Bilinear control and growth of Sobolev norms for the nonlinear Schr\"odinger equation

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


Introduction
In this paper, we study the controllability and the growth of Sobolev norms for the following nonlinear Schrödinger (NLS) equation on the torus T d = R d /2πZ d : We assume that V : T d → R is an arbitrary smooth potential, Q : T d → R q is a given smooth external field subject to some geometric condition, d, p ≥ 1 are arbitrary integers, and κ is an arbitrary real number.The role of the control (or the random perturbation) is played by R q -valued function (or random process) u which is assumed to depend only on time.Eq. (0.1) is equipped with the initial condition ψ(0, x) = ψ 0 (x) (0.2) belonging to a Sobolev space H s = H s (T d ; C) of order s > d/2, so that the problem is locally well-posed.The purpose of this paper is to study the NLS equation (0.1) when the driving force u acts multiplicatively through only few low Fourier modes.Referring the reader to the subsequent sections for the general setting, let us formulate in this Introduction particular cases of our main results.Let K ⊂ Z d * be the set of d vectors defined by K = {(1, 0, . . ., 0), (0, 1, . . ., 0), . . ., (0, 0, . . ., 1, 0), (1, . . ., 1)}, (0.3) and assume that the field Q = (Q 1 , . . ., Q q ) is such that {1, sin x, k , cos x, k : k ∈ K} ⊂ span{Q j : j = 1, . . ., q}. (0.4) Let s d be the least integer strictly greater than d/2.
More general formulation of this result is given in Theorem 2.2, where the controllability is proved under an abstract saturation condition for the field Q (see Condition (H 1 )).Note that the time T may depend on the initial condition ψ 0 , the target e iθ ψ 0 , and the parameters in the equation.In the second result, we show that, when V = 0 and ψ 0 is an eigenstate φ l (x) = (2π) −d/2 e i x,l , l ∈ Z d of the Laplacian, the system can be approximately controlled in any fixed time T > 0 to any target of the form e iθ φ m with m ∈ Z d .Theorem B. For any s ≥ s d , ε > 0, l, m ∈ Z d , θ ∈ C ∞ (T d ; R), and T > 0, there is a control u ∈ L 2 ([0, T ]; R q ) and a unique solution ψ ∈ C([0, T ]; H s ) of (0.1), (0.2) with V = 0 and ψ 0 = φ l such that ψ(T ) − e iθ φ m L 2 < ε.
The controllability of the Schrödinger equation with time-dependent bilinear (multiplicative) control has attracted a lot of attention during the last fifteen years.In the one-dimensional case, local exact controllability results are established by Beauchard,Coron,and Laurent [Bea05,BC06,BL10].There is a vast literature on the approximate controllability in the multidimensional case.For the first achievements, we refer the reader to the papers by Boscain et al. [CMSB09,BCCS12], Mirrahimi [Mir09], and the second author [Ner10].Except the paper [BL10], all the other works consider the linear Schrödinger equation, i.e., the one obtained by taking κ = 0 in Eq. (0.1); note that in that case the control problem is still nonlinear in u.
Theorems A and B are the first to deal with the problem of bilinear approximate controllability of the NLS equation.Let us emphasise that the controllability between any pair of eigenstates in arbitrarily small time is new even in the linear case κ = 0.It is interesting to note that Theorem B complements a result by Beauchard et al. [BCT18], which proves that, for some choices of the field Q, there is a minimal time for the approximate controllability to some particular states in the phase space.
The approach adopted in the proofs of Theorems A and B is quite different from those used in the literature for bilinear control systems.We proceed by developing Agrachev-Sarychev type arguments which were previously employed in the case of additive controls.Let us recall that Agrachev and Sarychev [AS05, AS06] considered the global approximate controllability of the 2D Navier-Stokes and Euler systems.Their approach has been further extended by many authors to different equations.Let us mention, for example, the papers [Shi06,Shi07] by Shirikyan who considered the approximate controllability of the 3D Navier-Stokes system and Sarychev [Sar12] who considered the case of the 2D defocusing cubic Schrödinger equation.The configuration we use in the present paper is closer to the one elaborated in the recent paper [Ner21], where parabolic PDEs are studied with polynomial nonlinearities.We refer the reader to the reviews [AS08,Shi18] and the paper [Ner21] for more references and discussions.
The present paper is the first to deal with Agrachev-Sarychev type arguments in a bilinear setting.To explain the scheme of the proof of Theorem A, let us denote by R t (ψ 0 , u) the solution of problem (0.1), (0.2) defined up to some maximal time.A central role in the proof is played by the limit 2 .Applying this limit with ϕ = 0 and using the assumption (0.4), we see that the equation can be controlled in small time from initial point ψ 0 arbitrarily close to e iθ ψ 0 for any θ in the vector space By applying again the limit (0.5) with functions ϕ = θ j ∈ H 0 , j = 1, . . ., n, we add more directions in θ.That is, we show that the system can be steered from ψ 0 close to e iθ ψ 0 , where θ now belongs to a larger vector space H 1 whose elements are of the form We iterate this argument and construct an increasing sequence of subspaces {H j } such that the equation can be approximately controlled to any target e iθ ψ 0 with any θ ∈ H j and j ≥ 1.Using trigonometric computations, we show that the union ∪ j=1 H j is dense in C k (T d , R) for any k ≥ 1 (in other words, H 0 is a saturating space for the NLS equation, see Definition 2.1).This completes the proof of Theorem A. Theorem B is derived from Theorem A by noticing that the eigenstate φ l can be approximated in L 2 by functions of the form e iθ φ m and that the eigenstates are constant solutions1 of Eq. (0.1) corresponding to some control.This allows to appropriately adjust the controllability time and choose it the same for any initial condition and target.
As an application of Theorem A, we study the large time behavior of the trajectories of the random NLS equation.We show that if a random process perturbes the same Fourier modes as in the above controllability results, then the energy is almost surely transferred to higher modes resulting in the unboundedness of the trajectories in regular Sobolev spaces.More precisely, we replace the control u by an R q -valued random process η of the form where is the indicator function of the interval [k − 1, k) and {η k } are independent identically distributed random variables in L 2 ([0, 1]; R q ) with nondegenerate law (see Condition (H 2 )).The solution ψ of the problem (0.1), (0.2), (0.6) will be itself a random process in H s .We prove the following result.
Theorem C. For any s > s d and any non-zero ψ 0 ∈ H s , the trajectory of (0.1), (0.2), (0.6) is almost surely unbounded in H s .
The idea of constructing unbounded solutions by using random perturbations is not new.Such results have been obtained by Bourgain [Bou99] and Erdogan et al. [EgKS03] for linear one-dimensional Schrödinger equations.They also provided polynomial lower bounds for the growth.Unboundedness of trajectories for multidimensional linear Schrödinger equations is obtained in [Ner09].In that paper, the assumptions on the law of the random perturbation are rather general and no estimates for the growth are given; Theorem C is a generalisation of that result to the case of the NLS equations.There are also examples of linear Schrödinger equations with various deterministic time-dependent potentials which admit unbounded trajectories: e.g., see the papers by Bambusi et al. [BGMR18], Delort [Del14], Haus and Maspero [HM20,Mas19], and the references therein.
There are only few results in the case of unperturbed NLS equations.For cubic defocusing Schrödinger equations on bounded domains or manifolds, the existence of unbounded trajectories in regular Sobolev spaces is a challenging open problem (see Bourgain [Bou00]).In different situations, existence of trajectories with arbitrarily large finite growth has been shown by Kuksin [Kuk97], Colliander et al. [CKS + 10], Guardia and Kaloshin [GK15], and others.Hani et al. [HPTV15] show the existence of unbounded trajectories in the case of the cubic defocusing Schrödinger equation on the infinite cylinder R × T d .In the case of the cubic Szegő equation on the circle, Gérard and Grellier [GG17] show that the trajectories are generically unbounded in Sobolev spaces.Moreover, they exhibit the existence of a family of solutions with superpolynomial growth.
Let us give a brief (and not entirely accurate) description of the main ideas of the proof of Theorem C. By starting from any initial point ψ 0 ∈ H s , Theorem A allows to increase the Sobolev norms by choosing appropriately the control.This, together with a compactness argument and the assumption that the law of the process η is non-degenerate, leads to a uniform estimate of the form for any M > 0. By combining the latter with the Markov property, we show that for any ψ 0 ∈ H s .Then, the Borel-Cantelli lemma implies that the norm of any trajectory becomes almost surely larger than M in some random time that is almost surely finite.As M is arbitrary, this proves the required result.
The paper is organised as follows.In Section 1, we discuss the local wellposedness and some stability properties of the NLS equation.In Section 2, we formulate more general versions of Theorems A and B and give their proofs.Section 3 is devoted to the derivation of limit (0.5).In Section 4, we establish a general criterion for the validity of the saturation property.Finally, in Section 5, we prove Theorem C.

Notation
In what follows, we use the following notation.
•, • is the Euclidian scalar product in R q and • is the corresponding norm.We write m⊥l when the vectors m, l ∈ R q are orthogonal and m ⊥ l when they are not.
and Lebesgue spaces of functions f : T d → C endowed with the norms • s and • L p .The space L 2 is endowed with the scalar product Let X be a Banach space.We denote by B X (a, r) the closed ball of radius r > 0 centred at a ∈ X.We write J T instead of [0, T ] and ⌈x⌉ is the least integer greater than or equal to x ∈ R. s d is the least integer strictly greater than d/2.
1 is the function identically equal to 1 on T d .

Preliminaries
In this section, we consider the NLS equation (0.1), where u is a deterministic R q -valued function and V : T d → R and Q : T d → R q are arbitrary smooth functions.In what follows, we shall always assume that the parameters d ≥ 1, p ≥ 1, and κ ∈ R are arbitrary.Here we formulate two propositions that will be used in the proofs of our main results.The first one gathers some well-known facts about the local well-posedness and stability of the NLS equation in regular Sobolev spaces.
(i) For any ψ 0 ∈ H s and u ∈ L 2 (J T ; R q ) satisfying the problem (0.1), (0.2) has a unique solution ψ ∈ C(J T ; H s ).
(ii) Let R be the resolving operator for Eq.(0.1), i.e., the mapping taking a couple (ψ 0 , u) satisfying (1.1) to the solution ψ.Then The proof of this proposition is rather standard, so we omit it (e.g., see Section 3.3 in [Tao06] or Section 4.10 in [Caz03] for similar results).Let S be the unit sphere in L 2 .As the functions V, Q, and u are real-valued, the solution ψ belongs to S throughout its lifespan, provided that ψ 0 ∈ S ∩ H s .
Before formulating the second proposition, let us introduce some notation.For any ψ 0 ∈ H s and T > 0, let Θ(ψ 0 , T ) be the set of functions u ∈ L 2 (J T ; R q ) such that the problem (0.1), (0.2) has a solution in C(J T ; H s ).By the previous proposition, the set We have the following asymptotic property in small time.
The proof of this proposition is postponed to Section 4. Limit (1.3) is a multiplicative version of a limit established in Proposition 2 in [Ner21] in the case of parabolic PDEs with additive controls.

Approximate controllability
In what follows, we assume that s ≥ s d and denote r = ⌈s⌉ + 2 as in Proposition 1.2.We start this section with a definition of a saturation property inspired by the papers [AS06,Shi06].Let H be a finite-dimensional subspace of C r (T d ; R), and let F (H) be the largest subspace of C r (T d ; R) whose elements can be represented in the form for some integer n ≥ 1 and functions θ j ∈ H, j = 0, . . ., n, where B is given by (1.2).As B is quadratic, F (H) is well-defined and finite-dimensional.Let us define a non-decreasing sequence {H j } of finite-dimensional subspaces by H 0 = H and H j = F (H j−1 ), j ≥ 1, and denote (2.1) We assume that the following condition is satisfied.
(H 1 ) The field Q = (Q 1 , . . ., Q q ) is saturating, i.e., the subspace is saturating in the sense of Definition 2.1 In this section, we prove the following result.As we will see below, it implies Theorems A and B formulated in the Introduction.
Proof.By using an induction argument in N , we show that the approximate controllability property in this theorem is true for any θ ∈ H N and N ≥ 0. Combined with the saturation hypothesis, this will lead to approximate controllability with any θ ∈ C r (T d ; R).
By iterating this argument with θ j ∈ H N −1 , j = 0, . . ., n, we obtain that for any ε > 0 and κ > 0, there is T n ∈ (0, κ) and As θ ∈ H N is arbitrary, this proves the required property for N .
Step 3. Conclusion.Finally, let θ ∈ C r (T d ; R) be arbitrary.By the saturation hypothesis, H ∞ is dense in C r (T d ; R).Hence, we can find N ≥ 1 and θ ∈ H N such that e iθ ψ 0 − e i θψ 0 s < ε.
Applying the controllability property proved in the previous steps for θ ∈ H N , we complete the proof.
As a consequence of this result, we have the following two theorems.
Proof.It suffices to apply Theorem 2.2 by choosing θ ∈ C r (T d ; R) such that e iθ ψ 0 s > M.
To find such θ, we take any θ 1 ∈ C r (T d ; R) verifying e iθ1 ψ 0 1 = 0, put θ = λθ 1 with sufficiently large λ > 0, and use the inequality Theorem 2.4.Assume that the conditions of Theorem 2.2 are satisfied and 1 ∈ span{Q j : j = 1, . . ., q} and V = 0. (2.3) Proof.Let us take any θ 1 ∈ C r (T d ; R).Applying Theorem 2.2, we find a time T 1 ∈ (0, T ) and a control u 1 ∈ Θ(φ l , T 1 ) such that Now, notice that φ l is a stationary solution of Eq. (0.1) corresponding a control u 0 ∈ L 2 loc (R + ; R q ) satisfying the relation Such a choice of u 0 is possible in view of assumption (2.3).Thus, u 0 ∈ Θ(φ l , t) and φ l = R t (φ l , u 0 ) for any t ≥ 0. Setting we complete the proof of the theorem.
Let us close this section with an example of a saturating subspace.Let I ⊂ Z d * be a finite set and let Recall that I is a generator if any vector of Z d is a linear combination of vectors of I with integer coefficients.The following proposition is proved in Section 4.
Proposition 2.5.The subspace H(I) is saturating in the sense of Definition 2.1 if and only if I is a generator and for any l, m ∈ I, there are vectors Clearly, the set K ⊂ Z d * defined by (0.3) satisfies the condition in this proposition.Therefore, the subspace H(K) is saturating, and Theorems A and B follow from Theorems 2.2 and 2.4, respectively.
We need to show that (a) there is a constant δ 0 > 0 such that T δ > δ for any δ < δ 0 ; (b) the following limit holds To prove these properties, we introduce the functions where For example, we can define ψ δ 0 by using the heat semigroup: ψ δ 0 = e δ 1/4 ∆ ψ 0 .In view of (3.1)-(3.4),we have Furthermore, v(t) is well-defined for t < δ −1 T δ and satisfies the equation and the initial condition where We take the scalar product of Eq. (3.7) with ∂ 2α v in L 2 and integrating by parts, we obtain (3.9) We estimate the terms I 1 , I 2 , I 3 , and I 5 by integrating by parts and by using (3.1), (3.5), and (3.6): We estimate I 4 as follows In the last relation, we used again the integration by parts, the identities (3.1), (3.5) and (3.6), and the equality Summing up inequalities (3.9) for all α ∈ N d , |α| ≤ s, combining the resulting inequality with the estimates for I j and the Young inequality, and recalling that δ ≤ 1, we obtain This inequality, together with (3.8) and the Gronwall inequality, implies that Let us take δ 0 ∈ (0, 1) so small that, for δ < δ 0 , and denote From (3.8) and (3.11) it follows that τ δ > 0 for δ < δ 0 .Let us show that Assume, by contradiction, that τ δ ≤ 1.Let t = τ δ in (3.10).By using (3.12) and (3.13), we obtain This contradiction shows that τ δ > 1 for δ < δ 0 , hence also 1 < δ −1 T δ .Thus, property (a) is proved.Taking t = 1 in (3.10), we arrive at This implies (b) and completes the proof in the case when s > d/2 is an integer.
To derive properties (a) and (b) in the general case, i.e., when s ≥ s d is an arbitrary number, we use inequality (3.10) for integer values of s and an interpolation argument.

Saturating subspaces
Proof of Proposition 2.5.The proof is divided into four steps.
Step 1. First, let us assume that I ⊂ Z d * is an arbitrary finite set, H 0 (I) = H(I) is the subspace defied by (2.4), H j (I) = F (H j−1 (I)) for j ≥ 1, and H ∞ (I) is defined by (2.1).
Step 1.1.Let us show that, if Indeed, assume that cos x, m , sin x, m ∈ H N (I) for some N ≥ 0. (4.1) The equalities the assumptions 1 ∈ H(I) and (4.1), and the definition of F imply that As a consequence of (4.2) and (4.3), we have which imply the required result.
Step 1.2.Let us show that, if cos x, m , sin x, m , cos x, l , sin x, l ∈ H ∞ (I) this follows immediately from the equalities for any a 1 , . . ., a N ∈ Z.
By the result of step 1.2, we have cos x, a 1 m 1 , sin x, a 1 m 1 , cos x, a 2 m 2 , sin x, a 2 m 2 ∈ H ∞ (I) for any a 1 , a 2 ∈ Z. Again, in view of step 1.2, this implies that cos x, a 1 m 1 + a 2 m 2 , sin x, a 1 m 1 + a 2 m 2 ∈ H ∞ (I) for any a 1 , a 2 ∈ Z.
Step 2.2.Assume that the required property is true if the cardinality of the set I is less or equal to N − 1.Let I ⊂ Z d * be such that N = card(I) and I = {m 1 , . . ., m N }.Without loss of generality, we can assume m N −1 ⊥ m N and the set {m 1 , . . ., m N −1 } satisfies the condition formulated in the beginning of step 2. Let us take any a 1 , . . ., a N ∈ Z and k ≥ 1 and write Then, As m N −1 = 0, for sufficiently large k ≥ 1, we have Relation (4.4) is proved by combining (4.5) and (4.6), the induction hypothesis, and the assumption that m N −1 ⊥ m N .
Step 3. We conclude from step 2 that, if I ⊂ Z d * is a set satisfying the conditions of Proposition 2.5, then cos x, m , sin x, m ∈ H ∞ (I) for any m ∈ Z d * .
This implies that H ∞ (I) is dense in C r (T d ; R) for any r ≥ 0, hence H(I) is saturating.
Step 4. Finally, let us assume that the conditions of the proposition are not satisfied for I ⊂ Z d * .We distinguish between two cases.
Step 4.1.If I is not a generator, we can find a vector n ∈ Z d * which does not belong to the set Ĩ of linear combinations of vectors of I with integer coefficients.It is easy to see that H ∞ (I) ⊂ span{sin x, m , cos x, m : m ∈ Ĩ}.Thus, the functions sin x, n and cos x, n are orthogonal to the vector space H ∞ (I) in the Sobolev spaces H j (T d ; R) for any j ≥ 0. We conclude that H ∞ (I) is not dense in C r (T d ; R), thus the subspace H(I) is not saturating.
Step 4.2.If I does not satisfy the second condition in the theorem, then it is of the form where k ≥ 2 and m j1 i1 ⊥m j2 i2 for any integers 1 ≤ j 1 < j 2 ≤ k, 1 ≤ i 1 ≤ n j1 , and 1 ≤ i 2 ≤ n j2 .By using the arguments of the steps 1 and 2, it is easy to verify that the function cos x, m j1 1 + m j2 2 is orthogonal to H ∞ (I) in H j (T d ; R) for any j ≥ 0. Thus, the space H ∞ (I) is not dense in C r (T d ; R).

Growth of Sobolev norms
Let us consider the NLS equation with potential V and parameters d, p, κ as in the previous sections.We assume that the field Q satisfies Condition (H 1 ) and η is a random process of the form (0.6) with the following condition satisfied for the random variables {η k }.
(H 2 ) {η k } are independent random variables in E common law ℓ such that E y 2 E ℓ(dy) < ∞ and supp ℓ = E.
For example, this condition is satisfied if the random variables {η k } are of the form where {b j } are non-zero real numbers verifying +∞ j=1 b 2 j < ∞, {e j } is an orthonormal basis in E, and {ξ jk } are independent real-valued random variables whose law has a continuous density ρ j with respect to the Lebesgue measure such that +∞ −∞ x 2 ρ j (x) dx = 1, ρ j (x) > 0 for all x ∈ R and j ≥ 1.
By Proposition 1.1, the problem (5.1), (5.2) is locally well-posed in H s for any s > d/2 up to some (random) maximal time T = T (ψ 0 , η) > 0. Let P ψ0 be the probability measure corresponding to the trajectories issued from ψ 0 (e.g., see Section 1.3.1 in [KS12]).Recall that S is the unit sphere in L 2 .Theorem 5.1.Under the Conditions (H 1 ) and (H 2 ), for any s > s d and any ψ 0 ∈ H s ∩ S, we have (5.3)By the blow-up alternative, equality (5.3) gives new information in the case T (ψ 0 , η) = +∞.
Proof.Step 1. Reduction.Together with Eq. (5.1), let us consider the following truncated NLS equation: be the σ-algebra generated by the family {η j } k j=1 .The problem (5.4), (5.2) is globally well-posed.The following proposition is proved at the end of this section.
Proposition 5.2.For any ψ 0 ∈ H s and R > 0, the problem (5.4), (5.2) has a unique solution ψ R ∈ C(R + ; H s ).Moreover, the family defines an C(J; H s )-valued Markov process with respect to the filtration F k+1 .
Let us fix any 0 < M < R and consider the stopping time where the minimum over an empty set is equal to +∞.Assume we have shown that P ψ0 {τ M,R < ∞} = 1, ψ 0 ∈ H s ∩ S. (5.5) Since R > M , this implies that where τ M = min k ≥ 0 : sup t∈J, k+t<T ψ(k + t) s > M and again the minimum over an empty set is +∞.As M > 0 is arbitrary, we conclude that (5.3) holds.
Step 2. Proof of (5.5).Assume that there is an integer l ≥ 1 such that c = c(M, R) = sup This, together with the Borel-Cantelli lemma, implies (5.5).
Step 3. Proof of (5.7).By Theorem 2.3, for any ψ 0 ∈ H s d ∩ S, there is a control u ∈ E such that sup This completes the proof of the theorem.
Combining this and (5.9), we arrive at the required result.
sin x, m ∓ cos x, l ) + B(cos x, m ∓ sin x, l ) − B(sin x, m ) − B(sin x, l ) − B(cos x, m ) − B(cos x, l ) and the result of step 1.1.Step 2. Now, let us suppose that I ⊂ Z d * is a finite set such that, for any l, m ∈ I, there are vectors {n j } k j=1 ⊂ I satisfying l ⊥ n 1 , n j ⊥ n j+1 , j = 1, . . ., k −1, and n k ⊥ m.Let N = card(I) and I = {m 1 , . . ., m N }.Arguing by induction on N , we show in this step that cos x, a 1 m 1 + . . .+ a N m N , sin x, a 1 m 1 + . . .+ a N m N ∈ H ∞ (I) (4.4)