SMALL-TIME LOCAL CONTROLLABILITY OF THE BILINEAR SCHR ¨ODINGER EQUATION WITH A NONLINEAR COMPETITION

. We consider the local controllability near the ground state of a 1D Schr¨odinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known [K. Beauchard and M. Morancey, Math. Control Relat. Fields 4 (2014) 125–160, M. Bournissou, J. Diff. Equ. 351 (2023) 324–360] that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann’s S ( θ ) sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.

(1.5) Define, for p = 1, 2 and 3, the following quadratic coefficients (with respect to µ), (H quad ) The coefficients (A p K ) p=1,2,3 satisfy A 2 K = 0, (1.8) (1.9) (H cub ) The following cubic coefficient (with respect to µ) does not vanish The assumption (H reg ) is quite technical.First, under (H reg ), integrations by parts and Riemann-Lebesgue's lemma lead to ∀q ∈ N * , ⟨µφ q , φ j ⟩ = 12q π 6 j 7 (−1) j+q µ (5) (1) − µ (5) (0) + o j→+∞ 1 j 7 . (1.11) Thus, all the series considered in (1.6) and (1.10) converge absolutely.Actually, one can prove that the boundary conditions (1.3) are a necessary and sufficient condition for the series defined in (1.6) with p = 3 to converge.Moreover, the regularity and the boundary conditions on µ are chosen to have the well-posedness of the Schrödinger equation and the controllability of the linearized equation in the same functional framework.This is complicated due to the bilinearity of the equation (see Sect. 4 or [12] for more details).
Notice that the Lie brackets (1.12) and (1.13) are exactly those along which the quadratic term induces a drift in the dynamics, denying W 3,∞ -STLC for finite-dimensional systems ẋ = f 0 (x) + uf 1 (x), as stated in [7, Theorem 3].All these computations can be made rigorous when µ is regular enough, but for the sake of simplicity, the details are not given.Moreover, if we denote by, for k ≥ 1, S k the linear subspace of L 2 (0, 1) spanned by the iterated Lie brackets of f 0 and f 1 , containing f 1 at most k times, evaluated at the equilibrium φ 1 , the assumptions (H lin ), (H quad ) and (H cub ) imply that S 2 ∋ [ad 2 f0 (f 1 ), ad 3 f0 (f 1 )]φ 1 is a linear combination of Lie brackets of S 1 and of [ad 1 f0 (f 1 ), [ad 1 f0 (f 1 ), ad 0 f0 (f 1 )]]φ 1 ∈ S 3 .
The existence of a function µ satisfying (H reg ), (H lin ), (H quad ) and (H cub ) is proved in Appendix A.

Main result
First, we state the notion of small-time local controllability (STLC) used in this paper, stressing the smallness assumption imposed on the control, as it plays a key role in the validity of controllability results.Definition 1.3.Let (E T , ∥ • ∥ E T ) T >0 be a family of normed vector spaces of real functions defined on [0, T ] for T > 0. The system (1.1) is said to be E-STLC around the ground state if there exists s ∈ N such that for every T > 0 and η > 0, there exists δ > 0 such that for every ψ f ∈ S ∩ H s (0) (0, 1) with ∥ψ f − ψ 1 (T )∥ H s (0) (0,1) < δ, there exists u ∈ L 2 ((0, T ), R) ∩ E T with ∥u∥ E T < η such that the solution ψ of (1.1) associated to the initial condition φ 1 and the control u satisfies ψ(T ) = ψ f .When the linearized system is controllable, using a fixed-point theorem, one can hope to prove STLC for the nonlinear system as explained in [19,Chapter 3.1] in finite dimension.When it is not the case, one needs to go further into the expansion of the solution in the spirit of [19,Chapter 8].For the Schrödinger equation, a few STLC results are already known.
Linear behavior.Since [6], it is known that when the coefficients (⟨µφ 1 , φ j ⟩) j∈N * satisfy there exists a constant c > 0 such that ∀j ∈ N for every k ∈ N, the Schrödinger equation is H k 0 -STLC around the ground state with targets in H 7+2k (0) , using the linear test.
Quadratic behavior.In [13], the author proved that when the condition (1.14) does not hold, and more precisely, when for some n ≥ 2 (resp.n = 1) (with enough regularity on µ so that the associated series converge), the Schrödinger equation is not The paper is organized as follows.First, in Section 2, the strategy of the proof of Theorem 1.4 is presented on a toy model in finite dimension.Then, a systematic approach to recover STLC when the linearized system misses a finite number of directions is described in Section 3. Before applying this method to the Schrödinger equation, we recall in Section 4 its well-posedness and the controllability result in projection of [12].Then, the power series expansion of the Schrödinger equation is computed in Section 5. Finally, Section 6 is dedicated to the proof of Theorem 1.4.

State of the art
In infinite dimension, bilinear systems have long been considered non exactly controllable due to a result of Ball, Marsden, and Slemrod [1] (see [14,36] for generalizations).But an exact controllability result [2], proved in a different functional setting, relaunched the research.
For the bilinear Schrödinger equation (1.1), exact controllability results were first demonstrated locally around the ground state, thanks to the linear test [6,12].These results require assumptions on the dipolar moment µ entailing that the linearized system around the ground state is exactly controllable, and they hold in arbitrarily small time, because so does the controllability of the linearized system (see [23,34] for generalizations).
When these assumptions on µ do not hold, the linearized system misses some directions, but one may try to use higher order terms to recover controllability: this is the power series expansion method (see [19], Chap. 3 or [20]).This method first made it possible to recover exact controllability in large time (see [3,5,9,15,16,30]), but the small-time local controllability was still open.
The contribution of the quadratic term, along a lost direction, can be a signed quadratic form, that induces a drift of the solution along this direction, preventing controllability.In the small time asymptotic, this always happens in finite dimension [7] and is also observed in infinite dimension [9,13,21,22] (without being systematic [8]).For these obstructions to hold, the control needs to be small in a norm adapted to the Lie bracket along which the system drifts; thus the small-time local controllability was still open without this smallness assumption, and this is the problem under study in this article.
We consider a dipolar moment µ for which the linearized system loses one direction, on which appears a competition between a quadratic and a cubic term.It is known [13] that the quadratic term prevents small-time local controllability in a certain regime (in which the cubic term is negligible compared to the quadratic term).In a different regime (in which the quadratic term is negligible compared to the cubic term), we prove that small-time local controllability holds.This is the first small-time local controllability result obtained on a PDE thanks to a competition between two nonlinear terms.Such mechanisms had already been understood for ODEs (see e.g.[35]) but the proof was not suitable for a transfer to PDEs.Thus, the first contribution of this article is a new proof strategy and the second one is to run this strategy in full on the PDE (1.1).Indeed, the infinite-dimensional nature of PDEs makes the job harder than in [35].

Notation
Let T > 0. For u ∈ L 1 (0, T ), the family (u n ) n∈N of the iterated primitives of u is defined by induction as, Sometimes, to uniformize the notations of the primitives and derivatives of u, we will write u (n) when n is a negative integer to denote u |n| , the |n|-th primitive of u.

Strategy
In short, when a control system satisfies assumptions of the form (H lin ), (H quad ) and (H cub ), the state space can be divided into two subspaces: the space of the directions controllable at a linear level (and thus at a nonlinear level by the means of the inverse mapping theorem) H = Span (φ j ; j ∈ N * − {K}); the space of the direction lost at the linear level M = Span (φ K ).
The proof of Theorem 1.4 consists first in moving along the lost direction, using a "good" cubic term, despite a "bad" quadratic term, by the means of oscillating controls.Then, one can use a fixed-point theorem to conclude.

A toy model
In this section, we illustrate the method used later on the Schrödinger equation (1.1) on the following polynomial toy model, (2.2) For this example, H = Span(e 1 , e 2 , e 3 ) and M = Span(e 4 ), where (e i ) i=1,...,4 denotes the canonical basis of R 4 .First, one can notice that the control system (2.2) is not H 3 -STLC.Indeed, solving explicitly (2.2), the fourth component is given by, (2.3) Moreover, using Cauchy-Schwarz and Gagliardo-Nirenberg inequalities [32], one gets the existence of C > 0 such that for all T > 0 and u ∈ H 3 (0, T ), Thus, the quadratic term prevails on the cubic term in (2.3) when controls are small in H 3 .This allows us to deny H 3 -STLC for (2.2) (see for instance [7,13] for more details on quadratic obstructions).Nonetheless, using the strategy described above, let us prove that (2.2) is H 2 0 -STLC.Namely, it consists in proving that, for all T > 0, there exist s > 0 and a family of controls (u b ) b∈R such that, for all b ∈ R small enough, More precise statements will be given in Section 3. Let T > 0. The estimates (2.5) can be obtained using for example oscillating controls of the form, for all b ∈ R * , These controls are fitted so that the cubic term can absorb the drift along the lost direction while remaining small in the H 2 0 -norm.Indeed, substituting these controls into (2.3) and performing the change of variables t = |b| Moreover, as ϕ is supported on (0, 1), one directly has, (x 1 (T ; u b , 0), x 2 (T ; u b , 0), x 3 (T ; u b , 0)) = (u 1 (T ), u 2 (T ), u 3 (T )) = (0, 0, 0). (2.7) Besides, for all b ∈ R * , ∥u ′′ b ∥ L 2 (0,T ) ≤ ∥ϕ (5) ∥ L 2 (0,1) |b| 1 41 .
Notice that the map b → u b from R * to H 2 0 (0, T ) can be extended at zero with u 0 = 0.In the end, (2.5) holds with s = 1 41 and using Brouwer's fixed-point theorem, one can conclude on the H 2 0 -STLC of (2.2).Remark 2.1.For more toy examples in finite dimension illustrating the method developed in this paper, the interested reader can refer to [11,Section 3].
Remark 2.2.At a heuristic level, assumptions (H reg ), (H lin ), (H quad ) and (H cub ) entail that, in the asymptotic of controls small in the H 2 0 -norm, the leading terms of the solution ψ of the Schrödinger equation (1.1) along the lost direction are given by (2.8) Thus, the behavior of the Schrödinger equation (1.1) can compare in some way to the polynomial toy model (2.2).However, let us state here the main difficulties we are going to face in the infinite-dimensional framework compared to this toy model.
Because the solution of the Schrödinger equation (1.1) is complex-valued, under the assumption (H lin ), two (real) directions are lost, φ K and iφ K , and not only one.The computations of the solution along the lost directions will be much more complicated than equation (2.3), even though still explicit (see Sect. 5).
Using the same oscillating controls (2.6) as for the polynomial toy model (2.2), we will prove in the same way that the component of the solution of the Schrödinger equation along the first lost direction iφ K can be moved, that is namely, see Proposition 6.1.Then, contrary to the finite-dimensional case (see (2.7)), the behavior of the solution along the remaining components (φ j ) j∈N * −{K} is, in general, not negligible.Thus, in a second time, we will correct exactly this infinite number of directions, using that there are controllable at the linear level, and thus at the nonlinear level.Therefore, one can find a control v b such that, The core of the paper is to prove that such a linear correction does not destroy the work done in (2.9).This is done in Proposition 6.6, using sharp and simultaneous estimates on the linear control v b obtained in [12,Theorem 1.2].
The motion along the second lost direction φ K will be built from the motion along iφ K , see Proposition 6.8.

Method of control variations
Under (H lin ), the linearized equation around the ground state of the Schrödinger equation (1.1) is not controllable: it misses one complex direction φ K .This situation can be called 'controllability up to (real) codimension two'.The goal of this section is to propose a systematic approach to deal with these situations, which is different from the ones already known for ODEs and better adapted to PDEs.
For finite-dimensional systems, the classical approach used by Kawski [28,Theorem 2.4] consists in, proving that any lost direction is a 'tangent vector' thanks to oscillating controls, and deducing STLC thanks to a time-iterative process that uses arbitrarily small time intervals.For PDEs, using small time intervals is not comfortable, due to the control-cost explosion when the time goes to zero.Therefore, in our new approach, Kawski's time-iterative process is replaced by Brouwer's fixed-point theorem.This is why our new notion of 'tangent vector' contains a continuity property.

Main result
To encompass finite and infinite-dimensional systems, STLC is discussed in terms of the surjectivity of the end-point map.Let us introduce first the functional setting.Let X be a Hilbert space over R. Let (E T , ∥ • ∥ E T ) T >0 be a family of normed vector spaces of functions defined on [0, T ] for T > 0. Assume that for all T 1 , T 2 > 0, u ∈ E T1 and v ∈ E T2 , the concatenation of the two functions u#v defined by is in E T1+T2 with moreover the following estimate: For example, for any positive integer k, the space H k 0 (0, T ) satisfies this property whereas H k (0, T ) does not.Finally, let (F T ) T >0 be a family of functions from X × E T to X for T > 0. Later, in our application, F T will denote the end-point map of a control system, but the following statements hold for general operators.Now, let us introduce the new definition of tangent vector used in this paper.Definition 3.1.A vector ξ ∈ X is called a small-time E-continuously approximately reachable vector if there exists a continuous map Ξ : [0, +∞) → X with Ξ(0) = ξ such that for all T > 0, there exist C, ρ, s > 0 and a continuous map b The family (u b ) b∈(−ρ,ρ) (resp.the map Ξ) is called the control variations (resp.the vector variations) associated with ξ.
Remark 3.2.Let us stress that, for finite-dimensional systems, in [28], Kawski (see also the work of Frankowska [25,26]) introduced rather the following definition: a vector ξ is said to be a tangent vector if there exist m > 0 and a family of controls (u T ) T >0 such that Definition 3.1 is different: the final time and the amplitude of the target are unrelated.This allows constants in (3.3) badly quantified with respect to the final time T , which is not possible in (3.4).
Our systematic approach is given in the following statement.
The space H := Ran dF T (0, 0).(0, •) does not depend on T , is closed and of finite codimension n.
(A 5 ) There exists M a complementary to H that admits a basis (ξ i ) i=1,...,n of small-time E-continuously approximately reachable vectors.
Then, for all T > 0, F T is locally onto from zero: for all η > 0, there exists δ > 0 such that for all x f ∈ X with ∥x f ∥ X < δ, there exists u ∈ E T with ∥u∥ E T < η such that Let us stress that similar results are already known for finite-dimensional systems (see [28], Thm.2.4 or [35] for instance).Theorem 3.3 provides a new method for these systems but is mostly a new tool to prove the STLC of infinite-dimensional systems when a finite number of directions is lost at the linear level.
Remark 3.4.If in addition to (A 1 ) − (A 5 ), we assume that then, for all T > 0, F T is locally onto: For all η > 0, there exists δ > 0 such that for all (x 0 , Indeed, let (x 0 , x f ) ∈ X 2 with ∥x 0 ∥ X + ∥x f ∥ X < δ.By Theorem 3.3, there exist u, v ∈ E T such that F T (0, u) = x 0 and F T (0, v) = x f .Then, using successively (A 3 ) and (A 6 ), one has ) is for convenience.Actually, in the proof of Theorem 3.3, one needs the following estimates: for all T > 0 and R > 0, there exists C > 0 such that for all (x, u) Both estimates follow from Taylor's formula when F T is of class C 2 with F T (0, 0) = 0 (expanding each term around (0, 0) in (3.6), one can check that there is no linear term in the expansion, the cross product on the right-hand side stems then from the bilinearity of the second-order differential).
Remark 3.6.When F T denotes the end-point map of a control system, (A 1 ) is linked to the well-posedness of the system; also, we assume that (0, 0) is an equilibrium of the system, the translation of our result to another equilibrium is not detailed (see Theorem 6.9 for instance); (A 2 ) is linked to the continuity of the solutions of the linearized system; (A 3 ) is related to the semigroup property of the equation; (A 4 ) means that the linearized system is controllable up to finite codimension; (A 5 ) means that the directions lost at the linear level can be recovered using oscillating controls; and (A 6 ) is linked to the time reversibility of the equation.

Proof of Theorem 3.3
The first tool is the local surjectivity of the nonlinear map F T up to finite codimension.
Proposition 3.7.Under the assumptions of Theorem 3.3, let T > 0, N be a complementary to H and P be the projection on H parallely to N .Then, F T is locally onto in projection on H: there exist δ 0 , C > 0 and a with the size estimate Proof.The proof follows by applying the inverse mapping theorem to which is of class C 1 on a neighborhood of (0, 0) between Banach spaces, and whose differential at (0, 0) is onto by construction of the space H.
Thus, (3.16) and (3.17) lead to Then, estimates (3.14), (3.15) and (3.18) lead to (3.13) for k + 1.This concludes the induction and (3.13) holds for all T > 0. Now, using the estimate (3.13), we prove the statement of Theorem 3.8.The following map is continuous and does not vanish at zero, (t 1 , . . ., t n , t1 , . . ., tn ) → det P M [dF t1 (0, 0).(Ξ 1 ( t1 ), 0)], . . ., P M [dF tn (0, 0).(Ξ n ( tn ), 0)] , where P M denotes the projection on M, defined in (A5), parallely to H. Thus, there exists T * > 0 such that for all T ∈ [0, T * ), the space is a complementary to H.Then, using (3.13), the proof is concluded with Notice that the continuity of the map z → u z stems from the ones of b → u i b .Remark 3.9.Proposition 3.8 is written this way to prepare the use of Brouwer's fixed-point theorem in the proof of Proposition 3.3.However, the proof of Proposition 3.8 gives also that every linear combination of smalltime E-continuously approximately reachable vectors is a small-time E-continuously approximately reachable vector.Indeed, if z = N i=1 z i ξ i is such a linear combination, then the estimate (3.13) entails the existence of two continuous maps with Ξ(0) = z (by construction of the maps Ξ i and thanks to (A 2 )) such that (3.3) holds.
Finally, one can prove Theorem 3.3: the higher order control variations constructed in Proposition 3.8 and the local surjectivity of F T up to finite codimension given in Proposition 3.7 are enough to gain back the controllability lost at the linear level.
Proof of Theorem 3.3.Let T > 0 the final time, T 1 ∈ (0, T ) an intermediate time and η > 0 the accuracy on the control.Define δ := min(δ 0 , η 4C ) where δ 0 and C are defined by Proposition 3.7.Let x f in X with ∥x f ∥ X < δ.Notice that, as F T (0, 0) = 0, it is enough to prove the theorem for T sufficiently small.
Step 1: Steering 0 almost to x f .Let M T1 be the complementary to H given by Proposition 3.8.The goal of this step is to construct a n-parameters family (v z ) z∈M T 1 such that, for all z ∈ M T1 small enough, one has ) ) where P M T 1 denotes the projection on M T1 parallely to H.By Proposition 3.8, there exist C, ρ, s > 0 and a continuous map z Denote by (e T1 i ) i=1,...,n a basis of M T1 .Then, the following map is continuous and non-vanishing at zero, Thus, for T small enough, P M T 1 [dF T −T1 (0, 0).(•, 0)] is invertible from M T1 to M T1 with a continuous inverse by the open mapping principle.Hence, there exists a linear continuous map h from M T1 to M T1 such that, Finally, for all z ∈ M T1 ∩ B X (0, ρ), we define where Γ T −T1 : B X (0, δ 0 ) × (B X (0, δ 0 ) ∩ H) is constructed in Proposition 3.7 with the complementary M T1 .As F T1 (0, u h(z) ) → 0 when z goes to 0, for ρ small enough, one has ∥F T1 (0, u h(z) )∥ X < δ 0 , and thus, the family (v z ) is well-defined.Size estimate.
Target almost reached.Moreover, using (A 3 ), one has, Yet, using the estimates (3.9) on Γ T −T1 , (3.22) on F T1 (0, u h(z) ) and the continuity of h, the last term of the right-hand side of (3.26) is estimated by C∥z∥ 2 + C∥Px f ∥ 2 .Besides, using the Taylor expansion (3.7), the second term of the right-hand side of (3.26) is estimated by using the estimate (3.22), the construction (3.23) and the continuity of h.Moreover, by definition of H in (A 4 ), Thus, using again the Taylor expansion (3.7), the first term of the right-hand side of (3.26) is estimated by Step 2: Steering 0 to x f .Thanks to (3.19) and (3.21), to conclude the proof, it remains to prove the existence To that end, we apply Brouwer's fixed-point theorem to the function First, by continuity of F T , Γ T −T1 , h and z → u z , the map z → v z defined in (3.24) is continuous from M T1 to E T .Thus, G x f is continuous.It remains to prove that it stabilizes a ball.Let ρ ′ ∈ (0, ρ) such that Cρ ′s < 1/2 and reduce δ > 0 so that Cδ 2 + δ < ρ ′ /2 where C is given in (3.20).Then, using estimate (3.20), one has, for all z ∈ M T1 ∩ B X (0, ρ ′ ), Thus, one can apply Brouwer's fixed-point theorem to G x f to conclude the proof.
Remark 4.3.The continuity results of Proposition 4.2 are sharp in the sense that, for instance, the operators (4.2), (4.3) and (4.5) do not map continuously H 7 (0) into itself.

Dependency of the solution with respect to the initial condition
We write ψ(•; u, ψ 0 ) to denote the solution of (1.1) associated with control u and initial data ψ 0 to keep track of such a dependency.Then, from Theorem 4.1, one can deduce the following result about the dependency of the solution of (1.1) with respect to the initial condition.

Controllability in projection with simultaneous estimates
Now, we recall a local controllability result in projection given in [12,Theorem 1.2], and obtained by the linear test.To that end, we denote by H := Span C (φ j , j ∈ N * − {K}) and the orthogonal projection on H by Pψ = ψ − ⟨ψ, φ K ⟩φ K for all ψ ∈ L 2 (0, 1).Besides, for any integer k ∈ N * , we introduce the following norm, where we recall that the family (u n ) n∈N of the primitives of u is defined in (2.1).One must be careful: these norms do not coincide with the usual H −k -norms (see [10], Lem.3.3.1 for more details).However, this notation is taken to simplify the following statement by being able to write ∥ • ∥ H k (0,T ) for positive and negative integers k. for all m ∈ {0, 1, 2} with the same control map.

Error estimates on the expansion of the solution
The goal of this section is to compute the power series expansion of the solution ψ of the Schrödinger equation (1.1), as it is the key to prove Theorem 1.4.Moreover, this expansion will be studied under the following asymptotic.Definition 5.1.Given two scalar quantities A(T, u) and B(T, u), we write A(T, u) = O(B(T, u)) if there exist C, T * > 0 such that for any T ∈ (0, T * ), there exists η > 0 such that for all u ∈ H 2 0 (0, T ) with ∥u∥ H 2 0 (0,T ) < η, we have |A(T, u)| ⩽ C|B(T, u)|.
Remark 5.2.The notation O introduced is not usual, but similar notations have already been used (see [8], Def.3.1 or [7], Def.10).Namely, it refers to the convergence ∥u∥ H 2 0 (0,T ) → 0.Moreover, this convergence holds uniformly with respect to the final time on a small time interval [0, T * ], but the smallness assumption on the control can depend on the final time T .For instance, the following estimates hold, ) (5.3)

The auxiliary system trick
More precisely, our goal is to prove that when µ satisfies (H lin ), (H quad ) and (H cub ), along the lost direction, the linear term of the expansion vanishes, and the cubic term prevails on the quadratic one (and on the terms of order higher than four).As the quadratic term namely behaves as the L 2 -norm of u 3 (the third time primitive of the control), classical error estimates on the expansion involving the L 2 -norm of the control u are not sharp enough.One can compute sharper estimates, involving rather the L 2 -norm of the primitive u 1 of the control u by introducing the new state (5.4) This new state satisfies the following equation, called the auxiliary system (5.5)This idea was introduced in [18] and later used in [9, Section 3.2] and [13, Section 4.2] for the Schrödinger equation.This strategy can also be found in [7,Section 4] to study the quadratic behavior of scalar-input systems or in [4, Section 7.1] to give refined error estimates for various expansions of scalar-input affine control systems.First, we state the well-posedness of the auxiliary system, stemming from Theorem 4.1.

Expansion of the auxiliary system
The first, second and third order terms of the expansion of the solution ψ of the auxiliary system (5.5)around the trajectory (ψ eq = ψ 1 , u eq = 0) are solutions of, ) ) Moreover, one has the following estimates, ) Besides, the following equalities hold in H 5 (0) for every t ∈ [0, T ], ) (5.17) From this, one can deduce the behavior of the first three terms of the expansion of the auxiliary system along the lost direction.Indeed, first, the solution of (5.9) can be computed explicitly as Thus, one can deduce directly the following result.
Proposition 5.5.When µ satisfies (1.4), Then, substituting the explicit form of Ψ given in (5.18) into (5.16), the quadratic term can also be explicitly computed as where, for all j ∈ N * , the quadratic kernel k quad,j is given by (5.21) Thanks to Remark 1.1, all the quadratic kernels k quad,j are bounded in C 4 (R 2 , C).This regularity is the key to perform integrations by parts and reveal a coercivity quantified by the H −3 -norm of the control, as stated in the following result.

Sharp error estimates for the auxiliary system
The goal of this section is to compute sharp error estimates on the expansion of the auxiliary system.
Proposition 5.7.If µ satisfies (H reg ), then the following error estimates on the expansion of the auxiliary system hold, (5.30) Proof.Proof of (5.29).First, by Proposition 5.3, the following equality holds in H 5 (0) for all t ∈ [0, T ], where every term under the integral belongs to H 5 ∩ H 3 (0) .Thus, the triangle inequality and the fact that for all s > 0, e iAs is an isometry from H 3 (0) to H 3 (0) give, using estimate (5.6) on ψ and (5.1).Then, using (5.7) and (5.15), the following equality holds in Once again, every term under the integral belongs to H 5 ∩ H 3 (0) thanks to Theorem 4.2, so using the triangle inequality, estimates (5.6) and (5.31), one has Proof of (5.30).Once again, the following equality holds in Moreover, as before, one can prove that (5.32) Therefore, (5.30) follows from (5.29) and (5.32).
To sum up, one has the following result.
Proof.The computations (5.19), (5.23), (5.25) and the error estimate (5.30) give that the right-hand side of (5.33) is estimated by Besides, for every control u such that u 2 (T ) = u 3 (T ) = 0, integrations by parts and Cauchy-Schwarz's inequality prove that recalling that we work in the asymptotic of controls small in H 2 0 (see Def. 5.1).

Expansion of the Schrödinger equation
From the expansion of the auxiliary system, one can deduce sharp error estimates on the expansion of the Schrödinger equation.The first, second and third order terms of the expansion of the solution ψ of the Schrödinger equation (1.1) around the trajectory (ψ eq = ψ 1 , u eq = 0) are solutions of, ) ) with Dirichlet boundary conditions, and the null function as initial data.Moreover, by identifying the same order terms in (5.4), one has the following links with the expansion of the auxiliary system, (5.39) Then, from the expansion of the auxiliary system, one can deduce sharp error estimates on the expansion of the Schrödinger equation (1.1).
To conclude on the error estimate on the expansion of the Schrödinger equation, one needs to estimate the boundary term u 1 (T ).This can be done for specific motions of the solution.
Corollary 5.11.Let µ satisfying (H reg ).Then, for every control u ∈ H 2 0 (0, T ) such that the solution of (1.1) satisfies (5.42), the following estimate holds, If (H quad ) is replaced by A 1 K ̸ = 0, by Lemma 5.6, the quadratic term is bounded by ∥u 1 ∥ 2 L 2 .Thus, for controls small in W −1,∞ , the quadratic term prevails on every cubic term, and the Schrödinger equation (1.1) is not W −1,∞ -STLC.If (H quad ) is replaced by A 1 K = 0 and A 2 K ̸ = 0, then Lemma 5.6 gives that the quadratic term is bounded by ∥u 2 ∥ 2 L 2 .However, the cubic term cannot absorb simultaneously such a quadratic term and the quartic term as by Cauchy-Schwarz's inequality.Thus, one cannot hope for a STLC result, unless one manages to prove a sharper error estimate than (5.44).
Under (H quad ), the quadratic term is bounded by ∥u 3 ∥ 2 L 2 (see (5.23)).This time, • in the asymptotic of controls small in H 2 0 , the cubic term can handle simultaneously such a quadratic term and the terms of order higher than four, as done in Section 6, leading to the H 2 0 -STLC of the Schrödinger equation; • in the asymptotic of controls small in H 3 0 , Gagliardo-Nirenberg inequalities prove that the quadratic term prevails on the cubic term (and on the higher-order terms), and thus one can deny H 3 0 -STLC as done in [13].The reader can refer to the toy model in Section 2.3 for more details on this competition between the quadratic and cubic terms.

Motions in the lost directions
The motions in the ±iφ K directions are done in two steps.
(a) First, we initiate the motion along ±iφ K by noticing that when µ satisfies (H lin ), (H quad ) and (H cub ), along iφ K , the solution is driven by a cubic term which enables us to move in both + and −iφ K directions.However, at the end of this step, along the other directions (φ j ) j∈N * − {K} , the error is possibly big (in a sense to precise).(b) Thus, this error is corrected using the local controllability in projection result given in Theorem 4.5.
However, one needs to make sure that such linear motions do not induce a too large error along iφ K and preserve the work done in the first step.
Then, the motions in the ±φ K directions are deduced from the motions along ±iφ K , exploiting a rotation phenomenon.
Expansion of the solution.As u 1 (T 1 ) = 0, by (5.4), the end-point of ψ and ψ are the same.Thus, it suffices to prove (6.1) with ψ instead of ψ.Moreover, Corollary 5.8 gives the following expansion of the auxiliary system when b goes to zero, because, using (6.2), one has the following estimates, and Then, substituting the explicit form of the control (6.4) and performing the change of variables (t, τ ) = (|b| 41 θ 2 ), the two integral terms of (6.5) are given by i sign(b)|b| Expanding the kernels when b goes to zero, as they are both bounded in C 1 (R 2 , C), one gets that (6.5) can be written as Moreover, looking at the definitions of C K , k 1 cub,K and k 2 cub,K given respectively in (1.10), (5.26) and (5.27), one can notice that Thus, the previous inequality leads to (6.1) by choice of ϕ given in (6.4).

Preparation of Step (b): Estimate on the evolution of the solution along the lost direction
To make sure that the linear correction of Step (b) will not destroy the motion done along the lost direction in Step (a) (see Proposition 6.1), one needs to quantify what happens along the lost direction on a time interval [T 1 , T ], that is to quantify First, this estimate is computed on the quadratic term of the expansion of the Schrödinger equation.Proposition 6.2.Let 0 < T 1 < T .If µ satisfies (H quad ), then for all w in H 2 0 (0, T ) such that w 1 (T 1 ) = w 2 (T 1 ) = w 3 (T 1 ) = w 2 (T ) = w 3 (T ) = 0, the solution of (5.35) satisfies |⟨ξ(T ; w, φ 1 ), ψ K (T )⟩ − ⟨ξ(T 1 ; w, φ 1 ), ψ K (T 1 )⟩| Proof.Using the link with the auxiliary system (5.38) and the explicit form of Ψ given in (5.18), one has, for every t ∈ [0, T ], where the quadratic kernel k quad,t is given by, Thus, if the control satisfies w 1 (T 1 ) = 0, then, Besides, using the estimate of ξ given in (5.23), one has As the kernel k quad,T is bounded (independently of T ), the third term of the right-hand side of (6.12) is naturally estimated by O(|w 1 (T )|∥w 1 ∥ L 1 (0,T ) ).However, this estimate will not be sharp enough to use in the sequel of this paper (and more precisely in the proof of Proposition 6.6).Thus, one computes a sharper estimate by performing one integration by parts in the integral to get an estimate by O(|w 1 (T )|∥w 2 ∥ L 1 (0,T ) ), noticing that k ′ quad,T is still bounded thanks to Proposition 1.1.
Then, the estimate is computed for the cubic term of the expansion of the auxiliary system.Proposition 6.3.Let 0 < T 1 < T .For all w in H 2 0 (0, T ) such that w 2 (T 1 ) = w 2 (T ) = 0, the solution of (5.11) satisfies Proof.Using the explicit form of ζ given in (5.25), one has, as the kernel k 3 cub,K defined in (5.28) is bounded in C 0 (R 3 , C).The first term of the right-hand side is bounded by ∥w 1 ∥ 2 L 2 (T1,T ) ∥w 1 ∥ L 1 (0,T ) as k 1 cub,K is also bounded.Moreover, it would seem natural to estimate the second term by ∥w 1 ∥ L 1 (T1,T ) ∥w 1 ∥ 2 L 2 (0,T ) .However, as before, it would not provide an estimate sharp enough to use in the sequel of this work.One can compute a sharper estimate by performing one integration by parts to get that the second term of the right-hand side of the previous equality is bounded by, as the kernel k 2 cub,K defined in (5.27) is bounded in Then, one can deduce the estimate for the cubic term of the expansion of the Schrödinger equation.
From the behavior of the quadratic and cubic terms given in (6.10) and (6.14), the error estimate (5.44) and the estimate (5.43) on the boundary term w 1 (T ), one can deduce the following estimate.Theorem 6.5.Let 0 < T 1 < T , µ satisfying (1.4) and (H quad ).For all w in H 2 0 (0, T ) such that w 1 (T 1 ) = w 2 (T 1 ) = w 3 (T 1 ) = w 2 (T ) = w 3 (T ) = 0, if the solution of (1.1) satisfies the specific motion (5.42), then one has Step (b): The vector iφ K is an approximately reachable vector Proposition 6.6.Let µ satisfying (H reg ), (H lin ), (H quad ) and (H cub ).Then, the vector iφ K is a small-time H 2 0continuously approximately reachable vector associated with vector variations Ξ(T ) = iψ K (T ).More precisely, for all T > 0, there exist C, ρ > 0 and a continuous map b → w b from R to H 2 0 (0, T ) such that, with the following size estimate, ∀b ∈ (−ρ, ρ), ∥w b ∥ H 2 0 (0,T ) ⩽ C|b| 1 41 .(6.17) Proof.Construction of the control.To move along the ±iφ K directions, we use non-overlapping controls.More precisely, let 0 < T 1 < T and define, for all b ∈ R, where (u b ) b∈R is the family of controls defined on [0, T 1 ], constructed in Theorem 6.1 and where Γ T1,T is a control operator such that the solution of (1.1) on [T 1 , T ] with control u = Γ T1,T (ψ 0 , ψ f ) and initial condition ψ(T 1 ) = ψ 0 satisfies (4.9) with the boundary conditions on the control (4.10) and the estimates (4.11) (the construction of this operator is deduced from Theorem 4.5 by a translation of controls and a change of global phase on the state, not detailed here for the sake of simplicity).
Size of the controls.Because we use non-overlapping controls, for all b ∈ R, one has using the size estimate (6.2) on (u b ) b∈R .Besides, using the linear estimates (4.11) on Γ T1,T and the estimates (6.3) on the end-point of the solution at time T 1 , for all ε ∈ (0, 3  4 ), one gets the existence of C, ρ > 0 such that for all b ∈ (−ρ, ρ) and m ∈ {−3, . . ., 2}, Taking ε < 1 4 , the estimates (6.19) and (6.20) imply (6.17).Motion along ±iφ K .By construction of Γ T1,T (see (4.9)), we already know that P(ψ(T ; w b , φ 1 )) = ψ 1 (T ) = P (ψ 1 (T ) + ibψ K (T )) , where P denotes the orthogonal projection on Span C (φ j , j ∈ N * − {K}).Thus, to prove (6.16), it only remains to prove that Then, it remains to prove that the linear correction used on the time interval [T 1 , T ] did not destroy this estimate, that is to prove that, for all b ∈ (−ρ, ρ), By Theorem 6.5, the left-hand side of the previous inequality is estimated by Therefore, it remains to use the estimates (6.2) on (u b ) b∈R and (6.20) on (v b ) b∈R to prove that the previous quantity is bounded by |b| 1+ 1 82 .For example, using the estimates in H −3 , one has Choosing ε < 1/24, for b small enough, these three terms are bounded by |b| 1+ 1 82 .Then, in (6.22), it only remains to estimate the last term.As (6.20) provides only estimates of (v b ) b∈R in L 2 -spaces, one can use a Gagliardo-Nirenberg inequality (see [32], Thm.p. 125) to estimate ∥v 2 ∥ L ∞ (T1,T ) .More precisely, there exists C > 0 such that Thus, thanks to (6.20) And, finally, one has Remark 6.7.The sharp estimates (4.11) and (6.15) on the control operator Γ T1,T and on the evolution of the solution along the lost direction are the key to move along the first lost direction iφ K .

The vector φ K is an approximately reachable vector
The second approximately reachable vector can be deduced from the first one using a proof inspired by the work [27,Theorem 6] of Hermes and Kawski, where the authors proved that for affine-control systems of the form ẋ = f 0 (x) + uf 1 (x), if for some Lie bracket V of f 0 and f 1 , V (0) is a tangent vector (in the sense of (3.4)), then [f 0 , V ](0) is also a tangent vector.Proposition 6.8.Let µ satisfying (H reg ), (H lin ), (H quad ) and (H cub ).Then, the vector φ K is a small-time H 2 0continuously approximately reachable vector associated with vector variations Ξ(T ) = ψ K (T ).More precisely, there exists T * > 0 such that for all T ∈ (0, T * ), there exist C, ρ > 0 and a continuous map b with the following size estimate, Proof.Denote by (u b ) b∈R the control variations associated with iφ K , constructed in Proposition 6.6.The goal is to prove the existence of C > 0 such that for all (α, β) ∈ R 2 small enough, one has one gets the existence of a family (v b ) b∈R satisfying (6.23) and (6.24).So, it remains to prove (6.25).First, by construction of (u b ) b∈R in Proposition 6.6, there exist C, ρ > 0 such that for all α ∈ (−ρ, ρ), Then, on [T, 2T ], no control is activated, so ψ(2T ) = e −iAT ψ(T ) and (6.26) becomes ⩽ C|α| 1+ 1 82 .(6.27) Then, using the semigroup property of the Schrödinger equation, one has, Together with Proposition 4.4 about the dependency of the solution of the Schrödinger equation with respect to the initial condition, this gives .
Using the estimates (6.17) on (u β ) β∈R and (6.27), the right-hand side of the previous inequality is bounded by |β| 1 41 |α|.Then, using once again the estimate (6.27) and the construction of (u β ) β given in (6.16), the previous inequality entails which gives (6.25) and this concludes the proof.
The proof of Theorem A.2 is in four steps.
First, using Baire theorem to deal with the infinite number of non-vanishing conditions, one can find a function µ ref satisfying (1.4), (1.10), (A.3) and (A.4).Notice that only the non-vanishing condition (1.9) is not treated at this stage as the strategy of the two following steps, relying on oscillating functions, would destroy this condition.Then, using some analyticity and the isolated zeros theorem, one constructs μref a perturbation of µ ref satisfying (1.7) while conserving all the previous properties already satisfied by µ ref .
Similarly, one constructs then µ ref a perturbation of μref satisfying (1.8) while conserving all the previous properties satisfied by μref .Finally, using the construction of a quadratic basis, from µ ref , one constructs a new function satisfying (1.9) in addition to the previous conditions.
The goal of Step 1 is to prove that U is not empty.As E is not empty, it suffices to prove that U is dense in E.Moreover, denoting by C := {µ ∈ E; C K (µ) ̸ = 0} , V := µ ∈ E; µ (5) (0) ̸ = 0 and U j := {µ ∈ E; ⟨µφ 1 , φ j ⟩ ̸ = 0} , U is the intersection of all the open subsets V, C and U j for j ∈ N * − {K}.Thus, as E is a complete space (because closed in H 11 ), by Baire theorem, to prove that U is dense in E, it suffices to prove that V, C and U j for j ∈ N * − {K} are dense in E. The density of V is clear.Let j ∈ N * − {K}.
We consider perturbations of µ ref of the following form, ) so that the quadratic and cubic forms can be seen as additive.Moreover, by construction, for all ε > 0 and λ ̸ = 0, ν ε,λ already satisfies (1.4), (A.2) and (A.3).

4. 1 .
Well-posedness of the Schrödinger equation First, we recall the result given in [12, Theorem 2.1] about the well-posedness of the following Cauchy problem, in a functional framework suitable with Theorem 1.4,

. 11 )
with Dirichlet boundary conditions, and the null function as initial data.Using the well-posedness result given in Theorem 4.1, Proposition 4.2 and the smoothing effect of Proposition 5.4, one can prove that Ψ, ξ, ζ are in C 2 ([0, T ], H 7 ∩ H 5 (0) ).

.44) Remark 5 . 12 .
This expansion of the solution of the Schrödinger equation (1.1) allows us to shed new light on the assumptions made on µ in Section 1.3.

for b small enough, choosing ε < 1 4 . 2 L 2 (
Similarly, one gets,∥w 1 ∥ 2 L 2 (0,T ) ∥w 2 ∥ L 1 (0,T ) T1,T ) ∥w 1 ∥ L 1 (0,T ) bounded by |b| 1+ 1 82 .This concludes the proof of (6.16).Continuity of b → w b .The map b → u b of Proposition 6.1 is continuous from R to H 2 0 (0, T 1 ).Besides, the continuity of b → v b from R to H 2 0 (T 1 , T ) stems from the regularity of Γ T1,T (see Theorem 4.5) and of the solution of the Schrödinger equation with respect to the control.This gives the continuity of the map b → w b from R to H 2 0 (0, T ).

Figure A. 1 .
Figure A.1.The supports of the functions used in Step 2 are depicted.