Mégane Bournissou

Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates

, under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the small-time local exact controllability in projection, around the ground state, with estimates on the control (depending linearly on the target) simultaneously in several spaces. These estimates are obtained at the level of the linearized system, thanks to a new result about trigonometric moment problems. Then, they are transported to the nonlinear system by the inverse mapping theorem, thanks to appropriate estimates of the error between the nonlinear and the linearized dynamics.

1 Introduction and main result

Description of the controlled system

Let T > 0. In this paper, we consider the 1D Schrödinger equation, i∂ t ψ(t, x) = -∂ 2

x ψ(t, x) -u(t)µ(x)ψ(t, x), (t, x) ∈ (0, T ) × (0, 1), ψ(t, 0) = ψ(t, 1) = 0, t ∈ (0, T ).

(

) 1
In quantum physics, this equation describes a quantum particle, in an infinite potential well, subjected to an electric field whose amplitude is given by u(t). The dipolar moment, µ : (0, 1) → R, depicts the interaction between the electric field and the particle. This equation is a bilinear control system where the state is the wave function ψ, such that ψ(t) L 2 (0,1) = 1 for all time and u : (0, T ) → R denotes a scalar control.

Functional settings

Unless otherwise specified, in space, we will work with complex valued functions. The Lebesgue space L 2 (0, 1) is equipped with the classical hermitian scalar product. Let S be the unit-sphere of L 2 (0, 1). The operator A is defined by D(A) := H 2 (0, 1) ∩ H 1 0 (0, 1), Aϕ := -

d 2 ϕ dx 2 .
Its eigenvalues and eigenvectors are given by ∀j ∈ N * , λ j := (jπ) 2 and ϕ j := √ 2 sin(jπ•).

The family of the eigenvectors (ϕ j ) j∈N * is an orthonormal basis of L 2 (0, 1). We also introduce, for all j ∈ N * , ψ j (t, x) := ϕ j (x)e -iλ j t for (t, x) ∈ R × [0, 1], which are solutions of the Schrödinger equation [START_REF] Avdonin | The method of moments in controllability problems for distributed parameter systems[END_REF] for u ≡ 0. When k = 1, ψ 1 is the ground state. We also introduce the normed spaces linked to the operator A, given by, for all s > 0,

H s (0) (0, 1) := D(A s 2 ), ϕ H s (0) (0,1) := ( ϕ, ϕ j ) j∈N * h s =   +∞ j=1 |j s ϕ, ϕ j | 2   1 2
.

If J is a subset of N * , then we define

H J := Span C (ϕ j , j ∈ J) ,
and we introduce the orthogonal projection on H J , given by,

P J : L 2 (0, 1) → H J ψ → ψ - j ∈J
ψ, ϕ j ϕ j .

For T > 0 and u ∈ L 1 (0, T ), the family (u n ) n∈N of the iterated primitives of u is defined by induction as, u 0 := u and ∀n ∈ N, u n+1 (t) := t 0 u n (τ )dτ, t ∈ [0, T ].

We will also consider, for any integer k ∈ N, H k ((0, T ), R), the usual integer-order real Sobolev spaces, equipped with the usual H k (0, T )-norm and H k 0 (0, T ) the adherence of C ∞ c (0, T ), the set of functions with compact support inside (0, T ), for the topology • H k (0,T ) . By Poincaré inequality, H k 0 (0, T ) can be equipped with the norm

u H k 0 (0,T ) := T 0 u (k) (t) 2 dt 1/2
.

For any integer k ∈ N * , the negative H -k (0, T )-norm is not defined by duality as usual but for every u ∈ L 2 (0, T ) by u H -k (0,T ) := |u 1 (T )| + u k L 2 (0,T ) ,

as such norms seem to arise naturally in both the nonlinear and linearized dynamics. For the sake of simplicity, we will sometimes omit (0, T ) or (0, 1) on the spaces.

Main result

The regularity assumptions play a crucial role in the validity of controllability results. Therefore, we define the following precise notion of small-time local controllability (STLC) used in this paper, stressing the regularity imposed on both the control and the data to be controlled.

Definition 1.1 (STLC around the ground state in X with controls in Y ). Let X be a vector space of complex-valued functions defined on [0, 1] and (Y T , • Y T ) be a family of normed vector spaces of real-valued functions defined on [0, T ], for T > 0. The system (1) is said to be STLC around the ground state in X with controls in Y if for every T > 0, for every ε > 0, there exists δ > 0 such that for every ψ * , ψ f in S ∩ X with ψ * -ψ 1 (0) X < δ and ψ f -ψ 1 (T ) X < δ, there exists u ∈ L 2 (0, T )∩Y T with u Y T < ε such that the solution ψ of (1) associated with the initial condition ψ * satisfies ψ(T ) = ψ f .

Since [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF], it is known that if there exists a constant c > 0 such that for all j ∈ N * , | µϕ 1 , ϕ j | ≥ c j 3 ,

then for any k ∈ N, the Schrödinger equation ( 1) is STLC around the ground state in H 2k+3 (0) with controls in H k 0 . However, in [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF], the associated control map (ψ 0 , ψ f ) → u depends on k. In this article, two goals are tackled:

• first, building a unique control map for the nonlinear system with simultaneous estimates in various control/data spaces,

• second, dealing with control in projection when an assumption of the type (3) holds only on a subset J of N * .

Our main result is the following one.

Theorem 1.2. Let (p, k) ∈ N 2 with p ≥ k, J a subset of N * and µ ∈ H 2(p+k)+3 ((0, 1), R) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1, such that there exists a constant c > 0 such that for all j ∈ J, | µϕ 1 , ϕ j | ≥ c j 2p+3 .

(4)

The Schrödinger equation [START_REF] Avdonin | The method of moments in controllability problems for distributed parameter systems[END_REF] is STLC in projection around the ground state in H 2(p+m)+3 (0) with controls in H m 0 (T 0 , T ), for every m ∈ {0, . . . , k} with the same control map.

More precisely, for all initial time T 0 ≥ 0 and final time T > T 0 , there exists C, δ > 0 and a C 1 -map Γ : Ω T 0 × Ω T → H k 0 ((T 0 , T ), R) where

Ω T 0 := {ψ 0 ∈ S ∩ H 2(p+k)+3 (0)
;

ψ 0 -ψ 1 (T 0 ) H 2(p+k)+3 (0) < δ}, (5)
Ω T := {ψ f ∈ H J ∩ H 2(p+k)+3 (0) ; ψ f -P J (ψ 1 (T )) H 2(p+k)+3 (0) < δ}, (6)
such that Γ(ψ 1 (T 0 ), ψ 1 (T )) = 0 and for every (ψ 0 , ψ f ) ∈ Ω T 0 × Ω T , the solution of (1) on [T 0 , T ] with control u := Γ(ψ 0 , ψ f ) and initial condition ψ 0 at t = T 0 satisfies

P J (ψ(T )) = ψ f ,
with the following boundary conditions

u 2 (T ) = . . . = u k+1 (T ) = 0, (7)
where here (u n ) n∈N denotes the iterated primitives of u vanishing at T 0 . Besides, for all m in {-(k + 1), . . . , k}, the following estimates hold

u H m 0 (T 0 ,T ) ≤ C ψ 0 -ψ 1 (T 0 ) H 2(p+m)+3 (0) + ψ f -P J ψ 1 (T ) H 2(p+m)+3 (0)
.

To simplify the notations, in all the following, we will take the initial time T 0 equal to 0, the proof when T 0 > 0 is deduced by translation of controls and a change of global phase on the state. Moreover, from now on, if not mentioned, T > 0 will denote the final time, p and k two integers, and J a subset of N * . Remark 1.3. Assume that J contains an infinite subset of 2N and one of 2N + 1. Then, for all µ in H 2p+3 (0, 1) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1, assumption (4) is equivalent to µ (2p+1) (0) ± µ (2p+1) (1) = 0 and ∀j ∈ J, µϕ 1 , ϕ j = 0, as, by integrations by parts and by Riemann-Lebesgue Lemma,

µϕ 1 , ϕ j = (-1) p 2(2p + 2) π 2p+2 j 2p+3 (-1) j+1 µ (2p+1) (1) -µ (2p+1) (0) + o j→+∞ 1 j 2p+3 .
This result is both a new control result and a toolbox for future works about nonlinear control of the Schrödinger equation [START_REF] Avdonin | The method of moments in controllability problems for distributed parameter systems[END_REF]. Indeed, such result can for example give a framework to prove positive controllability results on the Schrödinger equation, with nonlinear tools, when some of the coefficients µϕ 1 , ϕ j vanish. In that case, building a unique control map with estimates in simultaneous spaces can be useful to perform specific motions for the nonlinear solution. The proof of Theorem 1.2 is in three steps.

• In Section 2, we study the well-posedness of the Schrödinger equation and more precisely the regularity of the solutions with respect to the boundary conditions on the dipolar moment µ.

• In Section 3, we present a new result about the solvability of trigonometric moment problems in high regularity spaces with simultaneous estimates.

• This new moment result allows in Section 4 and more precisely in Subsection 4.1 to build a linear control operator, for the linearized system around the ground state, with simultaneous estimates in various control/data spaces. Then, in Subsection 4.3, we prove that the iterations of the inverse mapping theorem propagate these estimates to the nonlinear control operator of (1).

Remark 1.4. Actually, the question of building a control function that inherits the regularity of the data to be controlled has already been tackled by Ervedoza and Zuazua in [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF], for time-reversible linear systems. As the use of [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] is not straightforward in our case, we choose to present in this article a new result about trigonometric moment problems solving this question. However, in Section 5, we also explain how the controllability of the linearized system with simultaneous estimates can be proved using the results of [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF].

Bibliography

Local exact controllability results. From a general negative result on the controllability of bilinear control systems by Ball, Marsden and Slemrod [START_REF] Ball | Controllability for distributed bilinear systems[END_REF], Turinici in [START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF] deduced a negative control result for the Schrödinger equation ( 1): for a given initial data ψ 0 ∈ H 2 (0) (0, 1) ∩ S, the reachable set with controls in L r loc ((0, +∞), R), with r > 1, has an empty interior in H 2 (0) (0, 1) ∩ S. The case of controls in L 1 loc ((0, +∞), R) has been proved later in [START_REF] Boussaïd | Regular propagators of bilinear quantum systems[END_REF] by Boussaid, Caponigro, and Chambrion.

However, choosing more appropriate functional spaces, exact local controllability results for 1D models have been proved by Beauchard in [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF][START_REF] Beauchard | Controllability of a quantum particle in a 1D variable domain[END_REF], whose proofs have been later simplified by Beauchard and Laurent in [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF] by means of a hidden regularizing effect and an inverse mapping theorem (instead of Nash-Moser's one).

This strategy was later developed by Morancey and Nersesyan to control one Schrödinger equation with a polarizability term [START_REF] Morancey | Global exact controllability of 1D Schrödinger equations with a polarizability term[END_REF] or a finite number of equations with one control [START_REF] Morancey | Simultaneous local exact controllability of 1D bilinear Schrödinger equations[END_REF][START_REF] Morancey | Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations[END_REF]. This was also used by Puel [START_REF] Puel | Local exact bilinear control of the Schrödinger equation[END_REF] to prove the local exact controllability for a Schrödinger equation, in a bounded regular domain, in a neighborhood of an eigenfunction corresponding to a simple eigenvalue in dimension N ≤ 3.

Global approximate results.

With geometric techniques for the controllability of the Galerkin approximations, in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] Chambrion, Mason, Sigalotti, and Boscain prove the approximate controllability of Schrödinger in L 2 under hypotheses later refined by Boscain, Caponigro, Chambrion, and Sigalotti in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF]. In higher order Sobolev spaces, similar results were proved for one [START_REF] Boussaïd | Regular propagators of bilinear quantum systems[END_REF] or a finite number of equations [START_REF] Boscain | Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF]. Such types of results can also be proved from exact controllability results in infinite time [START_REF] Nersesyan | Global exact controllability in infinite time of Schrödinger equation: multidimensional case[END_REF] or from a variational argument [START_REF] Vahagn Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF].

About smooth controllability. The negative controllability result [START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF] and the positive controllability results [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF][START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF] proved on the same Schrödinger equation underline the importance of the regularity assumptions asked for the validity of controllability results, for a linear infinite dimensional equation. Nevertheless, even when the state lives within a finite dimensional space, Beauchard and Marbach proved in [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF] that the same nonlinear system, according to the functional setting, may or may not be small-time controllable. The same authors highlighted later the same phenomenon on a nonlinear infinite dimension parabolic equation in [START_REF] Beauchard | Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations[END_REF].

Moreover, for a controlled system that is already known to be controllable in a given setting, one can ask whether the control map preserves the smoothness of the data to be controlled. More precisely, if the data is smoother than expected, does the control constructed inherit from this smoothness? Generally, it is not the case. However, [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] gave a method to compute such controls. Such a question can be relevant to deal with nonlinear problems as in [START_REF] Dehman | Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time[END_REF] or to compute convergence rates for numerical approximation (see [START_REF] Burman | A fully discrete numerical control method for the wave equation[END_REF] for example).

About moment problems. The use of infinite moment theory for linear control problems was introduced by the work of Fattorini and Russel (see [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF]). For classical results about Riesz basis and moment problems, the reader can refer, for example, to the following works: Krabs in [START_REF] Krabs | On moment theory and controllability of one-dimensional vibrating systems and heating processes[END_REF], Avdonin and Ivanov in [START_REF] Avdonin | The method of moments in controllability problems for distributed parameter systems[END_REF], Komornik and Loreti in [START_REF] Komornik | Fourier series in control theory[END_REF], Haraux in [START_REF] Haraux | Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire[END_REF]. In this paper, the solvability of a moment problem at any time with estimates in simultaneous Sobolev spaces is investigated. For the Schrödinger equation, it has already been done by Beauchard in [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF] but only for a specific and not arbitrary small time. Moreover, the solvability of a moment problem in high-regularity spaces, but without simultaneous estimates, has been done in [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF] for Schrödinger, or in [START_REF] Beauchard | Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations[END_REF] for a parabolic equation, relying on the work [START_REF] Benabdallah | Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the N -dimensional boundary null controllability in cylindrical domains[END_REF].

Well-posedness of the Cauchy problem

This section is dedicated to the proof of the existence, uniqueness and bounds on the solution of the Cauchy problem

   i∂ t ψ(t, x) = -∂ 2 x ψ(t, x) -u(t)µ(x)ψ(t, x) -f (t, x), (t, x) ∈ (0, T ) × (0, 1), ψ(t, 0) = ψ(t, 1) = 0, t ∈ (0, T ), ψ(0, x) = ψ 0 (x), x ∈ (0, 1). ( 9
)
Our goal is to underline the link between the regularity of the solutions and the boundary conditions on the dipolar moment µ by proving the following statement.

Theorem 2.1. Let T > 0, (p, k) ∈ N 2 , µ ∈ H 2(p+k)+3 ((0, 1), R) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1 , u ∈ H k 0 ((0, T ), R), ψ 0 ∈ H 2(p+k)+3 (0) (0, 1) and f ∈ H k 0 ((0, T ), H 2p+3 ∩ H 2p+1 (0) (0, 1)
). There exists a unique weak solution of the Schrödinger equation, that is a function

ψ ∈ C k ([0, T ], H 2p+3 (0) (0, 1)) such that the following equality holds in H 2p+3 (0)
for every t ∈ [0, T ]:

ψ(t) = e -iAt ψ 0 + i t 0 e -iA(t-τ ) (u(τ )µψ(τ ) + f (τ )) dτ.
Moreover, for every R > 0, there exists C = C(T, µ, R) > 0 such that if u H k 0 < R, then this solution satisfies

ψ C k ([0,T ],H 2p+3 (0) ) ≤ C ψ 0 H 2(p+k)+3 (0) + f H k ((0,T ),H 2p+3 ∩H 2p+1 (0) ) . (10)
We will sometimes write ψ(•; u, ψ 0 ) to denote the solution of (9) associated with control u and initial data ψ 0 when we will need to recall the dependence with respect to the control or the initial condition. For p = 0, no boundary conditions are needed on µ. The proof of Theorem 2.1 is inspired by [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF]Proposition 2 and 5], where the authors dealt with the cases (p = 0, k = 0) and (p = 0, k = 1). Remark 2.2. Let T , p, k, µ, u, ψ 0 and f as in Theorem 2.1. Notice that as both the control and the source term vanish at the final time, the solution ψ ∈ C k ([0, T ], H 2p+3 (0) ) of ( 9) satisfies the following equality in H 2p+3 (0) (0, 1)

i∂ k t ψ(T ) = A k ψ(T ). Therefore, ψ(T ) ∈ H 2(p+k)+3 (0)
(0, 1).

However, in general, the solution does not belong to C 0 ([0, T ], H

2(p+k)+3 (0)
), as ψ(t) does not belong to H

2(p+k)+3 (0)
(0, 1) if u(t) = 0. Moreover, from [START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF], one deduces that for every R > 0, there exists

C = C(T, µ, R) > 0 such that if u H k 0 < R, then ψ(T ) H 2(p+k)+3 (0) ≤ C ψ 0 H 2(p+k)+3 (0) + f H k ((0,T ),H 2p+3 ∩H 2p+1 (0) ) . (11)

Smoothing effect

The main difficulty in this well-posedness result relies on the fact that, for a given τ in [0, T ], f (τ ) is not assumed to belong to H 2p+3 (0)

and moreover, the operator ψ → µψ is not bounded from

H 2p+3 (0) to H 2p+3 (0)
because µ (2p+1) does not vanish at x = 0 and x = 1. Therefore, the proof of Theorem 2.1 stems from the regularity of the function t → t 0 e iAτ f (τ )dτ in the spatial space H 2p+3 (0) even when f does not take values in such space. In the following proposition, for p = -1, we will write

H 2p+3 ∩ H 2p+1 (0)
to denote only the space H 1 to homogenize with the notation when p ∈ N. Proposition 2.3. Let p = -1 or p ∈ N. There exists a nondecreasing function C : [0, +∞) → (0, +∞) such that for all T ≥ 0 and for all f ∈ L 2 ((0, T ),

H 2p+3 ∩ H 2p+1 (0) (0, 1)), the function G : t → t 0 e -iA(t-τ ) f (τ )dτ belongs to C 0 ([0, T ], H 2p+3 (0) (0, 1)
) with the following estimate,

G C 0 ([0,T ],H 2p+3 (0) ) ≤ C(T ) f L 2 ((0,T ),H 2p+3 ∩H 2p+1 (0) ) . (12)
Proof.

Let T ≥ 0 and f ∈ L 2 ((0, T ), H 2p+3 ∩ H 2p+1 (0) ). Let t ∈ [0, T ].
By definition of the function G and of the norm H 2p+3 (0) , one seeks to estimate,

G(t) H 2p+3 (0) = +∞ j=1 t 0 f (τ ), ϕ j e -iλ j (t-τ ) dτ ϕ j H 2p+3 (0) = t 0 f (τ ), ϕ j e iλ j τ dτ j∈N * h 2p+3 . Yet, for almost every τ ∈ (0, T ), f (τ ) belongs to H 2p+3 ∩ H 2p+1 (0)
. Therefore, performing (2p + 3)integrations by parts, we get, for all j ∈ N * ,

f (τ ), ϕ j = √ 2 (jπ) 2p+3 (-1) j ∂ 2p+2 x f (τ, 1) -∂ 2p+2 x f (τ, 0) - 1 (jπ) 2p+3 ∂ 2p+3 x f (τ ), √ 2 cos(jπ•) ,
(with a minus added on each term if p = -1). Thus, there exists a constant C > 0, not depending on time, such that

G(t) H 2p+3 (0) ≤ C x 0 ∈{0,1} t 0 ∂ 2p+2 x f (τ, x 0 )e iλ j τ dτ j∈N * l 2 + C t 0 ∂ 2p+3 x f (τ ), √ 2 cos(jπ•) e iλ j τ dτ j∈N * l 2 . ( 13
)
Using the Cauchy-Schwarz inequality (in time) and then the orthonormality of the family ( √ 2 cos(jπ•)) j∈N in L 2 (0, 1), the square of the last term of the right-hand side of ( 13) is estimated by

+∞ j=1 t 0 ∂ 2p+3 x f (τ ), √ 2 cos(jπ•) e iλ j τ dτ 2 ≤ +∞ j=1 t t 0 ∂ 2p+3 x f (τ ), √ 2 cos(jπ•) 2 dτ ≤ t t 0 ∂ 2p+3 x f (τ ) 2 L 2 dτ, giving that t 0 ∂ 2p+3 x f (τ ), √ 2 cos(jπ•) e iλ j τ dτ j∈N * l 2 ≤ √ t f L 2 ((0,t),H 2p+3 ) . (14)
Moreover, the sum in the right-hand side of ( 13) is estimated using an Ingham inequality (see for example [5, Appendix B, Corollary 4]) which gives the existence of a nondecreasing function

C : t → C(t) > 0 such that, t 0 ∂ 2p+2 x f (τ, x 0 )e iλ j τ dτ j∈N * l 2 ≤ C(t) ∂ 2p+2 x f (•, x 0 ) L 2 (0,t) , for x 0 = 0 and 1. (15)
Therefore, ( 13), ( 14) and ( 15) together with the fact that

H 2p+3 (0, 1) is continuously embedded in C 2p+2 ([0, 1]) give G(t) H 2p+3 (0) ≤ C(t) f L 2 ((0,t),H 2p+3 ) ,
with a nondecreasing function t → C(t) > 0. This bound shows that G(t) belongs to H 2p+3 (0) (0, 1) for every t ∈ [0, T ] and that the map

t → G(t) ∈ H 2p+3 (0)
is continuous at t = 0 (as C(t) is uniformly bounded when t → 0 and f L 2 ((0,t),H 2p+3 ) → 0 when t → 0 thanks to the dominated convergence theorem). The continuity of G at any time t ∈ (0, T ] can be proved similarly.

The previous lemma stated the continuity of t → t 0 e iAτ f (τ )dτ and from this we can deduce, for all k ∈ N * , the C k -regularity of such function.

Proposition 2.4. Let (p, k) ∈ N 2 . There exists a nondecreasing function C : [0, +∞) → (0, +∞) such that for all T ≥ 0 and for all f ∈ H k 0 ((0, T ),

H 2p+3 ∩ H 2p+1 (0) (0, 1)), the function G : t → t 0 e -iA(t-τ ) f (τ )dτ belongs to C k ([0, T ], H 2p+3 (0) (0, 1)
) with the following estimate,

G C k ([0,T ],H 2p+3 (0) ) ≤ C f H k ((0,T ),H 2p+3 ∩H 2p+1 (0) ) . (16)
Proof. Let f ∈ H k 0 ((0, T ), H 2p+3 ∩ H 2p+1 (0)
). We will rather work with G written under the form

G(t) = t 0 e -iAτ f (t -τ )dτ, t ∈ [0, T ].
Step

1: Classical regularity. As f is in H k ((0, T ), H 2p+2 (0) ), the classical theory on semi-groups gives that G is in C k ([0, T ], H 2p+2 (0)
). Moreover, because f (0) = . . . = f (k-1) (0) = 0, the derivatives, for the H 2p+2 (0) -topology, are given by,

∀n = 0, . . . , k, G (n) (t) = t 0 e -iAτ f (n) (t -τ )dτ, t ∈ [0, T ]. (17)
Step 2: Higher regularity in space. We prove that G is in

C n ([0, T ], H 2p+3 (0) ) by induction on n ∈ {0, . . . k}. The initialization (n = 0) is proved in Proposition 2.3. Let n ∈ {0, . . . k -1} and assume that G is in C n ([0, T ], H 2p+3 (0) ). First, as f (n+1) is in L 2 ((0, T ), H 2p+3 ∩ H 2p+1 (0) ) (as n + 1 ≤ k), Proposition 2.3 and (17) give directly that G (n+1) is in C 0 ([0, T ], H 2p+3 (0) ). Then, for t ∈ [0, T ], with (17)
, one can write,

G (n) (t + h) -G (n) (t) h -G (n+1) (t) = 1 h t+h t e -iAτ f (n) (t + h -τ )dτ + t 0 e -iAτ f (n) (t + h -τ ) -f (n) (t -τ ) h -f (n+1) (t -τ ) dτ. ( 18
)
By Proposition 2.3, the H 2p+3 (0) -norm of the second term of the right-hand side of ( 18) is bounded by

c 1 (T ) f (n) (• + h) -f (n) h -f (n+1) L 2 ((0,T ),H 2p+3 ∩H 2p+1 (0) )
which goes to zero as h goes to zero, because

f (n) is in H 1 ((0, T ), H 2p+3 ∩ H 2p+1 (0)
). Besides, as f (n) (0) = 0, using successively a change of variables, that e -iA is an isometry from H 2p+3 (0) to H 2p+3 (0) , Proposition 2.3 and Cauchy-Schwarz inequality, one gets the following upper bound for the H 2p+3 (0)norm of the first term of the right-hand side of ( 18)

e iA(t+h) h h 0 e -iAs f (n) (s) -f (n) (0) ds H 2p+3 (0) ≤ c 1 (h) f (n) (•) -f (n) (0) h L 2 ((0,h),H 2p+3 ∩H 2p+1 (0) ) = c 1 (h) h • 0 ∂ t f (n) (τ )dτ L 2 ((0,h),H 2p+3 ∩H 2p+1 (0) ) ≤ c 1 (h) ∂ t f (n) L 2 ((0,h),H 2p+3 ∩H 2p+1 (0) ) ,
This bound goes to zero when h goes to zero as the function h → c 1 (h) given in Proposition 2.3 is nondecreasing and

lim h→0 ∂ t f (n) L 2 ((0,h),H 2p+3 ∩H 2p+1
(0) ) = 0 by the dominated convergence theorem. And this concludes the proof.

Proof of Theorem 2.1, the well-posedness

Let µ, ψ 0 , f and u satisfying the hypotheses of Theorem 2.1. We consider the map

F : C k ([0, T ], H 2p+3 (0) (0, 1)) → C k ([0, T ], H 2p+3 (0) (0, 1)) ψ → ξ, where ξ(t) := e -iAt ψ 0 + i t 0 e -iA(t-τ ) (u(τ )µψ(τ ) + f (τ )) dτ, t ∈ [0, T ],
so that ψ is a weak solution of (9) if and only if ψ is a fixed-point of F .

F is well-defined. For ψ ∈ C k ([0, T ], H 2p+3 (0)
) and τ ∈ [0, T ], by Leibniz formula and the algebra structure of H 2p+3 , the map τ → u(τ

)µψ(τ ) + f (τ ) belongs to H k 0 ((0, T ), H 2p+3 ∩ H 2p+1 (0) ) and thus, by Proposition 2.4, ξ = F (ψ) is in C k ([0, T ], H 2p+3 (0) ). F is a contraction. Let ψ, ψ in C k ([0, T ], H 2p+3 (0)
). Again, by Proposition 2.4 and the algebra structure of H 2p+3 (0, 1), we get, for all t ∈ [0, T ],

F (ψ)(t) -F ( ψ)(t) H 2p+3 (0) ≤ C(T ) u H k µ H 2p+3 ψ -ψ C k ([0,T ],H 2p+3 ) . (19)
Equation [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] proves that if u H k is small enough, F is a contraction and thus by the Banach fixedpoint theorem, admits a unique fixed-point

ψ in C k ([0, T ], H 2p+3 (0) ).
Computing the same estimates, by Proposition 2.4, we get that this fixed point satisfies

ψ C k ([0,T ],H 2p+3 (0) ) ≤ C(T, µ) ψ 0 H 2(p+k)+3 (0) + u H k ψ C k ([0,T ],H 2p+3 (0) ) + f H k ((0,T ),H 2p+3 ∩H 2p+1 (0) ) .
Therefore, for u ∈ H k 0 (0, T ) such that C(T, µ) u H k ≤ 1/2, we get [START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF]. If u in B H k (0,T ) (0, R) is not small enough in H k 0 , one can consider a subdivision 0 = T 0 < . . . < T N = T such that for all i ∈ {0, . . . , N -1}, u H k (T i ,T i+1 ) is small enough to apply the previous argument on [T i , T i+1 ]. Notice that as the constant T → C(T ) is nondecreasing, N only depends on R so that the constant in [START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF] does only depend on T , µ and R, as claimed in the theorem.

Solvability of a moment problem with simultaneous estimates

The goal of this section is the proof of a new result about trigonometric moment problems. More precisely, the aim is to prove the solvability of a moment problem in high-regularity spaces, with simultaneous estimates on the operator solving the moment problem. This will allow to build a control map for the linearized system with estimates in various control/data spaces in the following section.

Assumptions on the frequencies

Given an increasing sequence ω = (ω j ) j∈N of [0, +∞) with ω 0 = 0, we define, for all m ∈ N,

h 2m ω,r (N, C) := (d j ) j∈N ∈ l 2 (N, C); d 0 ∈ R and d 2 h 2m ω := +∞ j=0 δ j,0 + ω m j d j 2 < +∞ .
When m = 0, we will simply write l 2 r instead of h 0 ω,r . Moreover, we will say that a sequence (ω j ) j∈N satisfies an asymptotic gap if

ω j+1 -ω j → +∞ when j → +∞, (AsymptGap)
and satisfies a polynomial asymptotic gap if there exists ε > 0, N 0 ∈ N and c > 0 s.t. for all j ≥ N 0 , ω j+1 -ω j ≥ cj ε . (AsymptGapPoly)

If not mentioned, for all j ∈ Z, j < 0, we denote by ω j = -ω -j .

3.2 Solvability of a moment problem in L 2 (0, T ) with polynomial constraints

First, following some known results about trigonometric moment problems, one can prove the solvability in high-regularity spaces of such problems but without simultaneous estimates. This can be deduced from the solvability of a moment problem in L 2 (0, T ) with polynomial constraints, which is therefore the starting point of this section. The results presented in this subsection are a generalization of the work [5, Appendix B]. Therein, the full proofs are left to the reader. First, one can state that under an asymptotic gap condition, the family of complex exponentials, with an added finite number of polynomials, has a biorthogonal family. Lemma 3.1. Let T > 0 and (ω j ) j∈N an increasing sequence of [0, +∞) such that ω 0 = 0 and satisfying (AsymptGap).Then, for all n ∈ N * , the family Θ n := {t q ; q = 1, . . . n} ∪ e iω j t ; j ∈ Z is minimal in L 2 (0, T ) and thus admits a biorthogonal family.

From this result and the work of Haraux [START_REF] Haraux | Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire[END_REF], one can state the solvability of trigonometric moment problem in L 2 (0, T ) with a finite number of constraints on polynomial moments. Theorem 3.2. Let T > 0, n ∈ N * and (ω j ) j∈N an increasing sequence of [0, +∞) such that ω 0 = 0 and satisfying (AsymptGap). There exists a constant C > 0 and a continuous linear map

L T 0 : R n × l 2 r (N, C) → L 2 ((0, T ), R) such that for every sequence d = ((d -q ) q=1,...,n , (d j ) j∈N ) ∈ R n × l 2 r (N, C), the control u := L T 0 (d) ∈ L 2 ((0, T ), R) satisfies the moment problem ∀j ∈ N, T 0 u(t)e iω j t dt = d j and ∀q = 1, . . . , n, T 0 t q u(t)dt = d -q ,
and the following size estimate

u L 2 (0,T ) ≤ C   +∞ j=-n |d j | 2   1/2
.

Then, we deduce, for any integer k, the solvability of such moment problems in H k 0 (0, T ) with only an estimate in the most regular space.

Theorem 3.3. Let T > 0, (n, k) ∈ (N * ) 2 and (ω j ) j∈N an increasing sequence of [0, +∞) such that ω 0 = 0 and satisfying (AsymptGap). There exists a constant C > 0 and a continuous linear map

L T k : R n × h 2k ω,r (N, C) → H k 0 ((0, T ), R) such that for every sequence d = ((d -q ) q=1,...,n , (d j ) j∈N ) ∈ R n × h 2k ω,r (N, C), the control u := L T k (d) ∈ H k 0 ((0, T ), R) satisfies the moment problem ∀j ∈ N, T 0
u(t)e iω j t dt = d j and ∀q = 1, . . . , n,

T 0 t q u(t)dt = d -q ,
and the following size estimate

u H k 0 (0,T ) ≤ C   +∞ j=-n δ j,0 + ω k j d j 2   1/2 .
Proof. The proof follows with

L T k := B k • L T 0 • A k where L T 0 is defined in Theorem 3.2, A k : R n × h 2k ω,r (N, C) → R k+n × l 2 r (N, C) and B k : L 2 ((0, T ), R) → H k ((0, T ), R) are respectively given by A k ((d -q ) q=1,...,n , (d j ) j∈N ) := (-1) k q! (q -k)! d -q+k 1l q∈{k,...,k+n} q=1,...,k+n , (-iω j ) k d j 1l j∈N * j∈N , and
B k (v) := t → t 0 (t -τ ) k-1 (k -1)! v(τ )dτ . ( 20
)
3.3 Solvability of a moment problem in H k 0 (0, T ) with various estimates

Notice that Theorem 3.3 provides operators

L T k : ((d -q ) q=1,...,n , (d j ) j∈N ) ∈ R n × h 2k ω,r (N, C) → u ∈ H k 0 ((0, T ), R)
solving the moment problem which depend on k, preventing from having estimates on u, for a given sequence (d j ), simultaneously in various Sobolev spaces. Therefore, the goal of this subsection is to prove that one can solve trigonometric moment problems in H k 0 (0, T ) with simultaneous estimates on the control.

First, the result can be proved when dealing with only a finite number of moments.

Proposition 3.4. Let T > 0, (n, k, N ) ∈ N * × N 2 and (ω j ) j∈N an increasing sequence of [0, +∞) such that ω 0 = 0 and satisfying (AsymptGap). There exists a constant C N > 0 and a continuous linear map

L N,T lf : R n × R × C N -1 → H k 0 ((0, T ), R) such that for every sequence d = ((d -q ) q=1,...,n , (d j ) j=0,...,N -1 ) ∈ R n × R × C N -1 , the control u := L N,T lf (d) ∈ H k 0 ((0, T ), R) satisfies the moment problem ∀j = 0, . . . , N -1, T 0 u(t)e iω j t dt = d j ,
and ∀j ≥ N, T 0 u(t)e iω j t dt = 0, ∀q = 1, . . . , n,

T 0 t q u(t)dt = d -q ,
and the size estimates

u H m 0 (0,T ) ≤ C N   N -1 j=-n δ j,0 + ω m j d j 2   1/2
, ∀m = 0, . . . , k.

Proof. The proof follows with

L N,T lf ((d -q ) q=1,...,n , (d j ) j=0,...,N -1 ) := L T k ((d -q ) q=1,...,n , (d j 1l j=0,...,N -1 ) j∈N )
where L T k is defined in Theorem 3.3, using the equivalence of norms in finite dimension.

It remains to deal with the high frequencies. To that end, we will assume from now on that the sequence of frequencies satisfies the polynomial asymptotic gap (AsymptGapPoly). In other words, the goal is to prove that the map

L : u → T 0 u(t)e iω j t dt j∈N admits a continuous-right inverse P : h 2k ω,r (N, C) → H k 0 ((0, T ), R) which is still continuous from h 2m
ω,r (N, C) to H m 0 ((0, T ), R) for all m = 0, . . . , k. Usually, a continuous right inverse of L i.e. an operator solving the moment problem is sought under the form P (d) = d j ξ * j where {ξ * j , j ∈ Z} is the biorthogonal family of {e iω j • , j ∈ Z}. To conclude, one would need to be able to estimate such biorthogonal family simultaneously in all the Sobolev spaces H m 0 (0, T ), for m = 0, . . . , k. Such strategies have already been used. Explicit computations of the biorthogonal family with good estimates have, for example, been used: in [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF] to prove upper bounds for the control cost in the case of systems governed by the Schrödinger or the heat equation, in [START_REF] Lissy | The cost of the control in the case of a minimal time of control: the example of the one-dimensional heat equation[END_REF] to study the cost of the control in the case of a minimal time for the one-dimensional heat equation or in [START_REF] Benabdallah | Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the N -dimensional boundary null controllability in cylindrical domains[END_REF] to characterize the null controllability of a system of n parabolic equations in cylindrical domains. Sharp estimates for biorthogonal families of exponential functions without gap conditions have been given in [START_REF] González | Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions[END_REF] and used to prove new results on the cost of the boundary null controllability of parabolic systems. A new block resolution technique, together with sharp estimates, has also been used in [START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF] to characterize the minimal null control time for abstract linear control problem.

However, here we choose to not compute the biorthogonal family. As the exponentials are "almost orthogonal" for high frequencies, the main idea is to rather seek a solution of the moment problem under the form

P (d)(t) = +∞ j=-∞ d j e iω j t χ(t), (22)
with χ a weight function which allows to improve the decay of the coefficients e iω j • χ, e iωp• j =p at high frequencies. As such P will no longer exactly solve the moment problem, the right inverse of L will be constructed as an iteration of [START_REF] Komornik | Fourier series in control theory[END_REF], quantifying the error term. Besides, the explicit form of P will allow to easily estimate it in various spaces.

To implement such strategy, we start by introducing the operator giving the moment problem for high frequencies and the operator which will almost be its right-inverse.

L m N : H m 0 ((0, T ), R) → h 2m ω,r (N, C) u → T 0 u(t)e iω j t dt j N , P m N : h 2m ω,r (N, C) → H m 0 ((0, T ), R) (d j ) j N → |j|≥N d j ξ j ,
where for all j ∈ Z, j < 0, d j := d -j and for all j ∈ Z, for all t ∈ [0, T ] ξ j (t) := 1 T e -iω j t χ(t) with χ ∈ C ∞ c ((0, T ), R) such that T 0 χ(t)dt = 1. Then, for all m ∈ N and N ∈ N * , L m N and P m N are linear continuous applications.

Proof. Let m ∈ N and N ∈ N * . First, the continuity of L m N comes from that for all u ∈ H m 0 (0, T ), by integrations by parts,

+∞ j=N ω m j T 0 u(t)e iω j t dt 2 = +∞ j=N u (m) , e iω• 2 ≤ C u (m) 2 L 2 (0,T ) ,
as the family (e iω j • ) j∈Z is a Riesz basis. Secondly, the continuity of P m N stems from the fact that for all (d j ) j≥N in h 2m ω,r (N, C), by the algebra structure of H m 0 (0, T ),

|j|≥N d j ξ j H m 0 (0,T ) ≤ χ H m (0,T ) |j|≥N ω m j d j e -iω j • L 2 (0,T ) ≤ C ω m j d j |j|≥N l 2 (N,C) ,
as the family (e iω j • ) j∈Z is a Riesz basis. The reader can for example refer to [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF]Proposition 19] to find the results on Riesz basis used in this proof.

With these notations, our goal is to prove that, for N large enough, the application L m N has a common continuous right inverse for all m = 0, . . . , k. To that end, we start by quantifying in which way P m N is almost the right-inverse of L m N . Lemma 3.6. Let k ∈ N * . For all ε > 0, there exists N 1 ∈ N * such that for all N ≥ N 1 , for all m = 0, . . . , k, for all d ∈ h 2m ω,r (N, C),

L m N • P m N (d) -d h 2m ω,r (N,C) ≤ ε d h 2m ω,r (N,C) .
Proof. Let m ∈ {0, . . . , k} and N ≥ min(N 0 +1, 2) where N 0 is defined in (AsymptGapPoly). Notice that, performing integrations by parts (with no boundary terms as χ has a compact support), for all α ∈ N * , there exists a constant C = C(χ (α) , T ) > 0 such that, for all j, p ∈ N, T 0 ξ j (t)e iω j t dt = 1 and

T 0 ξ j (t)e iω k t dt ≤ C |ω k -ω j | α if j = k.
The coefficient α will be chosen later as large as needed. Using this remark, together with the following equality

(L m N • P m N (d)) p = T 0 P m N (d)(t)e iωpt dt = d p + |j|≥N, j =p d j T 0 ξ j (t)e iωpt dt, ∀p ∈ N,
we get, using Cauchy-Schwarz inequality,

L m N • P m N -d 2 h 2m ω,r = p≥N ω 2m p |j|≥N, j =p d j T 0 ξ j (t)e iωpt dt 2 ≤ C d 2 h 2m ω,r p≥N j≥N j =p ω 2m p ω 2m j |ω j -ω p | 2α .
Yet, by the triangular inequality, for all (j, p) ∈ N * , j = p,

ω 2m p ω 2m j |ω j -ω p | 2α ≤ C 1 ω 2m j |ω j -ω p | 2(α-m) + 1 |ω j -ω p | 2α .
Thus, as the sequence (ω j ) j∈N * is bounded by below by ω 1 , it is sufficient to prove that for β large enough, the series j,p 1 |ω j -ωp| β converges to get that choosing α large enough, for all ε > 0, there exists N 1 > 0 such that for all N ≥ N 1 , for all m = 0, . . . , k,

p≥N j≥N j =p ω 2m p ω 2m j |ω j -ω p | 2α ≤ ε,
which will conclude the proof. Yet, using the polynomial growth (AsymptGapPoly), for all j, p ≥ N , j = p,

|ω j -ω p | ≥ min (ω j -ω j-1 ) (ω p+1 -ω p ), (ω j+1 -ω j ) (ω p -ω p-1 ) ≥ c(j -1) ε/2 (p -1) ε/2 .
And thus, for all β > 2 ε , the series j,p 1 |ω j -ωp| β indeed converges.

By iterating the error estimate given in Lemma 3.6, we deduce a common continuous right-inverse for L m N , the operator of the moment problem.

Lemma 3.7. Let k ∈ N * . There exists N 1 ∈ N * such that for all N ≥ N 1 , L m N admits the same linear continuous right inverse for all m = 0, . . . , k. More precisely, for all N ≥ N 1 , there exists an operator M N such that for all m = 0, . . . , k, M m N :

h 2m ω,r (N, C) → H m 0 ((0, T ), R), d → M N (d) is continuous and satisfies L m N • M m N = Id h 2m .
Proof. By Lemma 3.6 with ε = 1 2 , set N 1 > 0 such that for all N ≥ N 1 , for all m = 0, . . . , k, we have,

L m N • P m N (d) -d h 2m ω,r (N,C) ≤ 1 2 d h 2m ω,r (N,C) , ∀d ∈ h 2m ω,r (N, C). (23)
Let N ≥ N 1 and m ∈ {0, . . . , k}. In the following proof, for the sake of clarity, we will forget all the exponents m on the name of the applications, that recall the spaces in which we work. First, if we define Σ := L N • P N -Id, an induction gives that

L N • n p=0 (-1) p P N • Σ p = Id +(-1) n Σ n+1 , ∀n ∈ N. (24)
Yet, Σ is a linear and continuous application from h 2m ω,r (N, C) to h 2m ω,r (N, C) with its operator norm satisfying Σ h 2m ,h 2m ≤ 1/2 by [START_REF] Krabs | On moment theory and controllability of one-dimensional vibrating systems and heating processes[END_REF]. Thus, the series p (-1) p P N • Σ p absolutely converges in the space L c (h 2m ω,r (N, C), H m 0 ((0, T ), R)). Therefore, passing to the limits [n → +∞] in the equality (24), we get,

L N • M N = Id with M N := +∞ p=0 (-1) p P N • Σ p ,
which is continuous from h 2m ω,r (N, C) to H m 0 ((0, T ), R) for all m = 0, . . . , k. Now, from Lemma 3.7, one can prove the solvability of a moment problem with simultaneous estimates on the control for high frequencies.

Proposition 3.8. Let T > 0, k ∈ N and (ω j ) j∈N an increasing sequence of [0, +∞) such that ω 0 = 0 and satisfying (AsymptGapPoly). There exists an integer N ∈ N * , a constant C > 0 and a continuous linear map L N,T hf :

h 2k ω,r (N, C) → H k 0 ((0, T ), R) such that for every sequence d = (d j ) j∈N ∈ h 2k ω,r (N, C), the control u := L N,T hf (d) ∈ H k 0 ((0, T ), R) satisfies the moment problem T 0 u(t)e iω j t dt = d j , ∀j ≥ N, (25)
and the following estimates

u H m 0 (0,T ) ≤ C   +∞ j=N ω m j d j 2   1/2
, ∀m = 0, . . . , k. Theorem 3.9. Let T > 0, k ∈ N, n ∈ N * and (ω j ) j∈N an increasing sequence of [0, +∞) such that ω 0 = 0 and satisfying (AsymptGapPoly). There exists a constant C > 0 and a continuous linear map

L T k : R n ×h 2k ω,r (N, C) → H k 0 ((0, T ), R) such that for every sequence d = ((d -q ) q=1,...,n , (d j ) j∈N ) ∈ R n × h 2k ω,r (N, C), the control u := L T k (d) ∈ H k 0 ((0, T ), R) satisfies the moment problem ∀j ∈ N, T 0
u(t)e iω j t dt = d j and ∀q = 1, . . . , n,

T 0 t q u(t)dt = d -q , ( 27
)
with the following size estimates,

u H m 0 (0,T ) ≤ C   +∞ j=-n δ j,0 + ω m j d j 2   1/2
, ∀m = 0, . . . , k.

Proof of Theorem 3.9. r (N,C). The proof follows with

Let T > 0, k ∈ N, n ∈ N * . Let (d -q ) q=1,...,n ∈ R n and (d j ) j∈N ∈ h 2k ω,
u = L T k ((d -q ) q=1,...,n , (d j ) j∈N ) := L N,T hf ((d j ) j∈N ) + L N,T lf d -q - T 0 t q L N,T hf (d j )(t)dt q=1,...,,n , d j - T 0 L N,T hf (d j )(t)e iω j t dt j=0,...,N -1
.

Indeed, by linearity and by construction of the operators L N,T hf and L N,T lf (given respectively in Proposition 3.8 and Proposition 3.4, the control u satisfies the moment problem [START_REF] Morancey | Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations[END_REF]. Furthermore, by ( 21) and ( 26), there exists a constant C > 0 such that for all m = 0, . . . , k,

u H m 0 (0,T ) ≤ C      +∞ j=N ω m j d j 2   1/2 +   N -1 j=-n (δ j,0 + ω j m ) d j -d j 2   1/2    . ( 29
)
where, if v := L N,T hf ((d j ) j∈N ), ∀j = 0, . . . , N -1, d j := T 0 v(t)e iω j t dt and ∀q = 1, . . . , n, d -q := T 0 t q v(t)dt.

Besides, using Cauchy-Schwarz inequality and the size estimate [START_REF] Morancey | Global exact controllability of 1D Schrödinger equations with a polarizability term[END_REF] on v (for m = 0), we get, for all j = 0, . . . , N -1, for all m = 0, . . . , k,

| d j | = T 0 v(t)e iω j t dt ≤ √ T v L 2 (0,T ) ≤ C   +∞ j=N |d j | 2   1/2 ≤ C   +∞ j=-n |(δ j,0 + ω j m ) d j | 2   1/2
, as the injection h 2m ω,r (N, C) ⊂ l 2 r (N, C) is continuous. The same estimates can be proved on ( d -q ) q=1,...,n . Together with [START_REF] Nersesyan | Global exact controllability in infinite time of Schrödinger equation: multidimensional case[END_REF], this gives [START_REF] Vahagn Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF].

Finally, from Theorem 3.9, one can deduce the solvability of moment problem with estimates on the function, some of its derivatives but also some of its primitives. Theorem 3.10. Let T > 0, k ∈ N * and (ω j ) j∈N an increasing sequence of [0, +∞) such that ω 0 = 0 and satisfying (AsymptGapPoly). There exists a constant C > 0 and a continuous linear map

M T k : h 2k ω,r (N, C) → H k 0 ((0, T ), R) such that for every d = (d j ) j∈N ∈ h 2k ω,r (N, C), the control u := M T k (d) ∈ H k 0 ((0, T ), R) satisfies the moment problem T 0 u(t)e iω j t dt = d j , ∀j ∈ N, (30)
with the boundary conditions

u 2 (T ) = u 3 (T ) = . . . = u k+1 (T ) = 0, (31)
and the following size estimates,

u H m 0 (0,T ) ≤ C d h 2m ω (N,C) , ∀m = -(k + 1), . . . , 0, . . . , k. ( 32
)
Proof of Theorem 3.10. Let (d j ) j∈N in h 2k ω,r (N, C). The operator M T k is given by,

M T k ((d j ) j∈N ) := D (k+1) • L T 2k+1 d j (-iω j ) k+1 1l j∈N * j∈N + L T k (T q d 0 ) q=1,...,k , (d 0 δ j,0 ) j∈N ,
where L T 2k+1 and L T k are defined in Theorem 3.9 and D (k+1) is the differential operator u → u (k+1) . Step 1. Solving the moment problem except for the first moment. Indeed, if we denote by

v := f (k+1) ∈ H k 0 (0, T ) with f := L T 2k+1 d j (-iω j ) k+1 1l j∈N * j∈N ∈ H 2k+1 0 (0, T ),
then, performing k + 1 integrations by parts, as f (m) has vanishing boundary terms for all m = 0, . . . , k, we get, for all j ∈ N * ,

T 0 v(t)e iω j t dt = (-iω j ) k+1 T 0 f (t)e iω j t dt = d j , (33)
by construction of f . However, for j = 1, we get

T 0 v(t)dt = f (k) (T ) = 0, (34)
and therefore, the first moment needs to be corrected in a second time. Moreover, by construction, we have the boundary conditions (31) on v and from estimates (28) on f , if we denote by u (m) = u -m when m < 0, we deduce

v H m = v (m) L 2 = f (k+1+m) L 2 ≤ C c h 2(k+1+m) ω,r ≤ C d h 2m ω,r , ∀m = -(k + 1), . . . , k. (35)
Step 2. Correcting the first component. If we denote by

w := L T k (T q d 0 ) q=1,...,k , (d 0 δ j,0 ) j∈N ∈ H k 0 (0, T ),
then by construction, together with (33)-(34), the control u := v + w in H k 0 (0, T ) solves the moment problem [START_REF] Puel | Local exact bilinear control of the Schrödinger equation[END_REF]. Moreover, by construction, w solves the polynomial moment

T 0 t q w(t)dt = T q d 0 , ∀q = 1, . . . , k
and thus, by integration by parts, as w 1 (T ) = d 0 by construction, we get the boundary conditions (31) on w. Then, by linearity, (31) holds for u. Moreover, by construction, w satisfies the size estimates

w H m (0,T ) ≤ C T |d 0 |, ∀m = 0, . . . , k. (36)
Thus, Cauchy-Schwarz inequality entails that

w m L 2 (0,T ) ≤ T m w L 2 (0,T ) ≤ T m C T |d 0 |, ∀m = 1, . . . , k + 1. (37)
Finally, estimates (35) on v and estimates (36)-(37) on w gives all the estimates (32) on u.

Nonlinear control in projection with simultaneous estimates

The goal of this section is the proof of Theorem 1.2. It relies on the controllability of the linearized system with simultaneous estimates, given in Subsection 4.1, which is then propagated to the nonlinear system through the iterations of the inverse mapping theorem thanks to estimates on the linear approximation of the end-point map given in Subsection 4.2. In this section, if ψ is in S, T S ψ the tangent space at ψ of S and Π ψ the orthogonal projection on T S ψ, are respectively given by T S ψ := {ξ ∈ L 2 (0, 1); ℜ ξ, ψ = 0} and Π ψ (ξ) := ξ -ℜ ξ, ψ ψ.

C 1 -regularity of the end-point map

Let T > 0. We consider the end-point map defined by,

Θ T : S ∩ H 2(p+k)+3 (0) × H k 0 → S ∩ H 2(p+k)+3 (0) × T S ψ 1 (T ) ∩ H J ∩ H 2(p+k)+3 (0) (ψ 0 , u) → ψ 0 , Π ψ 1 (T ) • P J [ψ(T )] ( 38
)
where ψ is the solution of

   i∂ t ψ(t, x) = -∂ 2 x ψ(t, x) -u(t)µ(x)ψ(t, x), (t, x) ∈ (0, T ) × (0, 1), ψ(t, 0) = ψ(t, 1) = 0, t ∈ (0, T ), ψ(0, x) = ψ 0 (x), x ∈ (0, 1).
Remark 4.1. To prove Theorem 1.2, we want estimates (8) on the control with respect to the final state but also to the initial data, that is why the initial data is added as an argument of the end-point map Θ T . Moreover, in the definition of Θ T , the end-point of the solution is composed with P J as we investigate exact controllability in projection. It is also composed with Π ψ 1 (T ) as only the imaginary part of the first component of the solution can be controlled (we gain back the real part of the first component as the solution lives in the L 2 (0, 1)-sphere S).

The C 1 -regularity of this end-point map is given in the following proposition. Proposition 4.2. Let µ ∈ H 2(p+k)+3 ((0, 1), R) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1. The map Θ T defined in (38) is C 1 . Moreover, for every (ψ 0 , u) in H 2(p+k)+3 (0) (0, 1) × H k 0 (0, T ), the differential at (ψ 0 , u) is given by,

dΘ T (ψ 0 , u).(Ψ 0 , v) = Ψ 0 , Π ψ 1 (T ) • P J [Ψ(T )] , (Ψ 0 , v) ∈ H 2(p+k)+3 (0) (0, 1) × H k 0 (0, T ),
where Ψ is the solution of the linearized system around the trajectory (u, ψ(•; u, ψ 0 )) given by

   i∂ t Ψ(t, x) = -∂ 2 x Ψ(t, x) -u(t)µ(x)Ψ(t, x) -v(t)µ(x)ψ(t; u, ψ 0 ), Ψ(t, 0) = Ψ(t, 1) = 0, t ∈ (0, T ), Ψ(0, x) = Ψ 0 , x ∈ (0, 1), (39)
Notice that Θ T is well-defined thanks to Remark 2.2. The proof of Proposition 4.2 is the same as [5, Proposition 3, Proposition 6] using Theorem 2.1 (and Remark 2.2) instead of their well-posedness result and thus is left to the reader.

In the following proposition, we state that one can build a right inverse of dΘ T (ϕ 1 , 0) which is continuous simultaneously in several spaces, meaning that one can control the linearized equation of (1) around the ground state with various estimates on the control. Proposition 4.3. Let µ in H 2(p+k)+3 ((0, 1), R) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1 and satisfying (4) . Then, the linear map

dΘ T (ϕ 1 , 0) : [T S ϕ 1 ∩ H 2(p+k)+3 (0) ] × H k 0 → [T S ϕ 1 ∩ H 2(p+k)+3 (0)
]

× [T S ψ 1 (T ) ∩ H J ∩ H 2(p+k)+3 (0)
] has a continuous right inverse

dΘ T (ϕ 1 , 0) -1 : [T S ϕ 1 ∩ H 2(p+k)+3 (0)
]

× [T S ψ 1 (T ) ∩ H J ∩ H 2(p+k)+3 (0)
]

→ [T S ϕ 1 ∩ H 2(p+k)+3 (0)
] × H k 0 , which satisfies that there exists a constant C T > 0 such that for all ψ 0 in T S ϕ 1 ∩ H

2(p+k)+3 (0) and ψ f in T S ψ 1 (T ) ∩ H J ∩ H 2(p+k)+3 (0)
, the control u ∈ H k 0 (0, T ) defined by (ψ 0 , u) := dΘ T (ϕ 1 , 0) -1 (ψ 0 , ψ f ) satisfies the following boundary conditions

u 2 (T ) = . . . = u k+1 (T ) = 0 (40)
and the following size estimates

u H m 0 (0,T ) ≤ C (ψ 0 , ψ f ) H 2(p+m)+3 (0) (0,1)×H 2(p+m)+3 (0) (0,1) , ∀m = -(k + 1), . . . , k. (41)
The proof follows from the solvability of a trigonometric moment problem with simultaneous estimates given in Theorem 3.10.

Proof of Proposition 4.3. By Proposition 4.2, for all u ∈ H k 0 (0, T ) and

ψ 0 ∈ H 2(p+k)+3 (0)
(0, 1),

dΘ T (ϕ 1 , 0).(ψ 0 , u) = (ψ 0 , P J [Ψ(T )]) ,
where Ψ is the solution of the linearized system (39) around (u, ψ 1 ) and can be computed as

Ψ(T ) = +∞ j=1 ψ 0 , ϕ j ψ j (T ) + i +∞ j=1 µϕ 1 , ϕ j T 0 u(t)e i(λ j -λ 1 )t dt ψ j (T ). (42) Let (ψ 0 , ψ f ) ∈ [T S ϕ 1 ∩ H 2(p+k)+3 (0) ] × [T S ψ 1 (T ) ∩ H J ∩ H 2(p+k)+3 (0)
]. The equality P J Ψ(T ) = ψ f is then equivalent to the trigonometric moment problem T 0 u(t)e i(λ j -λ 1 )t dt = ψ f , ψ j (T ) -ψ 0 , ϕ j i µϕ 1 , ϕ j := d j-1 (ψ 0 , ψ f ), ∀j ∈ J.

(43)

Applying Theorem 3.10 with (ω j := λ j+1 -λ 1 ) j∈N which satisfies (AsymptGapPoly), the proof of Proposition 4.3 follows with

dΘ T (ϕ 1 , 0) -1 (ψ 0 , ψ f ) := ψ 0 , M T k [d(ψ 0 , ψ f )] , (44)
where d(ψ 0 , ψ f ) := (d j (ψ 0 , ψ f )1 j∈J ) j∈N and M T k is defined in Theorem 3.10. Indeed, by construction, the control u := M T k [d(ψ 0 , ψ f )] satisfies the moment problem (43) after a shift in the indexes, entailing that the function defined by ( 44) is a right inverse of dΘ T (ϕ 1 , 0). Finally, Theorem 3.10 also gives that the control satisfies the boundary conditions (40) and gives the existence of a constant C T > 0 (not depending on ψ 0 nor on ψ f ) such that,

u H m 0 (0,T ) ≤ C d h 2m (N,C
) , ∀m = -(k + 1), . . . , k. Yet, as the function µ satisfies the hypothesis (4), we get for all m = -(k + 1), . . . , k,

u H m 0 (0,T ) ≤ C ( ψ f , ψ j (T ) -ψ 0 , ϕ j ) h 2(m+p)+3 (J,C) ≤ C ψ f H 2(p+m)+3 (0) (0,1) + ψ 0 H 2(p+m)+3 (0)
(0,1) .

Remark 4.4. The boundary conditions (40) allow to ease the propagation of estimates (41) to the nonlinear dynamics (see in the following Proposition 4.5 where those boundary conditions are useful to quantify the error between the nonlinear and the linearized dynamics). Notice that one can't add the boundary condition u 1 (T ) = 0 to (40) as the term u 1 (T ) drives the behavior of the first component of the linearized system (see (42)).

Error estimates between the nonlinear and linearized dynamics

The proof of Theorem 1.2 relies on the inverse mapping theorem: the control steering the solution of the Schrödinger equation ( 1) from Ψ 0 to Ψ f is constructed as the fixed point of the map

Φ : (ψ 0 , u) → (ψ 0 , u) -dΘ T (ϕ 1 , 0) -1 . [Θ T (ψ 0 , u) -(Ψ 0 , Ψ f )] .
Notice that such function can be rewritten as,

Φ (ψ 0 , u) -(ϕ 1 , 0) = -dΘ T (ϕ 1 , 0) -1 . [Θ T (ψ 0 , u) -dΘ T (ϕ 1 , 0).(ψ 0 -ϕ 1 , u) -(Ψ 0 , Ψ f )] .
Therefore, estimates on the nonlinear control map (Ψ 0 , Ψ f ) → u are closely linked to estimates on the linear approximation of the end-point map Θ T , given in the following proposition. (0, 1). For every R > 0, there exists a constant C = C(T, µ, R) > 0 such that if u H k 0 (0,T ) < R and u 2 (T ) = . . . = u k+1 (T ) = 0, then the following estimates hold

Θ T (ψ 0 , u) -Θ T (ϕ 1 , 0) -dΘ T (ϕ 1 , 0).(ψ 0 -ϕ 1 , u) H 2(p+m)+3 (0) ×H 2(p+m)+3 (0) ≤ C u H m ψ 0 -ϕ 1 H 2(p+k)+3 (0) + u H k 0 , ∀m = -(k + 1), . . . , k. ( 45
)
Remark 4.6. The assumption p ≥ k appears here to make sure that all the spaces H s (0) (0, 1) involved are positive Sobolev spaces.

To prove Proposition 4.5, we first show estimates on the solution of the Schrödinger equation in regular spaces. Proposition 4.7. Let T > 0, (p, k) ∈ N 2 , µ ∈ H 2(p+k)+3 ((0, 1), R) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1, u ∈ H k 0 ((0, T ), R),

ψ 0 ∈ H 2(p+k)+3 (0)
(0, 1). Then, if ψ := ψ(•; u, ψ 0 ),

ψ -ψ 1 ∈ C k ([0, T ], H 2p+3 (0) (0, 1)) ∩ H k+1 ((0, T ), H 2p+1 (0) (0, 1)).
Moreover, for all R > 0, there exists a constant C = C(T, µ, R) > 0 such that if u H k 0 (0,T ) < R then the following estimates hold

∂ n t (ψ -ψ 1 ) C 0 ([0,T ],H 2p+3 (0) ) ≤ C ψ 0 -ϕ 1 H 2(p+n)+3 (0)
+ u H n , ∀n = 0, . . . , k,

∂ k+1 t (ψ -ψ 1 ) L 2 ((0,T ),H 2p+1 (0) ) ≤ C ψ 0 -ϕ 1 H 2(p+k)+3 (0) + u H k . ( (46)
) 47
Proof. Let µ in H 2(p+k)+3 ((0, 1), R) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1 and

ψ 0 in H 2(p+k)+3 (0)
(0, 1). Let R > 0 and u ∈ H k 0 (0, T ) such that u H k 0 (0,T ) < R. First, as ψ is the solution of (1), ψ -ψ 1 is the solution of the following Cauchy problem

   i∂ t (ψ -ψ 1 ) = -∂ 2 x (ψ -ψ 1 ) -u(t)µ(x)ψ, (ψ -ψ 1 ) (t, 0) = (ψ -ψ 1 ) (t, 1) = 0, (ψ -ψ 1 ) (0, •) = ψ 0 -ϕ 1 . (48) By Theorem 2.1, ψ -ψ 1 , is in C k ([0, T ], H 2p+3
(0) (0, 1)) and there exists C = C(T, µ, R) > 0 such that

ψ -ψ 1 C n ([0,T ],H 2p+3 (0) (0,1)) ≤ C ψ 0 -ϕ 1 H 2(p+n)+3 (0)
(0,1) + u H n (0,T ) , ∀n = 0, . . . , k,

using that ψ is bounded in C k ([0, T ], H 2p+3 ∩ H 2p+1 (0) ).
To get (47), we differentiate (48) in a distribution sense, using Leibniz formula,

i∂ k+1 t (ψ -ψ 1 ) = A∂ k t (ψ -ψ 1 ) -µ k n=0 u (n) ∂ k-n t ψ. (49)
Yet, as ψ -ψ 1 and ψ belong to

C k ([0, T ], H 2p+3 (0) (0, 1)), u is in H k 0 (0, T ) and the space H 2p+1 (0)
is stable by multiplication by µ, the right-hand side of (49) belongs to L 2 ((0, T ), H 2p+1 (0) (0, 1)), giving that

∂ k+1 t (ψ -ψ 1 ) ∈ L 2 ((0, T ), H 2p+1 (0) (0, 1)),
with the following estimate,

∂ k+1 t (ψ -ψ 1 ) L 2 ((0,T ),H 2p+1 (0) ) ≤ ∂ k t (ψ -ψ 1 ) L 2 ((0,T ),H 2p+3 (0) ) + µ H 2p+1 u H k ψ C k ([0,T ],H 2p+1 (0) ) ≤ C ψ 0 -ϕ 1 H 2(p+k)+3 (0) + u H k , as ψ is bounded in C k ([0, T ], H 2p+3 (0) (0, 1
)) and using estimate (46) for n = k.

using Cauchy-Schwarz inequality in time and estimates ( 46)-(47) on ψ -ψ 1 . By inclusion of spaces, such bound still holds when we take the H 2(p-[k+1])+3 -norm of the left hand-side. For the term n = 0, for a.e. t ∈ (0, T ), µ (ψ -ψ

1 ) (t) is in H 2p+3 ∩H 2p+1 (0) , so A k+1 (µ (ψ -ψ 1 )) is in H 2(p-[k+1])+3 ∩ H 2(p-[k+1])+1 (0) . As p -[k + 1] ≥ -1, Proposition 2.3 gives the existence of a constant C > 0 such that T 0 u k+1 (t)e -iA(T -t) A k+1 µ (ψ -ψ 1 ) (t) dt H 2(p-[k+1])+3 (0) ≤ C u k+1 L 2 µ (ψ -ψ 1 ) C 0 ([0,T ],H 2p+3 ∩H 2p+1 (0) ) ≤ C u k+1 L 2 ψ 0 -ϕ 1 H 2p+3 (0) + u L 2 ,
using the algebra structure of H 2p+3 and estimate (46) on ψ -ψ 1 .

Proof of Theorem 1.2: STLC with simultaneous estimates on the control

Let p ≥ k and µ in H 2(p+k)+3 (0, 1) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1 and satisfying (4). By continuity with respect to the control (see Theorem 2.1), there exists R > 0 such that for all u in B R H k 0 (0, T ) and

ψ 0 ∈ S ∩ H 2(p+k)+3 (0) (0, 1) ℜ ψ(T ; u, ψ 0 ), ψ 1 (T ) > 0. Let η > 0 so that for all ψ f in S ∩ H 2(p+k)+3 (0) (0, 1) such that ψ f -ψ 1 (T ) H 2(p+k)+3 (0)
< η, we have

ℜ ψ f , ψ 1 (T ) > 0.
Finally, let R ∈ (0, R).

Step 1: Apply the inverse mapping theorem. By Proposition 4.2 and 4.3, the end-point map Θ T is C 1 on Banach spaces with its differential at (ϕ 1 , 0) that admits a continuous right inverse. Therefore, by the inverse mapping theorem (see for example [5, Subsection 2.3] for more details), there exists δ ∈ (0, η) and a C 1 -map Γ : Ω 0 × Ω T → H k 0 ((0, T ), R) (where Ω 0 and Ω T are respectively defined by ( 5) and ( 6)) such that for every (Ψ 0 , Ψ f ) ∈ Ω 0 × Ω T ,

P J ψ(T ; Γ(Ψ 0 , Ψ f ), Ψ 0 ) = Ψ f .
Step 2: Gaining the simultaneous estimates and boundary conditions on the nonlinear control. To prove that the estimates (8), true at the linear level (see Proposition 4.3), propagate to the nonlinear system, one must look inside the proof of the inverse mapping theorem. Let (Ψ 0 , Ψ f ) in Ω 0 × Ω T . At Step 1, the inverse mapping theorem gave the existence of (ψ 0 , u)

∈ Ω 0 × B R H k 0 (0, T ) such that P J ψ(T ; u, ψ 0 ) = ψ f and Ψ 0 = ψ 0 .
This antecedent is constructed as the fixed point of the following application

Φ (Ψ 0 ,Ψ f ) : Ω 0 × B R H k 0 (0, T ) → Ω 0 × B R H k 0 (0, T ) (ψ 0 , u) → (ψ 0 , u) -dΘ T (ϕ 1 , 0) -1 . Θ T (ψ 0 , u) -(Ψ 0 , Π ψ 1 (T ) Ψ f ) .
and therefore is given by

(ψ 0 , u) -(ϕ 1 , 0) = -dΘ T (ϕ 1 , 0) -1 . Θ T (ψ 0 , u) -dΘ T (ϕ 1 , 0).(ψ 0 -ϕ 1 , u) -(Ψ 0 , Π ψ 1 (T ) Ψ f ) . (51)
Step 2.1: Boundary conditions on the control. The linear control map is constructed in Proposition 4.3 so that any linear control defined as (ψ 0 , u)

:= dΘ T (ϕ 1 , 0) -1 (ψ 0 , ψ f ) satisfies the boundary Let T * ∈ (0, T ), δ > 0 such that T -2δ ≥ T * and η : R → [0, 1] in C 2k+1 (R) such that η(t) = 0 if t ∈ (0, T ) and η(t) = 1 if t ∈ [δ, T -δ].
As the system (58) is admissible by Proposition 2.3 and exactly controllable in projection in time T * by Proposition 5.3, by Theorem 5.1, there exists a constant C > 0 and a control V = V(ψ 00 ) in H 2k+1 0 ((0, T ), R) ∩ L 2 ((0, T ), dt η ) (with V a linear map) such that the solution Ψ(•; V, ψ 00 ) of (58) belongs to C 2k+1 ([0, T ], H 2p+3 (0) (0, 1)) and satisfies P J Ψ(T ; V, ψ 00 ) = 0 with the following estimates on the control,

T 0 V (t) 2 dt η(t) ≤ C T ψ 00 2 H 2p+3 (0) (0,1) , (61)
V H m 0 (0,T ) ≤ C ψ 00 H 2(p+m)+3 (0) (0,1) , ∀ m = 1, . . . 2k + 1. (62)
Then, if u := V (k+1) ∈ H k 0 (0, T ), by uniqueness, Ψ(•; u, ψ 0 -ψ 0 , ϕ 1 ϕ 1 ) = ∂ k+1 t Ψ(•; V, ψ 00 ) and thus satisfies P J Ψ(T ; u, ψ 0 -ψ 0 , ϕ 1 ϕ 1 ) = (-i A) k+1 P J Ψ(T ) = 0. By construction, u satisfies the boundary conditions (59) and the size estimates (60) on u are deduced from (61)-( 62) as for all m = 0, . . . , 2k + 1,

ψ 00 H 2(p+m)+3 (0) (0,1) = ψ 0 -ψ 0 , ϕ 1 ϕ 1 H 2(p+m-(k+1))+3 (0) (0,1) . Therefore, the proof is concluded with L := D (k+1) • V • C where D (k+1) : V → V (k+1) .
As stated in Lemma 5.4, the change of variables done in Remark 5.2 to work with a stationary equilibrium entails that we miss the first coordinate. However, we can correct it in a second time using a moment problem and thus prove Proposition 4.3.

Proof of Proposition 4.3. We prove Proposition 4.3 using the strategy of Remark 5.2, first for ψ f = 0 and then, by the time-reversibility of the system, for any target ψ f ∈ H J .

Step 1: Proof for ψ f = 0. More precisely, we prove that there exists a linear operator U such that for all ψ 0 in T S ϕ 1 ∩ H J ∩ H 2(p+k)+3 (0) (0, 1), the control U := U (ψ 0 ) ∈ H k 0 (0, T ) satisfies P J Ψ(T ; U, ψ 0 ) = 0 with the size estimates (41) (with ψ f = 0) and to prepare Step 2, solving the polynomial moment problem T 0 t m U (t)dt = 0, ∀m = 1, . . . , k.

(63)

By linearity, Ψ(T ; u + v, ψ 0 ) = Ψ(T ; u, ψ 0 , ϕ 1 ϕ 1 ) + Ψ(T ; v, ψ 0 -ψ 0 , ϕ 1 ϕ 1 ).

Yet, by Lemma 5.4, we have P J Ψ(T ; L(ψ 0 ), ψ 0 -ψ 0 , ϕ 1 ϕ 1 ) = 0 with every control L(ψ 0 ) satisfying the estimates (60) and the polynomial moment problem (63) thanks to the boundary conditions (59). Therefore, to prove Step 1, it is enough to prove the existence of u ∈ H k 0 (0, T ) such that P J Ψ(T ; u, ψ 0 , ϕ 1 ϕ 1 ) = 0 with the size estimates (41) and the polynomial moment problem (63). Besides, solving explicitly (58) with initial condition ψ 0 , ϕ 1 ϕ 1 , we get Ψ(T ; u, ψ 0 , ϕ 1 ϕ 1 ) = ψ 0 , ϕ 1 ϕ 1 + i +∞ j=1 µϕ 1 , ϕ j T 0 u(t)e -i(λ j -λ 1 )(T -t) dt ϕ j .

Thus, the equality P J Ψ(T ) = 0 is equivalent to the trigonometric moment problem T 0 u(t)e i(λ j -λ 1 )t dt = -ψ 0 , ϕ 1 δ j,1 i µϕ 1 , ϕ j , ∀j ∈ J.

(64) By Theorem 3.2, if d(ψ 0 ) := (-1) k+1 k! ψ 0 ,ϕ 1 i µϕ 1 ,ϕ 1 δ q,k q=1,...,2k , (0) j∈N * , the control w := L T 0 (d(ψ 0 )) in L 2 (0, T ) solves the following moment problem, T 0 w(t)e i(λ j -λ 1 )t dt = 0, ∀j ∈ N * , (65) T 0 t q w(t)dt = (-1) k+1 k! ψ 0 , ϕ 1 i µϕ 1 , ϕ 1 δ q,k , ∀q = 1, . . . , 2k,

with the size estimate, w L 2 (0,T ) ≤ C ψ 0 , ϕ 1 i µϕ 1 , ϕ 1 .

Let denote by u the k-th primitive of w with vanishing terms at t = 0, u(t) := meaning that u solves u (k) = w with u (m) (0) = 0 for all m = 0, . . . k -1.

Equation (65) for j = 1 and equations (66) for q = 1, . . . k -1 entail that u (m) (T ) = 0 for all m = 0, . . . k -1. Therefore, u ∈ H k 0 (0, T ) and using Poincaré inequality repeatedly, one gets the existence of C > 0 such that for all m = -(k + 1), . . . , k, u H m 0 (0,T ) ≤ C u H k 0 (0,T ) = C w L 2 (0,T ) ≤ C| ψ 0 , ϕ 1 |, giving the size estimates (41) as we deal with only one moment. Then, performing integrations by parts as u has vanishing boundary terms, one can use (66) for q = k and (65) for all j ≥ 2 to prove that u satisfies the moment problem (64). Finally, one can use (66) for q = k + 1, . . . , 2k to get (63). Therefore, Step 1 is concluded with

U := B k • L T 0 • d + L,
where B k is the operator primitiving k times, given in [START_REF] González | Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions[END_REF].

Step 2: Proof for any target.

Let (ψ 0 , ψ f ) in [T S ϕ 1 ∩ H 2(p+k)+3 (0) ] × [T S ϕ 1 ∩ H J ∩ H 2(p+k)+3 (0)
]. By Step 1, there exists U := U ψ f -P J e -i AT ψ 0 ∈ H k 0 (0, T ) (where U is constructed at Step 1) such that P J Ψ(T ; U, ψ f -P J e -i AT ψ 0 ) = 0, with the polynomial moment (63) and the size estimates (41) on U . Then, if we denote by ψ 0 := Ψ(T ; U, ψ f -P J e -i AT ψ 0 ) and V := U (T -•), by uniqueness, Ψ(t; V, ψ 0 ) = Ψ(T -t; U, ψ f -P J e -i AT ψ 0 ), and so, P J Ψ(T ; V, ψ 0 ) = ψ f -P J e -i AT ψ 0 .

And, finally, the solution associated with initial condition ψ 0 and control V is given by Ψ(t; V, ψ 0 ) = e -i At ψ 0 -ψ 0 + Ψ(t; V, ψ 0 ), and thus, as ψ 0 is in H ⊥ J by construction, P J Ψ(T ; v, ψ 0 ) = ψ f .

Besides, estimates (41) hold for U and so for V by translation. Moreover, the polynomial moments of U given in (63) entails the boundary conditions (40) on V because, for all m = 1, . . . , k, V m+1 (T ) = Remark 5.5. The key point to prove Proposition 4.3 from [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] is Remark 5.2 as the work [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] asks to work with a stationary equilibrium. Therefore, it seems like such strategy would not hold when linearizing around a more complicated trajectory than (ψ 1 , u = 0). For example, when linearizing around a linear combination of trajectories ψ j , for j ∈ N * , it would not be straightforward anymore to find a good change of variables allowing us to work equivalently with a stationary equilibrium. That is why in Subsection 4.1, we gave another proof of Proposition 4.3, relying on the solvability of a moment problem with simultaneous estimates, giving a strategy that could maybe work when linearizing around other trajectories, if needed. Notice that both strategy rely on the use of a weight function.

Lemma 3 . 5 .

 35 Define, for all m ∈ N and N ∈ N * ,

) Proof of Proposition 3 . 8 .

 38 Let k ∈ N and T > 0. Let N 1 ∈ N as in Lemma 3.7 and N ≥ N 1 . The proof follows with L N,T hf := M N where M N is defined in Lemma 3.7. Indeed, for all d ∈ h 2k ω,r (N, C), as L N •M N = Id, the function u := M N (d) ∈ H k 0 ((0, T ), R) satisfies the moment problem (25). And the estimates (26) hold by continuity of M N from h 2m ω,r (N, C) to H m 0 ((0, T ), R), for every m = 0, . . . , k. Now, from Proposition 3.4 dealing with low frequencies and Proposition 3.8 dealing with high frequencies, one can prove the main result.

Proposition 4 . 5 .

 45 Let T > 0, (p, k) ∈ N 2 with p ≥ k, µ ∈ H 2(p+k)+3 (0, 1) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1, u ∈ H k 0 (0, T ) and ψ 0 ∈ H

t k- 1 0

 1 w(t k )dt k dt k-1 . . . dt 1 , ∀t ∈ [0, T ],

  proof of Proposition 4.3 holds withdΘ T (ϕ 1 , 0) -1 (ψ 0 , ψ f ) := τ T • U ψ f e iλ 1 T -P J e -i AT ψ 0 ,where τ T : U → U (T -•) is the translation operator.

Now, we can prove Proposition 4.5.

Proof of Proposition 4.5. By Proposition 4.2, Θ T (ψ 0 , u) -Θ T (ϕ 1 , 0) -dΘ T (ϕ 1 , 0).(ψ 0 -ϕ 1 , u) = (0, (ψ -ψ 1 -Ψ)(T )) , where ψ := ψ(•; u, ψ 0 ) and Ψ is the solution of the linearized system (39) around (ϕ 1 , ψ 1 ). Then, ψ -ψ 1 -Ψ is the solution of the following Cauchy problem

Let R > 0 and a control in u ∈ H k 0 (0, T ) such that u H k 0 (0,T ) < R and u 2 (T ) = . . . = u k+1 (T ) = 0.

Step 1: m ∈ {0, . . . , k}. Estimate [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] on ψ -ψ 1 -Ψ and (46) on ψ -ψ 1 give C > 0 such that

Therefore, one gets (45) for all m ∈ {0, . . . , k} by inclusion of spaces.

Step 2: m ∈ {-(k + 1), . . . , -1}. For the sake of simplicity, we will only treat the worst case m = -(k + 1). The general case for any m ∈ {-(k + 1), . . . , -1} can be proved exactly the same. First, notice that, solving explicitly (50), the quadratic remainder is given by

To estimate (ψ -ψ 1 -Ψ)(T ) with respect to u k+1 , one can compute k + 1 integrations by parts in time to get, using Leibniz formula,

First, by definition (2) of the H -(k+1) (0, T )-norm and using (46

Besides, by Proposition 4.7 and as the space H 2p+1 (0) is invariant by multiplication by µ, for a.e.

and so (iA

. When n = 0, . . . , k, e iAs is an isometry from H

and thus the triangular inequality directly gives that

+ u H k , conditions [START_REF] Beauchard | Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations[END_REF]. Thus from (51), one deduces that the nonlinear control also satisfies [START_REF] Beauchard | Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations[END_REF].

Step 2.2: Simultaneous estimates on the control. Equation (51) together with Proposition 4.3 give the existence of a constant C = C(T ) > 0 such that, for all m = -(k + 1), . . . , k,

(0,1) .

Moreover, Proposition 4.5 gives C = C(T, µ, R) > 0 such that, for all m = -(k + 1), . . . , k,

Therefore, by the triangular inequality, as Θ T (ϕ 1 , 0) = (ϕ 1 , P J ψ 1 (T )), one gets,

Therefore, if δ > 0 and R > 0 are small enough so that, for example,

one deduces that, as Ψ 0 = ψ 0 by construction, for all m = -(k + 1), . . . , k,

(0,1) , ∀m = -(k + 1), . . . , k.

5 Linear control with simultaneous estimates with [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] The goal of this section is to explain another proof of Proposition 4.3, relying on the ideas of [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] instead of the moment result given in Theorem 3.10.

Building smooth controls for smooth data

In [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF], Ervedoza and Zuazua developed a method to construct a control map which preserves the regularity of the data to be controlled for time-reversible linear systems. However, to use such result in our case, we need to modify slightly the result of [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] as we want to deal with control in projection and not exact controllability. From [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF], we can deduce the following result.

Theorem 5.1. Let X, U two Hilbert spaces, H a closed subspace of X, P the orthogonal projection on H, (e At ) t∈R a strongly continuous group on X with generator A : D(A) ⊂ X → X and B in L(U, X -1 ) an admissible operator in the sense of [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF]Definition 1.1]. Assume that P • A = A • P and that the system

is exactly controllable in projection in some time T * in H: for all z 0 ∈ H, there exists a control u ∈ L 2 ((0, T * ), U ) such that the solution of (52) with the initial condition z 0 and control u satisfies

There exists a constant C = C(s, T, η) > 0 and a linear map V : D(A s ) ∩ H → H s 0 ((0, T ), U ) such that for every z 0 ∈ D(A s ) ∩ H, the solution of (52) with control V := V(z 0 ) and initial condition z 0 belongs to C s ([0, T ], X) and satisfies the requirement (53) with

Proof. This proposition is a consequence of the work [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] noticing that, as P • A = A • P, for every z 0 ∈ H and every control u ∈ L 2 ((0, T ), U )

where z is the solution of (52) and y is the solution of

Therefore, we apply [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] to the system (56) working on the Hilbert space H endowed with the scalar product of X, with the operators defined by D( A) = D(A)∩H, A = A which still generates a strongly continuous group and B = P • B still an admissible operator. From (55), the exact controllability in projection of (52) entails the exact controllability of (56). Therefore, [17, Proposition 1.3, Theorem 1.4 and Corollary 1.5] give the existence of a linear continuous map V : D( A s ) → H s 0 ((0, T ), U ) with the size estimates (54) such that for every z 0 ∈ D( A s ) = D(A s ) ∩ H, y(•; V(z 0 ), z 0 ) belongs to C s ([0, T ], H) with y(T ; V(z 0 ), z 0 ) = 0 and thus Pz(T ; V(z 0 ), z 0 ) = 0 by (55).

Proof of Proposition 4.3 with [17]

The goal of this subsection is to apply Theorem 5.1 to the linearized Schrödinger equation. To that end, we first state that with a change of global phase, one can work with a stationary equilibrium rather than around the ground state.

Remark 5.2. By Proposition 4.2, for all u ∈ H k 0 (0, T ) and

(0, 1),

where Ψ is the solution of the linearized system around the ground state,

To work with a stationary equilibrium, one can perform the change of function

(58) Such solutions will be denoted Ψ(•; u, ψ 0 ). To prove Proposition 4.3, it is then equivalent to prove that there exists a constant C T > 0 such that for all

], there exists u ∈ H k 0 (0, T ) (constructed as a linear function of ψ 0 and ψ f ) such that P J Ψ(T ; u, ψ 0 ) = ψ f with the boundary conditions (40) and the estimates (41).

To apply Theorem 5.1, we must check that (58) is controllable in projection in an appropriate functional setting. This is done by solving a trigonometric moment problem. Proposition 5.3. Let T > 0 and µ in H 2p+3 ((0, 1), R) satisfying (4). The linear equation (58) is exactly controllable in projection in H 2p+3 (0) (0, 1) ∩ T S ϕ 1 in time T : for all ψ 0 ∈ H 2p+3 (0) (0, 1) ∩ T S ϕ 1 , there exists a control u ∈ L 2 ((0, T ), R) such that P J Ψ(T ; u, ψ 0 ) = 0.

Proof. Let ψ 0 ∈ H 2p+3 (0) (0, 1) ∩ T S ϕ 1 . Solving explicitly (58), one gets

Therefore, the equality P J Ψ(T ) = 0 is equivalent to the following trigonometric moment problem

Yet, assumption (4) on µ together with the fact that ψ 0 belongs to H 2p+3 (0) (0, 1) entails that the sequence (d j 1l j∈J ) j∈N is in l 2 r (N, C) (as ψ 0 belongs to T S ϕ 1 ). Thus, the solvability of a trigonometric moment problem in L 2 (0, T ) given for example in Theorem 3.2 concludes the proof.

Then, we can apply Theorem 5.1 to our system. Lemma 5.4. Let T > 0 and µ in H 2(p+k)+3 ((0, 1), R) with µ (2n+1) (0) = µ (2n+1) (1) = 0 for all n = 0, . . . , p -1 and satisfying (4). There exists a linear map L : The operator A generates a strongly continuous group on X. Let ψ 0 ∈ H J ∩ H 2(p+k)+3 (0) (0, 1) and

(0, 1) the solution of the elliptic equation, (-i A) k+1 ψ 00 = ψ 0 -ψ 0 , ϕ 1 ϕ 1 with ψ 00 , ϕ 1 = 0.