Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations

HAL Id: hal-01481873

https://hal.archives-ouvertes.fr/hal-01481873v5

Submitted on 16 Jul 2020

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Distributed under a Creative Commons

Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations

Alessandro Duca

To cite this version:

Alessandro Duca. Simultaneous global exact controllability in projection of infinite 1D bilinear

Schrödinger equations. Dynamics of Partial Differential Equations, International Press, 2020, 17

(3), pp.275-306. �10.4310/DPDE.2020.v17.n3.a4�. �hal-01481873v5�

infinite 1D bilinear Schr¨ odinger equations

Abstract. The aim of this work is to study the controllability of infinite bilinear Schr¨odinger equations on a segment. We consider the equations (BSE) i∂tψ^j = −∆ψ^j+u(t)Bψ^j in the Hilbert space L²((0,1),C) for every j ∈ N^∗. The Laplacian−∆ is equipped with Dirichlet homogeneous boundary conditions,Bis a bounded symmetric operator andu∈L²((0, T),R) withT >

0. We prove the simultaneous local and global exact controllability of infinite (BSE) in projection. The local controllability is guaranteed for any positive time and we provide explicit examples of B for which our theory is valid.

In addition, we show that the controllability of infinite (BSE) in projection onto suitable finite dimensional spaces is equivalent to the controllability of a finite number of (BSE) (without projecting). In conclusion, we rephrase our controllability results in terms of density matrices.

Contents

1. Introduction 1

2. Auxiliary results 5

3. Simultaneous local exact controllability in projection 8 4. Simultaneous global exact controllability in projection 14 5. Global exact controllability in projection for density matrices 20

6. Conclusion 22

Appendix A. Moment problem 23

Appendix B. Analytic Perturbation 25

References 31

1. Introduction

1.1. The problem. In this work, we consider infinite particles constrained in a one-dimensional bounded region and subjected to an external control field. A

1991Mathematics Subject Classification. Primary 93C20, 93B05; Secondary 35Q41, 81Q15.

Key words and phrases. Schr¨odinger equation, simultaneous control, global exact controlla- bility, moment problem, perturbation theory, density matrices.

The author thanks Thomas Chambrion for suggesting him the problem and Nabile Boussa¨ıd for the periodic discussions. He is also grateful to Morgan Morancey for the explanation about the works [15] and [16].

1

suitable choice for such setting is to model the dynamics of these particles by infin- itely many bilinear Schr¨odinger equations in the Hilbert spaceH =L²((0,1),C)

(i∂tψj(t) =Aψj(t) +u(t)Bψj(t), t∈(0, T), T >0, ψj(0) =ψ⁰_j ∈L²((0,1),C), j∈N^∗.

(BSE)

The Laplacian A=−∆ is equipped with homogeneous Dirichlet boundary condi- tions such that

D(A) =H²((0,1),C)∩H₀¹((0,1),C).

The bounded symmetric operatorB models the action of the external field, while the control functionu∈L²((0, T),R) represents its intensity.

We study the controllability of the infinite bilinear Schr¨odinger equations (BSE) at the same timeT, with one unique controluand by projecting onto suitable finite dimensional subspaces ofH.

In order to detail the purpose of the work, we introduce the following notations.

We denote by Γ^u_t the unitary propagator in H generated by the dynamics of the (BSE) in a time interval [0, t] (when it is defined). Let Ψ := (ψ_j)_j∈N^∗ an orthonormal system ofH. We callπ_N(Ψ) withN ∈N^∗ the orthogonal projector

π_N(Ψ) :H −→span{ψk : k≤N}.

(1.1)

We say that two sequences of functions (ψ¹_j)_j∈N^∗,(ψ_j²)_j∈N^∗⊆H are unitarily equiv- alent when there exists Γ∈U(H) (the space of the unitary operators inH) such that

ψ¹_j = Γψ²_j, ∀j∈N^∗.

The aim of this work is to study the existence of orthonormal systems Ψ of H so that, for any N ∈N^∗ and for any suitable (ψ_j¹)_j∈N^∗ and (ψ_j²)_j∈N^∗ unitarily equivalent, there existT >0 andu∈L²((0, T),R) such that

πN(Ψ) Γ^u_Tψ¹_j =πN(Ψ)ψ_j², ∀j∈N^∗. (1.2)

If we denote byh·,·iL²the usualL²−scalar product, then the identities (1.2) become hψk,Γ^u_Tψ¹_ji_L2 =hψk, ψ²_ji_L2, ∀j, k∈N^∗, k≤N.

In order to achieve the result, we show that the simultaneous global exact con- trollability in projection onto suitable N dimensional spaces is equivalent to the controllability ofN problems (BSE) (without projecting).

1.2. Main results. Letk·k_L2be the norm of the Hilbert spaceH =L²((0,1),C) andh·,·i_L2 be the corresponding scalar product. Let

(φj)_j∈N^∗, (λj)_j∈N^∗

respectively be the eigenfunctions and the eigenvalues ofA such that φj(x) =√

2 sin(jπx), λj =π²j², ∀j∈N^∗. (1.3)

We notice that (φj)_j∈N^∗forms a complete orthonormal system ofH and we consider the spaces

H₍₀₎³ :=D(|A|³²), k · k₍₃₎:=k · k_H3

(0) =X^∞

k=1

|k³h·, φ_ki_L2|²¹₂ ,

`^∞(H₍₀₎³ ) =

(ψ_j)_j∈N^∗⊂H₍₀₎³ sup

j∈N^∗

kψ_jk₍₃₎<∞ .

Fors∈N^∗, we callH^s :=H^s((0,1),C),H₀¹:=H₀¹((0,1),C) and, forN ∈N^∗, we define

(1.4) I^N :={(j, k)∈N^∗× {1, . . . , N} : j > k}.

Assumptions I. The operator B is bounded and symmetric in the Hilbert spaceH =L²((0,1),C). In addition, it satisfies the following conditions.

(1) For any N ∈ N^∗, there exists CN >0 such that|hφk, Bφji_L2| ≥ ^C_k^N3 for everyj, k∈N^∗withj ≤N.

(2) Ran(B|_H2

(0))⊆H₍₀₎² andRan(B|_H3

(0))⊆H³∩H₀¹.

(3) For everyN ∈N^∗and for every (j, k),(l, m)∈I^N such that (j, k)6= (l, m) andj²−k²−l²+m²= 0, we have

hφ_j, Bφ_ji_L2− hφ_k, Bφ_ki_L2− hφ_l, Bφ_li_L2+hφ_m, Bφ_mi_L26= 0.

The first condition in Assumptions I quantifies how muchB mixes eigenstates, while the second fixes its regularity. The third condition instead is required in order to decouple, through perturbation theory techniques, the eigenvalues resonances appearing in the proof of the following statement.

The next theorem states one of the main results of the work that is the si- multaneous global exact controllability in projection of infinite (BSE). In order to keep this introduction as simple as possible, we postpone to Section 3 the second important result of the work which is the simultaneous local exact controllability in projection for any positive time (Theorem 3.1).

Theorem 1.1. Let Γ^u_t be the unitary propagator in H generated by the dy- namics of the (BSE) in the time interval [0, t] with B satisfying Assumptions I.

Assume that Ψ := (ψj)_j∈N^∗ ⊂H₍₀₎³ is an orthonormal system ofH. Let(ψ¹_j)_j∈N^∗ and (ψ²_j)j∈N^∗ ⊂ H₍₀₎³ be complete orthonormal systems of H and bΓ ∈ U(H) be the unitary operator such that (bΓψ²_j)_j∈N^∗= (ψ_j¹)_j∈N^∗. If the following condition is satisfied

(1.5) (bΓψ_j)_j≤N ⊂H₍₀₎³

withN ∈N^∗, then there existT >0,u∈L²((0, T),R)and(θk)_k≤N ⊂Rsuch that hψk,Γ^u_Tψ¹_ji_L2 =e^iθkhψk, ψ²_ji_L2, ∀j, k∈N^∗, k≤N.

(1.6)

Theorem 1.1 allows to control with a singleuand at the same timeT any finite number of components of infinitely many solutions of the problems (BSE). We notice that the statement is ensured up to phases in the components which prevents to formulate the result in terms of projectors. In addition, the orthonormal system (ψj)_j∈N^∗has to verify aH₍₀₎³ −compatibility condition exposed in (1.5). Despite this assumption may seem unusual, it spontaneously appears when we try to control in projection infinite (BSE). We provide further discussions on the subject in Remark 4.2 where we show that it is a natural constraint for this kind of problems.

When we want to control in projection with respect to the target orthonormal system by using Theorem 1.1, we choose Ψ≡Ψ². In this case, we notice that

(bΓψ_j)_j≤N = (bΓψ²_j)_j≤N = (ψ¹_j)_j≤N ⊂H₍₀₎³

and theH₍₀₎³ −compatibility condition (1.5) is trivially satisfied. In addition, e^iθkhψ_k², ψ_j²iL² =e^iθkδ_k,j=e^iθjhψ²_k, ψ²_jiL², ∀j, k∈N^∗.

Thus, the relations (1.6) become

(πN(Ψ²)Γ^u_Tψ¹_j =πN(Ψ²)e^iθjψ²_j, ∀j≤N, π_N(Ψ²)Γ^u_Tψ¹_j =π_N(Ψ²)ψ_j², ∀j > N.

(1.7)

As Ψ² is composed by orthonormal elements, the projector appearing in the first line of (1.7) acts as the identity operator and the right-hand side of the second line is equal to 0. These facts lead to the following corollary.

Corollary 1.2. Let Γ^u_t be the unitary propagator in H generated by the dynamics of the (BSE)in the time interval [0, t]with B satisfying Assumptions I.

Let Ψ¹ := (ψ_j¹)_j∈N^∗, Ψ² := (ψ²_j)_j∈N^∗ ⊂ H₍₀₎³ be complete orthonormal systems of H. For everyN ∈N^∗, there existT >0,u∈L²((0, T),R)and(θj)_j≤N ⊂Rsuch that

(Γ^u_Tψ_j¹=e^iθjψ_j², ∀j≤N, πN(Ψ²) Γ^u_Tψ_j¹= 0, ∀j > N.

Here, one can notice the parallelism between our results with the ones provided in the important work [16] by Morancey and Nersesyan. Indeed, Corollary 1.2 implies the controllability of any finite number of bilinear Schr¨odinger equations when B satisfies Assumptions I. Similar results are provided in [16] and here we rephrase the main one.

Theorem1.3. [16, Main Theorem]Let the bilinear Schr¨odinger equation(BSE) be considered with B=Mµ a multiplication operator for a function µ∈H⁴. Fixed N ∈ N^∗, there exists Q a residual set of H⁴ (an intersection of countably many subsets of H⁴ with dense interiors) such that, for every B =Mµ with µ∈ Q, the following result is satisfied. For any(ψ¹_k)_k≤N,(ψ²_k)_k≤N ⊂H₍₀₎³ unitarily equivalent, there existT >0 andu∈L²((0, T),R)such that Γ^u_Tψ_k¹=ψ_k² for everyk≤N.

As we show in Section 4.2, the controllability of infinite (BSE) in projection is equivalent to the controllability of a finite number of (BSE) (without projecting).

In view of this fact, similar statements to Theorem 1.1 can be provided by using the theory developed in Section 4.2 and the one from [16]. Even though such results can be really interesting, they are ensured with respect to abstract control operators B (of multiplicative type) and then the controllability is only generically verified.

From this perspective, our purpose is different. We aim to ensure the simultaneous global exact controllability when simple and explicit hypotheses on the problem, such as Assumptions I, are satisfied. This fact allows us to provide examples ofB for which the result is guaranteed, i.e. B : ψ∈ H 7→ x²ψ (we refer to Example 2.2 for further details on this case and for other examples). Our goal is achieved by using different techniques from [16] whose disadvantage is the loss of control on the phase terms appearing in Theorem 1.1 and Corollary 1.2.

The other main contributions of the work are the following. First, we prove the equivalence between the controllability of infinitely many (BSE) in projection and the controllability of a finite number of equations without projection. Second, we prove the local controllability in any positive timeT >0 which is stated in Section 3. Third, we use Theorem 1.1 and Corollary 1.2 in order to ensure the global exact controllability in projection for density matrices in Section 5.

1.3. A brief bibliography. Global approximate controllability results for the bilinear Schr¨odinger equation are provided with different techniques in literature.

For instance, adiabatic arguments are considered by Boscain, Chittaro, Gauthier, Mason, Rossi and Sigalotti in [6] and [7]. The controllability is achieved with Lyapunov techniques by Mirrahimi in [14] and by Nersesyan in [17]. Lie-Galerkin arguments are used by Boscain, Boussa¨ıd, Caponigro, Chambrion, Mason and Siga- lotti in [5], [8] and [10].

The exact controllability of the bilinear Schr¨odinger equation (BSE) is in gen- eral a more delicate matter as a consequence of the results provided in the work on bilinear systems [2] by Ball, Mardsen and Slemrod. There, they prove that the (BSE) is not exactly controllable in the Hilbert space where it is defined when B is a bounded operator andu∈L²_loc(R⁺,R) (even though it is well-posed).

Despite this non-controllability result, many authors have addressed the prob- lem for weaker notions of controllability by considering suitable subspaces ofD(A).

This idea was preliminarily introduced by Beauchard in [3] and popularized by the work in [4]. In [4], Beauchard and Laurent prove the local exact controllability of (BSE) in a neighborhood of the first eigenfunction ofA in S∩H₍₀₎³ when B is a suitable multiplication operator. The same kind of operators are considered in [15], where Morancey ensures the simultaneous local exact controllability inS∩H₍₀₎³ for at most three problems (BSE) and up to phases. In the work [16], Morancey and Nersesyan extend such result and prove Theorem 1.3.

1.4. Scheme of the work. In Section 2, we fix the notations considered in the work and we present some preliminary features of the problem such as the well-posedness of the (BSE) in the spaceH₍₀₎³ proved in [4].

In Section 3, we ensure Theorem 3.1 which states the simultaneous local exact controllability in projection for any positive time up to phases. In order to motivate the modification of the problem, we emphasize the obstructions to overcome.

In Section 4, we prove Theorem 1.1. First, we show that the simultaneous global exact controllability in projection is equivalent to the controllability of finite (BSE) in Proposition 4.1. Second, we ensure with Proposition 4.5 the simultaneous global exact controllability of finite (BSE) by using the theory from Section 3 and a global approximate controllability. The propositions 4.1 and 4.5 lead to Theorem 1.1.

In Section 5, we rephrase our results in terms of density matrices, while in Section 6, we provide some conclusive comments on the work.

In Appendix A, we briefly discuss the solvability of the so-called moment problems, while in Appendix B, we develop the perturbation theory techniques adopted in the work.

2. Auxiliary results

2.1. Notations and preliminaries. We denote by H the Hilbert space L²((0,1),C) equipped with the norm k · k_L2 and the scalar product h·,·i_L2 such that

hf, gi_L2 = Z 1

0

f(x)g(x)dx, ∀f, g∈H.

LetBbe a Banach space. We introduce fors >0,

H₍₀₎^s =D(|A|^s2), k · k_(s)=k · kH₍₀₎^s =X^∞

k=1

|k^sh·, φki_L2|²¹₂ ,

h^s(B) =n

(ψj)_j∈N^∗⊂B

∞

X

j=1

(j^skψjk_B)²<∞o ,

`^∞(B) =n

(ψ_j)_j∈N^∗⊂B sup

j∈N^∗

kψjk_B<∞o . (2.1)

We recall that (φj)j∈N^∗ is a complete orthonormal system of H composed by eigenfunctions ofAdefined in (1.3) and related to the eigenvalues (λ_j)_j∈N∗. Fixed

Ψ := (ψj)j∈N^∗⊂H, HN(Ψ) := span{ψj:j ≤N}, we defineπN(Ψ) the orthogonal projector such that

πN(Ψ) :H −→HN(Ψ).

(2.2)

Remark 2.1. If a bounded operator B satisfies Assumptions I, then B ∈ L(H₍₀₎² , H₍₀₎² ). Indeed, B is closed in H, so for every (u_n)_n∈N^∗ ⊂ H such that un −→H uandBun −→H v, we haveBu=v. Now, for every (un)n∈N^∗ ⊂H₍₀₎² such that un

H²₍₀₎

−→uandBun H₍₀₎²

−→v, the convergences with respect to the H-norm are implied andBu=v. Hence, the operatorBis closed inH₍₀₎² andB∈L(H₍₀₎² , H₍₀₎² ).

The same argument leads toB ∈L(H₍₀₎³ , H³∩H₀¹) sinceRan(B|_H³

(0))⊆H³∩H₀¹. Example 2.2. Assumptions I are satisfied for B : ψ 7→ x²ψ. Indeed, the condition 1) is guaranteed as

(hφ_j, x²φ_ki_L2 = _(j−k)^(−1)j−k_2π2 −_(j+k)^(−1)j+k_2π2, ∀j, k∈N, j6=k, hφk, x²φki_L2 = ¹₃−_2k¹2π², ∀k∈N^∗.

The point 2) of Assumptions I is trivially true, while the condition 3) is due to the following implication. For every N ∈ N^∗ and (j, k),(l, m) ∈ I^N such that (j, k)6= (l, m) andj²−k²−l²+m²= 0, we have

j⁻²−k⁻²−l⁻²+m⁻²6= 0.

We notice that the same properties are valid for other control operators. For in- stance, if we considerB :ψ∈H 7−→sin ^π₂x

ψ, then Assumptions I are satisfied thanks to the identities

(hφj, BφkiL²=−_π(16j4+16k⁴−8j^32jk2−8k²−32k²j²+1) ∀j, k∈N, j6=k, hφk, Bφ_kiL²= _π²+_(16k2²_−1)π, ∀k∈N^∗.

The same is true for the operator B : ψ ∈ H 7−→ x³ψ. Finally, an example of operatorB satisfying Assumptions I which is not of multiplicative type is

B :ψ∈H 7−→ X

j∈N^∗

φ_2j−1

φ_2j−1, x²ψ

+ X

j∈N^∗

φ2j

D

φ2j,sinπ 2x

ψE ψ.

2.2. Well-posedness. In the current subsection, we cite an important result of well-posedness for the following problem inH

(i∂tψ(t) =Aψ(t) +u(t)µψ(t), t∈(0, T), ψ(0) =ψ⁰∈L²((0,1),C).

(2.3)

Proposition 2.3. [4, Lemma 1; Proposition 2]Let µ∈H³,T >0,ψ⁰∈H₍₀₎³ and u ∈ L²((0, T),R). There exists a unique mild solution of (2.3) in H₍₀₎³ , i.e.

ψ∈C⁰([0, T], H₍₀₎³ )so that ψ(t) =e^−iAtψ⁰−i

Z t

0

e^−iA(t−s)u(s)µψ(s)ds, ∀t∈[0, T].

Moreover, for every R >0, there existsC=C(T, µ, R)>0 such that, if kukL²((0,T),R)< R, then, for everyψ⁰∈H₍₀₎³ , the solution satisfies

kψk_C0([0,T],H₍₀₎³ )≤Ckψ⁰k₍₃₎, kψ(t)k_L2 =kψ⁰k_L2, ∀t∈[0, T].

Remark2.4. The result of Proposition 2.3 is not only valid for multiplication operators, but also for other suitable operators B. Indeed, the same proofs of [4, Lemma 1] and [4, Proposition 2] lead to the well-posedness of the (BSE) when B is a bounded symmetric operator such that

B∈L(H₍₀₎³ , H³∩H₀¹), B∈L(H₍₀₎² ),

which are verified ifB satisfies Assumptions I, thanks to Remark 2.1.

Let Γ^u_t be the unitary propagator in H generated by the (BSE) in the time interval [0, t]. For any mild solution ψj in L²((0,1),C) of thej-th problem (BSE) withj∈N^∗, we have

Γ^u_tψ_j(0) =ψ_j(t).

As a consequence of Remark 2.4, it follows (Γ^u_Tψj)_j∈N^∗ ∈ `<sup>∞</sup>(H<sub>(0)</sub><sup>3</sup> ) for every (ψj)<sub>j∈N</sub><sup>∗</sup>∈`^∞(H₍₀₎³ ). We refer to (2.1) for the definition of the space`^∞(H₍₀₎³ ).

2.3. Time reversibility. An important feature of the bilinear Schr¨odinger equation is the time reversibility. If we consider ψ(t) = Γ^u_tψ⁰ and we substitute t withT−tforT >0 in a bilinear Schr¨odinger equation, then we have

(i∂_tΓ^u_T−tψ⁰=−AΓ^u_T−tψ⁰−u(T−t)BΓ^u_T−tψ⁰, t∈(0, T), Γ^u_T−0ψ⁰= Γ^u_Tψ⁰=ψ¹.

We define the operatoreΓ^u_t^esuch that Γ^u_T−tψ⁰=eΓ^eu_tψ¹ foreu(t) :=u(T−t) and (i∂tΓe^u_t^eψ¹= (−A−u(t)B)ee Γ^eu_tψ¹, t∈(0, T),

eΓ^u₀^eψ¹=ψ¹∈L²((0,1),C).

(2.4)

Asψ⁰=eΓ^eu_TΓ^u_Tψ⁰andψ¹= Γ^u_TeΓ^u_T^eψ¹, it followseΓ^eu_T = (Γ^u_T)⁻¹= (Γ^u_T)^∗.The operator Γe^u_t^edescribes the reversed dynamics of Γ^u_t induced by the system (2.4) and generated by the Hamiltonian (−A−u(t)B).e

3. Simultaneous local exact controllability in projection

3.1. Main result. In this section, we examine the simultaneous local exact controllability in projection stated by the following theorem.

Theorem 3.1. Let Γ^u_t be the unitary propagator in H generated by the dy- namics of the (BSE) in the time interval [0, t] with B satisfying Assumptions I.

Let N ∈ N^∗. For every T > 0, there exist an open set O in `^∞(H₍₀₎³ ) and an orthonormal system Ψ := (ψj)_j∈N^∗ ∈ O such that the following result is verified.

Let (ψ¹_j)_j∈N^∗ ∈ O be a complete orthonormal system and Γb ∈U(H)be such that (bΓψ¹_j)_j∈N^∗ = (ψ_j)_j∈N^∗. If (bΓψ_j)_j≤N ⊂ H₍₀₎³ , then there exist (θ_j)_j≤N ⊂ R and u∈L²((0, T),R) such that

(hψk,Γ^u_Tψji_L2 =e^iθjhψk, ψ_j¹i_L2, ∀j, k∈N^∗, j≤N, k≤N, hψk,Γ^u_Tψ_jiL² =hψk, ψ¹_jiL², ∀j, k∈N^∗, j > N, k≤N.

In other words, the following identities are satisfied (withπN(Ψ) defined in (1.1)):

(π_N(Ψ)Γ^u_Tψ_j=e^iθjπ_N(Ψ)ψ_j¹, ∀j∈N^∗, j≤N, πN(Ψ)Γ^u_Tψj=πN(Ψ)ψ¹_j, ∀j∈N^∗, j > N.

3.2. Introductive discussion. We start by explaining why we need to mod- ify the problem in order to prove Theorem 3.1. Let Φ = (φ_j)_j∈N∗ be a complete orthonormal system composed by eigenfunctions of A. For every j ∈ N^∗, we de- note φ_j(t, x) = e^−iλjtφ_j(x) with t > 0. From now on, we adopt the notation φ_j(t) =φ_j(t,·). Let >0 andT >0. We consider the set

O,T :=n

(ψj)j∈N^∗∈`^∞(H₍₀₎³ ) complete orthonormal system ofH such that sup

k≤N

X

j∈N^∗

k⁶|hψj, φ_k(T)iL²− hφj(T), φ_k(T)iL²|²< o (3.1) .

We would like to prove to validity of Theorem 3.1 in the neighborhood O,T for , T >0 with respect to the projector πN(Φ) (see the definition (2.2)). Then,

Γ^u_Tφ_j=

∞

X

k=1

φ_k(T)hφ_k(T),Γ^u_tφ_ji_L2, φ_j(T) =e^−iλjTφ_j, ∀j∈N^∗ is the solution of the j-th (BSE) with initial dataφj at time T >0. We consider the infinite matrixα(u) such that

αk,j(u) =hφk(T),Γ^u_Tφji_L2, ∀k, j∈N^∗, k≤N.

We would like to ensure the existence of > 0 and T > 0 such that for any (ψj)_j∈N^∗∈O,T, there existsu∈L²((0, T),R) such that

π_N(Φ)Γ^u_Tφ_j=π_N(Φ)ψ_j, ∀j∈N^∗.

This result can be proved by studying the local surjectivity ofαforT >0. To this purpose, we want to use the inverse mapping theorem and study the surjectivity of

the Fr´echet derivative ofαthe infinite matrixγ(v) := (duα(0))· vsuch that γk,j(v) : =

*

φk(T),−i Z T

0

e^−iA(T−s)v(s)Be^−iAsφjds +

L²

=−i Z T

0

v(s)e^{−i(λj−λk)s}dsBk,j, ∀j, k∈N^∗, k≤N,

with Bk,j =hφk, BφjiL² = hBφk, φjiL² =Bj,k. The surjectivity ofγ consists in proving the solvability of the moment problem

x_k,j Bk,j

=−i Z T

0

u(s)e^{−i(λj−λk)s}ds, ∀j, k∈N^∗, k≤N, (3.2)

for each infinite matrixx:= (xk,j)_j,k∈N^∗

k≤N

belonging to a suitable space. To this end, one would use Corollary A.9 which is consequence of the Haraux’s Theorem but an obstruction appears. The terms (λ_j−λ_k)_j,k∈N^∗

k≤N

in the moment problem (3.2) present the so-called eigenvalues resonances. Formally, for somej, k, n, m∈N^∗such that j6=k,n6=m, (j, k)6= (n, m) andk, m≤N, there holdsλj−λk =λn−λm, which implies

x_k,j Bk,j

=−i Z T

0

u(s)e^{−i(λj−λk)s}ds=−i Z T

0

u(s)e^{−i(λn−λm)s}ds= x_n,m Bn,m

. (3.3)

An example of eigenvalues resonance is λ₇−λ₁ =λ₈−λ₄, but many others can be listed. For instance, all the diagonal terms of γ since λ_j−λ_k = 0 for j = k.

The relation (3.3) represents a constraint on the considered matricesxwhich is not naturally satisfied in our framework.

In order to avoid this phenomenon, we adopt the following strategy. First, we consider the Hamiltonian characterizing the bilinear Schr¨odinger equations (BSE) and we use the following decomposition

A+u(t)B= (A+u₀B) +u₁(t)B, u₀∈R, u₁∈L²((0, T),R).

(3.4)

Second, we considerA+u0B instead ofA. We repeat the previous steps by consid- ering (φ^u_j⁰)_j∈N^∗ a complete orthonormal system ofH composed by eigenfunctions ofA+u0B and (λ^u_j⁰)_j∈N^∗ the corresponding eigenvalues. By usingu0B as a per- turbation inA+u0B, we modify the eigenvalues gaps

λ^u_j⁰−λ^u_k⁰, ∀j, k∈N^∗, k≤N

in order to remove all the non-diagonal resonances. Afterwards, we consider αb depending on the parameteru₀(instead ofα) such that it is defined by the elements αbk,j(u) = he^−iλuk0Tφ^u_k⁰,Γ^u_Tφ^u_j⁰i_L2 with k, j ∈ N^∗ and k ≤ N. Now, we rotate the terms ofαbin order to remove the resonances on the diagonal terms. We denote by α^u0 the obtained map, which is defined by the elements

α^u_k,j⁰(u) = αbj,j(u)

|αbj,j(u)|αbk,j(u), ∀j, k∈N^∗, k≤N.

In conclusion, we use the inverse mapping theorem with respect to the mapα^u0. The first step of our strategy is not so different from the techniques leading to [16, Main Theorem], however it presents an important difference. In our work, we seek for explicit conditions on the operatorB such that the perturbative argument is valid. On the contrary, the authors of [16] prove the existence ofQ, a residual

subset ofH⁴((0,1),R) (in the spirit of Theorem 1.3), such that the controllability holds whenB is a multiplication operator by a function µ∈ Q.

3.3. The modified problem. In this subsection, we rewrite the (BSE) by applying the decomposition (3.4) and we introduce the groundwork required to apply the strategy discussed in Section 3.2. Let u(t) = u0+u1(t) with u0 ∈ R, u₁∈L²((0, T),R) andT >0. We consider the following Cauchy problems

(i∂tψj(t) = (A+u0B)ψj(t) +u1(t)Bψj(t), t∈(0, T),

ψ_j⁰=ψj(0), j∈N^∗.

(3.5)

AsBis bounded,A+u0Bhas pure discrete spectrum. We recall that (λ^u_j⁰)_j∈N^∗are the eigenvalues ofA+u0Band Φ^u0:= (φ^u_j⁰)_j∈N^∗is a complete orthonormal system ofH made by corresponding eigenfunctions. FixedN ∈N^∗, for everyT >0 and 0>0, we denote

O^u0

0,T :=n

(ψ_j)_j∈N∗∈`^∞(H₍₀₎³ ) complete orthonormal system ofH such that sup

k≤N

X

j∈N^∗

k⁶|hψj, φ^u_k⁰(T)iL²− hφ^u_j⁰(T), φ^u_k⁰(T)iL²|²< ₀o (3.6) ,

with φ^u_j⁰(T) := e^−iλuj0Tφ^u_j⁰. We choose |u0| small such that λ^u_k⁰ 6= 0 for every k∈N^∗ (Lemma B.4). The modification of the problem imposes to define the space

He₍₀₎³ :=D(|A+u₀B|³²), k · k

He₍₀₎³ =X^∞

k=1

|λ^u_k⁰|³²h·, φ^u_k⁰iL²

2¹₂ . However, we consider from now on u0 in the neighborhood provided by Lemma B.6 so thatHe₍₀₎³ ≡H₍₀₎³ .As introduced in Section 3.2, we consider the mapαbwith elementsαbk,j(u1) =hφ^u_k⁰(T),Γ^u_T^0+u1φ^u_j⁰iL² fork≤N andj∈N^∗. The mapα^u0 is the infinite matrix with elements

(α^u_k,j⁰(u1) =_|^αbj,j(u1)

αb_j,j(u₁)|αbk,j(u1), ∀j, k∈N^∗, j, k≤N, α^u_k,j⁰(u₁) =αb_k,j(u₁), ∀j, k∈N^∗, j > N, k≤N.

(3.7)

Now, we study the space whereα^u0 takes value. LetΓe^eu_t be the propagator of the reversed dynamics defined in Section 2.3 fort∈[0, T],u∈L²((0, T),R) andT >0.

For everyk∈N^∗,u0∈Randu1∈L²((0, T),R), from Proposition 2.3, Remark 2.4 and Lemma B.6, there existsC >0 so that

+∞

X

j=1

j⁶|α^u_k,j⁰(u₁)|²=

+∞

X

j=1

j⁶|heΓ^u_T^0+eu1φ^u_k⁰, φ^u_j⁰iL²|²=keΓ^u_T^0+ue1φ^u_k⁰k²

He³₍₀₎

≤CkeΓ^u_T^0+eu1φ^u_k⁰k²₍₃₎<∞.

Thus, each (α^u_k,j⁰(u₁))_j∈N^∗∈h³(C) (defined in (2.1)). For every (ψ_j)_j∈N^∗ such that (ψj)j∈N^∗∈O^u0

0,T or such that (ψj)j∈N^∗ = (Γ^u_T^0+u1φ^u_j⁰)j∈N^∗, we have δj,k=hφ^u_j⁰, φ^u_k⁰iL² =D X

m∈N^∗

ψmhψm, φ^u_j⁰iL²,X

l∈N^∗

ψlhψl, φ^u_k⁰iL²

E

L²

=D

(hψm, φ^u_j⁰i_L2)_m∈N^∗,(hψm, φ^u_k⁰i_L2)_m∈N^∗E

`²

, ∀j, k≤N.

The last relations imply that α^u0 : u ∈ L²((0, T),R) 7−→ (α^u_k,j⁰(u))k,j∈N^∗

k≤N

∈ Q^N where

Q^N :=n

(xk,j)_k,j∈N^∗

k≤N

∈(h³(C))^N

xk,k∈R, (xj,l)_l∈N^∗,(xk,l)_l∈N^∗

`² =δj,k, ∀k, j≤No .

Now, Γ^(·)_T ψ : u ∈ L²((0, T),R) 7−→ Γ^u_Tψ ∈ H₍₀₎³ with ψ ∈ H₍₀₎³ is C¹ (see [3, Proposition 47] or [15, Section 2] for further details) and the same is true for Γe^(·)_T ψ:u∈L²((0, T),R)7−→Γe^u_Tψ∈H₍₀₎³ for everyψ∈H₍₀₎³ . Finally, the map

α^u0 :u∈L²((0, T),R)7−→(α^u_k,j⁰(u))k,j∈N^∗

k≤N

∈Q^N

isC¹thanks to the identitiesαbk,j(u1) =hφ^u_k⁰(T),Γ^u_T^0+u1φ^u_j⁰i=heΓ^u_T^0+,eu1φ^u_k⁰(T), φ^u_j⁰i for everyk, j∈N^∗withk≤N.We denote byγ^u0(v) = ((du1α^u0)(0))·vthe Fr´echet derivative ofα^u0. Definedbγ_k,j(v) = ((d_u1α)(0))b ·v, the elements ofγ^u0(v) are

(γ_k,j^u0 = bγ_j,jδ_k,j+bγ_k,j−δ_k,j<(bγ_j,j)

, ∀j, k∈N^∗, j, k≤N, γ_k,j^u0 =bγk,j, ∀j, k∈N^∗, k≤N, j > N and then, forB_k,j^u0 =hφ^u_k⁰, Bφ^u_j⁰iL² fork≤N andj∈N^∗,

(γ_k,j^u0 =bγk,j =−iRT

0 u1(s)e^{−i(λuj0−λuk0)s}dsB_k,j^u0, ∀j, k∈N^∗, k6=j, γ_k,k^u0 =<(bγk,k) = 0, ∀k∈N^∗.

(3.8)

The relation γ_k,k^u0 = 0 is due to the fact that (ibγ_k,k) ∈ R since bγ_k,j = −bγ_j,k for j, k≤N.Hence, the diagonal elements ofγ^u0are all equal to 0 due to the rotations adopted in the definitionα^u0. Since O^u0

0,T is composed by orthonormal elements, the tangent space ofO^u0

0,T in the point Φ^u0 is T_Φ^u0O^u0

0,T =n

(ψ_j)_j∈N^∗ ⊂`^∞(H₍₀₎³ )

hφ^u_k⁰, ψ_ji_L2 =−hφ^u_j⁰, ψ_ki_L2

o . The last relation implies that γ^u0 : u ∈ L²((0, T),R) 7−→ (γ^u_k,j⁰(u))_k,j∈N^∗

k≤N

∈ G^N where

G^N :=

(xk,j)_k,j∈N^∗

k≤N

∈(h³(C))^N

xk,j =−xj,k, xk,k= 0, ∀k, j≤N . Remark3.2. When the third point of Remark B.9 is valid, the controllability in O^u0

0,T (defined in (3.6)) with 0>0 ensures the controllability inO,T (defined in (3.1)) for suitable > 0. Let (ψ_j)_j∈N^∗ ∈ O_,T and bΓ ∈ U(H) be such that (bΓψj)_j∈N^∗ = (φ^u_j⁰)_j∈N^∗ and satisfying (bΓφ^u_j⁰)_j≤N ⊂H₍₀₎³ . There exists C > 0 so that, for everyk≤N,

X

j∈N^∗

|j³hφ^u_k⁰, ψjiL²|²= X

j∈N^∗

|j³hbΓφ^u_k⁰, φ^u_j⁰iL²|²≤CkbΓφ^u_k⁰k(3)<∞, (3.9)

thanks to Lemma B.4 and Lemma B.6. Now, fixede^iθj1:= ^hφ

u0

j (T),ψji_L2

|hφ^u_j⁰(T),ψ_ji_L2| forj ≤N and e^iθ1j := 1 for j > N, the relation (3.9) yields that (e^iθ1jhφ^u_k⁰(T), ψji_L2)_j,k∈N^∗

k≤N