Global controllability and stabilization for the nonlinear Schrödinger equation on an interval

DOI:10.1051/cocv/2009001 www.esaim-cocv.org

GLOBAL CONTROLLABILITY AND STABILIZATION FOR THE NONLINEAR SCHR ¨ ODINGER EQUATION ON AN INTERVAL

Camille Laurent

Abstract. We prove global internal controllability in large time for the nonlinear Schr¨odinger equa- tion on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result onL². We also get a regularity result about the control if the data are assumed smoother.

Mathematics Subject Classification.93B05, 93D15, 35Q55, 35A21.

Received May 23, 2008. Revised November 10, 2008.

Published online February 10, 2009.

Introduction

In this article, we study the stabilization and exact controllability for the periodic one-dimensional nonlinear Schr¨odinger equation (NLS):

i∂tu+∂_x²u = λ|u|²u on [0,+∞[×T¹

u(0) = u₀∈L²(T¹) (0.1)

withλ∈R.

The well posedness in such a low regularity was proved by Bourgain [3]. The proof uses the so called Bourgain spacesX^s,b to get local well posedness and the conservation of theL² mass for global existence.

The aim of this article is to prove exact internal controllability of system (0.1) in large time for a control supported in any small open subset of T¹. We also extend these results to ]0, π[ with Dirichlet or Neumann boundary conditions. The strategy follows the one of Dehman et al. [9] where exact controllability in H¹ is proved for defocusing NLS on compact surfaces. Our result differs from this one because we obtain a control at a lower regularity. This allows to consider the focusing and defocusing equation and to use a different stabilization term, which seems more natural. Moreover, if the Cauchy data are smoother, that isH^swiths≥0, the control we build on L² keeps that regularity, without any assumption on the size inH^s. Yet, in this low regularity, Strichartz inequality of [6] does not provide uniform well posedness, and this forces us to useX^s,bspaces. These spaces are also used in the prior and independent paper of Rosier and Zhang [20] where they obtain results of local controllability near 0 for the same problem.

Keywords and phrases. Controllability, stabilization, nonlinear Schr¨odinger equation, Bourgain spaces.

1 Universit´e Paris-Sud, Bˆatiment 425, 91405 Orsay, France. camille.laurent@math.u-psud.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2009

The strategy is first to prove stabilization and to combine it with local exact controllability near 0 to get null controllability. Then, we remark that the equation obtained by reversing time fulfills exactly the same properties and this allows to establish exact controllability.

Leta=a(x)∈L^∞(T¹) real valued, the stabilization system we consider is i∂tu+∂_x²u+ ia²u = λ|u|²u on [0, T]×T¹

u(0) = u₀∈L²(T¹). (0.2)

The well posedness of this system will be proved in Section2and we can check that it satisfies the mass decay:

u(t)²_L2− u(0)²_L2=−2 t

0 au(τ)²_L2dτ. (0.3) Our theorem states that we have an exponential decay.

Theorem 0.1. Assume that a(x)² > η >0 on some nonempty open set. Then, for every R₀>0, there exist C >0 andγ >0 such that inequality

u(t)L²≤Ce^−γtu₀L² t >0

holds for every solutionuof system (0.2)with initial datau₀ such thatu_0L2 ≤R₀.

Then, as a consequence of stabilization and local controllability near 0 established in Section 3, we obtain the following result.

Theorem 0.2. Let 1/2< b < 5/8. For any nonempty open set ω ⊂ T¹ and R₀ >0, there exist T > 0 and C >0 such that for every u₀ andu₁ inL²(T¹)with

u_0L2≤R₀ and u_1L2≤R₀

there exists a control g ∈C([0, T], L²) with gL^∞([0,T],L²) ≤C supported in [0, T]×ω, such that the unique solution uinX_T^0,bto the Cauchy problem

i∂tu+∂_x²u = λ|u|²u+g on [0, T]×T¹

u(0) = u₀∈L²(T¹) (0.4)

satisfiesu(T) =u₁.

Moreover, if u₀andu₁∈H^s, withs≥0, one can imposeg∈C([0, T], H^s).

We deduce the same results onL²(]0, π[) with the Dirichlet (respectively Neumann) Laplacian. To accomplish this, we use the identification ofD(−ΔD) (resp. D(−ΔN)) with the closed subspace ofH²(R/2πZ) of odd (resp.

even) functions. We only have to check along the proof that the control we build onT¹=R/2πZremains odd (resp. even) if u₀ is so. The propagation of regularity for the control takes the form: ifu₀ ∈ D(−Δ^sD), then one can chooseg∈C([0, T], D(−Δ^sD)) (and similarly for ΔN).

The continuity in time forgis obtained with time cutoff at each stage: the stabilization term is brought to 0 and the local control we build is identically zero at initial and final time. For example, ifu₀andu₁are assumed in C^∞, it allows to imposeuandgin C^∞([0, T]×T¹).

The independence of C, γ and the time of control T on the bound R₀ are an open problem. Yet, it is an interesting fact that even if we want a control in H^s, the time of controllability only depends on the size of the data in L². However, it is unknown whether there is really a minimal time of controllability. This is in strong contrast with the linear case where exact controllability occurs in arbitrary small time and the conditions are only geometric for the open set ω. For example, exact controllability is known to be true when Geometric Control Condition is realized, see Lebeau [15], but also for any open setωofTⁿ, see Jaffard [13] and Komornik

and Loreti [14]. Burq and Zworski [5] also proved the equivalence with a resolvent estimate. Moreover, some recent studies have analysed the explosion of the control cost when T tends to 0: Phung [19] by reducing to the heat or wave equation, Miller [17] with resolvent estimates, Tenenbaum and Tucsnak [22] with number theoretic arguments.

Let us now describe briefly the main arguments of the proof of Theorems 0.1and0.2. First, the functional spaces used are the Bourgain spaces which are especially suited for solving dispersive equations. In our problem, we use some multilinear estimates inX^s,b (see the definition in Sect.1). The first step is the following estimate forb≥3/8, uniformly forT ≤1

uL⁴([0,T]×T¹)≤Cu_X^0,b

T . (0.5)

This was first proved by Bourgain in [3]. A simpler proof, due to Tzvetkov, can be found in the book of Tao [21], p. 104. This allows to prove multilinear estimates inX^s,b, as follows.

Lemma 0.3. For every s≥0,b, b≥3/8, there exists Cs independent onT ≤1 such that foruandu˜∈X_T^s,b, we have

|u|²u

X_T^s,−b ≤ Cu²_X^0,b

T u_X^s,b

T (0.6)

|u|²u− |˜u|²˜u

X_T^s,−b ≤ C

u²_X^s,b

T +u˜ ²_X^s,b

T

u−u˜ _X^s,b

T . (0.7)

This type of multilinear estimates was introduced in [3], but we refer to [4], p. 107, where the estimates we need are stated during the proof of Theorem 2.1, Chapter V. In the Appendix, we recall the proof and precise some dependence insof the estimates.

We prove the control near 0 by a perturbative argument near the one of Zuazua [23]. We use the fixed point theorem of Picard to deduce our result from the linear control. The propagation of H^s regularity from the state to the control is obtained using this property for the linear control and a local linear behavior. The idea comes from the work of Dehman and Lebeau [7] about the wave equation where only some smallness on a finite number of harmonics is required. A notable fact in our case is that no assumption of smallness is made on the H^snorm. We only need theL²norm to be small. Yet, to obtain a bound independent ons, we have to make some estimates with constants independent on s. This will only be possible up to weaker terms, but this will be enough to conclude.

The proof of stabilization is more intricate. In a contradiction argument, following Dehman et al.[8,9], we are led to prove the strong convergence to zero in X_T^0,b of some weakly convergent sequence (un) of solutions to damped NLS. In [9], the authors use some linearisability property of NLS inH¹. Yet, this is false in theL² case. Moreover, as it was seen by Molinet in [18], a weak limituof solutions of NLS is in general not necessarily solution of the same equation. So, we have to proceed a little differently.

We first establish the strong convergence by some propagation of compactness. For a sequence (un) weakly convergent to 0 inX_T^0,bsatisfying

i∂tun+∂_x²un →0 inX_T^−1+b,−b un→0 inL²([0, T]×ω),

we prove that un → 0 in L²_loc([0, T]×T¹). As the geometric control assumption is fulfilled, the propagation of compactness could be proved using microlocal defect measure introduced by G´erard [10], adapting to X^s,b spaces the argument of [9] inspired by Bardos and Masrour [1]. In dimension 1, the microlocal analysis is much simpler and we have chosen, for the convenience of the reader, to prove it with elementary arguments (even if the ideas are the same).

Once we know that the convergence is strong, we infer that the limituis solution to NLS. We use a classical unique continuation theorem to infer that it is 0.

Proposition 0.4. For everyT >0 andω any nonempty open set of T¹, the only solution inC^∞([0, T]×T¹) to the system

i∂tu+∂_x²u=b(t, x)uon [0, T]×T¹ u= 0 on [0, T]×ω

whereb(t, x)∈C^∞([0, T]×T¹)is the trivial oneu≡0.

This was proved by Isakov [12] (see Cor. 6.1) using Carleman estimates.

Yet, the weak limita priori belongs toX_T^0,b. Therefore, to apply Proposition0.4, we needusmooth enough.

We prove that a solution of NLS withu∈C^∞([0, T]×ω) is actually smooth. The proof is an adaptation to the X^s,b spaces of propagation results of microlocal regularity coming from [9]. Again, we present it in such a way that no knowledge of microlocal analysis is necessary, even if the ideas deeply come from this theory.

Notation. Denote D^r the operator defined onD(T¹) by

D^ru(n) = sgn(n)|n|^ru(n) if n= 0

= u(0) if n= 0. (0.8)

In this article,bandbwill be two constants, fixed for the rest of the article, such that 1> b+b,b >1/2> b, and estimates (0.6) and (0.7) hold, see Lemma 1.3 below for the justification of these assumptions. Actually, we can check that for any 1/2< b <5/8, we can find a suitable constantb with the needed properties.

C will denote any absolute constant whose value could change along the article. It could actually depend on s. Yet, when the dependence on s will be needed, this will be announced and we will denote C if it is independent on sandCs otherwise.

1. Some properties of X

spaces

We equip the Sobolev spaceH^s(T¹) with the norm

u²H^s =D^su²L²=|u(0)|²+

k=0

|k|^2s|u(k)|².

The Bourgain spaceX^s,b is equipped with the norm u²_Xs,b = u(.,0)²_Hb(R)+

k

R|k|^2s

τ+k²₂bu(τ, k)²dτ

= u^#2

H^b(R,H^s(T¹))

where.=

1 +|.|²,u=u(t, x),t∈R,x∈T¹, andu^#(t) = e^−it∂x2u(t). u(τ, k) denotes the Fourier transform of u with respect to the time variable (indice τ) and space variable (indice k). u(t, k) denotes the Fourier transform in space variable.

X_T^s,b is the associated restriction space, with the norm u_Xs,b

T = inf

˜u_Xs,bu˜=uon [0, T]×T¹

·

Let us study the stability of theX^s,b spaces with respect to some particular operations.

Lemma 1.1. Let ψ∈C₀^∞(R)andu∈X^s,b thenψ(t)u∈X^s,b. If u∈X_T^s,b then we have ψ(t)u∈X_T^s,b.

Proof. We write

ψuX^s,b =e^−it∂x2ψ(t)u

H^b(H^s)=ψu^#

H^b(H^s)≤Cu^#

H^b(H^s)≤CuX^s,b.

We get the second result by applying the first one on any extension ofuand taking the infimum.

We easily get that D^r (using notation (0.8)) maps any X^s,b into X^s−r,b. In the case of multiplication by C^∞(T¹) function, we have to deal with a loss in X^s,b regularity compared to what we could expect. Some regularity in the index bis lost, due to the fact that multiplication does not keep the structure in time of the harmonics. This loss is unavoidable: takeun =ψ(t)e^inxe^in2t (where ψ∈ C₀^∞(R) equal to 1 on [−1,1]) which is uniformly bounded in X^0,bfor everyb≥0. Yet, if we consider the operator of multiplication by e^ix, we get e^ixun

X^0,b≈n^b. We can prove that our example is the worst one.

Lemma 1.2. Let −1≤b≤1,s∈Randϕ∈C^∞(T¹). Then, if u∈X^s,b we have ϕ(x)u∈X^s−|b|,b. Similarly, multiplication by ϕmaps X_T^s,b intoX_T^s−|b|,b.

Proof. We first deal with the two casesb= 0 and b= 1 and we will conclude by interpolation and duality.

Forb= 0,X^s,0=L²(R, H^s) and the result is obvious.

Forb= 1, we haveu∈X^s,1 if and only if

u∈L²(R, H^s) and i∂tu+∂_x²u∈L²(R, H^s) with the norm

u²_Xs,1 =u²_L2(R,H^s)+i∂tu+∂_x²u²

L²(R,H^s). Then, we have

ϕ(x)u²_Xs−1,1 = ϕu²_L2(R,H^s−1)+i∂t(ϕu) +∂²_x(ϕu)²

L²(R,H^s−1)

≤ C

u²_L2(R,H^s−1)+ϕ

i∂tu+∂_x²u²_L2

(R,H^s−1)

+ ϕ, ∂_x² u²

L²(R,H^s−1)

≤ C

u²L²(R,H^s−1)+i∂tu+∂_x²u²

L²(R,H^s−1)+u²L²(R,H^s)

≤ Cu²X^s,1. Here, we have used that

ϕ, ∂²_x

= −2(∂xϕ)∂x−(∂²_xϕ) is a differential operator of order 1. To conclude, we prove thatX^s,b spaces are in interpolation. For that, we considerX^s,b as a weightedL²(R×Z, μ⊗δ) spaces, whereμis the Lebesgue measure onRandδis the discrete measure onZ. Using the Fourier transform, we can interpretX^s,b as the weightedL² space

L²(R×Z, ws,b(τ, k)μ⊗δ) wherews,b(τ, k) =|k|^2s

τ+k²₂b

. Here, we denote

|k|=|k| ifk= 0 and 1 otherwise. (1.1) Then, we use the complex interpolation theorem of Stein-Weiss for weighted L^p spaces (see [2], p. 114): for 0< θ <1

X^s,0, X^s,1

[θ]≈L²

R×Z,|k|^{2s(1−θ)+2sθ}

τ+k²₂θ

μ⊗δ

≈X^{s(1−θ)+sθ,θ}.

SinceϕmapsX^s,0intoX^s,0andX^s,1intoX^s−1,1, we conclude that for 0≤b≤1,ϕmapsX^s,b=

X^s,0, X^s,1

[b]

into

X^s,0, X^s−1,1

[b]=X^s−b,b which yields theb loss of regularity as announced.

Then, by duality, this also implies that for 0 ≤ b ≤1, ϕ(x) maps X^−s+b,−b into X^−s,−b. As there is no assumption ons∈R, we also have the result for−1≤b≤0 with a loss−b=|b|.

To get the same result for the restriction spacesX_T^s,b, we write the estimate for an extension ˜uofu, which yields

ϕu_X^s−|b|,b

T ≤ ϕ˜u_Xs−|b|,b ≤Cu˜ _Xs,b.

Taking the infimum on all the ˜u, we get the claimed result.

We will also use (see [11] or [3]):

Lemma 1.3. Let Ψ∈C₀^∞(R)and(b, b)satisfying 0< b <1

2 < b, b+b≤1.

If we noteF(t) = Ψ_t

T t

0f(t)dt, we have forT ≤1

F_Hb≤CT^1−b−bf_H−b.

In the future aim of using a boot-strap argument, we will need some continuity inT of the X_T^s,b norm of a fixed function:

Lemma 1.4. Let 0< b <1 anduinX^s,b then the function f : ]0, T] −→ R

t −→ u_X^s,b

t

is continuous. Moreover, ifb >1/2, there existsCb such that limt→0f(t)≤Cbu(0)_Hs.

Proof. By reasoning on each component on the basis, we are led to prove the result inH^b(R). The most difficult case is the limit near 0. It suffices to prove that ifu∈H^b(R), withb >1/2, satisfiesu(0) = 0, and Ψ∈C₀^∞(R) with Ψ(0) = 1, then

Ψ t

T

u−→

T→00 in H^b. Such a functionucan be writtent

0f withf ∈H^b−1. Then, Lemma 1.3gives the result we want ifu∈H^b+ε. Nevertheless, if we only haveu∈H^b, Ψ(_T^t)uis uniformly bounded. We conclude by a density argument.

The following lemma will be useful to control solutions on large intervals that will be obtained by piecing together solutions on smaller ones. We state it without proof.

Lemma 1.5. Let 0< b <1. If

]ak, bk[is a finite covering of [0,1], then there exists a constantC depending only of the covering such that for every u∈X^s,b

u_X^s,b

[0,1] ≤C

k

u_X^s,b

[ak,bk].

Finally, we have the following Rellich type lemma:

Lemma 1.6. For every δ >0,η >0,s,b∈RandT >0, we have X_T^s+η,b+δ⊂X_T^s,b

with compact imbedding.

Proof. Using space Fourier transform and working with u^#_n = e^−it∂2xun, we are led to prove the compact embedding H^b+δ([0, T], l²_k2(s+η))⊂H^b([0, T], l²_k2s), wherel²_k2s is the discretel²space with the weightk^2s.

This is an easy adaptation of Rellich’s theorem.

2. Existence of a solution to NLS with source and damping term

Theorem 2.1. Let T >0,s≥0,λ∈Randa∈C^∞(T¹),ϕ∈C₀^∞(R)taking real values.

For every g∈L²([−T, T], H^s)andu₀∈H^s, there exists a unique solution uin X_T^s,b to i∂tu+∂_x²u+ iϕ(t)²a(x)²u = λ|u|²u+gon [−T, T]×T¹

u(0) = u₀∈H^s. (2.1)

Moreover the flow map

F : H^s(T¹)×L²([−T, T], H^s(T¹)) → X_[−^s,b_T,T] (u₀, g) → u is Lipschitz on every bounded subset.

The same results occur for s= 0 with the weaker assumptiona∈L^∞(T¹).

Proof. It is strongly inspired by Bourgain’s one (see [3,4,11]). First, we notice that if g ∈ L²([−T, T], H^s), it also belongs to X_T^s,−b as b ≥ 0. We restrict ourself to positive times. The solution on [−T,0] is ob- tained similarly. The distinction on the case s= 0 and s > 0 for the regularity assumption ona will appear along the proof with the following statement: with the assumptions of the theorem, multiplication by amaps X^s,0=L²([0, T], H^s) into itself.

We consider the functional

Φ(u)(t) = e^it∂x2u₀−i t

0 e^{i(t−τ)∂2x}

−ia²ϕ²u+λ|u|²u+g

(τ)dτ.

We will apply a fixed point argument on the Banach spaceX_T^s,b. Letψ∈C₀^∞(R) be equal to 1 on [−1,1]. Then by construction (see [11]):

ψ(t)e^it∂x2u₀

X^s,b=ψ_Hb(R)u0_Hs. Therefore, forT ≤1 we have

e^it∂x2u₀

X_T^s,b ≤Cu_0Hs. The one dimensional estimate of Lemma1.3implies

ψ(t/T) t

0 e^{i(t−τ)∂x2}F(τ)dτ

X^s,b ≤CT^1−b−bF_Xs,−b

and then t

0 e^{i(t−τ)∂x2}

−ia²ϕ²u+λ|u|²u+g

(τ)dτ X_T^s,b

≤CT^1−b−b−ia²ϕ²u+λ|u|²u+g

X_T^s,−b

≤CT^1−b−bϕ²a²u

X_T^s,0+|u|²u

X_T^s,−b +g_Xs,−b T

≤CT^1−b−bu_X^s,b

T

1 +u²_X^0,b

T

+CT^1−b−bg_Xs,−b

T .

(2.2) Thus

Φ(u)_X^s,b

T ≤Cu_0Hs+Cg_Xs,−b

T +CT^1−b−bu_X^s,b

T

1 +u²_X^0,b

T

(2.3) and similarly,

Φ(u)−Φ(˜u)_Xs,b

T ≤CT^1−b−bu−u˜ _Xs,b T

1 +u²_Xs,b

T +˜u²_Xs,b T

. (2.4)

These estimates imply that ifT is chosen small enough Φ is a contraction on a suitable ball ofX_T^s,b.

Moreover, we have uniqueness in the classX_T^s,bfor the Duhamel equation. To get the uniqueness inX_T^s,b for the Schr¨odinger equation itself, we prove that every solution uin X_T^s,b of equation (2.1) in the distributional sense is also solution of the integral equation. Let us put

w(t) = e^it∂x2u₀−i t

0 e^{i(t−τ)∂2x}

−iϕ²a²u+λ|u|²u+g

(τ)dτ.

Asu∈X_T^s,b, we have|u|²u∈X_T^s,−b and sinceb<1/2, we infer

∂t

t 0 e^{−iτ ∂x2}

−ia²ϕ²u+λ|u|²u+g

(τ)dτ

= e^−it∂2x

−ia²ϕ²u+λ|u|²u+g

(t) in the distributional sense which implies thatwis solution of

i∂tw+∂_x²w+ ia²ϕ²u=λ|u|²u+g.

Then, r = e^−it∂x2(u−w) is solution of ∂tr = 0 andr(0) = 0. Hence, r = 0 and uis solution of the integral equation. Actually, the above proof also gives that the solutionuof the integral equation is also solution in the distributional sense.

We also prove propagation of regularity.

If u₀ ∈ H^s, with s > 0, we have an existence time T for the solution in X_T^0,b and another time ˜T for the existence in X_T^s,b . By uniqueness in X_T^0,b, the two solutions are the same on [0,T˜]. If we assume ˜T < T, we have the explosion ofu(t, .)H^s as t tends to ˜T whereasu(t, .)L² remains bounded on this interval. Using local existence inL² and Lemma1.5, we easily get thatu_X^0,b

T is finite. Then, using tame estimate (2.3) on a subinterval [T−ε,T], with εsmall enough such thatCε^1−b−b

1 +u²_X^0,b

[T−ε,T]

<1/2, we obtain u_X^s,b

[T−ε,T] ≤Cu(T −ε)_Hs+g_Xs,−b [T−ε,T]

.

We conclude that u ∈ X_T^s,b , which contradicts the explosion of u(t, .)_Hs near ˜T. Therefore, the time of existence is the same for everys≥0.

Next, we useL²energy estimates to get global existence inX_T^0,band so inX_T^s,b. By multiplying equation (2.1) byu, taking imaginary part and integrating, we get

u(t)²L²− u(0)²L² =−2 t

0 aϕ(τ)u(τ)²L²dτ+ 2 t

0

T¹

gu¯

u(t)²L² ≤ u(0)²L²+C t

0 u(τ)²L²dτ+ t

0 u(τ)L²g(τ)L²dτ

≤ u(0)²_L2+C t

0 u(τ)²_L2dτ+Cg²_L2([−T,T],L²). Then, by Gronwall inequality, we have

u(t)²_L2 ≤C

u(0)²_L2+g²_L2([−T,T],L²)

e^C|t|. (2.5)

This ensures that theL² norm remains bounded and the solutionuis global in time.

For the continuity of the flow, we use a slight modification of estimate (2.4) for two solutionsuand ˜u u−u˜_X^s,b

T ≤Cu(0)−u(0)˜ H^s+Cg−˜g_Xs,−b

T +CT^1−b−bu−u˜_X^s,b

T

1 +u²_X^s,b

T +u˜²_X^s,b

T

.

Then, forT small enough (depending on the size ofu₀,u₀,gand ˜g), we get u−u˜ _X^s,b

T ≤Cu(0)−u(0)˜ _Hs+Cg−˜g_Xs,−b

T .

Then, we just have to piece solutions together on small intervals. Using the control of the X_T^s,b norm on L^∞([0, T], H^s) and Lemma1.5, we get thatF is Lipschitz on bounded sets for arbitraryT. After this point and until the end of the proof of local controllability, we will express the dependence onsof the constants by writing themCsorC(.) if some other dependence is considered. b,b,λ,aandϕbeing fixed, we will not write the dependence of constants in these variables.

The following propositions establish a linear behavior on bounded sets ofL².

Proposition 2.2. For every T > 0, η > 0 and s ≥ 0, there exists C(T, η, s) such that for every u ∈ X_T^s,b solution of(2.1)with u₀L²+gL²([0,T],L²)< η, we have the following estimate

u_X^s,b

T ≤C(T, η, s)

u_0Hs+g_L2([0,T],H^s)

.

Proof. AssumeT ≤1. Using (2.3), we obtain thatusatisfies u_X^s,b

T ≤C

u0_Hs+g_L2([0,T],H^s)

+CsT^1−b−bu_X^s,b

T

1 +u²_X^0,b

T

.

WithT such thatCsT^1−b−b <1/2, it yields u_Xs,b

T ≤C

u_0Hs+g_L2([0,T],H^s)

+CsT^1−b−bu_Xs,b

T u²_X0,b T .

First we use it withs= 0. As we have proved in Lemma1.4the continuity with respect toT ofu_X^0,b

T we are in position to apply a boot-strap argument: forT^1−b−b < ¹

2C₀( ^u0 L2+ g L2([0,T],L2))², we obtain:

u_X^0,b

T ≤C

u0_L2+g_L2([0,T],L²)

. (2.6)

The mass estimate (2.5) givesu(t)L²≤Cηe^C|t|≤C(η). Then, we have a constantε(η) such that (2.6) holds for every interval of length smaller thanε(η). Repeating the argument on every small interval, using thatX_T^0,b controlsL^∞(L²) and matching solutions with Lemma1.5, we get the same result for some large interval [0, T], withT ≤1, with a constantC dependent onη. It expresses a local linear behavior.

Then, returning to the cases >0 andCsT^1−b−b <1/2, we have the estimate CsT^1−b−bu²_X^0,b

T ≤CsT^1−b−bC(η)²η². Then, forT ≤ε(s, η), this can be bounded by 1/2 and we have

u_X^s,b

T ≤C

u0_Hs+g_L2([0,T],H^s)

. (2.7)

Again, piecing solutions together, we get the same result for large T ≤1 with C depending ons and η. The assumptionT ≤1 is removed similarly with a final constantC(s, η, T).

A notable consequence of this result is that NLS has a linear behavior in anyH^son any bounded set ofL². Yet, in the last estimate, the constants strongly depend on s. We will use the more precise estimates of the Appendix to eliminate this dependence ins, up to some weaker terms.

Proposition 2.3. For everyT >0,η >0, there existsC(T, η)such that for everys≥1, we can findC(T, η, s) such that for every u∈X_T^s,b solution of(2.1)with u₀L²+gL²([0,T],L²)< η, we have

u_X^s,b

T ≤C(η, T)

u₀H^s+gL²([0,T],H^s)

+C(s, η, T)u_X^s−1,b

T u_X^1,b

T u_X^0,b

T +C(s, η, T)u_X^s−1,b

T .

(2.8) Proof. First, we assumeT ≤1. Lemma1.3gives a constantC independent onssuch that

u_X^s,b

T ≤C

u_0Hs+g_L2([0,T],H^s)

+CT^1−b−ba²ϕ²u

L²([0,T],H^s)+|u|²u

X_T^s,−b

.

Estimate (A.5) of PropositionA.4and CorollaryA.2of the Appendix gives some constantCandCssuch that u_X^s,b

T ≤ C

u0_Hs+g_L2([0,T],H^s)

+T^1−b−b

Cu_X^s,b

T +Csu_X^s−1,b

T

+T^1−b−b

Cu²_X^0,b

T u_X^s,b

T +Csu_X^s−1,b

T u_X^1,b

T u_X^0,b

T

.

From the previous proposition, we have u_X^0,b

T ≤C(η, T)

u₀L²+gL²([0,T],L²)

≤C(η, T)η.

Actually,C(η, T) can be bounded byC(η) =C(η,1) ifT ≤1.

Again, forT small enough (depending only onη and not ons), we have u_X^s,b

T ≤C

u_0Hs+g_L2([0,T],H^s)

+Csu_X^s−1,b

T u_X^1,b

T u_X^0,b

T +Csu_X^s−1,b

T .

Then, piecing solutions together, we finally obtain the result on a large interval [0, T].

Remark 2.4. Ifg= 0, the solutionu∈X_T^0,b of (2.1) actually satisfies u(t)²L²− u(0)²L² =−2

t

0 aϕ(τ)u(τ)²L²dτ.

Remark 2.5. If ais even andu∈X_T^0,b solution of (2.1) with source term g, then±u(t,−x) is solution with source term±g(t,−x). As a conclusion, by uniqueness inX_T^0,b, we infer that ifu₀andg are odd (resp. even), then u is also odd (resp. even). This gives an existence and uniqueness theorem for Dirichlet and Neumann conditions ifa∈C₀^∞(]0, π[) (by identification it will becomea∈C^∞(T¹) even).

3. Controllability near 0

We know (see [9,15] or [16]) that any nonempty open setωsatisfies an observability estimate inL²in arbitrary small timeT >0. Namely, for anya(x)∈C^∞(T¹) andϕ(t)∈C₀^∞(]0, T[) real valued such thata≡1 onω and ϕ≡1 on [T /3,2T /3] (we add the cutoff in time to impose that the control g is zero at 0 andT), there exists C >0 such that

Ψ₀²L² ≤C T

0

a(x)ϕ(t)e^it∂x2Ψ₀²

L² dt (3.1)

for every Ψ₀∈L².

As a consequence, using the HUM method of Lions, this implies exact controllability in L² for the linear equation. More precisely, we can follow [9] to construct an isomorphism of controlS from L² toL². For every data Ψ₀ inL², there exists Φ₀=S⁻¹Ψ₀, Ψ₀=SΦ₀ such that if Φ is solution of the dual equation

i∂tΦ +∂_x²Φ = 0

Φ(x,0) = Φ₀(x) (3.2)

and Ψ solution of

i∂tΨ +∂_x²Ψ = a²(x)ϕ²(t)Φ

Ψ(T) = 0 (3.3)

we have Ψ(0) = Ψ₀.

Lemma 3.1. S is an isomorphism ofH^s for every s≥0.

Proof. We easily see thatSmapsH^sinto itself. So we just have to prove thatSΦ₀∈H^simplies Φ₀∈H^s,i.e.

D^sΦ₀∈L² (with notation (0.8) of the end of the Introduction). We use the formula SΦ₀= i

T

0 e^−it∂2xϕ²a²e^it∂x2Φ₀ dt.