1}\neq 0$, for sufficiently large $k\geq 1$, we have

$\displaystyle a_{1}m_{1}+\ldots+a_{N-2}m_{N-2}+(a_{N-1}-k)m_{N-1}\not\perp
km_{N-1}+a_{N}m_{N}.$ (4.6)

Relation (4.4) is proved by combining (4.5) and (4.6), the induction
hypothesis, and the assumption that $m_{N-1}\not\perp m_{N}$.

Step 3. We conclude from step 2 that, if ${\cal I}\subset{\mathbb{Z}}^{d}_{*}$
is a set satisfying the conditions of Proposition 2.5, then

$\cos\langle x,m\rangle,\,\sin\langle x,m\rangle\in{\cal H}_{\infty}({\cal
I})\quad\text{for any $m\in{\mathbb{Z}}^{d}_{*}$.}$

This implies that ${\cal H}_{\infty}({\cal I})$ is dense in
$C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$ for any $r\geq 0$, hence ${\cal
H}({\cal I})$ is saturating.

Step 4. Finally, let us assume that the conditions of the proposition are not
satisfied for ${\cal I}\subset{\mathbb{Z}}^{d}_{*}$. We distinguish between
two cases.

Step 4.1. If ${\cal I}$ is not a generator, we can find a vector
$n\in{\mathbb{Z}}^{d}_{*}$ which does not belong to the set $\tilde{\cal I}$
of linear combinations of vectors of ${\cal I}$ with integer coefficients. It
is easy to see that

${\cal H}_{\infty}({\cal I})\subset\text{span}\\{\sin\langle
x,m\rangle,\,\cos\langle x,m\rangle:\,\,m\in\tilde{\cal I}\\}.$

Thus, the functions $\sin\langle x,n\rangle$ and $\cos\langle x,n\rangle$ are
orthogonal to the vector space ${\cal H}_{\infty}({\cal I})$ in the Sobolev
spaces $H^{j}({\mathbb{T}}^{d};{\mathbb{R}})$ for any $j\geq 0$. We conclude
that ${\cal H}_{\infty}({\cal I})$ is not dense in
$C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$, thus the subspace ${\cal H}({\cal I})$
is not saturating.

Step 4.2. If ${\cal I}$ does not satisfy the second condition in the theorem,
then it is of the form

${\cal I}=\cup_{j=1}^{k}\\{m_{1}^{j},\ldots,m^{j}_{n_{j}}\\},$

where $k\geq 2$ and $m_{i_{1}}^{j_{1}}\bot m_{i_{2}}^{j_{2}}$ for any integers
$1\leq j_{1}<j_{2}\leq k,$ $1\leq i_{1}\leq n_{j_{1}},$ and $1\leq i_{2}\leq
n_{j_{2}}$. By using the arguments of the steps 1 and 2, it is easy to verify
that the function $\cos\langle x,m_{1}^{j_{1}}+m_{2}^{j_{2}}\rangle$ is
orthogonal to ${\cal H}_{\infty}({\cal I})$ in
$H^{j}({\mathbb{T}}^{d};{\mathbb{R}})$ for any $j\geq 0$. Thus, the space
${\cal H}_{\infty}({\cal I})$ is not dense in
$C^{r}({\mathbb{T}}^{d};{\mathbb{R}})$. ∎

##  5 Growth of Sobolev norms

Let us consider the NLS equation

$\displaystyle i\partial_{t}\psi$
$\displaystyle=-\Delta\psi+V(x)\psi+\kappa|\psi|^{2p}\psi+\langle\eta(t),Q(x)\rangle\psi,$
(5.1) $\displaystyle\psi(0)$ $\displaystyle=\psi_{0}$ (5.2)

with potential $V$ and parameters $d,p,\kappa$ as in the previous sections. We
assume that the field $Q$ satisfies Condition (H1) and $\eta$ is a random
process of the form (0.6) with the following condition satisfied for the
random variables $\\{\eta_{k}\\}$. We denote $J=[0,1]$ and ${\cal
E}=L^{2}(J;{\mathbb{R}}^{q})$.

(H2) $\\{\eta_{k}\\}$ are independent random variables in ${\cal E}$ with
common law $\ell$ such that

$\int_{E}\|y\|_{\cal
E}^{2}\,\ell({\textup{d}}y)<\infty\quad\quad\text{and}\quad\quad\mathop{\rm
supp}\nolimits\ell={\cal E}.$

For example, this condition is satisfied if the random variables
$\\{\eta_{k}\\}$ are of the form

$\eta_{k}(t)=\sum_{j=1}^{+\infty}b_{j}\xi_{jk}e_{j}(t),\quad t\in J,$

where $\\{b_{j}\\}$ are non-zero real numbers verifying
$\sum_{j=1}^{+\infty}b_{j}^{2}<\infty,$ $\\{e_{j}\\}$ is an orthonormal basis
in ${\cal E}$, and $\\{\xi_{jk}\\}$ are independent real-valued random
variables whose law has a continuous density $\rho_{j}$ with respect to the
Lebesgue measure such that

$\int_{-\infty}^{+\infty}x^{2}\rho_{j}(x)\,{\textup{d}}x=1,\quad\rho_{j}(x)>0\quad\text{for
all $x\in{\mathbb{R}}$ and $j\geq 1$.}$

By Proposition 1.1, the problem (5.1), (5.2) is locally well-posed in $H^{s}$
for any $s>d/2$ up to some (random) maximal time ${\cal T}={\cal
T}(\psi_{0},\eta)>0$. Let ${\mathbb{P}}_{\psi_{0}}$ be the probability measure
corresponding to the trajectories issued from $\psi_{0}$ (e.g., see Section
1.3.1 in [KS12]). Recall that ${\cal S}$ is the unit sphere in $L^{2}$.

######  Theorem 5.1.

Under the Conditions (H1) and (H2), for any $s>s_{d}$ and any $\psi_{0}\in
H^{s}\cap{\cal S}$, we have

${\mathbb{P}}_{\psi_{0}}\left\\{\limsup_{t\to{\cal
T}^{-}}\|\psi(t)\|_{s}=+\infty\right\\}=1.$ (5.3)

By the blow-up alternative, equality (5.3) gives new information in the case
${\cal T}(\psi_{0},\eta)=+\infty.$

###### Proof.

Step 1. Reduction. Together with Eq. (5.1), let us consider the following
truncated NLS equation:

$i\partial_{t}\psi=-\Delta\psi+V(x)\psi+\kappa\chi_{R}(\|\psi\|_{s})|\psi|^{2p}\psi+\langle\eta(t),Q(x)\rangle\psi,$
(5.4)

where $R>0$ and $\chi_{R}\in C^{\infty}_{0}({\mathbb{R}})$ is such that
$0\leq\chi_{R}(x)\leq 1$ for $x\in{\mathbb{R}}$ and $\chi_{R}(x)=1$ for
$|x|\leq R$. Let ${\cal F}_{k}$, $k\geq 1$ be the $\sigma$-algebra generated
by the family $\\{\eta_{j}\\}_{j=1}^{k}$. The problem (5.4), (5.2) is globally
well-posed. The following proposition is proved at the end of this section.

######  Proposition 5.2.

For any $\psi_{0}\in H^{s}$ and $R>0$, the problem (5.4), (5.2) has a unique
solution $\psi^{R}\in C({\mathbb{R}}_{+};H^{s})$. Moreover, the family

$\left\\{\psi^{R}(k+\cdot):J\to H^{s}\right\\}_{k\geq 0}$

defines an $C(J;H^{s})$-valued Markov process with respect to the filtration
${\cal F}_{k+1}$.

Let us fix any $0<M<R$ and consider the stopping time

$\tau_{M,R}=1+\min\left\\{k\geq
0:\|\psi^{R}(k+\cdot)\|_{C(J;H^{s})}>M\right\\},\quad\psi_{0}\in H^{s},$

where the minimum over an empty set is equal to $+\infty$. Assume we have
shown that

${\mathbb{P}}_{\psi_{0}}\\{\tau_{M,R}<\infty\\}=1,\quad\psi_{0}\in
H^{s}\cap{\cal S}.$ (5.5)

Since $R>M$, this implies that

${\mathbb{P}}_{\psi_{0}}\\{\tau_{M}<\infty\\}=1,\quad\psi_{0}\in
H^{s}\cap{\cal S},$ (5.6)

where

$\tau_{M}=\min\left\\{k\geq 0:\sup_{t\in J,\,k+t<{\cal
T}}\|\psi(k+t)\|_{s}>M\right\\}$

and again the minimum over an empty set is $+\infty$. As $M>0$ is arbitrary,
we conclude that (5.3) holds.

Step 2. Proof of (5.5). Assume that there is an integer $l\geq 1$ such that

$c=c(M,R)=\sup_{{\psi_{0}\in H^{s}\cap{\cal
S}}}{\mathbb{P}}_{\psi_{0}}\left\\{\tau_{M,R}>l\right\\}<1.$ (5.7)

Combining this with the Markov property, we obtain

$\displaystyle{\mathbb{P}}_{\psi_{0}}\left\\{\tau_{M,R}>nl\right\\}$
$\displaystyle={\mathbb{E}}_{\psi_{0}}\left({\mathbb{I}}_{\\{\tau_{M,R}>(n-1)l\\}}{\mathbb{P}}_{\phi}\\{\tau_{M,R}>l\\}|_{\phi={\psi^{R}((n-1)l)}}\right)$
$\displaystyle\leq
c\,{\mathbb{P}}_{\psi_{0}}\left\\{\tau_{M,R}>(n-1)l\right\\},$

where ${\mathbb{E}}_{\psi_{0}}$ is the expectation corresponding to
${\mathbb{P}}_{\psi_{0}}$. Iterating this inequality, we get

${\mathbb{P}}_{\psi_{0}}\left\\{\tau_{M,R}>nl\right\\}\leq c^{n}.$

This, together with the Borel–Cantelli lemma, implies (5.5).

Step 3. Proof of (5.7). By Theorem 2.3, for any $\psi_{0}\in
H^{s_{d}}\cap{\cal S}$, there is a control $u\in{\cal E}$ such that

$\sup_{t\in J,\,t<{\cal T}}\|\psi(t)\|_{s_{d}}>M.$ (5.8)

On the other hand, Condition (H2) implies that

${\mathbb{P}}\left\\{\|u-\eta\|_{{\cal E}}<\delta\right\\}>0$

for any $\delta>0$. Combining this with Proposition 1.1 and inequality (5.8),
we see that there is a number $\delta>0$ such that

$\inf_{\psi_{0}^{\prime}\in B_{H^{s_{d}}}(\psi_{0},\delta)\cap{\cal
S}}{\mathbb{P}}_{\psi_{0}^{\prime}}\left\\{\sup_{t\in J,\,t<{\cal
T}^{\prime}}\|\psi(t)\|_{s_{d}}>M\right\\}>0,$

where ${\cal T}^{\prime}={\cal T}(\psi_{0}^{\prime},\eta)$. As $R>M$, we also
have

$\inf_{\psi_{0}^{\prime}\in B_{H^{s_{d}}}(\psi_{0},\delta)\cap{\cal
S}}{\mathbb{P}}_{\psi_{0}^{\prime}}\left\\{\sup_{t\in
J}\|\psi^{R}(t)\|_{s_{d}}>M\right\\}>0.$

Since the ball $B_{H^{s}}(0,M)$ is compact in $H^{s_{d}}$ and
$\|\cdot\|_{s_{d}}\leq\|\cdot\|_{s}$, we derive that

$\inf_{\psi_{0}\in B_{H^{s}}(0,M)\cap{\cal
S}}{\mathbb{P}}_{\psi_{0}}\left\\{\sup_{t\in
J}\|\psi^{R}(t)\|_{s}>M\right\\}>0.$

The latter and the fact that

${\mathbb{P}}_{\psi_{0}}\left\\{\tau_{M,R}=1\right\\}=1\quad\text{if
$\|\psi_{0}\|_{s}>M$}$

imply (5.7) with $l=1$ and

$c=1-\inf_{\psi_{0}\in B_{H^{s}}(0,M)\cap{\cal
S}}{\mathbb{P}}_{\psi_{0}}\left\\{\sup_{t\in
J}\|\psi^{R}(t)\|_{s}>M\right\\}.$

This completes the proof of the theorem. ∎

###### Proof of Proposition 5.2.

The local well-posedness of (5.4), (5.2) is proved by standard arguments. As
the $H^{s}$-norm of the solution remains bounded on any bounded interval, it
can be extended to any $t>0$. For any $k\geq 1$, let us denote by
$\psi_{k}(\psi_{0},\eta_{1},\ldots,\eta_{k})$ the restriction of the solution
of (5.4), (5.2) to the interval $[k-1,k]$ (we skip the dependence on $R$).
Then $\\{\psi_{k}(\psi_{0},\eta_{1},\ldots,\eta_{k})\\}_{k\geq 1}$ is a Markov
process in $C(J,H^{s})$. Indeed, we have

$\psi_{k+n}(\psi_{0},\eta_{1},\ldots,\eta_{k+n})=\psi_{n}(\psi_{k}(\psi_{0},\eta_{1},\ldots,\eta_{k}),\eta_{k+1},\ldots,\eta_{k+n}).$

As $\\{\eta_{j}\\}_{j\geq k+1}$ is independent of ${\cal F}_{k}$ and
$\psi_{k}$ is ${\cal F}_{k}$-measurable, the following equality holds

${\mathbb{E}}\left(f(\psi_{k+n}(\psi_{0},\eta_{1},\ldots,\eta_{k+n}))|{\cal
F}_{k}\right)={\mathbb{E}}f(\psi_{n}(\psi,\eta_{k+1},\ldots,\eta_{k+n}))$
(5.9)

for any bounded measurable function $f:C(J,H^{s})\to{\mathbb{R}}$. Here,
$\psi$ is the value at time-1 of
$\psi_{k}(\psi_{0},\eta_{1},\ldots,\eta_{k})$. The vectors
$(\eta_{1},\ldots,\eta_{n})$ and $(\eta_{k+1},\ldots,\eta_{k+n})$ have the
same law, so

${\mathbb{E}}f(\psi_{n}(\psi,\eta_{k+1},\ldots,\eta_{k+n}))={\mathbb{E}}f(\psi_{n}(\psi,\eta_{1},\ldots,\eta_{n})).$

Combining this and (5.9), we arrive at the required result. ∎

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