Kévin Le Balc'h †
email: kevin.le-balc-h@inria.fr

Jérémy Martin
email: jeremy.a.martin@inria.fr.

Global stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus *

In this article, we prove the (uniform) global exponential stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus (R/2πZ) d , for d = 1, 2 or 3, with a linear damping localized in a subset of the torus satisfying some geometrical assumptions. In particular, this answers an open question of Dehman, Gérard and Lebeau from 2006. Our approach is based on three ingredients. First, we prove the well-posedness of the closed-loop system in Bourgain spaces. Secondly, we derive new Carleman estimates for the nonlinear equation by directly including the cubic term in the conjugated operator. Thirdly, by conjugating with energy estimates and Morawetz multipliers method, we then deduce quantitative observability estimates leading to the uniform exponential decay of the total energy of the system. As a corollary of the global stabilization result, we obtain an upper bound of the minimal time of the global null-controllability of the nonlinear equation by using a stabilization procedure and a local null-controllability result.

Introduction 1.Control of Schrödinger equations on compact manifolds

Let (M, g) be a compact smooth connected boundaryless Riemannian d-dimensional manifold, for d ∈ {1, 2, 3} and ∆ g be the Laplace Beltrami operator on M associated to the metric g. Very quickly, we will restrict ourselves to M = T d = (R/2πZ) d .

We are interested in the cubic defocusing nonlinear Schrödinger equation

i∂ t u = -∆ g u + |u| 2 u in (0, +∞) × M, u(0, •) = u 0 in M. (1.1)
This equation arises naturally in nonlinear optics, as a model of wave propagation in fiber optics.

The function u(t, x) ∈ C represents a wave and the nonlinear Schrödinger equation describes the propagation of the wave through a nonlinear medium. In this context, the metric g can be interpreted as an inhomogeneity of the optical index. Formally, in (1.1), two quantities are conserved. First, by multiplying by u the equation (1.1) and by taking the imaginary part, we observe that the L 2 -energy is conserved, i.e. Concerning the (global) well-posedness of (1.1) for u 0 ∈ H 1 (M), in the 1-dimensional case, this comes from energy estimates and Sobolev embeddings, see for instance [START_REF] Thierry | Semilinear Schrödinger equations[END_REF]Corollary 3.5.2]. However, this strategy fails in the d-dimensional case (d ≥ 2), see for instance [START_REF] Thierry | Semilinear Schrödinger equations[END_REF]Corollary 3.5.2]. In order to obtain global existence results for initial data in H s (M) for s ≥ 1, one needs to use Strichartz-type estimates. For M = T d , we have that (1.1) is globally well-posed for u 0 ∈ H s (T d ) for every s ≥ 1, up to dimension d = 3, see [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations[END_REF] and [START_REF] Bourgain | Global solutions of nonlinear Schrödinger equations[END_REF]Chapter 5]. For M = S d , (1.1) is globally well-posed for u 0 ∈ H s (T d ) for every s ≥ 1, up to dimension d = 3, see [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces[END_REF] for surfaces and [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF] for d = 3.

The goal of the paper is to analyse controllability and stabilization properties of (1.1) by mean of a force h localized in ω, a nonempty open subset of M, satisfying some geometrical assumptions, see below.

We first introduce the controlled linear Schrödinger equation i∂ t u = -∆ g u + h1 ω in (0, +∞) × M, u(0, •) = u 0 in M.

(1.4)

In (1.4), at time t ∈ (0, +∞), u(t, •) : M → C is the state and h(t, •) : ω → C is the control. Controllability for the linear Schrödinger equation has been started to be strongly investigated in the 1990's. We recall the definitions of controllability and its dual notion, called observability.

Let s ≥ 0 and T > 0.

The equation (1.4) is exactly controllable in H s (M) at time T > 0 if for every u 0 ∈ H s (M) and u 1 ∈ H s (M), there exists h ∈ L 2 (0, T ; H s (M)) such that the mild solution u of (1.4) belongs to C([0, T ]; H s (M)) and satisfies u(T, •) = u 1 .

The linear Schrödinger equation is observable in H -s (M) at time T > 0 if there exists a constant C = C(M, ω, T ) > 0 such that for every u 0 ∈ H -s (M), u 0 2

H -s (M) ≤ C T 0 e it∆ u 0 1 ω 2 H -s (M) dt.
The Hilbert Uniqueness Method (H.U.M.) relates the two previous notions, see for instance [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF]Theorem 2.42]. The controlled linear Schrödinger equation (1.4) is exactly controllable in H s (M) at time T > 0 if and only if the the linear Schrödinger equation is observable in H -s (M) at time T > 0.

In this direction, one of the most breakthrough result is from [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF] that guarantees that the socalled Geometric Control Condition (GCC) for the wave equation equation is sufficient for the exact controllability of the Schrödinger equation in any time T > 0. The proof of this result is based on microlocal analysis. The GCC can be, roughly, formulated as follows, the subdomain ω is said to satisfy the GCC in time T > 0 if and only if all rays of Geometric Optics that propagate inside the domain reach the control set ω in time less than T . A particular case of this result was proved previously by [START_REF] Machtyngier | Exact controllability for the Schrödinger equation[END_REF] by multiplier technics. One can also see [START_REF] Lasiecka | Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. II. L2(Ω)estimates[END_REF] for the obtention of such results by using Carleman estimates. In [START_REF] Phung | Observability and control of Schrödinger equations[END_REF], the author establishes the connections between the heat, wave and Schrödinger equations through suitable integral transformations called Fourier-Bros-Iagonitzer (FBI) transformations. This allows him to get, for instance, estimates on the cost of approximate controllability for the Schrödinger equation when the GCC is not satisfied and also on the dependence of the size of the control with respect to the control time. For the sphere S d , by using explicit quasimodes that concentrate on the equator, one can prove that the GCC is necessary and sufficient for the observability. On the other hand, there a number of results showing that, in some situations in which the GCC is not fulfilled in any time, one can still achieve very satisfactory results for the Schrödinger equation. For instance, any nonempty open subset of T d is sufficient for the observability then for the exact controllability of the Schrödinger equation, see [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF] or [START_REF] Komornik | Fourier series in control theory[END_REF] for a proof using Ingham's estimates or [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] for a proof using semi-classical measures. Note that it has been recently extended to any nontrivial measurable subset of T 2 in [START_REF] Burq | Rough controls for Schrödinger operators on 2-tori[END_REF] with the crucial use of dispersive properties of the Schrödinger equation. For the unit disk D with Dirichlet boundary conditions, explicit eigenfunctions concentrate near the boundary so one can prove that the observability holds if and only if ω contains a (small) part of the boundary ∂D, see [START_REF] Nalini Anantharaman | Wigner measures and observability for the Schrödinger equation on the disk[END_REF]. For a survey of these results up to 2002, one can read [START_REF] Zuazua | Remarks on the controllability of the Schrödinger equation[END_REF].

For λ ∈ R * , the controlled cubic focusing (λ < 0) or defocusing (λ > 0) Schrödinger equation writes as follows

i∂ t u = -∆u + λ|u| 2 u + h1 ω in (0, +∞) × M, u(0, •) = u 0 in M. (1.5) In (1.5), at time t ∈ (0, +∞), u(t, •) : M → C is the state and h(t, •) : ω → C is the control.
Controllability and stabilization properties have been started to be investigated at the beginning of the 2000 s. These notions are usually split into local, semiglobal and global. We just give the definitions for the stabilization in H s (M), for a feedback operator P ∈ L(H s (T d )), to illustrate the differences between them and because we will mainly focus on it after. Other precise definitions, in particular concerning controllability, can be found in references mentioned below.

Let s ≥ 0 and assume that there exists a family of Hilbert spaces (E T,s ) T >0 ⊂ C([0, T ], H s (M)) such that for all T > 0, the equation (1.5) posed on the time interval [0, T ] with h = 0 and u 0 ∈ H s (M) is well-posed on the space E T,s .

The equation (1.5) is locally exponentially stabilizable in H s (M) if there exists P ∈ L(H s (M)) such that the equation (1.5) with h = P u is well-posed on E T,s for all T > 0, and there exist δ > 0, C > 0 and γ > 0 such that for every u 0 ∈ H s (M) satisfying u 0 H s (M) ≤ δ, the solution u of (1.5) satisfies u(t, •) H s (M) ≤ Ce -γt for every t ≥ 0.

The equation (1.5) is semiglobally exponentially stabilizable in H s (M) if there exists P ∈ L(H s (M)) such that the equation (1.5) with h = P u is well-posed on E T,s for all T > 0, and for every R > 0, there exist C = C(R) > 0 and γ = γ(R) > 0 such that for every u 0 ∈ H s (M) satisfying u 0 H s (M) ≤ R, the solution u of (1.5) satisfies u(t, •) H s (M) ≤ Ce -γt for every t ≥ 0.

The equation (1.5) is globally exponentially stabilizable in H s (M) if there exists P ∈ L(H s (M)) such that the equation (1.5) with h = P u is well-posed on E T,s for all T > 0, and there exist C > 0 and γ > 0 such that for every u 0 ∈ H s (M), the solution u of (1.5) satisfies u(t, •) H s (M) ≤ Ce -γt u 0 H s (M) for every t ≥ 0.

We consider a ∈ C ∞ c (ω) such that a(x) ≥ a 0 > 0 in ω ⊂⊂ ω where ω is a nonempty open subset of M. Local exact controllability in H 1 (T) for (1.5) has been first obtained in [START_REF] Illner | A note on the exact internal control of nonlinear Schrödinger equations[END_REF]. It has then been extended in H s (T) in [START_REF] Rosier | Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval[END_REF] for every s ≥ 0 by using moments theory and Bourgain analysis for the treatment of the semilinearity seen as a small perturbation of the linear case. Note that the local stabilization with the feedback h = -ia(x)u has also been obtained in [START_REF] Rosier | Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval[END_REF]. This type of result has been generalized to any d-dimensional torus T d , d ≥ 2, in [START_REF] Rosier | Control and stabilization of the nonlinear Schrödinger equation on rectangles[END_REF]. In [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF], for M = T 2 or M = S 2 , the authors prove that (1.5) for λ > 0 is semiglobally exponentially stabilizable in H 1 (M) with h = a(x)(1 -∆) -1 a(x)∂ t u by using propagation of singularities and Strichartz-type estimates from [START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF], assuming that ω contains the union of a neighborhood of the generator circle and a neighborhood of the largest exterior circle of T 2 for M = T 2 or ω contains a neighborhood of the equator for M = S 2 . The authors in [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF] also deduce that (1.5) is semiglobally exactly controllable in H 1 (M), by using the semiglobal stabilization and a local exact controllability result. This type of result has been generalized to the situations M = T 3 and M = S 3 , with the same kind of assumptions, in [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3[END_REF]. It is worth mentioning that [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3[END_REF] uses the Bourgain analysis to handle the semilinearity and also proposes another approach for obtaining the semiglobal exact controllability. Furthermore, [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on an interval[END_REF] also obtains semiglobal controllablility and stabilizability results for (1.5), both for focusing and defocusing cases, working at L 2 (T)-regularity, with the feedback h = -ia(x)u. More recently, [START_REF] Cavalcanti | Asymptotic behavior of cubic defocusing Schrödinger equations on compact surfaces[END_REF] and [START_REF] Yang | Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold[END_REF] generalize among other things [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF] and [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3[END_REF] with the feedback laws h = -ia(x)(-∆) 1/2 u and h = -ia(x)u. For a survey of these results up to 2014, one can read [START_REF] Laurent | Internal control of the Schrödinger equation[END_REF].

Main results

The goal of this paper is to prove the global exponential stabilization of the equation (1.5) on the specific case M = T d for d ∈ {1, 2, 3}. Before stating our main results, it is worth mentioning that this work only deals with such dimensions because of the well-posedness result of Proposition 2.1, see below. More precisely, the Section 2 is devoted to well-posedness results in Bourgain spaces.

For d ∈ {1, 2, 3}, we now consider

i∂ t u = -∆u + |u| 2 u + h1 ω in (0, +∞) × T d , u(0, •) = u 0 in T d . (1.6)
The stabilization property will be established in the energy space associated to the total energy given by the sum of the L 2 -energy and the nonlinear energy,

E(t) = 1 2 T d |u(t, x)| 2 dx L 2 -energy + 1 2 T d |∇u(t, x)| 2 dx + 1 4 T d |u(t,
I ε = (0, ε) ∪ (2π -ε, 2π) + 2πZ ⊂ T, (1.9)
we have

   ω 0 := I ε ⊂ ω when d = 1, ω 0 := (I ε × T) ∪ (T × I ε ) ⊂ ω when d = 2, ω 0 := I ε × T 2 ∪ (T × I ε × T) ∪ T 2 × I ε ⊂ ω when d = 3.
(1.10) When d = 1, up to a translation, (1.10) corresponds to the fact that ω contains an open interval, therefore, we can only assume that ω is a non-open open subset of T 1 . When d = 2, (1.10) corresponds to the fact where ω contains an union of a neighborhood of the generator circle and a neighborhood of the largest exterior circle of T 2 . When d = 3, (1.10) corresponds to the fact that ω contains a neighborhood of each face of the cube, the fundamental volume of T 3 . It is worth mentioning that such a ω satisfies in particular GCC.

We consider a ∈ C ∞ c (ω) such that a(x) ≥ a 0 > 0 in ω 0 . We then look at

i∂ t u = -∆u + |u| 2 u -ia(x)u in (0, +∞) × T d , u(0, •) = u 0 in T d . (1.11)
The main result of the paper is the (uniform) global stabilization of (1.11).

Theorem 1.1. Let d ∈ {1, 2, 3}. There exist C, γ > 0 such that for all u 0 ∈ H 1 (T d ), the solution u of (1.11) belongs to C([0, +∞); H 1 (T d )) and satisfies

E(t) ≤ Ce -γt E(0) ∀t ≥ 0.
(1.12)

From Theorem 1.1, one can obtain an estimate of the minimal time of the null-controllability for (1.6).

From [Lau10a, Theorem 0.2] in 1-d, [DGL06, Theorem 2] in 2-d and [Lau10b, Theorem 0.1] in 3-d, we have that for every u 0 ∈ H 1 (T d ), there exists a time T > 0 and a control h ∈ L 2 (0, T ; H 1 (T d )) such that the solution u of (1.6) belongs to C([0, T ]; H 1 (T d )) and satisfies u(T, •) = 0. So, one can define, for u 0 ∈ H 1 (T d ), the associated minimal time of controllability, i.e.

T (u 0 ) = inf{T > 0 ; (1.6) is null-controllable at time T > 0 from u 0 }.

(1.13) Then, we define the following time of controllability

τ (R) = sup{T (u 0 ) > 0 ; E(u 0 ) ≤ R} ∀R ≥ 0. (1.14)
The second main result of the paper is an estimate from above of τ .

Theorem 1.2. Let d ∈ {1, 2, 3}. There exists C > 0 sufficiently large such that

τ (R) ≤ C log(R + 1) ∀R ≥ 0.
(1.15)

Comments. Theorems 1.1 and 1.2 differ from the existing literature. Indeed, Theorem 1.1 states the (uniform) global exponential stabilization of (1.6) by the feedback h = -ia(x)u in the energy space while Theorem 1.2 states the global null-controllability of (1.6) with an explicit estimate of the (possible) minimal time in function of the size of the initial data in the energy space. Up to our knowledge, they are the first results in this direction for nonlinear Schrödinger equations. The key point is hidden in the exponential decay (1.12) where the constants C > 0, γ > 0 do not depend on the initial data, but only on the geometry of the torus T d and the observation set ω. In particular, this estimate answers by the affirmative the open problem stated in [DGL06, Remark 1].

For proving Theorem 1.1, we develop a new method in comparison to the above mentioned references. Our strategy is based on new quantitative observability inequalities for the cubic defocusing Schrödinger equation with the internal damping (1.11). The first ingredient for obtaining such inequalities is a new Carleman estimate for (1.11). Instead of seeing |u| 2 u as V (t, x)u with V a time/spacedependent potential then performing a Carleman estimate in a linear Schrödinger-type equation, we directly include the cubic semilinearity in the symmetric part of the Carleman conjugated operator. The internal linear damping -ia(x)u is then treated as a (local) source term that can be absorbed in the Carleman estimate. First, note that such a strategy only enables us to treat defocusing cases, that is strongly in contrast with [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on an interval[END_REF] where the author can manage to deal with focusing cases. Second, one cannot handle internal linear dampings like a(x)(1-∆) -1 a(x)∂ t u or -ia(x)(-∆) 1/2 u that are nonlocal. The second ingredient is a combination of energy estimates and Morawetz multipliers method for obtaining the exponential decay of the energy of the solution to the damped equation. As a consequence, we completely bypass the classical use of propagation of compactness-regularity for tackling such a question. We strongly believe that such a method can have other applications to the problem of exponential stabilization of partial differential equations, for instance for semilinear wave equations as considered in [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF].

The proof of Theorem 1.2 is a corollary of the global exponential stabilization from Theorem 1.1 and local controllability results given by [Lau10a, Theorem 3

.2] in 1-d, [DGL06, Proof of Theorem 2] in 2-d and [Lau10b, Theorem 0.3] in 3-d.
Extensions. Our main results, i.e. Theorems 1.1 and 1.2, can be extended into two directions, that are the geometry of (M, ω) and the semilinearity. For the first point, the key tool is the adaptation of the Carleman estimate in such a setting. For the second point, the key ingredient is the well-posedness of the Cauchy problem (1.1) in C([0, +∞); H s (M)) for every s ≥ 1. We mention below the following situations that can be treated with our method:

• d = 1, M = T, ω a nonempty open subset of T, replacing |u| 2 u by |u| p-1 u for every p > 1, see [Bou99, Theorem 2.3] for the well-posedness, • d = 2, M = T 2 , ω as in (1.10), replacing |u| 2 u by |u| p-1 u for every p > 1, see [Bou99, Chapter V]
for the well-posedness, However, we decide for simplicity to focus on the toy model of the cubic defocusing Schrödinger equation on the d-dimensional torus.

• d = 2, M = S 2 , ω a nonempty open subset of S 2 containing a neighborhood of {x 3 = 0} ⊂ R 3 , replacing |u| 2 u by |u| p-1 u for every p > 1, see [DGL06,
Open questions. We finish this part by mentioning some open problems related to Theorems 1.1 and 1.2.

From (1.12) and the Sobolev embedding H 1 (T d ) → L 4 (T d ), we can deduce that there exist C > 0 and γ > 0 such that for every u 0 ∈ H 1 (T d ), the solution u of (1.11) satisfies

u(t, •) 2 H 1 (T d ) ≤ Ce -γt u 0 2 H 1 (T d ) + u 0 4 H 1 (T d ) ,
(1.16) But, we do not know if (1.16) can be replaced with u(t, •)

2 H 1 (T d ) ≤ Ce -γt u 0 2 H 1 (T d ) .
Last but not least, from (1.16), we obtain in particular the exponential decay of the L 2 -energy of the solution, but for H 1 (T d )-initial data. In the 1-d case, where (1.11) is well-posed for initial data in L 2 (T), we may wonder in the spirit of [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on an interval[END_REF]

if u(t, •)
On the other hand, as said previously, one can extend our main results to the subcritical semilinearities, i.e. |u| p-1 u for p ∈ (1, 5) in 3-d by using the corresponding well-posedness results. Concerning the critical case, it is known from [IP12] that (1.1) replacing the cubic semilinearity |u| 2 u by the quintic semilinearity |u| 4 u is globally well-posed in H 1 (T 3 ). An interesting open question to adapt our strategy is to prove that the closed-loop equation (1.11) replacing |u| 2 u by |u| 4 u is globally well-posed in H s (T 3 ) for every s ≥ 1. It seems that it is probably the case but details remain to be written.

Finally, concerning the null-controllability adressed in Theorem 1.2, one may ask the question of the uniform large-time controllability, respectively small-time null-controllability, that is there exists a time T > 0, respectively for every time T > 0, for every initial data u 0 ∈ H 1 (T d ), (1.6) is null-controllable at time T > 0 from u 0 .

Organization of the paper

The article is organized as follows. In Section 2, we state the well-posedness of (1.11) for initial data in H s (T d ) and source terms in L 2 (0, T ; H s (T d )) for s ≥ 1 with the use of Bourgain spaces and we also present energy estimates. In Section 3, we prove new Carleman estimates then quantitative observability estimates for (1.11) by the use of energy estimates and Morawetz multipliers. In Section 4, we prove the main results of the paper i.e. Theorems 1.1 and 1.2; in Subsection 4.1, we deduce the exponential decay of the total energy of the solution to (1.11) then in Subsection 4.2, we obtain the upper bound on the minimal time of the global null-controllability of (1.6). Finally, the Appendix is devoted to the proof of the results of Section 2.

Well-posedness results

Let d ∈ {1, 2, 3}. This section is devoted to present the well-posedness in Bourgain spaces X s,b

T for Cauchy problems associated to

i∂ t u = -∆u + |u| 2 u -ia(x)u + g in (0, T ) × T d , u(0, •) = u 0 in T d . (2.1)
where

T > 0, s ≥ 1, b ∈ (1/2, 1) (depending on d and s), a ∈ C ∞ (T d ; R), u 0 ∈ H s and g ∈ L 2 (0, T ; H s (T d )).
In the first part, we introduce the so-called Bourgain spaces X s,b T and the well-posedness result. In the second part, we present energy identities for solutions of (2.1). This section contains only the statements and the proofs are given in the Appendix.

Bourgain spaces and well-posedness result

This subsection aims at presenting the Bourgain spaces and the well-posedness results for (2.1). The proofs are given in the Appendix. This idea of defining Bourgain spaces for the well-posedness of cubic nonlinear Schrödinger equation was first introduced in [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations[END_REF], see for instance [Bou99, Chapter 5] for a detailed account of these techniques.

In all the following, we use the notation

x = 1 + x 2 ∀x ∈ R.
For s ∈ R, we equip the Sobolev space H s (T d ) with the norm

u 2 H s (T d ) = k∈Z d |k| 2s |û(k)| 2 ∀u ∈ H s (T d ). (2.2)
For s, b ∈ R, we define the Bourgain space by

X s,b := {u ∈ L 2 (R × T d ), u X s,b < +∞} (2.3) with u 2 X s,b = k∈Z d R |k| 2s τ + |k| 2 2b | û(τ, k)| 2 dτ = u 2 H b (R;H s (T d )) , (2.4)
where u (t) = e -it∆ u(t) and û(τ, k) denotes the Fourier transform with respect to the time variable and the spatial variable.

For T > 0, the restricted Bourgain space X s,b T is the associated restriction space with the norm

u X s,b T = inf{ ũ X s,b ; ũ = u in (0, T ) × T d }.
(2.5) More generally, for I an interval in R, one can define X s,b I the associated restriction space. One can readily show that, for b > 1/2, the space

X s,b T is continously embedded in C([0, T ]; H s (T d )).
One of the main interests of the Bourgain spaces comes from the fact that these spaces are suitable to study the well-posedness of (2.1). The following proposition ensures that (2.1) is well-posed in some Bourgain spaces as soon as the initial data belongs to some Sobolev spaces.

Proposition 2.1. Let T > 0, s ≥ 1 and a ∈ C ∞ (T d ; R). Then there exists b ∈ (1/2, 1) such that for every u 0 ∈ H s , g ∈ L 2 (0, T ; H s (T d )), there exists a unique solution u ∈ X s,b T to (2.1). Moreover, the flow map

F : H s (T d ) × L 2 (0, T ; H s (T d )) → X s,b T (u 0 , g) → u, (2.6)
is Lipschitz on every bounded subset.

This proposition is instrumental in this work. It provides in particular that the solution of (2.1) belongs to C([0, T ], H s (T d )) as soon as the initial data belongs to H s (T d ) and the source term belongs to L 2 (0, T ; H s (T d )) and enables us to work with the energy defined in (1.7). Let us insist on the fact that this well-posedness result is only proved for d ∈ {1, 2, 3}. This limitation comes from the trilinear estimations given by Proposition A.3, which are only known for these dimensions, up to our knowledge. Consequently, our stabilisation and control results Theorem 1.1 and 1.2 are limited to the same dimensions.

Energy estimates

The purpose of this section is to present energy identities and energy estimates which play a key role to prove the stability of the equation (1.11). The first proposition states energy identities and multipliers identities for solutions of (2.1) in

X s,b T , with s ≥ 2. Proposition 2.2. Let T > 0, s ≥ 2, a ∈ C ∞ (T d ; R), u 0 ∈ H s and g ∈ L 2 (0, T ; H s (T d )). Assume that u ∈ X s,b
T is a solution of (2.1) for some b ∈ (1/2, 1), then

1 2 T d |u(t , x)| 2 dx - 1 2 T d |u(t, x)| 2 = - t t T d a(x)|u(s, x)| 2 dxds + t t T d (g(s, x)u(s, x))dxds ∀0 ≤ t ≤ t ≤ T, (2.7) 1 2 T d |∇u(t , x)| 2 dx + 1 4 T d |u(t , x)| 4 dx - T d |∇u(t, x)| 2 dx - T d 1 4 |u(t, x)| 4 dx = - t t T d a(x) (u(s, x)∂ t u(s, x))dxds - t t T d (g(s, x)∂ t u(s, x)) ∀0 ≤ t ≤ t ≤ T, (2.8
)

and for P ∈ C ∞ (T d ; R), t t T d ( (u∂ t u) -|∇u| 2 -|u| 4 )P (x)dxds = 1 2 t t T d (∇P (x) • ∇)(|u| 2 )dxds + t t T d (g(s, x)P u(s, x))dxds ∀0 ≤ t ≤ t ≤ T. (2.9)
Note that the equation (2.9) is inspired from Morawetz multipliers strategy.

The next result establishes energy estimates. Let us recall that the energy considered in this work is E defined by (1.7).

Proposition 2.3. Let T > 0, s ≥ 1, a ∈ C ∞ (T d ; R), u 0 ∈ H s and g ∈ L 2 (0, T ; H s (T d )). There exists a positive constant C = C T,d,a > 0 such that if u ∈ X s,b
T is a solution of (2.1) for some b ∈ (1/2, 1), then we have

E(t) ≤ C E(0) + g 2 L 2 (0,T ;H 1 (T d )) + g 4 L 2 (0,T ;H 1 (T d )) ∀t ∈ [0, T ].
(2.10) Althought Proposition 2.2 is stated for sufficiently smooth solutions, it is worth mentionning that Proposition 2.3 allows to consider solutions in X s,b T , with s ≥ 1.

Carleman estimate on the nonlinear equation

The goal of this part is to obtain a Carleman estimate for the nonlinear Schrödinger equation (2.1) and to deduce from it an observability inequality. In order to do this, we closely follow the approach of [START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF] for establishing Carleman estimates for linear Schrödinger equation. We want to highlight the fact that the main difference is the presence of the cubic defocusing nonlinearity -|u| 2 u, that we include in our operator. Note that such a strategy has been proposed for instance in the context of dissipative nonlinear parabolic equations in [START_REF] Boulakia | Some inverse stability results for the bistable reaction-diffusion equation using Carleman inequalities[END_REF] but up to our knowledge, this strategy seems to be new in the context of nonlinear Schrödinger equation.

Definition of Carleman weights and main properties

Recall the definition of ω 0 in (1.10) and let us define ω 1 ⊂⊂ ω 0 ⊂⊂ ω such that by denoting

I ε0 = (0, ε 0 ) ∪ (2π -ε 0 , 2π) + 2πZ ⊂ T ε 0 ∈ (0, ε), (3.1)
we have  



ω 1 := I ε0 ⊂ ω 0 when d = 1, ω 1 := (I ε0 × T) ∪ (T × I ε0 ) ⊂ ω 0 when d = 2, ω 1 := I ε0 × T 2 ∪ (T × I ε0 × T) ∪ T 2 × I ε0 ⊂ ω 0 when d = 3. (3.2)
First, we have the following easy lemma.

Lemma 3.1. There exists η ∈ C ∞ (T d ; R + ) such that for some c > 0 |∇η(x)| ≥ c > 0 ∀x ∈ T d \ ω 0 , (3.3) D 2 η(x)(ξ, ξ) + |∇η(x) • ξ| 2 ≥ c|ξ| 2 ∀(x, ξ) ∈ (T d \ ω 0 ) × R d . (3.4) Proof. First, let us define χ ∈ C ∞ c (T d ) such that χ = 1 on T d \ ω 0 and χ = 0 in ω 1 ⊂⊂ ω 0 . The function, defined by η(x) = χ(x)|x| 2 ∀x ∈ (0, 2π) d , (3.5)
can be extended to a smooth function in T d satisfying the two expected properties (3.3), (3.4).

Let us define the Carleman weights for λ ≥ 1 a parameter,

α(t, x) = e 2λm η ∞ -e λ(η(x)+m η ∞ ) t(T -t) , β(t, x) = e λ(η(x)+m η ∞ ) t(T -t) ∀(t, x) ∈ (0, T ) × T d , (3.6)
where m > 1 is a fixed number.

The Carleman estimate

The main result of this part is the following Carleman estimate.

Proposition 3.2. There exist positive constants

C = C(Ω, ω) > 0, C 1 = C 1 (Ω, ω) > 0 and b ∈ (1/2, 1) such that for all T > 0, λ ≥ C 1 , s ≥ C 1 (T + T 2 + T 2 |a| ∞ ), u 0 ∈ H 1 (T d ) and g ∈ L 2 (0, T ; H 1 (T d )), the solution u ∈ X 1,b T of (2.1) satisfies s 3 λ 4 T 0 T d e -2sα β 3 |u| 2 dxdt + sλ T 0 T d e -2sα β|∇u| 2 dxdt + s 2 λ 2 T 0 T d e -2sα β 2 |u| 4 dxdt ≤ C T 0 T d e -2sα |g| 2 dxdt + s 3 λ 4 T 0 ω0 e -2sα β 3 |u| 2 dxdt + sλ T 0 ω0 e -2sα β|∇u| 2 dxdt + s 2 λ 2 T 0 ω0 e -2sα β 2 |u| 4 dxdt . (3.7)
Proof. By using a standard regularization argument using Proposition 2.1, we just need to consider the case where u ∈ X 2,b T so in particular (2.1) is satisfied in the strong sense. Denote ψ = e -sα u, Γ = e -sα g.

(3.8)

Let us recall that we have

i∂ t u + ∆u = |u| 2 u -ia(x)u + g = e 2sα
|ψ| 2 e sα ψ -ia(x)e sα ψ + e sα g.

We then have

P ψ := i∂ t ψ +isα t ψ +∆ψ +2s∇α•∇ψ +s(∆α)ψ +s 2 |∇α| 2 ψ -e 2sα |ψ| 2 ψ = -ia(x)ψ +Γ =: Γ ψ,g . (3.9)
We decompose P = P 1 + P 2 with

P 1 ψ = isα t ψ + 2s∇α • ∇ψ + s(∆α)ψ, (3.10)
P 2 ψ = i∂ t ψ + ∆ψ + s 2 |∇α| 2 ψ -e 2sα |ψ| 2 ψ. (3.11)
For the rest of the proof, we denote Q T = (0, T ) × T d and q T = (0, T ) × ω 0 . We have

P 1 ψ + P 2 ψ 2 L 2 (Q T ) = P 1 ψ 2 L 2 (Q T ) + P 2 ψ 2 L 2 (Q T ) + 2 P 1 ψ, P 2 ψ L 2 (Q T ) = Γ ψ,g 2 L 2 (Q T ) , (3.12) therefore 2 P 1 ψ, P 2 ψ L 2 (Q T ) ≤ Γ ψ,g 2 L 2 (Q T ) . (3.13)
We then decompose 2

P 1 ψ, P 2 ψ L 2 (Q T ) = I 1 + I 2 + I 3 , (3.14)
with

I 1 = 2 Q T (2s∇α • ∇ψ + s(∆α)ψ)(-i∂ t ψ + ∆ψ + s 2 |∇α| 2 ψ -e 2sα |ψ| 2 ψ) , (3.15)
I 2 = 2 Q T i(∂ t α)ψ(-i∂ t ψ + ∆ψ) , (3.16
)

I 3 = 2 Q T i(∂ t α)ψ(s 2 |∇α| 2 ψ -e -2sα |ψ| 2 ψ)) = 0. (3.17)
We first deal with I 1 , decomposing as follows

I 1 = 2 Q T (2s∇α • ∇ψ + s(∆α)ψ)(∆ψ + s 2 |∇α| 2 ψ -e -2sα |ψ| 2 ψ) (3.18) -2 Q T i(2s∇α • ∇ψ + s(∆α)ψ)∂ t ψ (3.19) = I 1 1 + I 2 1 . (3.20)
By integration by parts, we have

J := Q T (∇α • ∇ψ)∆ψ = - Q T ∇ψ • ∇(∇α • ∇ψ)). (3.21) Moreover we have ∇ψ • ∇(∇α • ∇ψ)) = D 2 (α)(∇ψ, ∇ψ) + D 2 (ψ)(∇ψ, ∇α), (3.22) 2 D 2 (ψ)(∇α, ∇ψ) = ∇α • ∇|∇ψ| 2 . (3.23)
Therefore, from (3.21), (3.22), (3.23) and an integration by parts, we have

2 J = -2 Q T D 2 (α)(∇ψ, ∇ψ) -2 Q T D 2 (ψ)(∇α, ∇ψ) = -2 Q T D 2 (α)(∇ψ, ∇ψ) + Q T ∆α |∇ψ| 2 . (3.24)
We can now expand I 1 1 as follows, using ∇|ψ| 2 = 2 (ψ∇ψ) and ∇|ψ| 4 = 4 (|ψ| 2 ψ∇ψ),

I 1 1 = 2 2sJ + Q T s(∆α)ψ∆ψ + Q T 2s 3 (∇α • ∇ψ)|∇α| 2 ψ + Q T s 3 (∆α)|ψ| 2 |∇α| 2 - Q T 2s(∇α • ∇ψ)e 2sα |ψ| 2 ψ -s Q T (∆α)e 2sα |ψ| 4 = 4s J -2s Q T ((∇∆α)ψ + ∆α∇ψ) • ∇ψ -2 Q T s 3 ∇ • (|∇α| 2 ∇α)|ψ| 2 + 2 Q T s 3 (∆α)|ψ| 2 |∇α| 2 + Q T s(∆α)e 2sα |ψ| 4 + 2 Q T s 2 |∇α| 2 e 2sα |ψ| 4 -2 Q T s(∆α)e 2sα |ψ| 4 = -4s Q T D 2 (α)(∇ψ, ∇ψ) + s Q T (∆ 2 α)|ψ| 2 -2s 3 Q T ∇α • ∇(|∇α| 2 )|ψ| 2 + 2s 2 Q T |∇α| 2 e 2sα |ψ| 4 -s Q T (∆α)e 2sα |ψ| 4 . (3.25)
We now compute I 2 1 , using 2 z = z + z, we get by integration by parts

-I 2 1 = Q T i(2s∇α • ∇ψ + s(∆α)ψ)ψ t -i Q T (2s∇α • ∇ψ + s(∆α)ψ)ψ t = Q T -i [2s∇α t • ∇ψ + 2s∇α • ∇ψ t + s(∆α t )ψ + s(∆α)ψ t ] ψ -i Q T 2s(∇α • ∇ψ)ψ t -i Q T s(∆α)ψψ t .
The second term in the right hand side of the previous computation becomes

-i Q T 2s(∇α • ∇ψ)ψ t = 2is Q T (∆α)ψψ t + 2is Q T (∇α • ∇ψ t )ψ. (3.26)
As a consequence, we get

-I 2 1 = Q T -i2s(∇α t • ∇ψ)ψ -is Q T (∆α t )|ψ| 2 = Q T -i2s(∇α t • ∇ψ)ψ + is Q T ∇α t • ∇|ψ| 2 = i Q T s∇α t • (ψ∇ψ -ψ∇ψ)) = 2s i Q T ∇α t • (ψ∇ψ)) . (3.27)
Finally, we obtain from (3.25) and (3.27)

I 1 = -4s Q T D 2 (α)(∇ψ, ∇ψ) + s Q T (∆ 2 α)|ψ| 2 -2s 3 Q T ∇α • ∇(|∇α| 2 )|ψ| 2 + 2s 2 Q T |∇α| 2 e 2sα |ψ| 4 -s Q T (∆α)e 2sα |ψ| 4 -2s i Q T ∇α t • (ψ∇ψ)) (3.28)
We now turn to the other term I 2 , we have

I 2 = 2 Q T isα t ψ(-iψ t + ∆ψ) = s Q T α t ∂ t |ψ| 2 + 2s i Q T α t ψ∆ψ = -s Q T α tt |ψ| 2 -2s i Q T (∇α t ψ + α t ∇ψ) • ∇ψ = -s Q T α tt |ψ| 2 -2s Q T i(∇α t • ∇ψ)ψ. (3.29)
Consequently, we get from (3.14), (3.28), (3.29) and using ∇α • ∇|∇α| 2 = 2D 2 (α)(∇α, ∇α) that

2 (P 1 ψ, P 2 ψ) = Q T -4s 3 D 2 (α)(∇α, ∇α) -sα tt + s(∆ 2 α) |ψ| 2 -4s Q T D 2 (α)(∇ψ, ∇ψ) + Q T [s 2 |∇α| 2 e 2sα -s(∆α)e 2sα ]|ψ| 4 -4s Q T iψ∇α t • ∇ψ. (3.30)
The following identities and estimates will be useful in the reminder of the proof ∇α = -λβ∇η, (3.31)

D 2 (α)(X, Y ) = -βλ D 2 (η)(X, Y ) + λ(∇η • X)(∇η • Y ) ∀X, Y ∈ R d . (3.32) |∇α t | ≤ CT λβ 2 , |α tt | ≤ CT 2 β 3 , (3.33) |∆α| ≤ Cλ 2 β, |∆ 2 α| ≤ Cλ 4 β. (3.34)
Then, we have from the properties of the weight (3.3), (3.4) and (3.31), (3.32),

-4s 3 D 2 (α)(∇α, ∇α) = 4s 3 λβ D 2 (η)(∇α, ∇α) + λ |∇η • ∇α| 2 ≥ cs 3 λ 4 β 3 in T d \ ω 0 , (3.35) -4sD 2 (α)(X, X) = sλβ D 2 (η)(X, X) + λ|∇η • X| 2 ≥ csλβ|X| 2 in T d \ ω 0 , ∀X ∈ R d , (3.36) s 2 |∇α| 2 ≥ cs 2 λ 2 β 2 in T d \ ω 0 . (3.37)
We then have from (3.30), (3.9), (3.13) and (3.35), (3.36), (3.37),

s 3 λ 4 Q T β 3 |ψ| 2 + sλ Q T β|∇ψ| 2 + s 2 λ 2 Q T β 2 e 2sα |ψ| 4 ≤ C Q T |Γ| 2 + Q T |a| 2 |ψ| 2 + Q T -sα tt + s(∆ 2 α) |ψ| 2 + Q T s(∆α)e 2sα |ψ| 4 + 4s Q T iψ∇α t • ∇ψ + s 3 λ 4 q T β 3 |ψ| 2 + sλ q T β|∇ψ| 2 + s 2 λ 2 q T β 2 e 2sα |ψ| 4 . (3.38)
Now, we will absorb some right hand side terms in (3.38). Take ε > 0 a small positive number and C ε > 0 a positive constant depending only of ε that can vary from one line to another, we have from (3.33), (3.34) that

Q T |a| 2 |ψ| 2 ≤ |a| 2 ∞ Q T |ψ| 2 ≤ εs 3 λ 4 Q T β 3 |ψ| 2 for s ≥ C ε T 2 |a| 2/3 ∞ , (3.39) s Q T |α tt ||ψ| 2 ≤ CsT 2 Q T β 3 |ψ| 2 ≤ εs 3 λ 4 Q T β 3 |ψ| 2 for s 2 ≥ C ε T 2 i.e. s ≥ C ε T, (3.40) s Q T |∆ 2 α||ψ| 2 ≤ Csλ 4 Q T β|ψ| 2 ≤ εs 3 λ 4 Q T β 3 |ψ| 2 for s 2 ≥ C ε β -2 i.e. s ≥ C ε T 2 , (3.41) s Q T |∆α|e 2sα |ψ| 4 ≤ Csλ 2 Q T βe 2sα |ψ| 4 ≤ εs 2 λ 2 Q T β2e 2sα |ψ| 4 for s ≥ C ε β -1 i.e. s ≥ C ε T 2 , (3.42) 4s Q T iψ∇α t • ∇ψ ≤ CT sλ Q T β 2 |∇ψ||ψ| ≤ εsλ Q T β|∇ψ| 2 + C ε sλT 2 Q T β 3 |ψ| 2 ≤ εsλ Q T β|∇ψ| 2 + εs 3 λ 4 Q T β 3 |ψ| 2 for s ≥ C ε T. (3.43)
We finally get from (3.39), (3.40), (3.41), (3.42), (3.43) and (3.38) that

s 3 λ 4 Q T β 3 |ψ| 2 + sλ Q T β|∇ψ| 2 + s 2 λ 2 Q T β 2 e 2sα |ψ| 4 ≤ C Q T |Γ| 2 + s 3 λ 4 q T β 3 |ψ| 2 + sλ q T β|∇ψ| 2 + s 2 λ 2 q T
β 2 e 2sα |ψ| 4 .

(3.44)

Now we reuse the expression of ψ in function of u and Γ in function of g given in (3.8) to get the desired Carleman estimate (3.7).

From the Carleman estimate to the observability inequality

The goal of this part is to obtain an observability inequality for (2.1), starting from the Carleman estimate previously obtained in Proposition 3.2 and energy and multipliers estimates stated in Proposition 2.2. Proposition 3.3. There exist a positive constant C = C(Ω, ω, |a| ∞ ) > 0 and b ∈ (1/2, 1) such that for every T > 0 and

u 0 ∈ H 1 (T d ), the solution u ∈ X 1,b T of (2.1) with g = 0 satisfies ∀t ∈ [0, T ], E(t) ≤ exp C 1 + 1 T T 0 T d (|u| 2 + |∇u| 2 + |u| 4 )a(x)dxds. (3.45)
Proof. From the properties of the weights (3.6) and from the choices of λ, s in Proposition 3.2, we deduce that

e -2sα (β + β 2 + β 3 ) ≥ exp -C 1 + 1 T in (T /4, 3T /4) × T d , e -2sα (1 + β 3 + β + β 2 ) ≤ C 1 + 1 T 6 in (0, T ) × T d .
We then obtain from the Carleman estimate (3.7) and the property of a in ω 0 that

3T /4 T /4 E(t)dt ≤ exp C 1 + 1 T T 0 T d (|u| 2 + |∇u| 2 + |u| 4 )a(x)dxdt. (3.46)
As a first step, let us show that ∀t ∈ [0, T ],

T d |u(t, x)| 2 dx ≤ exp C 1 + 1 T T 0 T d (|u| 2 + |∇u| 2 + |u| 4 )a(x)dxds. (3.47)
Indeed, thanks to (2.7), we have for all t, t ∈ [0, T ],

T d |u(t, x)| 2 dx ≤ T 0 T d a(x)|u(s, x)| 2 dxds + T d |u(t , x)| 2 dx.
By integrating on {T /4 ≤ t ≤ 3T /4}, we deduce from the above estimate together with (3.46) that

T d |u(t, x)| 2 dx ≤ 1 + 2 T exp C 1 + 1 T T 0 T d (|u| 2 + |∇u| 2 + |u| 4 )a(x)dxds,
which proves (3.47) for a suitable constant C > 0. By now, let us deal with the whole energy E(t). Notice that from the identities (2.7) and (2.8), we have for all 0 ≤ t ≤ t ≤ T ,

E(t ) -E(t) = - t t T d a(x)|u(x, s)| 2 dxds - t t T d a(x)Im(u∂ t u)dsdx.
Moreover, by using (2.9) with P = a, this leads to

E(t ) -E(t) = - t t T d a(x)|u(x, s)| 2 dxds - 1 2 t t T d ∇a(x) • ∇(|u| 2 )dxds - t t T d a(x)(|∇u| 2 + |u| 4 )dxds = - t t T d a(x)|u(x, s)| 2 dxds + 1 2 t t T d ∆a(x)|u| 2 dxds - t t T d a(x)(|∇u| 2 + |u| 4 )dxds.
We then deduce that for all t, t ∈ [0, T ],

E(t) ≤ T 0 T d a(x)(|u| 2 + |∇u| 2 + |u| 4 )dxds + ∆a L ∞ 2 T 0 T d |u(s, x)| 2 dxds + E(t ). (3.48)
After integrating on {T /4 ≤ t ≤ 3T /4}, the conclusion of Proposition 3.3 follows from (3.46) and (3.47).

From Proposition 3.3, we finally obtain the following useful result.

Corollary 3.4. There exist a positive constant C = C(Ω, ω, |a| ∞ ) > 0 and b ∈ (1/2, 1) such that for every u 0 ∈ H 1 (T d ), the solution u ∈ X 1,b T of (2.1) with g = 0 satisfies

E(0) + T 0 E(t)dt ≤ exp C 1 + 1 T T 0 T d a(x)(|u(t, x)| 2 + |∇u(t, x)| 2 + |u(t, x)| 4 )dxdt. (3.49)
4 Proof of the main results

Exponential decay of the solution to the nonlinear equation

The goal of this part is to prove Theorem 1.1. We first state a technical lemma that would be useful in the sequel, it comes from [YNC21, Lemma 4.4].

Lemma 4.1. Let a ∈ C 1 (T d ) be a non-negative real function. For all ε > 0, there exists a positive constant

C ε > 0 such that ∀x ∈ T d , |∇a(x)| 2 ≤ C ε a(x) + ε.
Proof. Let us proceed by contradiction and assume that there exist ε 0 > 0 and a sequence (

x n ) n∈N ⊂ T d such that for all n ∈ N, |∇a(x n )| 2 ≥ na(x n ) + ε 0 . (4.1)
Up to a subsequence, we can assume that (x n ) n∈N tends to some x ∞ ∈ T d . Since a is non-negative and ε 0 > 0, (4.1) implies that a(x ∞ ) = 0. In particular, x ∞ minimizes a and we obtain ∇a(x ∞ ) = 0. The contradiction then follows from the fact that 0

< ε 0 ≤ |∇a(x ∞ )| 2 = 0.
Proof of Theorem 1.1. Once again, we only deal with the case where u 0 ∈ H 2 (T d ). The general case follows from a standard regularization argument and Proposition 2.1. We first express the right hand side of the observability estimate (3.49) thanks to the total energy of the system. We proceed as follows. From (2.7), we have

T 0 T d a(x)|u(s, x)| 2 dxds = 1 2 T d |u(0, x)| 2 dx - 1 2 T d |u(T, x)| 2 dx,
From (2.9) with P = a together with (2.8), we have

T 0 T d a(x)(|∇u| 2 + |u| 4 )dxds = T 0 T d a(x)( (u∂ t u))dxds - 1 2 T 0 T d (∇a(x) • ∇)(|u| 2 )dxds = 1 2 T d |∇u(0, x)| 2 dx + 1 4 T d |u(0, x)| 4 dx - 1 2 T d |∇u(T, x)| 2 dx - 1 4 T d |u(T, x)| 4 dx - 1 2 T 0 T d (∇a(x) • ∇)(|u| 2 )dxds
We sum the last two previous identities and we use the observability estimate (3.49) to get that

E(0) + T 0 E(t)dt ≤ C T 2 T 0 T d |∇a(x)||∇u(t, x)||u(t, x)|dxdt + C T (E(0) -E(T )). (4.2)
Let ε > 0 to be chosen later. According to Lemma 4.1, there exists a positive constant

C ε > 0 such that ∀x ∈ T d , |∇a(x)| 2 ≤ C ε a(x) + ε.
We therefore deduce from (4.2) and the L 2 -identity (2.7) that there exists a new constant C ε > 0 such that

E(0) + T 0 E(t)dt ≤ C T 2 C ε T 0 T d a(x)|u(t, x)| 2 dxdt + E(0) -E(T ) + ε T 0 E(t)dt ≤ C T 2 C ε 2 u(0, •) 2 L 2 (T d ) -u(T, •) 2 L 2 (T d ) + E(0) -E(T ) + ε T 0 E(t)dt . (4.3)
By now, we choose ε = C -1 T . This readily provides

E(0) + T 0 E(t)dt ≤ C T CT u(0, •) 2 L 2 (T d ) -u(T, •) 2 L 2 (T d ) + E(0) -E(T ) , (4.4)
where CT > 0 is a new positive constant depending only on T .

Let us define an auxiliary energy by

∀t ≥ 0, Ẽ(t) = E(t) + CT u(t, •) 2 L 2 (T d ) , which satisfies for all t ≥ 0, E(t) ≤ Ẽ(t) ≤ (1 + CT )E(t).
From (4.4), we have

Ẽ(0) ≤ ĈT ( Ẽ(0) -Ẽ(T )), (4.5)
where ĈT = (1 + CT )C T . This last inequality directly implies that

Ẽ(T ) ≤ ĈT -1 ĈT Ẽ(0).
Thanks to the Gronwall's inequality from Proposition 2.3, one can readily obtain that there exists a positive constant M T > 0 such that

∀0 ≤ t ≤ T, Ẽ(t) ≤ M T Ẽ(0).
Finally, we obtain that there exists two positive constants K, γ > 0 such that for all t ≥ 0,

Ẽ(t) ≤ Ke -γt Ẽ(0) and then, ∀t ≥ 0, E(t) ≤ (1 + CT )Ke -γt E(0).
This concludes the proof of Theorem 1.1.

Global null-controllability of the nonlinear equation

This section is devoted to the proof of Theorem 1.2. We adopt the classical strategy (see for example [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on an interval[END_REF], [START_REF] Laurent | Internal control of the Schrödinger equation[END_REF]) which consists in using our stabilisation result Theorem 1.1 after proving a local controllability result near to 0. Let ϕ ∈ C ∞ c (0, T ) be a nonnegative function different from zero.

4.2.1

First step: study of the linear system.

Before studying the local controllability of nonlinear equation, let us consider the linear system

i∂ t Ψ = -∆Ψ + a 2 (x)ϕ 2 (t)e it∆ φ 0 in (0, +∞) × T d , Ψ(T, •) = 0 in T d , (4.6)
for φ 0 ∈ L 2 (T d ). Let us define the linear operator

S : L 2 (T d ) -→ L 2 (T d ) φ 0 -→ Ψ(0, •).
where Ψ is the mild solution of (4.6). One can easily check that S is an injective continuous map. Let us highlight that the surjectivity of S would lead to the exact controllability of the linear system (4.6). Thanks to the Hilbert Uniqueness Method, the question of its surjectivity is equivalent to the observability estimates

∃C a,ϕ > 0, ∀u 0 ∈ L 2 (T d ), u 0 2 L 2 (T d ) ≤ C a,ϕ R ϕ(t) 2 ae it∆ u 0 2 L 2 (T d ) dt,
which are known to hold in any dimension, see [AM14, Theorem 4]. The linear map S is therefore an isomorphism from L 2 (T d ) to L 2 (T d ). Actually, the following proposition states that S is also an isomorphism from

H 1 (T d ) to H 1 (T d ). Proposition 4.2 ([Lau10a, Lemma 3.1]). The Sobolev space H 1 (T d ) is S invariant and S : H 1 (T d ) -→ H 1 (T d ) is an isomorphism.
Let us mention that Proposition 4.2 is proved in [Lau10a, Lemma 3.1] in the one-dimensional setting. However the strategy adopted by the author [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on an interval[END_REF] can be easily adapted in any dimension d ≥ 1.

Second step: controlability near to 0.

As a first step, we prove the following proposition: Proposition 4.3. Let T > 0 and b ∈ (1/2, 1) be the parameter provided by Proposition 2.1. There exists ε > 0 such that for all

u 0 ∈ H 1 (T d ) satisfying u 0 H 1 (T d ) ≤ ε, there exists g ∈ C([0, T ], H 1 (T d )) supported in [0, T ] × ω so that the unique solution u ∈ X 1,b T of i∂ t u = -∆u + |u| 2 u + g1 ω in (0, +∞) × T d , u(0, •) = u 0 in T d , (4.7
)

satisfies u(T, •) = 0. Proof. For φ 0 ∈ H 1 (T d ), we consider u ∈ X 1,b
T the unique solution of

i∂ t u = -∆u + |u| 2 u + a 2 (x)ϕ 2 (t)e it∆ φ 0 in (0, +∞) × T d , u(T, •) = 0 in T d , (4.8) v ∈ X 1,b
T the unique solution of

i∂ t v = -∆v + |u| 2 u in (0, +∞) × T d , v(T, •) = 0 in T d , (4.9)
and define Lφ 0 = u(0) and Kφ 0 = v(0). We therefore have

∀φ 0 ∈ H 1 (T d ), Lφ 0 = Kφ 0 + Sφ 0 .
Our goal is to show that there exists η > 0 such that B H 1 (0, η) ⊂ Im(L). Notice that the equation u 0 = Lφ 0 is equivalent to φ 0 = S -1 u 0 -S -1 Kφ 0 and this question is then equivalent to find a fixed point of

Bφ 0 := S -1 u 0 -S -1 Kφ 0 ,
for u 0 sufficiently small in H 1 (T d ). Let 0 < η, ε ≤ 1 be two small parameters to be chosen later and u 0 ∈ B H 1 (0, η). Without loss of generality, we can assume T ≤ 1. Since S : H 1 (T d ) -→ H 1 (T d ) is an isomorphism, we have that for all φ 0 ∈ H 1 (T d ),

Bφ 0 H 1 (T d ) ≤ C( u 0 H 1 (T d ) + Ku 0 H 1 (T d ) ) = C( u 0 H 1 (T d ) + v(0, •) H 1 (T d ) ).
Moreover, we have thanks to the continuous embedding of X 1,b T in C([0, T ], H 1 (T d )), Lemma A.2 and the trilinear estimate (A.6), that there exists b ∈ (0, 1/2) such that

v(0, •) H 1 (T d ) ≤ C v X 1,b T ≤ CT 1-b-b |u| 2 u X 1,-b ≤ C u 3 X 1,b T ≤ u 3 X 1,b T .
Furthermore, by using the fact that the flow map, defined by (2.6), is Lipschitz on the bounded set B H 1 (T d ) (0, 1) × B L 2 (0,T ;H 1 (T d )) (0, 1), we obtain for all

φ 0 ∈ B H 1 (T d ) (0, ε), u X 1,b T ≤ C φ 0 H 1 (T d ) ≤ Cε.
As a consequence, we deduce that for all φ 0 ∈ B H 1 (T d ) (0, ε),

Bφ 0 H 1 (T d ) ≤ C(η + ε 3 ),
for some positive constant C > 0 independent on ε and η. We can therefore choose ε 0 > 0 such that

ε 3 0 ≤ ε 0 2C ,
and η = ε0 2C , and we obtain that the closed ball B H 1 (T d ) (0, ε 0 ) is B invariant. It remains to check that B is a contraction mapping on this ball. Let φ 0 , φ 1 ∈ B H 1 (T d ) (0, ε 0 ). We obtain thanks to the continuous embedding of X 1,b T in C([0, T ], H 1 (T d )) and Lemma A.2, that for all b ∈ (0, 1/2) satisfying b + b ≤ 1,

Bφ 0 -Bφ 1 H 1 (T d ) = S -1 (Kφ 0 -Kφ 1 ) H 1 (T d ) ≤ C v 0 (0, •) -v 1 (0, •) H 1 (T d ) ≤ C v 0 -v 1 X 1,b T ≤ CT 1-b-b |u 0 | 2 u 0 -|u 1 | 2 u 1 X 1,-b T ,
where v 0 (respectively v 1 ) is solution to (4.9) with φ 0 (respectively with φ 1 ). It follows from the last inequality, together with the trilinear estimates (A.7), that if b is the parameter provided by Proposition A.3, then

Bφ 0 -Bφ 1 H 1 (T d ) ≤ C u 0 2 X 1,b T + u 1 2 X 1,b T u 0 -u 1 X 1,b T ≤ C u 0 2 X 1,b T + u 1 2 X 1,b T u 0 -u 1 X 1,b T .
By using once again the fact that the flow map given by (2.6) is Lipschitz on bounded set, we obtain a new constant C > 0 independant on ε 0 such that

Bφ 0 -Bφ 1 H 1 (T d ) ≤ C φ 0 2 H 1 (T d ) + φ 1 2 H 1 (T d ) φ 0 -φ 1 H 1 (T d ) ≤ 2Cε 2 0 φ 0 -φ 1 H 1 (T d )
.

By now, we set

ε 0 ≤ 1 2 √ C and B : B H 1 (T d ) (0, ε 0 ) -→ B H 1 (T d ) (0, ε 0
) is a contraction mapping and admits an unique fixed point, according to the Banach fixed point Theorem.

Third step: application of the stabilization result.

According to Proposition 4.3, there exists ε > 0 such that for all

w 0 ∈ H 1 (T d ) satisfying w 0 H 1 (T d ) ≤ ε there exists g ∈ C([0, 1], H 1 (T d )) supported in [0, 1] × ω so that the unique solution w ∈ X 1,b 1 of (4.7) satisfies w(1, •) = 0. Let R ≥ 1 and u 0 ∈ H 1 (T d ) such that E(u 0 ) ≤ R.
By the stabilization result stated by Theorem 1.1, there exists a control h 1 ∈ C([0, +∞), H 1 (T d )) such that the solution of (1.6) satisfies ∀t ≥ 0, E(u(t)) ≤ Ce -γt E(u 0 ), where C, γ are positive constants only depending on ω. In particular, for

T = γ -1 ln R + γ -1 ln C ε 2 , we have u(T ) 2 H 1 (T d ) ≤ E(u(T )) ≤ Ce -γT R ≤ ε 2 .
On the other hand, thanks to Proposition 4.3, there exists a control h 2 ∈ C([0, 1], H 1 (T d )) supported in [0, 1] × ω such that the solution ũ of (1.6) started from u(T ) satisfies ũ(1, •) = 0.

To conclude, it suffices to define the control by h(t, •) = h 1 (t, •) on [0, T ] and h(t, •) = h 2 (t -T, •) on [T, T + 1]. With this choice of control, the solution u of (1.6) satisfies u(T + 1, •) = 0 and T + 1 is a controllability time independent on u 0 . In particular, τ (R) ≤ T + 1 ≤ C ln(R + 1) where C is a positive constant depending on γ and C.

A Appendix

Let d ∈ {1, 2, 3}. This appendix is devoted to establish the well-posedness in Bourgain spaces X s,b T for Cauchy problems associated to

i∂ t u = -∆u + |u| 2 u -ia(x)u + g in (0, T ) × T d , u(0, •) = u 0 in T d , (A.1)
where T > 0, s ≥ 1, b ∈ (1/2, 1) (depending on d and s), a ∈ C ∞ (T d ; R), u 0 ∈ H s and g ∈ L 2 (0, T ; H s (T d )). Of course, this is an adaptation of the breakthrough idea introduced in [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations[END_REF], see for instance [Bou99, Chapter 5] for a detailed account of these techniques. However, since we did not find the exact result we needed in the literature, we present here the main steps of the proofs for obtaining such a result. We will mainly follow and adapt the presentation given by [Lau10b, Section 1 and Section 2] for the treatment of the 3-d case to our d-dimensional case with d ∈ {1, 2, 3}. Note that in the one-dimensional case, one can directly use [Lau10a, Section 1 and 2] since we consider the same damping terms than the author.

In the first subsection, we introduce the so-called Bourgain spaces X s,b T , recall their main properties and present trilinear estimates that will be one of the key points for the proof of the well-posedness result. In the second part, we present a priori energy identities and estimates for solutions to (A.1). In the two last parts, we prove the well-posedness result stated by Proposition 2.1.

A.1 Properties of the Bourgain spaces

This subsection recalls basic properties of Bourgain spaces, as well as trilinear estimates which are instrumental in this work. The definition of Bourgain spaces is given in Section 2.1.

Let us begin by enumerate several useful properties without proof:

• The Bourgain spaces X s,b and X s,b T are Hilbert spaces. • If s 1 ≤ s 2 and b 1 ≤ b 2 , then X s2,b2 is continously embedded in X s1,b1 . • For all s ∈ R and b > 1/2, the space X s,b T is continously embedded in C([0, T ]; H s (T d )). • For all s 1 < s 2 and b 1 < b 2 , the space X s2,b2 T is compactly embedded in X s1,b1 T .

• The dual space of X s,b T is X -s,-b T . • For θ ∈ (0, 1), the complex interpolation space (X s1,b1 , X s2,b2 ) θ is X

(1-θ)s1+θs2,(1-θ)b1+θb2 . • If s ∈ R, b ∈ ( 1 2 , 1), 0 < T 1 < T 2 , u 1 ∈ X s,b (0,T1) and u 2 ∈ X s,b (T1,T2) with u 1 (T 1 ) = u 2 (T 1 ), then the function u defined by u(t, •) = u 1 (t, •), t ∈ [0, T 1 ] u 2 (t, •), t ∈ [T 1 , T 2 ] belongs to X s,b (0,T2) .
The following result studies the stability of the Bourgain spaces with respect to multiplication operators.

Lemma A.1. Let ϕ ∈ C ∞ c (R), ψ ∈ C ∞ (T d ), s ∈ R, b ∈ [-1, 1] and T > 0.
The following linear mappings

Φ : u ∈ X s,b → ϕ(t)u ∈ X s,b , Φ T : u ∈ X s,b T → ϕ(t)u ∈ X s,b T , (A.2) Ψ : u ∈ X s,b → ψ(x)u ∈ X s-|b|,b , Ψ T : u ∈ X s,b T → ψ(x)u ∈ X s-|b|,b T , (A.3) are continuous.
Proof. We only prove the first parts of (A.2) and (A.3). By using the commutation of e -it∆ with ϕ(t), we have

ϕu X s,b = e -it∆ [ϕ(t)u] H b t (H s x ) = ϕu # H b t (H s x ) ≤ C u # H b t (H s x ) ≤ C u X s,b , (A.4)
which concludes the proof of (A.2). For (A.3), we first treat the two cases b = 0 and b = 1. For b = 0, we notice that X s,0 = L 2 (R, H s ) and the result is obvious.

For b = 1, we have u ∈ X s,1 if and only if u ∈ L 2 (R, H s ) and i∂

t u + ∆u ∈ L 2 (R, H s ), with the norm u 2 X s,1 = u 2 L 2 (R,H s ) + i∂ t u + ∆u 2 L 2 (R,H s )
. Then, we have, by using that the commutator [ψ, ∆] is an operator of order 1 in space,

ψ(x)u 2 X s-1,1 = ψu 2 L 2 (R,H s-1 ) + i∂ t (ψu) + ∆(ψu) 2 L 2 (R,H s-1 ) ≤ C u 2 L 2 (R,H s-1 ) + ψ (i∂ t u + ∆u) 2 L 2 (R,H s-1 ) + [ψ, ∆] u 2 L 2 (R,H s-1 ) ≤ C u 2 L 2 (R,H s-1 ) + i∂ t u + ∂ 2 x u 2 L 2 (R,H s-1 ) + u 2 L 2 (R,H s ) ≤ C u 2 X s,1 ,
that concludes the proof in the case b = 1. We finally conclude by interpolation and duality.

The following elementary lemma holds, see [Gin96, Lemma 3.2].

Lemma A.2. Let ϕ ∈ C ∞ c (R), b, b ∈ R such that 0 < b < 1/2 < b, b + b ≤ 1 and T > 0. If f ∈ H -b (R), then t → ϕ t T t 0 f (τ )dτ H b (R) ≤ CT 1-b-b f H -b (R) . (A.5)
One of the key points for establishing well-posedness results associated to the cubic defocusing nonlinear Schrödinger equation consists in establishing the following trilinear estimates.

Proposition A.3. For every s 0 > 1/2, for every s 2 ≥ s 1 ≥ s 0 , there exist b ∈ (0, 1/2) and C > 0 such that for every T ∈ (0, 1),

u, v ∈ X s2,b T , |u| 2 u X s 2 ,-b T ≤ C u 2 X s 1 ,b T u X s 2 ,b T , (A.6) |u| 2 u -|v| 2 v X s 2 ,-b T ≤ C u 2 X s 2 ,b T + v 2 X s 2 ,b T u -v X s 2 ,b T . (A.7)
Proposition A.3 is part of the "folklore" for the study of nonlinear Schrödinger equation with periodic boundary conditions. This is a straightforward corollary of [Lau10a, Lemma 0.3] in 1-d (one can even take s 0 = 0 and b = 3/8), [BGT05a, Proposition 2.5 and Proposition 3.5] in 2-d (one can even take s 0 > 0) and [Lau10b, Assumption 3, Lemma 1.1] in 3-d.

A.2 Energy estimates for strong solutions

In this section, we establish energy identities given by Proposition 2.2.

Proof of Proposition

2.2. Let u ∈ X s,b
T be a solution of the equation (A.1) with s ≥ 2 and for some

b ∈ ( 1 2 , 1). Since X 2,b T ⊂ C([0, T ]; H 2 (T d ))
, u satisfies (A.1) in the strong sense. For the identity (2.7), we multiply (A.1) by u and integrate on T d . By integration by parts, we get

T d i∂ t u(s, x)u(s, x)dx- T d |∇u(s, x)| 2 = T d |u(s, x)| 4 dx-i T d a(x)|u(s, x)| 2 dx+ T d g(s, x)u(s, x)dx.
Then, by taking the imaginary part and by using the fact that 2 (i∂ t uu) = 2 (∂ t uu) = ∂ t |u| 2 , we obtain 1 2

d ds T d |u(s, x)| 2 dx = - T d a(x)|u(s, x)| 2 dx + T d (g(s,
x)u(s, x))dx.

We then integrate for s ∈ (t , t) to get the result.

For the identity (2.8), we multiply (A.1) by ∂ t u and integrate on T d . By integration by parts, we get Then, by taking the real part and by using the facts that 2 (∇u∂ t ∇u) = ∂ t |∇u| 2 , 4 (|u| 2 u∂ t u) = ∂ t |u| 4 and (iz) = -(z), we obtain 1 2 it follows that u satisfies (2.9).

d
The next result deals with a particular case of Proposition 2.3 when the solution belongs to X s,b T , with s ≥ 2. The general case will be made in Section A.4, thanks to a regularization argument. Proof. First, from (2.7), we deduce from a Gronwall's estimate that

u(t, •) 2 L 2 (T d ) ≤ C u(0, •) 2 L 2 (T d ) + g 2 L 2 (0,T ;L 2 (T d ))
∀t ∈ [0, T ]. (A.9)

We then use (2.9) with P = a to estimate the first term in the right hand side of (2.8), we then have for every t ∈ [0, T ], We then estimate the second term in the right hand side of (2.8) by using the equation (A.1), integration by parts, Hölder's estimate and the Sobolev embedding H 1 (T d ) → L 4 (T d ) and Young's inequality, we

The estimate (A.5) from Lemma A.2 then implies that ψ(t/T ) t 0 e i(t-τ )∆ F (τ )dτ e i(t-τ )∆ [(|u| 2 u -|v| 2 v) -ia(x)(u -v) + (g 1 -g 2 )](τ )dτ

X s,b T ≤ CT 1-b-b |u| 2 u -|v| 2 v -ia(x)(u -v) + g 1 -g 2 X s,-b T ≤ CT 1-b-b |u| 2 u -|v| 2 v X s,-b T + a(x)(u -v) X s,-b T + g 1 -g 2 X s,-b T ≤ CT 1-b-b |u| 2 u -|v| 2 v X s,-b T + a(x)(u -v) X s,0 T + g 1 -g 2 X s,-b T ≤ CT 1-b-b |u| 2 u -|v| 2 v X s,-b T + u -v X s,b T + g 1 -g 2 X s,-b T ≤ CT 1-b-b u -v X s,b T 1 + u 2 X 1,b T + v 2 X 1,b T + CT 1-b-b g 1 -g 2 X s,-b T . (A.17) Then, Φ u0,g1 (u) -Φ v0,g2 (v) X s,b T ≤ C u 0 -v 0 H s (T d ) + C g 1 -g 2 X s,-b T + CT 1-b-b u -v X s,b T 1 + u 2 X 1,b T + u 2 X 1,b
T , (A.18) that exactly gives (A.13) recalling the bound on u and v in the Bourgain spaces X s,b T . Proposition A.6 (Local existence). Let a ∈ C ∞ (T d , R) and S ≥ 1. There exists b ∈ ( 1 2 , 1) such that for every 1 ≤ s ≤ S, u 0 ∈ H s (T d ) and g ∈ L 2 (0, T ; H s (T d )), there exists T > 0 and a unique solution u ∈ X s,b T to (A.1). Moreover, u satisfies ∀t ∈ (0, T ), u(t) = Φ u0,g (u)(t).

Proof. Let us show that Φ u0,g admits a unique fixed point in X s,b T provided that T > 0 is sufficiently small. Let 0 < T ≤ 1. According to (A.13), we have for all u, v ∈ X (0, M ) to itself and then, admits an unique fixed point. Proposition A.6 allows us to define the unique maximal solution starting from u 0 ∈ H s (T d ). We define for u 0 ∈ H s (T d ) and g ∈ L 2 (0, T ), H s (T d )) and s ≥ 1, T (u 0 , g) := sup{T > 0; ∃u ∈ X s,b T solution to (2.1) starting from u 0 }.

Thanks to Proposition A.6, together with Lemma A.5, there exists a unique maximal solution u ∈ X s,b T , for all 0 < T < T (u 0 , g).

  multiplying by ∂ t u the equation (1.1) and by taking the real part, we also observe that the nonlinear-energy (or H 1 -energy) is conserved, i.e.

  Section 2] for the well-posedness and [Lau10b, Appendix B.1] for the Carleman part, • d = 3, M = T 3 , ω as in (1.10), replacing |u| 2 u by |u| p-1 u for every p ∈ (1, 5), see [Bou99, Chapter V] for the well-posedness, • d = 3, M = S 3 , ω a nonempty open subset of S 2 containing a neighborhood of {x 4 = 0} ⊂ R 3 , replacing |u| 2 u by |u| p-1 u for every p ∈ (1, 5), see [BGT05b, Theorem 1] for the well-posedness and [Lau10b, Appendix B.1] for the Carleman part.

  T d i|∂ t u(s, x)| 2 dx -T d ∇u(s, x) • ∂ t ∇u(s, x) = T d |u(s, x)| 2 u(s, x)∂ t u(s, x)dx -i T d a(x)u(s, x)∂ t u(s, x)dx + T d g(s, x)∂ t u(s, x)dx.

Proposition A. 4 .

 4 Let T > 0, s ≥ 2, a ∈ C ∞ (T d ; R), u 0 ∈ H s and g ∈ L 2 (0, T ; H s (T d )). There exists a positive constant C = C T,d,a > 0 such that if u ∈ X s,bT is a solution of (A.1) for some b ∈ (1/2, 1), then we haveE(t) ≤ C E(0) + g 2 L 2 (0,T ;H 1 (T d )) + g 4 L 2 (0,T ;H 1 (T d ))∀t ∈ [0, T ].(A.8)

X s, b T≤

 b CT 1-b-b F X s,-b T . (A.16) Then, by using the trilinear estimate (A.6) from Lemma A.3, the multiplication estimate (A.3) from Lemma A.1 and Bourgains spaces embeddings, we get t 0

  s,b T , with u X s,b T ≤ M and v X s,b T ≤ M , Φ u0,g (u) -Φ u0,g (v) X s,b T ≤ CT 1-b-b u -v X s,b T 1 + 2M 2 and Φ u0,g (u) X s,b T ≤ C( u 0 H s (T d ) + g L 2 ((0,T ),H s (T d ) ) + CT 1-b-b u X s,b T M = C( u 0 H s (T d ) + g L 2 ((0,T ),H s (T d ) )+1 and T > 0 such that T 1-b-b CM (1+2M 2 ) < 1 2 .With this choice, we readily show that Φ u0,g is a contraction map from B X s,b T

  We then integrate for s ∈ (t , t) to get the result.Let us multiply (A.1) by P u and integrate on (t , t) × T d . By taking the real part and since

	ds T d	|∇u(s, x)| 2 dx +	1 4	d ds T d	|u(s, x)| 4 dx
						= -	a(x) (u(s, x)∂ t u(s, x))dxds -	(g(s, x)∂ t u(s, x))
						T d	T d
	t
	Re		∆uP udxds = -
	t	T d

t t T d Re(∇u • ∇P (x)u) + |∇u| 2 P (x)dxds = -t t T d 1 2 (∇P (x) • ∇)(|u| 2 ) + |∇u| 2 P (x)dxds,

L 2 (T) holds true.In comparison to what is known for the linear case, where an arbitrary open set of the torus T d is sufficient for the control, we may wonder to what extent one can weaken the geometrical assumption on ω, that satisfies (1.10) in our situation. Actually, in the Carleman part, one can consider ω, containing only a neighborhood of the generator circle in 2-d by taking Carleman weights satisfying only weak pseudoconvexity assumptions as done in[START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF]. However, this only leads to an observability inequality with a L 2 (T d )-left hand side. This is not sufficient for obtaining the exponential decay of the total energy with our multipliers strategy.

* Both authors are partially supported by the Project TRECOS ANR-20-CE40-0009 funded by the ANR (2021-2024).

then have for every t ∈ [0, T ],

We plug (A.9), (A.10), (A.11) together with (2.8) to obtain

Nonlinear Gronwall's estimate leads to (A.8).

A.3 Well-posedness results for the nonlinear Schrödinger equation

Now we can state the local well-posedness result for Cauchy problems associated to (A.1). Let us consider the functional

where

T is a solution to the distributional sense of (A.1) then u coincides with Φ u0,g , see for instance [Lau10a, Proof of Theorem 2.1]. The following lemma is instrumental in this section.

Proof. By Proposition A.3, for all 1 ≤ s 1 ≤ s 2 ≤ S, we have some parameter b s1,s2 ∈ (0, 1 2 ) such that (A.6) and (A.7) hold. By choosing b = max 1≤s1≤s2≤S b s1,s2 , we obtain that (A.6) and (A.7) hold for all 1 ≤ s 1 ≤ s 2 ≤ S with the same parameter b ∈ (0, 1 2 ). First, we notice that if

Therefore, for T ≤ 1, we have

Let us show that T (u 0 , g) does not depend on s when u 0 ∈ H 1 (T d ). Let us take here s > 1. By using that

T1 and a maximal solution u 2 ∈ X s,b T2 for some T 1 , T 2 > 0. We clearly have T 2 ≤ T 1 . Moreover, by uniqueness in X 1,b T1 , we also have

We deduce that u 2 C([T2-δ,T2);H 1 (T d )) ≤ C. By using the local existence in H 1 (T d ) and gluing of solutions, we then get that there exists C > 0 such that

Then by using (A.13

we obtain (recalling that a solution of (A.1) in the distribution sense is necessarily a solution in the Duhamel sense),

Therefore, by using gluing of solutions, we obtain that u 2 ∈ X s,b T2 contradicting (A.20). The following proposition shows that when u 0 ∈ H 2 (T d ), the solution is actually defined at any time.

Proof. Let u 0 ∈ H 2 (T d ) and assume by contradiction that T (u 0 , g) < +∞. Let us consider the maximal solution u ∈ X 2,b T , for all 0 < T < T (u 0 , g) starting from u 0 . According to Proposition A.6, u is also the maximal solution starting from u 0 seen as a H 1 (T d )-function. We therefore have lim t→T (u0,g)

However, thanks to the energy estimates (A.8), the energy E(t) is bounded on (0, T (u 0 , g)) and yields

By using the local existence in H 1 (T d ) and gluing of solutions, we then get that there exists

This is a contradiction.

A.4 Energy estimates and global existence for less regular data

Thanks to the global existence of solutions for H 2 (T d )-data, we are now in position to establish energy estimates given by Proposition 2.3 for

) N a sequence tending to g in L 2 ((0, T ), H 1 (T d )). Associated to u 0,n and g n , we can define the solution u n ∈ X 2,b T , which satisfies

∀t ∈ [0, T ], (A.23) thanks to Proposition A.4. Then by using (A.13), one can prove that for T * small enough depending on u 0 H 1 (T d ) and g L 2 (0,T ;H 1 (T d )) that

We then just have to piece solutions together in small intervals by using the fact that X 1,b T * -norm controls the L ∞ (0, T * ; H 1 (T d ))-norm. We obtain

This allows us to pass to the limit in (A.23).

By proceeding in the same manner as in the proof of Proposition A.7, the energy estimates (2.10) ensure that the solution, associated to H 1 -data, are global:

We are now in position to give the proof of Proposition 2.1 which states the global well-posedness of (A.1). Actually, it remains to show that data in H s (T d ) lead to global solutions in X s,b , for any s ≥ 1 and to prove that the flow map is Lipschitz on every bounded subset.

Proof of Proposition 2.1. The local existence given by Proposition A.6 with s = 1 and the a priori energy estimate (A.8) implies therefore global existence in X 1,b T . This implies the global existence in X s,b T for s ≥ 1.

For the local Lipschitz estimate on the flow, we know from (A.13) that this is true in small time intervals. By gluing solutions together, we then deduce that it is true in the time interval [0, T ].