Controllability of localised quantum states on infinite graphs through bilinear control fields

TL;DR: In this article, the bilinear Schrodinger equation (BSE) is considered in the Hilbert space L2(G,C) with G an infinite graph. And the Laplacian −Δ is equipped with self-adjoint boundary cond...

Abstract: In this work, we consider the bilinear Schrodinger equation (BSE) i∂tψ=−Δψ+u(t)Bψ in the Hilbert space L2(G,C) with G an infinite graph. The Laplacian −Δ is equipped with self-adjoint boundary cond...

Summary

1. INTRODUCTION

• The operator A is a self-adjoint Laplacian, while the action of the controlling external field is given by the bounded symmetric operator B and by the function u, which accounts its intensity.
• The author exhibits the global exact controllability of the bilinear Schrödinger equation between eigenstates via explicit controls and explicit times in [Duc19].
• In [Duc18a], another weaker result is introduced, the so-called energetic controllability.

2. INFINITE TADPOLE GRAPH

• Γut the unitary propagator generated by the operator −∆ + u(t)B. 2.1. Well-posedness.
• The statement is proved by using the techniques developed in the proof of [Duc18b, Proposition 4.1], which generalize the ones of [BL10, Lemma 1; Proposition 2].
• The generalization for u ∈ L2((0, T ),R) follows from a classical density argument.

3. GENERIC GRAPHS

• When the (BSE) is well-posed, the authors call Γut the unitary propagator generated byA+u(t)B.
• In addition, the external vertices Ve are equipped with Dirichlet or Neumann type boundary conditions.

4. EXAMPLE

• As in [Duc18a, Section 6], the techniques just developed are valid when G contains suitable sub-graphs denoted “uniform chains”.
• If {Lj}j≤Ñ ∈ AL(Ñ), then the energetic controllability can be guaranteed in{.
• The second author has been financially supported by the ISDEEC project by ANR-16-CE40-0013.

1) (d) Infinite dimensional estimates.

• N∗. 1) (e) Global approximate controllability with respect to the L2-norm.
• Vice versa, thanks to the time reversibility, there exists a control steering ψ close to ϕ1.

Figures

Figure 3. The parametrization of the infinite tadpole graph and its symmetry axis r.

Figure 4. Parametrization of a star graph with N = 4 edges.

Figure 1. An infinite graph is an one-dimensional domain composed by vertices (points) connected by edges (segments and half-lines).

Figure 6. Uniform chains contained in a generic graph.

Figure 2. Graph described in Section 4.

Figure 5. Example of star graph described in Section 4.

Full text

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Controllability of Localized Quantum States on Innite
Graphs through Bilinear Control Fields
Kaïs Ammari, Alessandro Duca
To cite this version:
Kaïs Ammari, Alessandro Duca. Controllability of Localized Quantum States on Innite Graphs
through Bilinear Control Fields. International Journal of Control, Taylor & Francis, In press,
�10.1080/00207179.2019.1680868�. �hal-02164419v2�

CONTROLLABILITY OF LOCALIZED QUANTUM STATES ON INFINITE
GRAPHS THROUGH BILINEAR CONTROL FIELDS
KA
¨
IS AMMARI AND ALESSANDRO DUCA
ABSTRACT. In this work, we consider the bilinear Schr
¨
odinger equation (BSE) i∂
t
ψ =
−∆ψ + u(t)Bψ in the Hilbert space L
2
(G , C) with G an infinite graph. The Laplacian
−∆ is equipped with self-adjoint boundary conditions, B is a bounded symmetric operator
and u ∈ L
2
((0, T ), R) with T > 0. We study the well-posedness of the (BSE) in suitable
subspaces of D(|∆|
3/2
) preserved by the dynamics despite the dispersive behaviour of
the equation. In such spaces, we study the global exact controllability and the “energetic
controllability”. We provide examples involving for instance infinite tadpole graphs.
Online published https://doi.org/10.1080/00207179.2019.1680868
In this version, the references of the preprint articles cited in the original work were updated.
CONTENTS
1. Introduction 2
1.1. Novelties of the work 3
2. Infinite tadpole graph 4
2.1. Well-posedness 6
2.2. Global exact controllability 7
3. Generic graphs 9
3.1. Interpolation properties and well-posedness 10
3.2. Controllability results 14
4. Example 14
Appendix A. Analytic perturbation 17
Appendix B. Global approximate controllability 20
References 23
2010 Mathematics Subject Classification. 35Q40, 93B05, 93C05.
Key words and phrases. Bilinear control, infinite graph.
1

2 KA
¨
IS AMMARI AND ALESSANDRO DUCA
1. INTRODUCTION
We study the evolution of a particle confined in an infinite graph structure and subjected
to an external field that plays the role of a control.
Figure 1. An infinite graph is an one-dimensional domain composed by
vertices (points) connected by edges (segments and half-lines).
Its dynamics is described by the so-called bilinear Schr
¨
odinger equation
(1) i∂
t
ψ(t) = (A + u(t)B)ψ(t), t ∈ (0, T ),
in L
2
(G , C), where G is the graph. The operator A is a self-adjoint Laplacian, while the
action of the controlling external field is given by the bounded symmetric operator B and
by the function u, which accounts its intensity. We call Γ
u
t
the unitary propagator generated
by A + u(t)B (when it is defined).
It is natural to wonder whether, given any couple of states ψ
1
and ψ
2
, there exists u
steering the bilinear quantum system from ψ
1
into ψ
2
. The bilinear Schr
¨
odinger equation
is said to be exactly controllable when the dynamics reach precisely the target.
We denote it approximately controllable when it is possible to approach the target as close
as desired. If it is possible to control (either exactly, or approximately) more initial states at
the same time with the same u, then the equation is said to be simultaneously controllable.
The controllability of finite-dimensional quantum systems (i.e. modeled by an ordinary
differential equation) is currently well-established. If we consider the bilinear Schr
¨
odinger
equation (1) in C
N
such that A and B are N ×N Hermitian matrices and t 7→ u(t) ∈ R is
the control, then the controllability of the problem is linked to the rank of the Lie algebra
spanned by A and B (we refer to [Alt02] by Altafini and [Cor07] by Coron). Nevertheless,
the Lie algebra rank condition can not be used for infinite-dimensional quantum systems
(see [Cor07] for further details). Thus, different techniques were developed in order to deal
with this type of problems.
Regarding the linear Schr
¨
odinger equation, the controllability and observability prop-
erties are reciprocally dual (often referred to the Hilbert Uniqueness Method). One can
therefore address the control problem directly or by duality with various techniques: mul-
tiplier methods ([Lio83]), microlocal analysis ([BLR92]), Carleman estimates ([MOR08]).
Even though the linear Schr
¨
odinger equation is widely studied in the literature, the
bilinear Schr
¨
odinger equation in a generic Hilbert space H can not be approached with
the same techniques since it is not exactly controllable in H . We refer to the work on
bilinear systems [BMS82] by Ball, Mardsen and Slemrod, where the well-posedness and
the non-controllability are provided. Despite they prove the well-posedness of the bilinear
Schr
¨
odinger equation in H when u ∈ L
1
((0, T), R) and T > 0, they also show that it is
not exactly controllable in H for u ∈ L
2
loc
((0, ∞), R) (see [BMS82, Theorem 3.6]).
Because of the Ball, Mardsen and Slemrod result, many authors have considered weaker
notions of controllability when G = (0, 1). Let
D(A
D
) = H
2
((0, 1), C) ∩ H
1
0
((0, 1), C)), A
D
ψ := −∆ψ, ∀ψ ∈ D(A
D
).

BILINEAR CONTROL 3
In [BL10], Beauchard and Laurent prove the well-posedness and the local exact control-
lability of the bilinear Schr
¨
odinger equation in H
s
(0)
:= D(A
s/2
D
) for s = 3, when B is a
multiplication operator for suitable µ ∈ H
3
((0, 1), R).
In [Mor14], Morancey proves the simultaneous local exact controllability of two or three
(1) in H
3
(0)
for suitable operators B = µ ∈ H
3
((0, 1), R).
In [MN15], Morancey and Nersesyan extend the previous result. They achieve the si-
multaneous global exact controllability of finitely many (1) in H
4
(0)
for a wide class of
multiplication operators B = µ with µ ∈ H
4
((0, 1), R).
In [Duc20], the author ensures the simultaneous global exact controllability in projection
of infinite (1) in H
3
(0)
for bounded symmetric operators B.
The author exhibits the global exact controllability of the bilinear Schr
¨
odinger equation
between eigenstates via explicit controls and explicit times in [Duc19].
The global approximate controllability of the bilinear Schr
¨
odinger equation is proved
with many different techniques in literature as the following. The outcome is achieved
with Lyapunov techniques by Mirrahimi in [Mir09] and by Nersesyan in [Ner10]. Adia-
batic arguments are considered by Boscain, Chittaro, Gauthier, Mason, Rossi and Sigalotti
in [BCMS12] and [BGRS15]. Lie-Galerking methods are used by Boscain, Boussa
¨
ıd,
Caponigro, Chambrion and Sigalotti in [BdCC13] and [BCS14].
Control problems involving networks have been very popular in the last decades, how-
ever the bilinear Schr
¨
odinger equation on compact graphs has been only studied in [Duc18b]
and [Duc18a]. In the mentioned works, the well-posedness and the global exact controlla-
bility of the (1) are provided in some spaces D(|A|
s/2
) with s ≥ 3. In [Duc18a], another
weaker result is introduced, the so-called energetic controllability. In particular, a bilinear
quantum system is said to be energetically controllable with respect to some energy levels
when there exist corresponding bounded states {ϕ}
j∈N
∗
such that
∀m, n ∈ N
∗
, ∃T > 0, u ∈ L
2
((0, T), R) : ϕ
n
= Γ
u
T
ϕ
m
.
The peculiarity of the bilinear Schr
¨
odinger equation on compact graphs is that, even though
A admits purely discrete spectrum {λ
k
}
k∈N
∗
(see [Kuc04, Theorem 18]), the uniform gap
condition inf
k∈N
∗
|λ
k+1
− λ
k
| ≥ 0 is satisfied if and only if G = (0, 1). This hypothesis
is crucial for the classical arguments adopted in the previous works as [BL10], [Duc20],
[Duc19] and [Mor14]. To this purpose, new techniques are developed in [Duc18b] and
[Duc18a] in order to achieve controllability results.
1.1. Novelties of the work. Up to our knowledge, the controllability of the bilinear
Schr
¨
odinger equation on infinite graphs is still an open problem. The main reason can be
found on the dispersive phenomena characterizing the equation on infinite graphs (not con-
sidering the difficulties already appearing on compact graphs; see [Duc18b] and [Duc18a]).
A characteristic feature of the Schr
¨
odinger equation is the loss of localization of the wave
packets during the evolution, the dispersion. This effect can be measured by L
∞
-time
decay, which implies a spreading out of the solutions, due to the time invariance of the L
2
-
norm. In [AAN17], Ali Mehmeti-Ammari-Nicaise prove that the free Schr
¨
odinger group
on the tadpole graph satisfies the standard L
1
− L
∞
dispersive estimate and that it is in-
dependent of the length of the circle (compact part of the graph) (see also [AAN15, Ali
Mehmeti-Ammari-Nicaise] for the case of the star-shaped network and with potential). The
proof of this result is based on an appropriate decomposition of the kernel of the resolvent.
This technique gives a full characterization of the spectrum made of the point spectrum
and of the absolutely continuous one, while the singular continuous spectrum is empty.

4 KA
¨
IS AMMARI AND ALESSANDRO DUCA
Our strategy can be resumed as follows.
• When A has discrete spectrum, we construct some eigenfunctions of A in L
2
(G , C)
denoted {ϕ
k
}
k∈N
∗
. The flow of the Schr
¨
odinger equation i∂
t
ψ = Aψ preserves
f
H = span{ϕ
k
: k ∈ N
∗
}
L
2
.
• When B stabilizes the space
f
H , the bilinear Schr
¨
odinger equation is well-posed
in
f
H and in D(|A|
s
2
) ∩
f
H for suitable s > 0 when B is sufficiently regular.
• In such space, we study the global exact controllability and the energetic control-
lability with respect to {ϕ
k
}
k∈N
∗
by adapting the techniques developed for the
compact graphs in [Duc18b] and [Duc18a].
In the first part of the work, we consider a specific B localized on the “head” of an
infinite tadpole G . The chosen B is symmetric with respect to the natural symmetry axis r
of G and we denote
f
H the space of those L
2
(G , C)-functions that are antisymmetric with
respect to r (see Figure 3). We prove the global exact controllability in D(|A|
3
2
) ∩
f
H .
In the second part, we generalize the results for generic graphs and we apply them for
those G containing a star graph (Section 4).
Figure 2. Graph described in Section 4.
In presence of suitable substructures in an infinite graph G , it is possible to construct
eigenfunctions of A. For instance, when G contains a self-closing edge e of length 1, the
functions
{ϕ
k
}
k∈N
∗
: ϕ
k


e
=
√
2 sin

2kπx

, ϕ
k


G \{e}
≡ 0, ∀k ∈ N
∗
,
are eigenfunctions of A. If B preserves the span of {ϕ
k
}
k∈N
∗
, then the controllability
could be achieved. The same argument is true for graphs containing more self-closing
edges or other suitable substructures (see Remark 4.3 for few examples).
2. INFINITE TADPOLE GRAPH
Let T be an infinite tadpole graph composed by two edges e
1
and e
2
. The self-closing
edge e
1
, the “head”, is connected to e
2
in the vertex v and it is parametrized in the clock-
wise direction with a coordinate going from 0 to 1 (the length of e
1
). The “tail” e
2
is a
half-line equipped with a coordinate starting from 0 in v and going to +∞.

Frequently asked questions

Q1. What is the function of the self-closing edge e1?

The self-closing edge e1, the “head”, is connected to e2 in the vertex v and it is parametrized in the clockwise direction with a coordinate going from 0 to 1 (the length of e1). 

Q2. What is the peculiarity of the bilinear Schrödinger equation on compact graphs?

The peculiarity of the bilinear Schrödinger equation on compact graphs is that, even though A admits purely discrete spectrum {λk}k∈N∗ (see [Kuc04, Theorem 18]), the uniform gap condition infk∈N∗ |λk+1 − λk| ≥ 0 is satisfied if and only if G = (0, 1). 

Q3. What is the function of the self-closing edge?

For instance, when G contains a self-closing edge e of length 1, the functions{ϕk}k∈N∗ : ϕk ∣∣ e = √ 2 sin ( 2kπx ) , ϕk ∣∣ G\\{e} ≡ 0, ∀k ∈ N ∗,are eigenfunctions of A. 

Q4. What are the main reasons for the bilinear Schrödinger equation on infinite graphs?

Control problems involving networks have been very popular in the last decades, however the bilinear Schrödinger equation on compact graphs has been only studied in [Duc18b] and [Duc18a]. 

Q5. How can the authors measure the effect of the bilinear Schrödinger equation on infinite graph?

This effect can be measured by L∞-time decay, which implies a spreading out of the solutions, due to the time invariance of the L2norm. 

Q6. what is the controllability of the (BSE*) in H3T?

AsΓut ϕ1 = ∑ k∈N∗ ϕk(t)〈ϕk(t),Γut ϕ1〉, T > 0, u ∈ L2((0, T ),R),the controllability is equivalent to the local surjectivity of α.