Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time

We consider a linear Schr\"odinger equation, on a bounded interval, with bilinear control. Beauchard and Laurent proved that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. Coron proved that a positive minimal time is required for this controllability, on a particular degenerate example. In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.


The problem
Let us consider the 1D Schrödinger equation x ψ(t, x) − u(t)µ(x)ψ(t, x), (t, x) ∈ R × (0, 1), ψ(t, 0) = ψ(t, 1) = 0, t ∈ R. (1.1) Such an equation arises in the modelization of a quantum particle, in an infinite square potential well, in a uniform electric field with amplitude u(t). The function µ : (0, 1) → R is the dipolar moment of the particle. The system (1.1) is a bilinear control system in which the state is the wave function ψ, with ψ(t) L 2 (0,1) = 1, ∀t ∈ R and the control is the real valued function u.
In this article, we study the minimal time required for the local controllability of (1.1) around the ground state. Before going into details, let us introduce several notations. The operator A is defined by The family (ϕ k ) k∈N * is an orthonormal basis of L 2 ((0, 1), C) and ψ k (t, x) := ϕ k (x)e −iλ k t , ∀k ∈ N * is a solution of (1.1) with u ≡ 0 called eigenstate, or ground state, when k = 1. We denote by S the unit L 2 ((0, 1), C)-sphere. In this article, we consider two types of initial conditions for (1.1): the ground state ψ(0, x) = ϕ 1 (x), x ∈ (0, 1), (1.4) or an arbitrary one ψ(0, x) = ψ 0 (x), x ∈ (0, 1). (1.5) Now, let us define the concept of local controllability used in this article.
Definition 1 Let T > 0, X and Y be normed spaces such that X ⊂ L 2 ((0, 1), C) and Y ⊂ L 2 ((0, T ), R). The system (1.1) is controllable in X, locally around the ground state, with controls in Y , in time T , if, for every ǫ > 0, there exists δ > 0 such that, for every ψ f ∈ S ∩ X with ψ f − ψ 1 (T ) X < δ, there exists u ∈ Y with u Y < ǫ such that the solution of the Cauchy problem (1.1)-(1.4) satisfies ψ(T ) = ψ f .
In particular, this definition requires that arbitrarily small motions may be done with arbitrarily small controls.

A first previous result
First, let us introduce the normed spaces The following result, proved in [10], emphasizes that the local controllability holds in any positive time when the dipolar moment µ satisfies an appropriate non-degeneracy assumption.
Note that the function spaces in Theorem 1 are optimal. Indeed, they are the same as for the well posedness of the Cauchy problem (1.1)-(1.4) (see Proposition 1).
Finally, let us summarize the proof of Theorem 1 in [10]. This proof relies on the linear test (see [16,Chapter 3.1]), the inverse mapping theorem and a regularizing effect. In particular, the assumption (1.7) is necessary for the linearized system to be controllable in H 3 (0) ((0, 1), C) with controls in L 2 ((0, T ), R). When one of the coefficients µϕ 1 , ϕ k vanishes, then the linearized system is not controllable anymore and the strategy of [10] fails.

A second previous result
The first article in which a positive minimal time is proved, for the local controllability of system similar to (1.1), is [15]. In this reference, Coron considers the control system    i∂ t ψ(t, x) = −∂ 2 x ψ(t, x) − u(t)(x − 1/2)ψ(t, x), (t, x) ∈ R × (0, 1), ψ(t, 0) = ψ(t, 1) = 0, t ∈ R, s ′ (t) = u(t), d ′ (t) = s(t), t ∈ R, (1.9) where the state is (ψ, s, d) and the control is the real valued function u. This system represents a quantum particle in a moving box: u, s, d are the acceleration, the speed and the position of the box. Note that, here, the relation (1.7) is not satisfied: thus Theorem 1 does not apply. On one hand, it is proved in [9] that this system is controllable in H 7 (0) ((0, 1), C) × R × R, locally around the ground state (ψ = ψ 1 , s = 0, d = 0), with controls u ∈ L ∞ ((0, T ), R), in time T large enough.
On the other hand, Coron proved in [15] that this local controllability does not hold in arbitrary time: contrary to Theorem 1, a positive minimal time is required for the local controllability. Precisely, Coron proved the following statement.
The goal of this article is to go further in this analysis: 1. we propose a general context for the minimal time to be positive (in particular, the variables s and d are not required anymore in the state), 2. we propose a sufficient condition for the local controllability to hold in large time; this assumption is compatible with the previous context and weaker than (1.7), 3. we work in an optimal functional frame, for instance, our non controllability result requires u small in H −1 -norm, not in L ∞ -norm as in Theorem 2, 4. we perform a first step toward the characterization of the minimal time.

Main results of this article
The first result of this article is the following one. 10) and α K ∈ {−1, +1} be defined by There exists T * K > 0 such that, for every T < T * K , there exists ǫ > 0 such that, for every u ∈ L 2 ((0, T ), R) with First, let us remark that the assumption (1.10) holds, for example, with µ(x) = (x− 1/2) and K = 1. In particular, Theorem 3 applies to the particular case studied by Coron in [15]. Thus, the variables (s, d) are not required in the state for the minimal time to be positive. Moreover, the control u does not need to be small in L ∞ as in Theorem 2: a smallness assumption in H −1 (0, T ) is sufficient.
Note that the validity of the same result without the assumption 'A K = 0' is an open problem (see remark 2 for technical reasons). A possible (but not optimal) value of T * K is given in (3.18). The proof of Theorem 3 relies on an expansion of the solution to the second order.
The second result of this article is the following one.
A direct consequence of Theorems 3 and 4 is the following result.
Note that an explicit upper bound T ♯ for the minimal time T min is proposed in the proof (see (4.13)).
Finally, let us summarize the proof of Theorem 4. Under assumption (1.13), only a finite number of the coefficients µϕ 1 , ϕ k vanish (see (1.8)). Thus, the linearized system around the ground state is not controllable along a finite number of directions. We will see that all of these directions are recovered at the second order. Moreover, all these directions excepted one, present a rotation phenomena in the complex plane, for the null input solution. Thus, our proof is an adaptation of [13].
Under a weaker assumption than (1.13) and still in the framework (ψ ∈ H 3 (0) , u ∈ L 2 ), we prove the following result. (1.14) N ∈ N * and P N be the orthogonal projection from L 2 ((0, 1), C) to V N := Span{ϕ k ; k is odd and N or k is even } (resp. V N := Span{ϕ k ; k is even and N or k is odd } ). Then, for every ǫ > 0, there exists T > 0 and δ > 0 such that, for Under assumption (1.14), we prove that 1. an infinite number of directions are controlled at the first order, in any positive time, 2. all the lost directions are recovered either at the second order, or at the third order, 3. any direction corresponding to vanishing first and second orders, are recovered at the third order in arbitrary time.

A review about control of bilinear systems
The first controllability result for bilinear Schrödinger equations such as (1.1) is negative and proved by Turinici [27], as a corollary of a more general result by Ball, Marsden and Slemrod [2]. Because of this noncontrollability result, such equations have been considered as non controllable for a long time. However, progress have been made and this question is now better understood. Let us also mention that this negative result has been adapted to nonlinear Schrödinger equations in [19] by Ilner, Lange and Teismann.
Concerning exact controllability issues, local results for 1D models have been proved in [6,7] by Beauchard; almost global results have been proved in [9], by Coron and Beauchard. In [10], Beauchard and Laurent proposed an important simplification of the above proofs. In [15], Coron proved that a positive minimal time may be required for the local controllability of the 1D model. In [8], Beauchard studied the minimal time for the local controllability of 1D wave equations with bilinear controls. In this reference, the origin of the minimal time is the linearized system, whereas in the present article, the minimal time is related to the nonlinearity of the system. Let us emphasize that exact controllability has also been studied in infinite time by Nersesyan and Nersisian in [29,30]. Now, let us quote some approximate controllability results. In [11] Mirrahimi and Beauchard proved the global approximate controllability, in infinite time, for a 1D model and in [23] Mirrahimi proved a similar result for equations involving a continuous spectrum. Approximate controllability, in finite time, has been proved for particular models by Boscain and Adami in [1], by using adiabatic theory and intersection of the eigenvalues in the space of controls. Approximate controllability, in finite time, for more general models, have been studied by 3 teams, with different tools: by Boscain, Chambrion, Mason, Sigalotti [14,28,24], with geometric control methods; by Nersesyan [25,26] with feedback controls and variational methods; and by Ervedoza and Puel [18] thanks to a simplified model.
Optimal control techniques have also been investigated for Schrödinger equations with a non linearity of Hartee type in [3,4] by Baudouin, Kavian, Puel and in [17] by Cances, Le Bris, Pilot. An algorithm for the computation of such optimal controls is studied in [5] by Baudouin and Salomon.
Finally, let us quote some references concerning bilinear wave equations. In [22,21,20], Khapalov considers nonlinear wave equations with bilinear controls. He proves the global approximate controllability to nonnegative equilibrium states.

Notations
Let us introduce some conventions and notations valid in all this article. Unless otherwise specified, the functions considered are complex valued and, for example, we write H 1 0 (0, 1) for H 1 0 ((0, 1), C). When the functions considered are real valued, we specify it and we write, for example, L 2 ((0, T ), R). The same letter C denotes a positive constant, that can change from one line to another one. If (X, . ) is a normed vector space, x ∈ X and R > 0, B X (x, R) denotes the open ball {y ∈ X; x − y < R} and B X (x, R) denotes the closed ball {y ∈ X; x − y R}. We denote by ., . the L 2 (0, 1)-scalar product and by T S ϕ := {ξ ∈ L 2 (0, 1); ℜ ϕ, ξ = 0} the tangent space to S at any point ϕ ∈ S.

Structure of this article
In Section 2, we recall a well posedness result concerning system (1.1). In Section 3, we prove Theorem 3. In Section 4, we prove Theorem 4 thanks to power series expansions to the second order as in [13] (see also ([16, Chapter 8])). In Section 5, we prove Theorem 6 thanks to power series expansions to the order 2 and 3. In Section 6, we perform a first step toward the characterization of the minimal time, in a favorable situation. Finally, in Section 7, we gather several concluding remarks and perspectives.

Well posedness
This section is dedicated to the well posedness of the Cauchy problem   x ∈ (0, 1).

Heuristic
Since we are interested in small motions around the trajectory (ψ = ψ 1 , u = 0), with small controls, it is natural to try to do them, in a first step, with the first and the second order terms. Let us consider a control u of the form u = 0 + ǫv + ǫ 2 w, then, formally, the solution ψ of (1. x ∈ (0, 1), x ∈ (0, 1). where Let us assume that (1.10) holds for some K ∈ N * . By adapting the choice of v ∈ L 2 ((0, T ), R), Ψ(T ) can reach any target in the closed subspace Adh (see Proposition 19 in Appendix); but the complex direction Ψ(T ), ψ K (T ) is lost. Let us show that, when T is small, the second order term imposes a sign on the component along this lost direction, preventing the local exact controllability around the ground state.
Thanks to (3.2) and (3.3), we have where Integrations by part show that for some constant C = C(µ) > 0, thus h 2 K ∈ C 0 (R 2 , C) and the quadratic form Q 2 K,T is well defined on L 2 ((0, T ), R). In particular, Let us try to move ǫΨ(T ) and ξ(T ) = iδα K ϕ K e −iλ1T for some δ > 0. Thus the sign of Q 2 K,T (v) has to be α K . The following lemmas show that this is not possible when T is small, Proof of Lemma 1: Let T > 0 and v ∈ V T − {0}. Integrations by parts show that, for every j ∈ J, give the conclusion.
Lemma 2 Let µ ∈ H 3 ((0, 1), R) be such that (1.10) holds for some K ∈ N * . There exists T * K > 0 such that, for every T < T * Remark 2 This statement enlightens the importance of the assumption A K = 0 in Theorem 3. Indeed, if A K vanishes then we do not know whether the quadratic form Q 2 K,T has a sign on V T in small time T . Note that another integration by parts (leading to a quadratic form in σ(t) := t 0 S) is not possible, because of problems of divergence in infinite sums. Proof of Lemma 2: One may assume that A K > 0, α K = 1. We define the quantity Thanks to (3.16) and (1.10), there exists j ∈ N * − {1, K} such that µϕ 1 , ϕ j µϕ K , ϕ j = 0. Thus, C K > 0. We introduce (3.14), (3.15) and Cauchy-Schwarz inequality we get, for every S ∈ L 2 ((0, T ), R), With additional arguments, one may prove that, for T < T * K , The non existence of a positive constant c(T ) > 0 such that prevents from proving the non controllability in the framework (ψ ∈ H 3 (0) (0, 1), u ∈ L 2 ) (such an inequality would allow to deal with the nonlinear terms). Our solution relies on the fact that, for T small, the quadratic form (3.17)). Thus, for the negative result, we need to work in a framework (ψ ∈ H 1 . This is why our analysis relies on an auxiliary system studied in the next section.

Auxiliary system
Let us consider the system It is a control system in which the state is ψ, with ψ(t) L 2 ≡ 1 and the control is the real valued function s. The system (3.19) results from (1.1) through the transformation which is also used in [15]. We will work with initial conditions The well posedness of the Cauchy problem (3.19)-(3.21) may be proved similarly to [10,Proposition 3].
s L 2 (0,T ) < R, then this weak solution satisfies The proof of Theorem 3 is a direct consequence of the following result. (3.25) The proof of Theorem 7 requires several steps, thus, it is developed in Section 3.4.

Proof of Theorem 3 thanks to Theorem 7
Let T < T * K . Let ǫ > 0 be as in Theorem 7. Let u ∈ L 2 ((0, T ), R) be such that the function s(t) := t 0 u(τ )dτ satisfies s L 2 (0,T ) < ǫ. Let us assume that the solution of the Cauchy problem (1.1)- Thanks to Theorem 7, this is impossible.

Proof of Theorem 7
The proof of Theorem 7 requires the following preliminary result.
First step: Let us prove that Thanks to (3.23), there exists C > 0 such that Moreover, thanks to (3.29) and the assumption µϕ 1 , ϕ K = 0, we have We get (3.30) thanks to the 2 previous relations.
Second step: Let us prove that Thanks to (3.29) and the assumption µϕ 1 , ϕ K = 0, we have We get (3.31) thanks to (3.30).

Preliminaries
The goal of this section is the proof of the following result.
We recall that Q 2 Kj,T is defined in (3.7)-(3.8), and V T in (3.13). For the proof of Proposition 4, we need the following preliminary result.
Proof of Proposition 5: To simplify the notation of this proof, we write Q T and h, instead of Q 2 K,T and h 2 K . Let us assume that Q T ≡ 0 on V T , for every T < T * . Then ∇Q T (v) ⊥ V T , for every v ∈ V T and T < T * . Easy computations show that, for v ∈ V T , (4.2)) Thanks to Ingham inequality on (T, T 1 ) we get α k = 0, ∀k (see Proposition 19 in Appendix).
In the right hand side of the 2 previous equalities, the frequencies (λ j − λ 1 ) are 0 for every j ∈ J, while the frequencies (λ K − λ k ) are negative for every k > K. Thus, for every k > K the frequency (λ K − λ k ) appears only one time in the right hand side of each equality. The uniqueness of the decomposition on a Riesz basis gives Thus, b k = 0, ∀k ∈ J − {k * } with k > K. Coming back to (4.4), we only have a finite sum in the right hand side, over j ∈ J with j K and over k ∈ J − {k * } with k K. We deduce the existence of a unique j * ∈ J with j * K such that , satisfies h(t, τ ) = h(τ, t) and ∇Q T ≡ 0 on V T , for every T > 0. By linearity, the same conclusion holds when h is a finite sum of such terms.
Proof of Proposition 4: Thanks to three integrations by part and the Riemann-Lebesgue Lemma, we get

Strategy for the proof of Theorem 4
Until the end of Section 4, we fix µ ∈ H 3 ((0, 1), R) such that µ ′ (1) ± µ ′ (0) = 0, N ∈ N and K 1 , ..., K N ∈ N * as in Proposition 4. To simplify the notations, we assume that K 1 = 1. We define the space and, for j = 1, ..., N the space The global strategy relies on power series expansion of the solutions to the order 2 as in [13] (see also [16]). In Section 4.3, we prove the local exact controllability 'in H', with a first order strategy. Then, in Section 4.4, we prove that any direction in M is reached with the second order term. Finally, in Section 4.5, we conclude thanks to a fixed point argument.

Controllability in H in arbitrarily small time
Let us introduce the orthogonal projection The goal of this section is the proof of the following result.
Theorem 8 Let T 1 , T > 0 be such that T 1 < T . There exists δ 1 > 0 and a C 1 -map where , P T [ψ 1 (T )]) = 0 and for every (ψ 0 , ψ f ) ∈ Ω T1 × Ω T , the solution of (1.1) with initial condition ψ(T 1 ) = ψ 0 and control u : This theorem may be proved exactly as Theorem 1 in [10]. We recall the main steps of the proof because several intermediate results will also be used in the end of this article. To simplify the notations, we take T 1 = 0.
Thanks to Proposition 1, we consider the map where ψ is the solution of (1.1)-(1.5). Then Theorem 8 corresponds to the local surjectivity of the nonlinear map Θ T around the point (ϕ 1 , 0), that will be proved thanks to the inverse mapping theorem. Thus, the first property required is the C 1 -regularity of Θ T , which is a consequence of [10, Proposition 3].
The second property required for the application of the inverse mapping theorem is the the existence of a continuous right inverse for dΘ T (ϕ 1 , 0), that may be proved exactly as [10, Proposition 4] (it is a consequence of Proposition 19 in Appendix).

Reaching the missed directions, at the second order, in large time.
The goal of this section is the proof of the following result.

Proposition 8 Let T > T ♯ where
There exists a continuous map such that, for every z ∈ M , the solutions Ψ and ξ of (3.1) and (3.2) satisfy Ψ(T ) = 0 and ξ(T ) = z.
In this statement, the quantity T 2 min is defined as follows.

Lemma 3
The quantity is well defined and belongs to (0, 2/π]. Let us recall that Q 2 1,T and V T are defined in (3.11), (3.13).

Preliminaries
Our proof of Proposition 8 requires 3 preliminary results. The first one consists in proving the existence of controls such that the projections of the second order term on the lost directions are non zero.
The second preliminary result for the proof of Proposition 8 is a measure of the rotation of the null input solution, precised in the next statement.

Proof of Lemma 4: We have
The same relations hold for ξ θ .
The third preliminary result for the proof of Proposition 8 is the non overlapping principle.

Proof of Proposition 8
The strategy for the proof of Proposition 8 is the same as in [13]. It relies strongly on the rotation of the lost directions, emphasized in Lemma 4. However, it needs to be adapted because there is no rotation phenomenon on our first lost direction. In order to simplify the notations, in a first step, we prove Proposition 8 in the case We will explain later how it can be adapted for N 3 and K 1 , ..., K N arbitrary.

any vector in M is a linear combination of 3 of theses vectors, with only non negative
coefficients before f 2 j +f 2 j .
Now, let us explain the adaptation of this strategy for N ≥ 3. As previously, we denote by K 1 < · · · < K N the directions missed at the first order and we explain how to reach a basis of missed directions on the second order (3.2), iteratively.
The first step consists in reaching a R + basis of M N , the projections on M 1 , . . . , M N −1 being possibly non zero. This is done as in the first step of the proof of Proposition 8, by designing four controls with non overlapping supports. It is done in any time T 1 > π λK N −λ1 . The (k + 1) th step consists in reaching a R + basis of M N −k while driving to zero the projections on M j , for j = N − k + 1, . . . , N . This can be done iteratively in the following way. Let (v (0) , w (0) ) be as in Proposition 9 for a sufficiently small time and for j = N − k. Then, the controls drive the projection on M N to zero while the projection on M N −k is still non zero, thanks to Lemma 4. We iterate this construction Then the controls v, w = v (k) , w (k) drive the projection on M N , . . . , M N −k+1 to zero while the projection on M N −k is still non zero. Finally, we can find T θ sufficiently small such that v, w and v T θ , w T θ have non overlapping supports and the four pairs of control v, w , v T θ , w T θ , v p , w p and v p+T θ , w p+T θ with p = π λK N −k −λ1 allows to conclude the (k + 1) th step. This can be done in any time T > π λK N −k −λ1 + · · · + π λK N −λ1 . The final step depends on the value of K 1 . If K 1 ≥ 2, we end with the same strategy. If K 1 = 1, the elementary brick of control cannot be designed in arbitrary small time but in time greater than T 2 min . This is why the expression of T ♯ changes when K 1 = 1. Figure 1 illustrates the distribution of controls during the 4 th step with p j := π λK N −j −λ1 .
Let T 1 ∈ (T ♯ , T ) and δ 1 > 0 associated to the map Γ [T1,T ] of Theorem 8. From now on, we assume that One may assume that δ 1 small enough so that condition (4.24) implies ℜ ψ f , ψ 1 (T ) > 0. We introduce the map where • ρ ∈ (0, 1) will be chosen later on, • ψ z is the solution of (1.1)-(1.4) associated to the control u z defined by where . is the L 2 (0, 1)-norm and Note that, for every z, P T [ψ z (T )] = P T [ψ f ]. Thus, our goal is to find z * such that First, let us check that the map F ψ f is well defined when ρ is small enough.

Proof of Proposition 11:
In order to prove that F ψ f is well defined, it is sufficient to find ρ > 0 such that Thanks to Proposition 1, there exists C 1 , C ′ 1 > 0 such that, for every z ∈ M , Thus, (4.25) holds with ρ := min{1; (δ 1 /C ′ 1 ) 2 }. The continuity of F ψ f is a consequence of the continuity of Γ [T1,T ] and the continuity of the solutions of (1.1)-(1.5) with respect to the control u and the initial condition ψ 0 (see (2.3)).
One may assume ρ small enough so that The goal of this section is the proof of the following result, which proves Theorem 4.
To pass from the first line to the second one, we use the commutativity between e iAt and P M , the isometry on L 2 (0, 1) of e iAt . To pass from the second line to the third one, we use the relation P M [µψ 1 (t)] ≡ 0 (that holds because µϕ 1 , ϕ Kj = 0 for j = 1, ..., N ).

Proof of Theorem 6
In this section, we prove Theorem 6 when µ ′ (0) = µ ′ (1) = 0. The case µ ′ (0) = −µ ′ (1) = 0 may be proved similarly. The strategy is similar to the one of the previous section, excepted that, for some lost directions, the second order may vanish and thus, we need to go to a higher order. We prove that the third order is sufficient.
Proof: The proof relies on the equality (4.5). If K is odd, we are in the first case. If K is even and if we are not in the first situation, then, the second situation holds: consider n 1 odd, n 2 even, both large enough.
The previous and next propositions show that any lost direction (at the first order) is recovered either at the second order, or at the third order.
Proof of Proposition 15: To simplify the notations, we write Q T and h instead of Q 3

K,T
and h 3 K . Working by contradiction, we assume that Q T ≡ 0 on V T , for every T < T * . Then ∇Q T (v) ⊥ V T , for every v ∈ V T and T < T * . Easy computations show that, for v ∈ V T , We know that ∇Q T (v) belongs to Adh L 2 (0,T ) (Span{e ±i(λj −λ1)t ; j ∈ J}) because ∇Q T (v) ⊥ V T . The uniqueness of the decomposition on a Riesz basis ensures that Notice that the frequencies (λ K − λ k1 ) in the left hand side are < 0 when k 1 > K, and the frequencies (λ k2 − λ 1 ) in the right hand side are 0. The functions compactly supported on (0, T ) are dense in V T , thus Let n 1 , n 2 > K be such that µϕ K , ϕ n1 µϕ n1 , ϕ n2 µϕ n2 , ϕ 1 = 0. In particular, µϕ K , ϕ k1 = 0 and Q 2 n1,T = 0 on V T , for every T > 0 thanks to Proposition 5. This is in contradiction with (5.2).
In particular, when J = N * −{1}, the minimal time T min required for the local controllability satisfies T min ∈ [T 1 min , T 2 min ].

Remark 6
The equality between T 1 min and T 2 min is an open problem, equivalent to the question addressed in the next paragraph.
Let P T be the orthogonal projection from L 2 ((0, T ), R) to the closed subspace V T and K T be the compact self adjoint operator on L 2 ((0, T ), R) defined by We know that • for any T < T 1 min all the eigenvalues of K T are < A 1 (see the first statement of Theorem 9), • for any T > T 1 min , the largest eigenvalue of K T is > A 1 . (by definition of T 1 min ). For T > T 1 min , does the associated eigenvector belong to H 1 0 ((0, T ), R)?
The proof of the second statement of Theorem 9 may be done exactly as in Section 4. Indeed, when J = N * − {1}, then 1. the vector space M of lost directions (at the first order) is iRψ 1 (T ), 2. for any T 1 ∈ (0, T ), the controls S ± ∈ V T1 ∩H 1 0 (0, T ) allow to reach the states ±iψ 1 (T ) with the second order term; moreover, (iψ 1 (T ), −iψ 1 (T )) is an 'R + -basis' of M Thus, in this section, we focus only on the proof of the first statement of Theorem 9, which is a direct consequence of the following result.
In section 6.1, we state several preliminary results for the proof of Theorem 10, which is detailled in section 6.2.

Preliminaries
For T > 0 and η > 0, we introduce the sets (see (3.5) for the definition of J).
Proposition 18 For every T < T 1 min , there exists λ = λ(T ), η = η(T ) > 0 such that This proposition may be proved with the formalism of Legendre quadratic forms (see [12]). For this article to be self contained, we propose an elementary proof.

Conclusion, open problems, perspectives
In Theorem 3, we have proposed a general context for the local controllability of the system (1.1) to require a positive minimal time. This statement extends Coron's previous result in [15] because: 1. it does not use the variables (s, d) in the state, 2. the control u has to be small in H −1 (not in L ∞ ), 3. µ(x) is not necessarily (x − 1/2).
The validity of the conclusion without the assumption A K = 0 is an open problem.
In Theorem 4, we have proposed a sufficient condition for the system (1.1) to be controllable around the ground state in large time. This sufficient condition is compatible with the general context of Theorem 3, thus there exists a large class of functions µ for which local controllability holds in large time, but not in small time.
The existence of a positive minimal time for the controllability is closely related to a second order approximation of the solution. When a direction is not controllable neither at the first order, nor at the second one, then it is recovered at the third one, and no minimal time is required.
The characterization of the minimal time for the local controllability around the ground state is essentially an open problem. A first step has been done in this article, when only the first direction is lost.
In [13], Crépeau and Cerpa prove the local controllability of the KdV equation, with boundary control. When the length of the domain is critical, the linearized system is not controllable along a finite number of directions, but all of them are recovered at the second order. The existence of a positive minimal time, required for the local controllability is an open problem. The technics developed in this article may be helpful for this question.

A Trigonometric moment problems
In this article, we use several times the following result (see, for instance [10, Corollary 1 in Appendix B] for a proof).