R A Capistrano-Filho

A F Pazoto

L Rosier

INTERNAL CONTROLLABILITY OF THE KORTEWEG-DE VRIES EQUATION ON A BOUNDED DOMAIN

Keywords: 2000 Mathematics Subject Classification. Primary: 35Q53, Secondary: 37K10, 93B05, 93D15 KdV equation, Carleman estimate, null controllability, exact controllability

This paper is concerned with the control properties of the Korteweg-de Vries (KdV) equation posed on a bounded interval with a distributed control. When the control region is an arbitrary open subdomain, we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, the state function being controlled on the left part of the complement of the control region. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L 2 -space is also established.

Introduction

The Korteweg-de Vries (KdV) equation can be written

u t + u xxx + u x + uu x = 0,
where u = u(t, x) is a real-valued function of two real variables t and x, and u t = ∂u/∂t, etc. The equation was first derived by Boussinesq [START_REF] Boussinesq | Essai sur la théorie des eaux courantes[END_REF] and Korteweg-de Vries [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[END_REF] as a model for the propagation of water waves along a channel. The equation furnishes also a very useful approximation model in nonlinear studies whenever one wishes to include and balance a weak nonlinearity and weak dispersive effects. In particular, the equation is now commonly accepted as a mathematical model for the unidirectional propagation of small amplitude long waves in nonlinear dispersive systems.

The KdV equation has been intensively studied from various aspects of mathematics, including the well-posedness, the existence and stability of solitary waves, the integrability, the long-time behavior, etc. (see e.g. [START_REF] Kenig | A bilinear estimate with applications to the KdV equation[END_REF][START_REF] Miura | The Korteweg-de Vries equation: A survey of results[END_REF]). The practical use of the KdV equation does not always involve the pure initial value problem. In numerical studies, one is often interested in using a finite interval (instead of the whole line) with three boundary conditions.

Here, we shall be concerned with the control properties of KdV, the control acting through a forcing term f incorporated in the equation:

(1.1)

u t + u x + u xxx + uu x = f, t ∈ [0, T ], x ∈ [0, L], + b.c.
Our main purpose is to see whether one can force the solutions of (1.1) to have certain desired properties by choosing an appropriate control input f . The focus here is on the controllability issue: Given an initial state u 0 and a terminal state u 1 in a certain space, can one find an appropriate control input f so that the equation (1.1) admits a solution u which equals u 0 at time t = 0 and u 1 at time t = T ?

If one can always find a control input f to guide the system described by (1.1) from any given initial state u 0 to any given terminal state u 1 , then the system (1.1) is said to be exactly controllable. If the system can be driven, by means of a control f , from any state to the origin (i.e. u 1 ≡ 0), then one says that system (1.1) is null controllable.

The study of the controllability and stabilization of the KdV equation started with the works of Russell and Zhang [START_REF] Russell | Exact controllability and stabilizability of the Korteweg-de Vries equation[END_REF] for a system with periodic boundary conditions and an internal control. Since then, both the controllability and the stabilization have been intensively studied. (We refer the reader to [START_REF] Rosier | Control and stabilization of the Korteweg-de Vries equation: Recent progresses[END_REF] for a survey of the results up to 2009.) In particular, the exact boundary controllability of KdV on a finite domain was investigated in e.g. [START_REF] Cerpa | Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain[END_REF][START_REF] Cerpa | Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain[END_REF][START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with a critical length[END_REF][START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF][START_REF] Glass | Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition[END_REF][START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF][START_REF] Rosier | Control of the surface of a fluid by a wavemaker[END_REF][START_REF] Zhang | Exact boundary controllability of the Korteweg-de Vries equation[END_REF]. Most of those works were concerned with the following system (1.2) u t + u x + u xxx + uu x = 0 in (0, T ) × (0, L), u(t, 0) = g 1 (t), u(t, L) = g 2 (t), u x (t, L) = g 3 (t) in (0, T ) in which the boundary data g 1 , g 2 , g 3 can be chosen as control inputs. System (1.2) was first studied by Rosier [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] considering only the control input g 3 (i.e. g 1 = g 2 = 0). It was shown in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] that the exact controllability of the linearized system holds in L 2 (0, L) if, and only if, L does not belong to the following countable set of critical lengths

(1.3) N := 2π √ 3 k 2 + kl + l 2 : k, l ∈ N * .
The analysis developed in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] shows that when the linearized system is controllable, the same is true for the nonlinear one. Note that the converse is false, as it was proved in [START_REF] Cerpa | Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain[END_REF][START_REF] Cerpa | Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain[END_REF][START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with a critical length[END_REF] that the (nonlinear) KdV equation is controllable even when L is a critical length. The existence of a discrete set of critical lengths for which the exact controllability of the linearized equation fails was also noticed by Glass and Guerrero in [START_REF] Glass | Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition[END_REF] when g 2 is taken as control input (i.e. g 1 = g 3 = 0). Finally, it is worth mentioning the result by Rosier [START_REF] Rosier | Control of the surface of a fluid by a wavemaker[END_REF] and Glass and Guerrero [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF] for which g 1 is taken as control input (i.e. g 2 = g 3 = 0). They proved that system (1.2) is then null controllable, but not exactly controllable, because of the strong smoothing effect. By contrast, the mathematical theory pertaining to the study of the internal controllability in a bounded domain is considerably less advanced. As far as we know, the null controllability problem for system (1.1) was only addressed in [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF] when the control acts in a neighborhood of the left endpoint. On the other hand, the exact controllability results in [START_REF] Laurent | Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain[END_REF][START_REF] Russell | Exact controllability and stabilizability of the Korteweg-de Vries equation[END_REF] were obtained on a periodic domain.

The aim of this paper is to address the controllability issue for the KdV equation on a bounded domain with a distributed control. Our first main result is a null controllability result valid for any localization of the control region. Actually, a controllability to the trajectories is established: Theorem 1.1. Let ω = (l 1 , l 2 ) with 0 < l 1 < l 2 < L, and let T > 0. For ū0 ∈ L 2 (0, L), let ū ∈ C 0 ([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)) denote the solution of (1.4)    ūt + ūx + ū ūx + ūxxx = 0 in (0, T ) × (0, L), ū(t, 0) = ū(t, L) = ūx (t, L) = 0 in (0, T ), ū(0, x) = ū0 (x) in (0, L).

Then there exists δ > 0 such that for any u 0 ∈ L 2 (0, L) satisfying u 0 -ū0 L 2 (0,L) ≤ δ, there exists f ∈ L 2 ((0, T ) × ω) such that the solution u ∈ C 0 ([0, T ]; L 2 (0, L)) ∩ L 2 (0, T, H 1 (0, L)) of (1.5)

   u t + u x + uu x + u xxx = 1 ω f (t, x) in (0, T ) × (0, L), u(t, 0) = u(t, L) = u x (t, L) = 0
in (0, T ), u(0, x) = u 0 (x) in (0, L), satisfies u(T, •) = ū(T, •) in (0, L).

The null controllability is first established for a linearized system (1.6)    u t + (ξu) x + u xxx = 1 ω f in (0, T ) × (0, L) , u (t, 0) = u (t, L) = u x (t, L) = 0 in (0, T ) , u (0, x) = u 0 (x) in (0, L) , by following the classical duality approach (see [START_REF] Dolecki | A general theory of observation and control[END_REF][START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]), which reduces the null controllability of (1.6) to an observability inequality for the solutions of the adjoint system. To prove the observability inequality, we derive a new Carleman estimate with an internal observation in (0, T ) × (l 1 , l 2 ) and use some interpolation arguments inspired by those in [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF], where the authors derived a similar result when the control acts on a neighborhood on the left endpoint (that is, l 1 = 0). The null controllability is extended to the nonlinear system by applying Kakutani fixed-point theorem.

The second problem we address is related to the exact internal controllability of system (1.1). As far as we know, the same problem was studied only in [START_REF] Laurent | Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain[END_REF][START_REF] Russell | Exact controllability and stabilizability of the Korteweg-de Vries equation[END_REF] in a periodic domain T with a distributed control of the form

f (x, t) = (Gh)(x, t) := g(x)(h(x, t) - T g(y)h(y, t)dy),
where g ∈ C ∞ (T) was such that {g > 0} = ω and T g(x)dx = 1, and the function h was considered as a new control input. Here, we shall consider the system (1.7)

   u t + u x + uu x + u xxx = f in (0, T ) × (0, L), u(t, 0) = u(t, L) = u x (t, L) = 0 in (0, T ), u(0, x) = u 0 (x) in (0, L).
As the smoothing effect is different from those in a periodic domain, the results in this paper turn out to be very different from those in [START_REF] Laurent | Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain[END_REF][START_REF] Russell | Exact controllability and stabilizability of the Korteweg-de Vries equation[END_REF]. First, for a controllability result in L 2 (0, L), the control f has to be taken in the space L 2 (0, T, H -1 (0, L)). Actually, with any control f ∈ L 2 (0, T, L 2 (0, L)), the solution of (1.7) starting from u 0 = 0 at t = 0 would remain in H 1 0 (0, L) (see [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF]). On the other hand, as for the boundary control, the localization of the distributed control plays a role in the results.

When the control acts in a neighborhood of x = L, we obtain the exact controllability in the weighted Sobolev space L 2 1 L-x dx defined as

L 2 1 L-x dx := {u ∈ L 1 loc (0, L); L 0 |u(x)| 2 L -x dx < ∞}.
More precisely, we shall obtain the following result:

Theorem 1.2. Let T > 0, ω = (l 1 , l 2 ) = (Lν, L) where 0 < ν < L. Then, there exists δ > 0 such that for any u 0 ,

u 1 ∈ L 2 1 L-x dx with u 0 L 2 1 L-x dx ≤ δ and u 1 L 2 1 L-x dx ≤ δ, one can find a control input f ∈ L 2 (0, T ; H -1 (0, L)) with supp(f ) ⊂ (0, T ) × ω such that the solution u ∈ C 0 ([0, L], L 2 (0, L)) ∩ L 2 (0, T, H 1 (0, L)) of (1.7) satisfies u(T, .) = u 1 in (0, L) and u ∈ C 0 ([0, T ], L 2 1 L-x dx ). Furthermore, f ∈ L 2 (T -t)dt (0, T, L 2 (0, L)).
Actually, we shall have to investigate the well-posedness of the linearization of (1.7) in the space L 2 1 L-x dx and the well-posedness of the (backward) adjoint system in the "dual space" L 2 (L-x)dx . To do this, we shall follow some ideas borrowed from [START_REF] Goubet | On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval[END_REF], where the well-posedness was investigated in the weighted space L 2 x L-x dx . The needed observability inequality is obtained by the standard compactness-uniqueness argument and some unique continuation property. The exact controllability is extended to the nonlinear system by using the contraction mapping principle.

When the control is acting far from the endpoint x = L, i.e. in some interval ω = (l 1 , l 2 ) with 0 < l 1 < l 2 < L, then there is no chance to control exactly the state function on (l 2 , L) (see e.g. [START_REF] Rosier | Control of the surface of a fluid by a wavemaker[END_REF]). However, it is possible to control the state function on (0, l 1 ), so that a "regional controllability" can be established:

Theorem 1.3. Let T > 0 and ω = (l 1 , l 2 ) with 0 < l 1 < l 2 < L. Pick any number l ′ 1 ∈ (l 1 , l 2 ).
Then there exists a number δ > 0 such that for any u 0 , u 1 ∈ L 2 (0, L) satisfying

||u 0 || L 2 (0,L) ≤ δ, ||u 1 || L 2 (0,L) ≤ δ, one can find a control f ∈ L 2 (0, T, H -1 (0, L)) with supp(f ) ⊂ (0, T ) × ω such that the solution u ∈ C 0 ([0, T ], L 2 (0, L)) ∩ L 2 (0, T, H 1 (0, L)) of (1.7) satisfies (1.8) u(T, x) = u 1 (x) if x ∈ (0, l ′ 1 ); 0 if x ∈ (l 2 , L).
The proof of Theorem 1.3 combines Theorem 1.1, a boundary controllability result from [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF], and the use of a cutt-off function. Note that the issue whether u may also be controlled in the interval (l

′ 1 , l 2 ) is open.
The paper is outlined as follows. In Section 2, we review some linear estimates from [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF][START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] that will be used thereafter. Section 3 is devoted to the proof of Theorems 1.1 and 1.3. It contains the proof of a new Carleman estimate for the KdV equation with some internal observation (Proposition 3.1). In Section 4 we prove the well-posedness of KdV in the weighted spaces L 2 xdx and L 2 1 L-x dx by using semigroup theory, and derive Theorem 1.2.

Linear estimates

We review a series of estimates for the system (2.1)

   u t + (ξu) x + u xxx = f (t, x) in (0, T ) × (0, L), u(t, 0) = u(t, L) = u x (t, L) = 0 in (0, T ), u(0, x) = u 0 (x)
in (0, L) and its adjoint system. Here f = f (t, x) is a function which stands for the control of the system, and ξ = ξ(t, x) is a given function.

2.1. The linearized KdV equation. It was noticed in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] that the operator A = -

∂ 3 ∂x 3 - ∂ ∂x with domain D(A) = w ∈ H 3 (0, L); w(0) = w(L) = w x (L) = 0 ⊆ L 2 (0, L)
is the infinitesimal generator of a strongly continuous semigroup of contractions in L 2 (0, L). More precisely, the following result was established in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF].

Proposition 2.1. Let u 0 ∈ L 2 (0, L), ξ ≡ 1 and f ≡ 0. There exists a unique (mild) solution u of (2.1) with

(2.2) u ∈ C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T, H 1 0 (0, L))
. Moreover, there exist positive constants c 1 and c 2 such that for all u 0 ∈ L 2 (0, L)

u L 2 (0,T ;H 1 (0,L)) + u x (., 0) L 2 (0,T ) ≤ c 1 u 0 L 2 (0,L) , (2.3) u 0 2 L 2 (0,L) ≤ 1 T u 2 L 2 (0,T ;L 2 (0,L)) + c 2 u x (., 0) 2 L 2 (0,T ) . (2.4)
If in addition u 0 ∈ D(A), then (2.1) has a unique (classical) solution u in the class

(2.5) u ∈ C([0, T ]; D(A)) ∩ C 1 ([0, T ]; L 2 (0, L)).
2.2. The modified KdV equation. We introduce a system related to the adjoint system to (2.1), namely

(2.6)    -v t -ξv x -v xxx = f in (0, T ) × (0, L), v(t, 0) = v(t, L) = v x (t, 0) = 0 in (0, T ), v(T, x) = 0 in (0, L),
for which we review some estimates borrowed from [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF].

2.2.1. Energy Estimates. We introduce the following spaces (2.7)

X 0 := L 2 (0, T ; H -2 (0, L)), X 1 := L 2 (0, T ; H 2 0 (0, L)), X0 := L 1 (0, T ; H -1 (0, L)), X1 := L 1 (0, T ; (H 3 ∩ H 2 0 )(0, L)), and
(2.8) Y 0 := L 2 ((0, T ) × (0, L)) ∩ C 0 ([0, T ] ; H -1 (0, L)), Y 1 := L 2 (0, T ; H 4 (0, L)) ∩ C 0 ([0, T ] ; H 3 (0, L)).
The spaces X 0 , X 1 , X0 , X1 , Y 0 , and Y 1 are equipped with their natural norms. For instance, the spaces Y 0 and Y 1 are equipped with the norms

w Y0 := w L 2 ((0,T )×(0,L)) + w L ∞ (0,T ;H -1 (0,L))
and w Y1 := w L 2 (0,T ;H 4 (0,L)) + w L ∞ (0,T ;H 3 (0,L)) .

For θ ∈ [0, 1], we define the complex interpolation spaces (see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF] and [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF])

X θ = (X 0 , X 1 ) [θ] , Xθ = ( X0 , X1 ) [θ] and Y θ = (Y 0 , Y 1 ) [θ] .
Then, (2.9)

X 1/4 = L 2 (0, T ; H -1 (0, L)), X1/4 = L 1 (0, T ; L 2 (0, L))

and

(2.10)

Y 1/4 = L 2 (0, T ; H 1 (0, L)) ∩ C 0 ([0, T ] ; L 2 (0, L)).
Furthermore,

(2.11)

X 1/2 = L 2 ((0, T ) × (0, L)), X1/2 = L 1 (0, T ; H 1 0 (0, L)) and (2.12) Y 1/2 = L 2 (0, T ; H 2 (0, L)) ∩ C 0 ([0, T ] ; H 1 (0, L)). Proposition 2.2. ([8, Section 2.2.2]) Let ξ ∈ Y 1 4 and f ∈ X 1 4 ∪ X 1 4 = L 2 (0, T ; H -1 (0, L))∪L 1 (0, T ; L 2 (0, L)).
Then the solution v of (2.6) belongs to Y 1 4 , and there exists some constant

C = C(||ξ|| Y 1 4
) > 0 such that

(2.13) v L ∞ (0,T,L 2 (0,L)) + v L 2 (0,T ;H 1 (0,L)) + v x (•, L) L 2 (0,T ) ≤ C( ξ Y 1/4 ) f L 2 (0,T ;H -1 (0,L))
and

(2.14) v L ∞ (0,T,L 2 (0,L)) + v L 2 (0,T ;H 1 (0,L)) + v x (•, L) L 2 (0,T ) ≤ C( ξ Y 1/4 ) f L 1 (0,T ;L 2 (0,L)) .
More can be said when ξ ≡ 0. Consider the following system (2.15)

   -v t -v xxx = g in (0, T ) × (0, L), v(t, 0) = v(t, L) = v x (t, 0) = 0 in (0, T ), v(T, x) = 0 in (0, L). Proposition 2.3. ([8, Section 2.3.1]. If g ∈ X 1 ∪ X1 , then v ∈ Y 1
and there exists a constant C > 0 such that

(2.16) v Y1 + v x (•, L) H 1 (0,T ) ≤ C g X1

and

(2.17)

v Y1 + v x (•, L) H 1 (0,T ) ≤ C g X1 . Proposition 2.4. ([8, Section 2.3.2]. If g ∈ X 1/2 ∪ X1/2 , then v ∈ Y 1/2
, and there exists some constant C > 0 such that

(2.18) v Y 1/2 + v x (•, L) H 1/3 (0,T ) + v xx (•, 0) L 2 (0,T ) + v xx (•, L) L 2 (0,T ) ≤ C g X 1/2 and (2.19) v Y 1/2 + v x (•, L) H 1/3 (0,T ) + v xx (•, 0) L 2 (0,T ) + v xx (•, L) L 2 (0,T ) ≤ C g X1/2 .

Null controllability results

This section is devoted to the proof of Theorems 1.1 and 1.3.

3.1.

Null controllability of a linearized equation. We first consider the system (3.1)

   u t + (ξu) x + u xxx = 1 ω f (t, x) in (0, T ) × (0, L), u(t, 0) = u(t, L) = u x (t, L) = 0 in (0, T ), u(0, x) = u 0 (x) in (0, L),
where ξ = ξ(t, x) is a given function in Y 1 4 , and ω = (l 1 , l 2 ) ⊂ (0, L). Our aim is to prove the null controllability of (3.1). To this end, we shall establish an observability inequality for the corresponding adjoint system

(3.2)    -v t -ξ(t, x)v x -v xxx = 0 in (0, T ) × (0, L), v(t, 0) = v(t, L) = v x (t, 0) = 0 in (0, T ), v(T, x) = v T (x)
in (0, L) by using some Carleman inequality.

3.1.1. Carleman inequality with internal observation. Assume that ω = (l 1 , l 2 ) with 0 < l 1 < l 2 < L.

Pick any function

ψ ∈ C 3 ([0, L]) with ψ > 0 in [0, L]; (3.3) |ψ ′ | > 0, ψ ′′ < 0, and ψ ′ ψ ′′′ < 0 in [0, L] \ ω; (3.4) ψ ′ (0) < 0 and ψ ′ (L) > 0; (3.5) min x∈[l1,l2] ψ(x) = ψ(l 3 ) < max x∈[l1,l2] ψ(x) = ψ(l 1 ) = ψ(l 2 ), max x∈[0,L] ψ(x) = ψ(0) = ψ(L) (3.6) ψ(0) < 4 3 ψ(l 3 ), (3.7) for some l 3 ∈ (l 1 , l 2 ). A convenient function ψ is defined on [0, L] \ ω as ψ(x) = εx 3 -x 2 -x + c 1 if x ∈ [0, l 1 ], -εx 3 + ax + c 2 if x ∈ [l 2 , L]
with ε, a, c 1 , c 2 > 0 conveniently chosen. Note first that ψ(l 1 ) = ψ(l 2 ) and ψ(0) = ψ(L) if, and only if,

a = (L -l 2 ) -1 (l 2 1 + l 1 -εl 3 2 -εl 3 1 + εL 3 ), c 1 = c 2 -εL 3 + aL.
Then a > 0, c For f ∈ L 2 (0, T ; L 2 (0, L)) and q 0 ∈ L 2 (0, L), let q denote the solution of the system q t + q xxx = f, t ∈ (0, T ), x ∈ (0, L), (3.9) q(t, 0) = q(t, L) = q x (t, L) = 0, t ∈ (0, T ), (3.10) q(0, x) = q 0 (x), x ∈ (0, L). (3.11) Then the following Carleman inequality holds. Proposition 3.1. Pick any T > 0. There exist two constants C > 0 and s 0 > 0 such that any f ∈ L 2 (0, T ; L 2 (0, L)), any q 0 ∈ L 2 (0, L) and any s ≥ s 0 , the solution q of (3.9)-(3.11) fulfills (3.12) ) and

T 0 L 0 [sϕ|q xx | 2 +(sϕ) 3 |q x | 2 +(sϕ) 5 |q| 2 ]e -2sϕ dxdt+ T 0 [(sϕ|q xx | 2 +(sϕ) 3 |q x | 2 )e -2sϕ ] |x=0 +[sϕ|q xx | 2 e -2sϕ ] |x=L dt ≤ C T 0 L 0 |f | 2 e -2sϕ dxdt + T 0 ω [sϕ|q xx | 2 + (sϕ) 3 |q x | 2 + (sϕ)
C = C(T, ||ξ|| Y 1 4
) such that for all s ≥ s0 and all v T ∈ L 2 (0, L), the solution v of (3.2) fulfills

(3.13) T 0 L 0 [s φ|v xx | 2 + (s φ) 3 |v x | 2 + (s φ) 5 |v| 2 ]e -2s φdxdt ≤ C T 0 ω [s φ|v xx | 2 + (s φ) 3 |v x | 2 + (s φ) 5 |v| 2 ]e -2s φdxdt.
Proof of Proposition 3.1. We first assume that q 0 ∈ D(A) and that f ∈ C([0, T ]; D(A)), so that q ∈ C([0, T ]; D(A)) ∩ C 1 ([0, T ]; L 2 (0, L)). This will be sufficient to legitimate the following computations. The general case (q 0 ∈ L 2 (0, L) and f ∈ L 2 (0, T ; L 2 (0, L))) follows by density. Indeed, if we set p(t, x) := ϕ(t, l 3 )e -sϕ(t,l3) q(t, x) then p solves (3.9)-(3.11) with q 0 replaced by 0, and f replaced by

f = ϕ(t, l 3 )e -sϕ(t,l3) f + 1 2 ϕ t (t, l 3 )ϕ -1 2 (t, l 3 ) -sϕ t (t, l 3 ) ϕ(t, l 3 ) e -sϕ(t,l3) q, so that (with different constants C) T 0 L 0 ϕ|q xx | 2 e -2sϕ dxdt ≤ C||p|| 2 L 2 (0,T,H 2 (0,L)) ≤ C|| f || 2 L 2 (0,T,L 2 (0,L)) ≤ C ||f || 2 L 2 (0,T,L 2 (0,L)) + ||q 0 || 2 L 2 (0,L) . Since ||q|| 2 L 2 (0,T,H 1 (0,L)) ≤ C ||f || 2 L 2 (0,T,L 2 (0,L)) + ||q 0 || 2 L 2 (0,L)
we conclude that we can pass to the limit in each term in (3.12), if we take a sequence

{(q n 0 , f n )} n≥0 in D(A) × C([0, T ], D(A)) such that q n 0 → q 0 in L 2 (0, L) and f n → f in L 2 (0, T, L 2 (0, L)).
Assume from now on that q 0 ∈ D(A) and that f ∈ C([0, T ]; D(A)). Let q denote the solution of (3.9)- (3.11), and let u = e -sϕ q, w = e -sϕ L(e sϕ u), where

(3.14) L = ∂ t + ∂ 3 x . Straightforward computations show that (3.15) w = M u := u t + u xxx + 3sϕ x u xx + (3s 2 ϕ 2 x + 3sϕ xx )u x + (s 3 ϕ 3 x + 3s 2 ϕ x ϕ xx + s(ϕ t + ϕ xxx ))u.
Let M 1 and M 2 denote the (formal) self-adjoint and skew-adjoint parts of the operator M . We readily obtain that

M 1 u := 3s(ϕ x u xx + ϕ xx u x ) + [s(ϕ t + ϕ xxx ) + s 3 ϕ 3 x ]u, (3.16) M 2 u := u t + u xxx + 3s 2 (ϕ 2 x u x + ϕ x ϕ xx u). (3.17)
On the other hand

(3.18) ||w|| 2 = ||M 1 u|| 2 + ||M 2 u|| 2 + 2(M 1 u, M 2 u)
where (u, v) = T 0 L 0 uvdxdt and ||w|| 2 = (w, w). From now on, for the sake of simplicity, we write u (resp.

u L 0 ) instead of T 0 L 0 u(t, x)dxdt (resp. T 0 u(t, x) L x=0 dt).
The proof of the Carleman inequality follows the same pattern as in [START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF][START_REF] Rosier | Null controllability of the complex Ginzburg-Landau equation[END_REF]. The first step provides an exact computation of the scalar product (M 1 u, M 2 u), whereas the second step gives the estimates obtained thanks to the (pseudoconvexity) conditions (3.3)-(3.7).

Step 1. Exact computation of the scalar product in (3.18). Write

2(M 1 u, M 2 u) = 2 [s(ϕ t + ϕ xxx ) + s 3 ϕ 3 x ]uM 2 u + 2 3s(ϕ x u xx + ϕ xx u x )M 2 u =: I 1 + I 2 . Let (3.19) α := s(ϕ t + ϕ xxx ) + s 3 ϕ 3
x . Using (3.17), we decompose I 1 into

I 1 = 2αuu t + 2αuu xxx + 3s 2 2αu(ϕ 2 x u x + ϕ x ϕ xx u).
Integrating by parts with respect to t or x, noticing that u |x=0 = u |x=L = u x|x=L = 0, and that u |t=0 = u |t=T = 0 by (3.3), we obtain that

I 1 = - α t u 2 + (3 α x u 2 x - α xxx u 2 -αu 2 x L 0 ) -3s 2 ϕ 2 x α x u 2 = - (α t + α xxx + 3s 2 ϕ 2 x α x )u 2 + 3 α x u 2 x -αu 2 x L 0 . (3.20)
Next, we compute

I 2 = 2 3s(ϕ x u xx + ϕ xx u x )(u t + u xxx + 3s 2 (ϕ 2 x u x + ϕ x ϕ xx u)).
Performing integrations by parts, we obtain successively

2 (ϕ x u xx + ϕ xx u x )u t = ϕ xt u 2 x , 2 (ϕ x u xx + ϕ xx u x )u xxx = -3 ϕ xx u 2 xx + ϕ 4x u 2 x + (ϕ x u 2 xx -ϕ 3x u 2 x + 2ϕ xx u xx u x ) L 0 , and
2 (ϕ x u xx + ϕ xx u x )(ϕ 2 x u x + ϕ x ϕ xx u) = -3 ϕ 2 x ϕ xx u 2 x + [(ϕ 2 x ϕ xx ) xx -(ϕ x ϕ 2 xx ) x ]u 2 + ϕ 3 x u 2 x L 0 . Thus (3.21) I 2 = -9s ϕ xx u 2 xx + [-27s 3 ϕ 2 x ϕ xx + 3s(ϕ xt + ϕ 4x )]u 2 x + 9s 3 [(ϕ 2 x ϕ xx ) xx -(ϕ x ϕ 2 xx ) x ]u 2 + [3s(ϕ x u 2 xx -ϕ 3x u 2 x + 2ϕ xx u x u xx ) + 9s 3 ϕ 3 x u 2 x ] L 0 Gathering together (3.20)-(3.21), we infer that 2(M 1 u, M 2 u) = [-(α t + α xxx + 3s 2 ϕ 2 x α x ) + 9s 3 ((ϕ 2 x ϕ xx ) xx -(ϕ x ϕ 2 xx ) x )]u 2 + [3α x -27s 3 ϕ 2 x ϕ xx + 3s(ϕ xt + ϕ 4x )]u 2 x -9s ϕ xx u 2 xx + [3sϕ x u 2 xx + (9s 3 ϕ 3 x -3sϕ xxx -α)u 2 x + 2ϕ xx u x u xx ] L 0 (3.22)
Step 2. Estimation of each term in (3.22). The estimates are given in a series of claims. Claim 1. There exist some constants s 1 > 0 and C 1 > 1 such that for all s ≥ s 1 , we have

[-(α t + α xxx + 3s 2 ϕ 2 x α x ) + 9s 3 ((ϕ 2 x ϕ xx ) xx -(ϕ x ϕ 2 xx ) x )]u 2 ≥ C -1 1 (sϕ) 5 u 2 -C 1 T 0 ω (sϕ) 5 u 2 .
From (3.19), we see that the term in s 5 in the brackets reads

-3s 5 ϕ 2 x (ϕ 3 x ) x = -9s 5 ϕ 4 x ϕ xx = -9s 5 (ψ ′ ) 4 ψ ′′ t 5 (T -t) 5 •
We infer from (3.4) that for some κ 1 > 0 and all s > 0

-9s 5 ϕ 4 x ϕ xx ≥ κ 1 (sϕ) 5 (t, x) ∈ (0, T ) × ([0, L] \ ω).
On the other hand, we have for some κ 2 > 0 and all s > 0

|α t | + |α xxx | + |9s 3 ((ϕ 2 x ϕ xx ) xx -(ϕ x ϕ 2 xx ) x )| ≤ κ 2 s 3 ϕ 4 (t, x) ∈ (0, T ) × (0, L), |3s 2 ϕ 2 x α x | ≤ κ 2 (sϕ) 5 (t, x) ∈ (0, T ) × ω.
Claim 1 follows then for all s > s 1 with s 1 large enough and some C 1 > 1. Claim 2. There exist some constants s 2 > 0 and C 2 > 1 such that for all s ≥ s 2 , we have

(3.23) [3α x -27s 3 ϕ 2 x ϕ xx + 3s(ϕ xt + ϕ 4x )]u 2 x ≥ C -1 2 (sϕ) 3 u 2 x -C 2 T 0 ω (sϕ) 3 u 2 x .
Indeed, the term in s 3 in the brackets is found to be

-18s 3 ϕ 2 x ϕ xx ≥ κ 3 (sϕ) 3 (t, x) ∈ (0, T ) × ([0, L] \ ω)
for some κ 3 > 0 and all s > 0, by (3.4). On the other hand, we have for some κ 4 > 0 and all s > 0

|6s(ϕ tx + ϕ 4x )| ≤ κ 4 sϕ 2 (t, x) ∈ (0, T ) × (0, L), |18s 3 ϕ 2 x ϕ xx | ≤ κ 4 (sϕ) 3 (t, x) ∈ (0, T ) × ω.
Claim 2 follows for all s ≥ s 2 with s 2 large enough and some C 2 > 1. Claim 3. There exist some constants s 3 > 0 and C 3 > 1 such that for all s ≥ s 3 , we have

(3.24) -9s ϕ xx u 2 xx ≥ C -1 3 sϕu 2 xx -C 3 T 0 ω sϕu 2 xx .
Claim 3 is clear, for ψ ′′ < 0 on [0, L] \ ω. Claim 4. There exist some constants s 4 > 0 and C 4 > 1 such that for all s ≥ s 4 , we have

[3sϕ x u 2 xx + (9s 3 ϕ 3 x -3sϕ xxx -α)u 2 x + 2ϕ xx u x u xx ] L 0 ≥ C -1 4 T 0 [(sϕu 2 xx ) |x=0 + (sϕu 2 xx ) |x=L + (s 3 ϕ 3 u 2 x ) |x=0 ]dt.
Since u x|x=L = 0 and

[(9s 3 ϕ 3 x -3sϕ xxx -α)u 2 x ] |x=0 = [(8s 3 ϕ 3 x -s(ϕ t + 4ϕ xxx ))u 2
x ] |x=0 , we obtain with (3.5) for s ≥ s 4 with s 4 large enough,

[(9s 3 ϕ 3 x -3sϕ xxx -α)u 2 x ] L 0 ≥ κ 5 [(sϕ) 3 u 2 x ] |x=0 and 3sϕ x u 2 xx | L 0 ≥ κ 6 ([sϕu 2 xx ] |x=0 + [sϕu 2 xx ] |x=L ) for some constant κ 5 , κ 6 > 0. Finally |[2sϕ xx u x u xx ] x=0 | ≤ κ 6 2 [sϕu 2 xx ] |x=0 + κ 7 [sϕu 2 x ] |x=0
for some constant κ 7 > 0. Since sϕ(t, 0) ≪ (sϕ) 3 (t, 0) for s ≫ 1, Claim 4 follows.

We infer from Claims 1, 2, 3, and 4 that for some positive constants s 0 , C and all s ≥ s 0

(3.25) [(sϕ) 5 |u| 2 + (sϕ) 3 |u x | 2 + sϕ|u xx | 2 ] + T 0 [(sϕu 2 xx ) |x=0 + (sϕu 2 xx ) |x=L + (s 3 ϕ 3 u 2 x ) |x=0 ]dt ≤ C( |w| 2 + T 0 ω [(sϕ) 5 |u| 2 + (sϕ) 3 |u x | 2 + sϕ|u xx | 2 ] ).
Replacing u by e -sϕ q yields (3.12). Proof of Corollary 3.2. Note first that for ξ ∈ Y 1 4 and v T ∈ L 2 (0, L), one can prove that (3.2) has a unique solution v ∈ Y 1 4 , by using the contraction mapping principle for the integral equation. Corollary 3.2 follows from Proposition 3.1 by taking q 0

(x) = v T (L -x), q(t, x) = v(T -t, L -x), and f (t, x) = -ξ(T -t, L -x)q x (t, x), assuming first that ξ ∈ Y 1 4 ∩ L ∞ (Q) (so that f ∈ L 2 (Q)). Indeed, with u = e -sϕ q, w = e -sϕ L(e sϕ u) = -ξ(T -t, L -x)(u x + sϕ x u), so that |w| 2 dxdt ≤ C T 0 L 0 |ξ(T -t, L -x)| 2 (|u x | 2 + |sϕ x u| 2 )dxdt ≤ C T 0 ||ξ(T -t)|| 2 L 2 (0,L) ||u x || 2 L ∞ (0,L) + ||sϕ x u|| 2 L ∞ (0,L) dt ≤ C||ξ|| 2 L ∞ (0,T,L 2 (0,L)) T 0 L 0 [u 2 x + u 2 xx + s 2 t 2 (T -t) 2 (u 2 + u 2
x )]dx. The result for ξ ∈ Y 1 4 follows by density. 3.1.2. Internal observation. We go back to the adjoint system (3.2). Our next goal is to remove the terms v xx and v x from the r.h.s. of (3.13). In addition to the weight φ(t, x) = 1 t(T -t) ψ(Lx), we introduce the functions

(3.27) φ(t) = 1 t(T -t) max x∈[0,L] ψ(x) = ψ(0) t(T -t) and φ(t) = 1 t(T -t) min x∈[0,L] ψ(x) = ψ(l 3 ) t(T -t) ,
where we used (3.6). By (3.7), we have

(3.28) φ(t) < 4 3 φ(t), t ∈ (0, T ). Lemma 3.3. Let 0 < l 1 < l 2 < L, ξ ∈ Y 1 4
, and s0 be as in Corollary 3.2. Then there exists a constant

C = C(T, ||ξ|| Y 1 4
) > 0 such that for any s ≥ s0 and any v T ∈ L 2 (0, L), the solution v of (3.2) satisfies

(3.29) Q (s φ) 5 |v| 2 + (s φ) 3 |v x | 2 + s φ|v xx | 2 e -2s φdxdt ≤ C 1 s 10 T 0 e s(6 φ-8 φ) φ31 v(t, •) 2 L 2 (ω) dt,
where Q = (0, T ) × (0, L) and ω = (l 1 , l 2 ) ⊂ (0, L).

Proof. We follow the same approach as in [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF]. From (3.13) and (3.27)-(3.28), we first obtain

(3.30) Q s 5 φ5 |v| 2 + s 3 φ3 |v x | 2 + s φ|v xx | 2 e -2s φdxdt ≤ C T 0 ω s 5 φ5 |v| 2 + s 3 φ3 |v x | 2 + s φ|v xx | 2 e -2s φdxdt =: C(I 0 + I 1 + I 2 ).
Since φ and φ do not depend on x, we clearly have that

(3.31) I 1 ≤ s 3 T 0 φ3 e -2s φ v(t, •) 2 H 1 (ω) dt and (3.32) I 2 ≤ s T 0 φe -2s φ v(t, •) 2 H 2 (ω) dt.
Using interpolation in the Sobolev spaces H s (ω) (s ≥ 0), we obtain for some positive constants

K 1 , K 2 (3.33) v(t, •) H 1 (ω) ≤ K 1 v(t, •) 3/8 H 8/3 (ω) v(t, •) 5/8 L 2 (ω) and (3.34) v(t, •) H 2 (ω) ≤ K 2 v(t, •) 3/4 H 8/3 (ω) v(t, •)
I 2 ≤ Cs T 0 φe -2s φ v(t, •) 3/2 H 8/3 (ω) v(t, •) 1/2 L 2 (ω) dt.
Next, an application of Young inequality in (3.35) and (3.36) gives

I 1 ≤ Cs 3 T 0 φ3 e -2s φe -3 4 s φe 3 4 s φ φ-27 8 φ 27 8 v(t, •) 3/4 H 8/3 (ω) v(t, •) 5/4 L 2 (ω) dt ≤ C ǫ s 6 T 0 e s( 6 5 φ-16 5 φ) φ51/5 v(t, •) 2 L 2 (ω) dt + ǫs -2 T 0 e -2s φ φ-9 v(t, •) 2 H 8/3 (ω) dt (3.37)
and

I 2 ≤ Cs T 0 e -2s φe -3 2 s φe 3 2 s φ φ-27 4 φ 31 4 v(t, •) 3/2 H 8/3 (ω) v(t, •) 1/2 L 2 (ω) dt ≤ C ǫ s 10 T 0 e s(6 φ-8 φ) φ31 v(t, •) 2 L 2 (ω) dt + ǫs -2 T 0 e -2s φ φ-9 v(t, •) 2 H 8/3 (ω) dt, (3.38)
for any ǫ > 0. Note that (3.39)

I 0 + s 6 T 0 e s( 6 5 φ-16 5 φ) φ51/5 v(t, •) 2 L 2 (ω) dt ≤ Cs 10 T 0 e s(6 φ-8 φ) φ31 v(t, •) 2 L 2 (ω) dt.
Gathering together (3.30) and (3.37)-(3.39), we obtain

(3.40) Q s 5 φ5 |v| 2 + s 3 φ3 |v x | 2 + s φ|v xx | 2 e -2s φdxdt ≤ Cs 10 T 0 e s(6 φ-8 φ) φ31 v(t, •) 2 L 2 (ω) dt + 2ǫs -2 T 0 e -2s φ φ-9 v(t, •) 2 H 8/3 (ω) dt.
It remains to estimate the integral term

T 0 e -2s φ φ-9 v(t, •) 2 H 8/3 (ω) dt. Let v 1 (t, x) := θ 1 (t)v(t, x) with θ 1 (t) = exp(-s φ) φ-1 2 . Then v 1 satisfies the system (3.41)    -v 1t -v 1xxx = f 1 := ξθ 1 v x -θ 1t v in (0, T ) × (0, L), v 1 (t, 0) = v 1 (t, L) = v 1x (t, 0) = 0 in (0, T ), v 1 (T, x) = 0 in (0, L). Now, observe that, since v x (t, 0) = 0, ξ ∈ L ∞ (0, T, L 2 (0, L)) and |θ 1t | ≤ Cs φ 3 2 exp(-s φ), we have f 1 2 L 2 ((0,T )×(0,L)) ≤ C||ξ|| 2 L ∞ (0,T,L 2 (0,L)) T 0 e -2s φ||v x || 2 L ∞ (0,L) dt + C Q e -2s φs 2 φ3 |v| 2 dxdt ≤ C Q s 2 φ3 |v| 2 + s|v x | 2 + s -1 |v xx | 2 e -2s φdxdt (3.42)
for some constant C > 0 and all s ≥ s 0 . Moreover, by Proposition 2.4, v 1 ∈ Y 1/2 . Then, interpolating between L 2 (0, T ; H 2 (0, L)) and L ∞ (0, T ; H 1 (0, L)), we infer that v 1 ∈ L 4 (0, T ; H 3/2 (0, L)) and

(3.43) v 1 L 4 (0,T ;H 3/2 (0,L)) ≤ C f 1 L 2 ((0,T )×(0,L)) . Let v 2 (t, x) := θ 2 (t)v(t, x) with θ 2 = exp(-s φ) φ-5 2 .
Then v 2 satisfies system (3.41) with f 1 replaced by

f 2 := ξθ 2 θ -1 1 v 1x -θ 2t θ -1 1 v 1 . Observe that θ 2 θ -1 1 + θ 2t θ -1 1 ≤ Cs.
On the other hand, since ξ ∈ L 4 (0, T ; H 1 2 (0, L)) and v 1x ∈ L 4 (0, T ; H 1 2 (0, L)) by (3.43), we infer that ξv 1x ∈ L 2 (0, T ; H 1/3 (0, L)) (the product of two functions in H 1 2 (0, L) being in H 1 3 (0, L)). Thus, we obtain

(3.44) f 2 L 2 (0,T ;H 1/3 (0,L)) ≤ Cs v 1 L 4 (0,T ;H 3/2 (0,L)) .
Interpolating between (2.16) and (2.18), we have that v 2 ∈ L 2 (0, T ;

H 7/3 (0, L)) ∩ L ∞ (0, T ; H 4/3 (0, L)) with (3.45) v 2 L 2 (0,T ;H 7/3 (0,L))∩L ∞ (0,T ;H 4/3 (0,L)) ≤ C f 2 L 2 (0,T ;H 1/3 (0,L)) .
Finally, let v 3 := θ 3 (t)v(t, x) with

θ 3 (t) = exp(-s φ) φ-9 2 .
Then v 3 satisfies system (3.41) with f 1 replaced by

f 3 := ξθ 3 θ -1 2 v 2x -θ 3t θ -1 2 v 2 . Again θ 3 θ -1 2 + θ 3t θ -1 2 ≤ Cs.
Interpolating again between (2.16) and (2.18), we have that

(3.46) v 3 L 2 (0,T ;H 8/3 (0,L))∩L ∞ (0,T ;H 5/3 (0,L)) ≤ C f 3 L 2 (0,T ;H 2/3 (0,L)) . Since ξ ∈ Y 1 4 , we have that ξ ∈ L 3 (0, T ; H 2 3 (0, L)).
On the other hand, by (3.45),

v 2x ∈ L 2 (0, T ; H 4/3 (0, L)) ∩ L ∞ (0, T ; H 1/3 (0, L)).
It follows that v 2x ∈ L 6 (0, T, H Thus we infer from (3.42)-(3.47) that for some constants C 1 , C 2 > 0 and all s ≥ s 0

v 3 2 L 2 (0,T ;H 8/3 (0,L)) ≤ C 1 s 4 ||f 1 || 2 L 2 ((0,T )×(0,L)) ≤ C 2 Q s 6 φ3 |v| 2 + s 5 |v x | 2 + s 3 |v xx | 2 e -2s φdxdt. (3.48) Hence, replacing v 3 = exp(-s φ) φ-9 2 v in (3.48) yields for some constant C 3 > 0 (3.49) T 0 e -2s φ φ-9 v(t, •) 2
H 8/3 (ω) dt ≤ C 3 s 2 Q (s φ) 5 |v| 2 + (s φ) 3 |v x | 2 + s φ|v xx | 2 e -2s φdxdt.
Then, picking ǫ = 1/(4C 3 ) in (3.40) results in

Q s φe -2s φ s 4 φ4 |v| 2 + s 2 φ2 |v x | 2 + |v xx | 2 dxdt ≤ C 4 s 10 T 0 e s(6 φ-8 φ) φ31 v(t, •) 2 L 2 (ω) dt
for all s ≥ s0 and some positive constant

C 4 = C 4 (T, ||ξ|| Y 1 4
).

We are in a position to prove the null controllability of system (3.1).

Theorem 3.4. Let T > 0. Then there exists δ > 0 such that for any ξ ∈ Y 1/4 with ||ξ|| L 2 (0,T,H 1 (0,L)) ≤ δ and any u 0 ∈ L 2 (0, L), one may find a control f ∈ L 2 ((0, T ) × ω) such that the solution u of (3.1) fulfills u(T, •) = 0.

Proof. Scaling in (3.2) by v and (Lx)v, we obtain after some computations the estimate

||v|| 2 L ∞ (0,T,L 2 (0,L)) + 2||v x || 2 L 2 (0,T,L 2 (0,L)) ≤ C(L) ||v T || 2 L 2 (0,L) + ||ξ|| 2 L 2 (0,T,H 1 (0,L)) ||v x || 2 L 2 (0,T,L 2 (0,L))
for some constant C(L) > 0. It follows that if ||ξ|| L 2 (0,T,H 1 (0,L)) ≤ δ := 1/ C(L), then we have

(3.50) max t∈[0,T ] ||v(t)|| 2 L 2 (0,L) + ||v x || 2 L 2 (0,T,L 2 (0,L)) ≤ C(L)||v T || 2 L 2 (0,L) .
Replacing v(t) by v(0) and v T by v(τ ) for T /3 < τ < 2T /3 in (3.50), and integrating over τ ∈ (T /3, 2T /3), we obtain that

(3.51) ||v(0)|| 2 L 2 (0,L) ≤ 3C(L) T 2T 3 T 3 ||v(τ )|| 2 L 2 (0,L) dτ.
Combining (3.51) with Lemma 3.3 for a fixed value of s ≥ s0 , we derive the following observability inequality

(3.52) L 0 |v(0, x)| 2 dx ≤ C * T 0 v(t, •) 2 L 2 (ω) dt
where C * = C * (T, ||ξ|| Y 1/4 ) > 0. Using (3.52), we can deduce the existence of a function v ∈ L 2 ((0, T ) × ω) as in Theorem 3.4 proceeding as follows.

On L 2 (0, L), we define the norm

v T B := v L 2 ((0,T )×ω) ,
where v is the solution of (3.2) associated with v T . The fact that || • || B is a norm comes from (3.52) applied on (t, T ) for 0 < t < T . Let B denote the completion of L 2 (0, L) with respect to the above norm. We define a functional J on B by

J(v T ) := 1 2 v T 2 B + L 0 v(0, x)u 0 (x)dx.
From (3.52) we infer that J is well defined and continuous on B. As it is strictly convex and coercive, it admits a unique minimum v * T , characterized by the Euler-Lagrange equation

(3.53) T 0 ω v * wdxdt + L 0 w(0, x)u 0 (x)dx = 0, ∀w T ∈ B,
where w (resp. v * ) denotes the solution of (3.2) associated with

w T ∈ B (resp. v * T ∈ B). Define f ∈ L 2 ((0, T ) × ω) by (3.54) f := 1 ω v * ,
and let u denote the solution of (3.1) associated with u 0 and f . Multiplying (3.1) by w(t, x) and integrating by parts, we obtain for all w T ∈ L 2 (0, L)

(3.55) L 0 u(T, x)w T dx = L 0 u 0 (x)w(0, x)dx + T 0 ω v * wdxdt = 0,
where the second equality follows from (3.53). Therefore u(T, •) = 0. Finally, letting w T = v * T in (3.53) and using (3.52), we obtain

(3.56) T 0 ω |f | 2 dxdt ≤ C * L 0 |u 0 (x)| 2 dx.

3.2.

Null controllability of the nonlinear equation. In this section we prove Theorem 1.1. This is done by using a fixed-point argument.

3.2.1. Proof of Theorem 1.1. Consider u and ū fulfilling system (1.5) and (1.4), respectively. Then q = uū satisfies (3.57)

   q t + q x + ( q 2 2 + ūq) x + q xxx = 1 ω f (t, x)
in (0, T ) × (0, L), q(t, 0) = q(t, L) = q x (t, L) = 0 in (0, T ), q(0, x) = q 0 (x) := u 0 (x) -ū0 (x) in (0, L).

The objective is to find f such that the solution q of (3.57) satisfies q(T, •) = 0.

Given ξ ∈ Y 1 4 and q 0 := u 0 -ū0 ∈ L 2 (0, L), we consider the control problem q t + q x + (ξq) x + q xxx = 1 ω f (t, x) in (0, T ) × (0, L), (3.58) q(t, 0) = q(t, L) = q x (t, L) = 0 in (0, T ), (3.59) q(0, x) = q 0 (x) in (0, L). (3.60)

We can prove the following estimate

(3.61) ||q|| 2 L ∞ (0,T,L 2 (0,L)) + 2||q x || 2 L 2 (0,T,L 2 (0,L)) ≤ C(L) ||q 0 || 2 L 2 (0,L) + ||ξ|| 2 L 2 (0,T,H 1 (0,L)) ||q x || 2 L 2 (0,T,L 2 (0,L)) + ||f || 2 L 2 ((0,T )×ω)
Let δ = min(δ, 1/ C(L)). We introduce the space

E := C 0 ([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)) ∩ H 1 (0, T ; H -2 (0, L))
endowed with its natural norm

z E := ||z|| Y 1/4 + ||z|| H 1 (0,T,H -2 (0,L)) .
We consider in L 2 ((0, T ) × (0, L)) the following set

B := z ∈ E; z E ≤ 1 and ||z|| L 2 (0,T,H 1 (0,L)) ≤ δ .
B is compact in L 2 ((0, T ) × (0, L)), by Aubin-Lions' lemma. We will limit ourselves to controls f fulfilling the condition

(3.62) ||f || 2 L 2 ((0,T )×ω) ≤ C * ||q 0 || 2 L 2 (0,L)
where

C * := C * (T, ||ū|| Y 1/4 + 1 2
). We associate with any z ∈ B the set T (z) := q ∈ B; ∃f ∈ L 2 ((0, T ) × ω) such that f satisfies (3.62) and q solves (3.58)-(3.60) with ξ = ū + z 2 and q(T, •) = 0 . Note that ||ū|| L 2 (0,T,H 1 (0,L)) < δ/2 for T ≪ 1. By Theorem 3.4 and (3.61), we see that if q 0 L 2 (0,L) and T are sufficiently small, then T (z) is nonempty for all z ∈ B. We shall use the following version of Kakutani fixed point theorem (see e.g. [START_REF] Zeidler | Nonlinear functional analysis and its applications I[END_REF]Theorem 9.B]): Theorem 3.5. Let F be a locally convex space, let B ⊂ F and let T : B -→ 2 B . Assume that (1) B is a nonempty, compact, convex set;

(2) T (z) is a nonempty, closed, convex set for all z ∈ B;

(3) The set-valued map T : B -→ 2 B is upper-semicontinuous; i.e., for every closed subset

A of F , T -1 (A) = {z ∈ B; T (z) ∩ A = ∅} is closed.
Then T has a fixed point, i.e., there exists z ∈ B such that z ∈ T (z).

Let us check that Theorem 3.5 can be applied to T and

F = L 2 ((0, T ) × (0, L)).
The convexity of B and T (z) for all z ∈ B is clear. Thus (1) is satisfied. For (2), it remains to check that T (z) is closed in F for all z ∈ B. Pick any z ∈ B and a sequence q k k∈N in T (z) which converges in F towards some function q ∈ B. For each k, we can pick some control function f k ∈ L 2 ((0, T ) × ω) fulfilling (3.62) such that (3.58)-(3.60) are satisfied with ξ = ū + z 2 and q k (T, •) = 0. Extracting subsequences if needed, we may assume that as k → ∞

f k → f in L 2 ((0, T ) × ω) weakly, (3.63) q k → q
in L 2 (0, T ; H 1 (0, L)) ∩ H 1 (0, T ; H -2 (0, L)) weakly, (3.64) By (3.64), the boundedness of ||q k || L ∞ (0,T,L 2 (0,L)) and Aubin-Lions' lemma, {q k } k∈N is relatively compact in C 0 ([0, T ], H -1 (0, L)). Extracting a subsequence if needed, we may assume that q k → q strongly in C 0 ([0, T ], H -1 (0, L)).

In particular, q(0, x) = q 0 (x) and q(T, x) = 0. On the other hand, we infer from (3.64) that ξq k → ξq in L 2 ((0, T ) × (0, L)) weakly.

Therefore, (ξq k ) x → (ξq) x in D ′ ((0, T ) × (0, L)). Finally, it is clear that

||f || 2 L 2 ((0,T )×ω) ≤ C * ||q 0 || 2 L 2 (0,L)
and that q satisfies (3.58) with ξ = ū + z 2 and q(T, •) = 0. Thus q ∈ T (z) and T (z) is closed. Now, let us check (3). To prove that T is upper-semicontinuous, consider any closed subset A of F and any sequence z k k∈N in B such that (3.65)

z k ∈ T -1 (A), ∀k ≥ 0, and
(3.66) z k → z in F
for some z ∈ B. We aim to prove that z ∈ T -1 (A). By (3.65), we can pick a sequence q k k∈N in B with q k ∈ T (z k ) ∩ A for all k, and a sequence f k k∈N in L 2 ((0, T ) × ω) such that

(3.67)      q k t + q k x + ((ū + z k 2 )q k ) x + q k xxx = 1 ω f k (t, x) in (0, T ) × (0, L), q k (t, 0) = q k (t, L) = q k x (t, L) = 0 in (0, T ), q k (0, x) = q 0 (x) in (0, L), (3.68) q k (T, x) = 0, in (0, L), and (3.69)
f k 2 L 2 ((0,T )×ω) ≤ C * q 0 2 L 2 (0,L)
. From (3.69) and the fact that z k , q k ∈ B, extracting subsequences if needed, we may assume that as k → ∞,

f k → f in L 2 ((0, T ) × ω) weakly, q k → q in L 2 (0, T ; H 1 (0, L)) ∩ H 1 (0, T ; H -2 (0, L)) weakly, q k → q in C 0 ([0, T ], H -1 (0, L)) strongly, q k → q in F strongly, z k → z in F strongly,
where f ∈ L 2 ((0, T ) × ω) and q ∈ B. Again, q(0, x) = q 0 (x) and q(T, x) = 0. We also see that (3.59) and (3.62) are satisfied. It remains to check that (3.70)

q t + q x + ((ū + z 2 )q) x + q xxx = 1 ω f (t, x).
Observe that the only nontrivial convergence in (3.67) is those of the nonlinear term (z k q k ) x . Note first that

||z k q k || L 2 (0,T,L 2 (0,L)) ≤ ||z k || L ∞ (0,T,L 2 (0,L)) ||q k || L 2 (0,T,L ∞ (0,L)) ≤ C,
so that, extracting a subsequence, one can assume that z k q k → f weakly in L 2 ((0, T ) × (0, L)). To prove that f = zq, it is sufficient to observe that for any ϕ ∈ D(Q),

T 0 L 0 z k q k ϕdxdt → T 0 L 0 zqϕdxdt,
for z k → z and q k ϕ → qϕ in F . Thus

z k q k → zq in L 2 ((0, T ) × (0, L)) weakly.
It follows that (z k q k ) x → (zq) x in D ′ ((0, T ) × (0, L)). Therefore, (3.70) holds and q ∈ T (z). On the other hand, q ∈ A, since q k → q in F and A is closed. We conclude that z ∈ T -1 (A), and hence T -1 (A) is closed. Il follows from Theorem 3.5 that there exists q ∈ B with q ∈ T (q), i.e. we have found a control f ∈ L 2 ((0, T ) × ω) such that the solution of (3.57) satisfies q(T, •) = 0 in (0, L). The proof of Theorem 1.1 is complete.

With Theorem 1.1 at hand, one can prove Theorem 1.3 about the regional controllability.

3.3. Proof of Theorem 1.3. By Theorem 1.1, if δ is small enough one can find a control input f ∈ L 2 (0, T /2, L 2 (0, L)) with supp(f ) ⊂ (0, T ) × ω such that the solution of (1.7) satisfies u(T /2, .) ≡ 0 in (0, L). Pick any number l ′ 2 ∈ (l ′ 1 , l 2 ) with l ′ 2 ∈ N . (This is possible, the set N being discrete.) By [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]Theorem 1.3], if δ is small enough one can pick a function h ∈ L 2 (T /2, T ) such that the solution

y ∈ C 0 ([T /2, T ], L 2 (0, l ′ 2 )) ∩ L 2 (T /2, T, H 1 (0, l ′ 2 )) of the system    y t + y xxx + y x + yy x = 0 in (T /2, T ) × (0, l ′ 2 ), y(t, 0) = y(t, l ′ 2 ) = 0, y x (t, l ′ 2 ) = h(t) in (T /2, T ), y(T /2, x) = 0 in (0, l ′ 2 ) satisfies y(T, x) = u 1 (x) for 0 < x < l ′ 2 . We pick a function µ ∈ C ∞ ([0, L]) such that µ(x) = 1 if x < l ′ 1 , 0 if x > l ′ 1 +l ′ 2 2
and set for

T /2 < t ≤ T u(t, x) = µ(x)y(t, x) if x < l ′ 2 , 0 if x > l ′ 2 . Note that, for T /2 < t < T , u t + u xxx + u x + uu x = f with f = µ(µ -1)yy x + (µ xxx y + 3µ xx y x + 3µ x y xx + µ x y) + µµ x y 2 . Since ||y|| 4 L 4 (0,T,L 4 (0,l ′ 2 )) ≤ C||y|| 2 L ∞ (0,T,L 2 (0,L)) ||y|| 2 L 2 (0,T,H 1 (0,L)) , it is clear that f ∈ L 2 (0, T, H -1 (0, L)) with supp(f ) ⊂ (0, T ) × (l 1 , l 2 ). Furthermore, u ∈ C([0, T ], L 2 (0, L)) ∩ L 2 (0, T, H 1 (0, L)) solves (1.

Exact controllability results

Pick any function

ρ ∈ C ∞ (0, L) with (4.1) ρ(x) = 0 if 0 < x < L -ν, 1 if L -ν 2 < x < L, for some ν ∈ (0, L).
This section is devoted to the investigation of the exact controllability of the system (4.2)

   u t + u x + uu x + u xxx = f = (ρ(x)h) x in (0, T ) × (0, L), u(t, 0) = u(t, L) = u x (t, L) = 0 in (0, T ), u(0, x) = u 0 (x)
in (0, L).

More precisely, we aim to find a control input h ∈ L 2 (0, T ; L 2 (0, L)) (actually, with (ρ(x)h(t, x)) x in some space of functions) to guide the system described by (4.2) in the time interval [0, T ] from any (small) given initial state u 0 in L 2 1 L-x dx to any (small) given terminal state u T in the same space. We first consider the linearized system, and next proceed to the nonlinear one. The results involve some weighted Sobolev spaces. 4.1. The linear system. For any measurable function w : (0, L) → (0, +∞) (not necessarily in L 1 (0, L)), we introduce the weighted L 2 -space

L 2 w(x)dx = {u ∈ L 1 loc (0, L); L 0 u(x) 2 w(x)dx < ∞}.
It is a Hilbert space when endowed with the scalar product

(u, v) L 2 w(x)dx = L 0 u(x)v(x)w(x)dx.
We first prove the well-posedness of the linear system associated with (4.2), namely

(4.3)    u t + u x + u xxx = 0 in (0, T ) × (0, L), u(t, 0) = u(t, L) = u x (t, L) = 0 in (0, T ), u(0, x) = u 0 (x) in (0, L),
in both the spaces L 2 xdx and L 2 1 L-x dx , following [START_REF] Goubet | On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval[END_REF] where the well-posedness was established in L 2

x L-x dx . We need the following result.

Theorem 4.1. (see [START_REF] Goubet | On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval[END_REF]) Let W ⊂ V ⊂ H be three Hilbert spaces with continuous and dense embeddings. Let a(v, w) be a bilinear form defined on V × W that satisfies the following properties: (i) (Continuity)

(4.4) a(v, w) ≤ M ||v|| V ||w|| W , ∀v ∈ V, ∀w ∈ W ; (ii) (Coercivity) (4.5) a(w, w) ≥ m||w|| 2 V , ∀w ∈ W ; Then for all f ∈ V ′ (the dual space of V ), there exists v ∈ V such that (4.6) a(v, w) = f (w), ∀w ∈ W.
If, in addition to (i) and (ii), a(v, w) satisfies (iii) (Regularity) for all g ∈ H, any solution v ∈ V of (4.6) with f (w) := (g, w) H belongs to W , then (4.6) has a unique solution v ∈ W . Let D(A) denote the set of those v ∈ W when g ranges over H, and set Av = -g. Then A is a maximal dissipative operator, and hence it generates a continuous semigroup of contractions (e tA ) t≥0 in H.

Well-posedness in

L 2 xdx . Theorem 4.2. Let A 1 u = -u xxx -u x with domain D(A 1 ) = {u ∈ H 2 (0, L) ∩ H 1 0 (0, L); u xxx ∈ L 2 xdx , u x (L) = 0} ⊂ L 2 xdx . Then A 1 generates a strongly continuous semigroup in L 2 xdx . Proof. Let H = L 2 xdx , V = H 1 0 (0, L), W = {w ∈ H 1 0 (0, L), w xx ∈ L 2
x 2 dx }, be endowed with the respective norms

||u|| H := || √ xu|| L 2 (0,L) , ||v|| V := ||v x || L 2 (0,L) , ||w|| W := ||xw xx || L 2 (0,L) .
Clearly, V ⊂ H with a continuous (dense) embedding between two Hilbert spaces. On the other hand, we have that (4.7)

||w x || L 2 ≤ C||xw xx || L 2 ∀w ∈ W.
First, we note that we have for w

∈ T := C ∞ ([0, L]) ∩ H 1 0 (0, L) and p ∈ R 0 ≤ L 0 (xw xx +pw x ) 2 dx = L 0 (x 2 w 2 xx +2pxw x w xx +p 2 w 2 x )dx = L 0 x 2 w 2 xx dx+(p 2 -p) L 0 w 2 x dx+pLw 2 x (L).
Taking p = 1/2 results in

(4.8) L 0 w 2 x dx ≤ 4 L 0 x 2 w 2 xx dx + 2L|w x (L)| 2 .
The estimate (4.8) is also true for any w ∈ W , since T is dense in W . Let us prove (4.7) by contradiction. If (4.7) is false, then there exists a sequence {w n } n≥0 in W such that

1 = ||w n x || L 2 ≥ n||xw n xx || L 2 ∀n ≥ 0.
Extracting subsequences, we may assume that

w n → w in H 1 0 (0, L) weakly xw n xx → 0 in L 2 (0, L) strongly
and hence xw xx = 0, which gives w(x) = c 1 x + c 2 . Since w ∈ H 1 0 (0, L), we infer that w ≡ 0. Since w n is bounded in H 2 (L/2, L), extracting subsequences we may also assume that w n x (L) converges in R. We infer then from (4.8) that w n is a Cauchy sequence in H 1 0 (0, L), so that w n → w in H 1 0 (0, L) strongly, and hence ||w x || L 2 = lim n→∞ ||w n

x || L 2 = 1. This contradicts the fact that w ≡ 0. The proof of (4.7) is achieved.

Thus || • || W is a norm in W , which is clearly a Hilbert space, and W ⊂ V with continuous (dense) embedding. Let

a(v, w) = L 0 v x [(xw) xx + xw]dx, v ∈ V, w ∈ W.
Let us check that (i), (ii), and (iii) in Theorem 4.1 hold. For v ∈ V and w ∈ W ,

|a(v, w)| ≤ ||v x || L 2 ||xw xx + 2w x + xw|| L 2 ≤ ||v x || L 2 ||xw xx || L 2 + C||w x || L 2 ≤ C||v|| V ||w|| W
where we used Poincaré inequality and (4.7). This proves that the bilinear form a is well defined and continuous on V × W . For (ii), we first pick any w ∈ T to obtain

a(w, w) = L 0 w x (xw xx + 2w x + xw)dx = 3 2 L 0 w 2 x dx + [x w 2 x 2 ]| L 0 - 1 2 L 0 w 2 dx ≥ 3 2 L 0 w 2 x dx - 1 2 L 0 w 2 dx. By Poincaré inequality L 0 w 2 (x)dx ≤ ( L π ) 2 L 0 w 2 x (x)dx, and hence a(w, w) ≥ ( 3 2 - L 2 2π 2 ) L 0 w 2 x dx.
This shows the coercivity when L < π √ 3. When L ≥ π √ 3, we have to consider, instead of a, the bilinear form a λ (v, w) := a(v, w) + λ(v, w) H for λ ≫ 1. Indeed, we have by Cauchy-Schwarz inequality and Hardy inequality

||w|| 2 L 2 ≤ ||x 1 2 w|| L 2 ||x -1 2 w|| L 2 ≤ √ L||w|| H ||x -1 w|| L 2 ≤ ε||w x || 2 L 2 + C ε ||w|| 2 H
and hence

a λ (w, w) ≥ ( 3 2 - ε 2 )||w|| 2 V + (λ - C ε 2 )||w|| 2
H . Therefore, if ε < 3 and λ > C ε /2, then a λ is a continuous bilinear form which is coercive.

Let us have a look at the regularity issue. For given g ∈ H, let v ∈ V be such that

a λ (v, w) = (g, w) H ∀w ∈ W, i.e. (4.9)
L 0 v x ((xw) xx + xw)dx + λ L 0 v(x)w(x)xdx = L 0 g(x)w(x)xdx.
Picking any w ∈ D(0, L) results in (4.10)

x(v xxx + v x + λv), w D ′ ,D = xg, w D ′ ,D ∀w ∈ D(0, L),
and hence (4.11)

v xxx + v x + λv = g in D ′ (0, L). Since v ∈ H 1 0 (0, L) and g ∈ L 2 xdx , we have that v ∈ H 3 (ε, L) for all ε ∈ (0, L) and v xxx ∈ L 2 xdx .
Picking any w ∈ T and ε ∈ (0, L), and scaling in (4.11) by xw yields

L ε v x ((xw) xx + xw)dx + [v xx (xw) -v x (xw) x ]| L ε = L ε (g -λv)xwdx.
Letting ε → 0 and comparing with (4.9), we obtain (4.12)

-Lv x (L)w x (L) = lim ε→0 εv xx (ε)w(ε) -v x (ε)(w(ε) + εw x (ε)) . Since v xxx ∈ L 2 xdx
, we obtain successively for some constant C > 0 and all ε ∈ (0, L)

|v xx (ε) -v xx (L)| ≤ ( L ε x|v xxx | 2 dx) 1 2 ( L ε x -1 dx) 1 2 ≤ C| log ε| (4.13) |v x (ε)| ≤ C. (4.14)
We infer from (4.13) that v ∈ H 2 (0, L), and hence v ∈ W . Furthermore, letting ε → 0 in (4.12) and using (4.13)-(4.14) yields v x (L) = 0, since w x (L) was arbitrary. We conclude that v ∈ D(A 1 ). Conversely, it is clear that the operator A 1λ maps D(A 1 ) into H, and actually onto H from the above computations. Hence A 1λ generates a strongly semigroup of contractions in H.

Well-posedness in

L 2 (L-x) -1 dx . Theorem 4.3. Let A 2 u = -u xxx -u x with domain D(A 2 ) = {u ∈ H 3 (0, L) ∩ H 1 0 (0, L); u xxx ∈ L 2 1 L-x dx and u x (L) = 0} ⊂ L 2 1 L-x dx . Then A 2 generates a strongly continuous semigroup in L 2 1 L-x dx .
Proof. We will use Hille-Yosida theorem, and (partially) Theorem 4.1. Let (4.15)

H = L 2 1 L-x dx , V = {u ∈ H 1 0 (0, L), u x ∈ L 2 1 (L-x) 2 dx }, W = H 2 0 (0, L),
be endowed respectively with the norms (4. [START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF])

||u|| H = ||(L -x) -1 2 u|| L 2 , ||u|| V = ||(L -x) -1 u x || L 2 , ||u|| W = ||u xx || L 2 .
From [START_REF] Goubet | On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval[END_REF], we know that V endowed with || • || V is a Hilbert space, and that (4.17)

||(L -x) -2 u|| L 2 ≤ 2 3 ||(L -x) -1 u x || L 2 ∀u ∈ V, and hence (4.18
)

||u|| H ≤ ( L 0 L 3 (L -x) 4 u 2 (x)dx) 1 2 ≤ 2 3 L 3 2 ||u|| V ∀u ∈ V.
Thus V ⊂ H with continuous embedding. From Poincaré inequality, we have that || • || W is a norm on W equivalent to the H 2 -norm. On the other hand, from Hardy inequality

(4.19) L 0 v 2 (L -x) 2 dx ≤ C L 0 v 2 x dx ∀v ∈ H 1 (0, L) with v(L) = 0, we have that (4.20) ||v|| V ≤ C||v|| W ∀v ∈ W.
Thus W ⊂ V with continuous embedding. It is easily seen that D(0, L) is dense in H, V , and W . Let

a(v, w) = L 0 [v x ( w L -x ) xx + v x w L -x ]dx (v, w) ∈ V × W. Then |a(v, w)| ≤ | L 0 v x ( w xx L -x + 2 w x (L -x) 2 + 2 w (L -x) 3 + w L -x )dx| ≤ ||w xx || L 2 || v x L -x || L 2 + 2|| w x L -x || L 2 || v x L -x || L 2 + || v x L -x || L 2 2|| w (L -x) 2 || L 2 + ||w|| L 2 ≤ C||v|| V ||w|| W
by (4.17), (4.18), and (4.20). This shows that a is well defined and continuous. Let us look at the coercivity of a. Pick any w ∈ D(0, L). Then

a(w, w) = L 0 w x w xx L -x + 2 w x (L -x) 2 + 2 w (L -x) 3 + w L -x dx = 3 2 L 0 w 2 x (L -x) 2 dx -3 L 0 w 2 (L -x) 4 dx - 1 2 L 0 w 2 (L -x) 2 dx ≥ 1 6 L 0 w 2 x (L -x) 2 dx - 1 2 L 0 w 2 (L -x) 2 dx
where we used (4.17) for the last line. Note that, using Cauchy-Schwarz inequality and (4.17), we have that

|| w L -x || 2 L 2 ≤ ||(L -x) -1 2 w|| L 2 ||(L -x) -3 2 w|| L 2 ≤ 2 √ L 3 ||w|| H ||w|| V ≤ ε||w|| 2 V + L 9ε ||w|| 2 H . (4.21)
If we pick ε ∈ (0, 1/3), we infer that for all w ∈ D(0, L)

(4.22) a(w, w) + L 18ε ||w|| 2 H ≥ 1 6 - ε 2 ||w|| 2 V ≥ C||w|| 2 V .
The result is also true for any w ∈ W , by density. This shows that the continuous bilinear form

a λ (v, w) = a(v, w) + λ(v, w) H
is coercive for λ > L/6. Let g ∈ H be given. By Theorem 4.1, there is at least one solution v ∈ V of (4.23)

a λ (v, w) = (g, w) H ∀w ∈ W.
Pick such a solution v ∈ V , and let us prove that v ∈ D(A 2 ). Picking any w ∈ D(0, L) in (4.23) yields (4.24)

v xxx + v x + λv = g in D ′ (0, L).
As g ∈ L 2 (0, L) and v ∈ H 1 (0, L), we have that v xxx ∈ L 2 (0, L), and v ∈ H 3 (0, L). Pick finally w of the form w

(x) = x 2 (L -x) 2 w(x), where w ∈ C ∞ ([0, L]) is arbitrary chosen. Note that w ∈ W and that w/(L -x) ∈ H 1 0 (0, L) ∩ C ∞ ([0, L]).
Multiplying in (4.24) by w/(Lx) and integrating over (0, L), we obtain after comparing with (4.23)

0 = -v x ( w L -x ) x | L 0 = -v x (2xL -3x 2 )w + x 2 (L -x)w x | L 0 = v x (L)L 2 w(L).
As w(L) can be chosen arbitrarily, we conclude that v x (L) = 0. Using (4.19) twice, we infer that v x +λv ∈ H, and hence v xxx = g -(v x + λv) ∈ H. Therefore v ∈ D(A 2 ). Thus, for λ > L/6 we have that A 2λ : D(A 2 ) → H is onto. Let us check that A 2λ is also dissipative in H. Pick any w ∈ D(A 2 ). Then we obtain after some integrations by parts that

(A 2 w, w) H = - 3 2 L 0 w 2 x (L -x) 2 dx + 3 L 0 w 2 (L -x) 4 dx + 1 2 L 0 w 2 (L -x) 2 dx - w 2 x (0) 2L and (A 2 w -λw, w) H ≤ -( 1 6 - ε 2 )||w|| 2 V - w 2 x (0) 2L ≤ 0
for ε < 1/3 and λ = L/(18ε). We conclude that A 2λ is maximal dissipative for λ > L/6, and thus it generates a strongly continuous semigroup of contractions in H by Hille-Yosida theorem.

A global Kato smoothing effect as in [START_REF] Goubet | On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval[END_REF][START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] can as well be derived. Proof. We proceed as in [START_REF] Goubet | On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval[END_REF]. First, we notice that D(A 2 ) is dense in H, so that it is sufficient to prove the result when u 0 ∈ D(A 2 ). Note that the estimate ||u|| L ∞ (0,T,H) ≤ C||u 0 || H is a consequence of classical semigroup theory. Assume u 0 ∈ D(A 2 ), so that u t = A 2 u in the classical sense. Taking the inner product in H with u yields

(u t , u) H = -a(u, u) ≤ -C||u|| 2 V + L 18ε ||u|| 2

H

where we used (4.22). An integration over (0, T ) completes the proof of the estimate of ||u|| L 2 (0,T,V ) .

4.4. Non-homogeneous system. In this section we consider the nonhomogeneous system

u t + u x + u xxx = f (t, x) in (0, T ) × (0, L), (4.26) u(t, 0) = u(t, L) = u x (t, L) = 0
in (0, T ), (4.27) u(0, x) = u 0 in (0, L). (4.28)

We need the prove the existence of a "reasonable" solution when solely f ∈ L 2 (0, T, H -1 (0, L)). ., . H -1 ,H 1 0 denoting the duality pairing between H -1 (0, L) and H 1 0 (0, L), we have that for all ε > 0 (4.31)

τ 0 L 0 xuf dxdt = τ 0 f, xu H -1 ,H 1 0 ≤ ε 2 τ 0 L 0 u 2 x dxdt + C ε τ 0 ||f || 2 H -1 dt.
The last term in the l.h.s. of (4.30) is decomposed as

1 2 τ 0 L 0 |u| 2 dxdt = 1 2 τ 0 √ ε 0 |u| 2 dxdt + 1 2 τ 0 L √ ε |u| 2 dxdt =: I 1 + I 2 .
We claim that

I 1 ≤ ε 2 τ 0 L 0 |u x | 2 dxdt, (4.32)
I 2 ≤ 1 2 √ ε τ 0 L 0 x|u| 2 dxdt. (4.33)
For (4.32), since u(0, t) = 0 we have that for (t, x) ∈ (0, T ) × (0, √ ε)

|u(x, t)| ≤ √ ε 0 |u x |dx ≤ ε 1 4 √ ε 0 |u x | 2 dx 1 2
and hence

√ ε 0 |u| 2 dx ≤ ε √ ε 0 |u x | 2 dx
which gives (4.32) after integrating over t ∈ (0, τ ). (4.33) is obvious.

Gathering together (4.30)-(4.33), we obtain

1 2 L 0 x|u(τ, x)| 2 dx + ( 3 2 -ε) τ 0 L 0 |u x | 2 dxdt ≤ 1 2 L 0 x|u 0 (x)| 2 dx + 1 2 √ ε τ 0 L 0 x|u| 2 dxdt + C ε τ 0 ||f || 2 H -1 dt.
Letting ε = 1 and applying Gronwall's lemma, we obtain

||u|| 2 L ∞ (0,T,L 2 xdx ) + ||u x || 2 L 2 (0,T,L 2 (0,L)) ≤ C(T ) ||u 0 || 2 L 2 xdx + ||f || 2 L 2 (0,T,H -1 (0,L))
. This gives (4.29) for u 0 ∈ D(A 1 ) and f ∈ C 0 ([0, T ], D(A 1 )). A density argument allows us to construct a solution u ∈ C([0, T ], L 2 xdx ) ∩ L 2 (0, T, H 1 (0, L)) of (4.26)-(4.28) satisfying (4.29) for u 0 ∈ L 2 xdx and f ∈ L 2 (0, T, H -1 (0, L)). The uniqueness follows from classical semigroup theory.

Our goal now is to obtain a similar result in the spaces H and V introduced in (4.15)-(4.16). To do that, we limit ourselves to the situation when f = (ρ(x)h) x with h ∈ L 2 (0, T, L 2 (0, L)). Proposition 4.6. Let u 0 ∈ H and h ∈ L 2 (0, T, L 2 (0, L)), and set f := (ρ(x)h) x . Then there exists a unique solution u ∈ C([0, T ], H) ∩ L 2 (0, T, V ) to (4.26)-(4.28). Furthermore, there is some constant C > 0 such that

(4.34) ||u|| L ∞ (0,T,H) + ||u|| L 2 (0,T,V ) ≤ C ||u 0 || H + ||h|| L 2 (0,T,L 2 (0,L)) .
Proof. Assume that u 0 ∈ D(A 2 ) and h ∈ C ∞ 0 ((0, T ) × (0, L)), so that f ∈ C 1 ([0, T ], H). Taking the inner product of u t -A 2 uf = 0 with u in H yields

(4.35) (u t , u) H = -a(u, u) + (f, u) H ≤ -C||u|| 2 V + L 18ε ||u|| 2 H + (f, u) H ,
where we used (4.22). Then

|(f, u) H | = | L 0 (ρ(x)h) x u L -x dx| = | L 0 ρ(x)h u x L -x + u (L -x) 2 dx| ≤ C||h|| L 2 (|| u x L -x || L 2 + || u (L -x) 2 || L 2 ) ≤ C||h|| L 2 ||u|| V ,
where we used (4.17) in the last line. Thus, we have that

|(f, u) H | ≤ C 2 ||u|| 2 V + C ′ ||h|| 2 L 2
which, when combined with (4.35), gives after integration over (0, τ ) for 0 < τ < T

||u(τ )|| 2 H + C τ 0 ||u|| 2 V dt ≤ ||u 0 || 2 H + C ′′ τ 0 ||u|| 2 H dt + τ 0 L 0 |h| 2 dxdt .
An application of Gronwall's lemma yields (4.34) for u 0 ∈ D(A 2 ) and h ∈ C ∞ 0 ((0, T ) × (0, L)). A density argument allows us to construct a solution u ∈ C([0, T ], H) ∩ L 2 (0, T, V ) of (4.26)-(4.28) satisfying (4.34) for u 0 ∈ H and h ∈ L 2 (0, T, L 2 (0, L)). The uniqueness follows from classical semigroup theory. 4.5. Controllability of the linearized system. We turn our attention to the control properties of the linear system

u t + u xxx + u x = f = (ρ(x)h) x , (4.36)
u(t, 0) = u(t, L) = u x (t, L) = 0, (4.37) u(0, x) = u 0 (x). (4.38) Theorem 4.7. Let T > 0 , ν ∈ (0, L) and ρ(x) as in (4.1). Then there exists a continuous linear operator

Γ : L 2 1 L-x dx → L 2 (0, T, L 2 (0, L)) ∩ L 2
(T -t)dt (0, T, H 1 (0, L)) such that for any u 1 ∈ L 2 1 L-x dx , the solution u of (4.36)-(4.38) with u 0 = 0 and h = Γ(u 1 ) satisfies u(T, x) = u 1 (x) in (0, L).

Note that the forcing term

f = (ρ(x)h) x is actually a function in L 2 (T -t)dt (0, T, L 2 (0, L)) supported in (0, T ) × (L -ν, L).
Proof. We use the Hilbert Uniqueness Method (see e.g. [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]). Introduce the adjoint system

-v t -v xxx -v x = 0, (4.39) v(t, 0) = v(t, L) = v x (t, 0) = 0, (4.40) v(T, x) = v T (x). (4.41)
If u 0 ≡ 0, v T ∈ D(0, L), and h ∈ D((0, T ) × (0, L)), then multiplying in (4.36) by v and integrating over (0, T ) × (0, L) gives

L 0 u(T, x)v T (x)dx = T 0 L 0 (ρ(x)h) x vdxdt = - T 0 L 0 ρ(x)hv x dxdt.
The usual change of variables x → Lx, t → Tt, combined with Proposition 4.5, gives

||v|| L ∞ (0,T,L 2 (L-x)dx ) + ||v|| L 2 (0,T,H 1 (0,L)) ≤ C||v T || L 2 (L-x)dx
. By a limiting argument, we obtain that for all h ∈ L 2 (0, T, L 2 (0, L)) and all

v T ∈ L 2 (L-x)dx , u(T, .), v T L 2 1 L-x dx ,L 2 (L-x)dx = - T 0 (h, ρ(x)v x ) L 2 dt,
where u and v denote the solutions of (4.36)-(4.38) and (4.39)-(4.41), respectively, and

•, • L 2 1 L-x dx ,L 2 (L-x)dx
denotes the duality pairing between L 2 1 L-x dx and L 2 (L-x)dx . We have to prove the following observability inequality

(4.42) ||v T || 2 L 2 (L-x)dx ≤ C T 0 L 0 |ρ(x)v x | 2 dxdt or, equivalently, letting w(t, x) = v(T -t, L -x), (4.43) ||w 0 || 2 L 2 xdx ≤ C T 0 L 0 |ρ(L -x)w x | 2 dxdt
where w solves (4.44)

   w t + w xxx + w x = 0, w(t, 0) = w(t, L) = w x (t, L) = 0, w(0, x) = w 0 (x).
From [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF], we know that for any q

∈ C ∞ ([0, T ] × [0, L]) - T 0 L 0 (q t + q xxx + q x ) w 2 2 dxdt + L 0 (q w 2 2 )(T, x)dx - L 0 (q w 2 2 )(0, x)dx + 3 2 T 0 L 0 q x w 2 x dxdt + T 0 (q w 2 x 2 )(t, 0)dt = 0. We pick q(t, x) = (T -t)b(x), where b ∈ C ∞ ([0, L]) is nondecreasing and satisfies b(x) = x if 0 < x < ν/4, 1 if ν/2 < x < L.
This yields

||w 0 || 2 L 2 xdx ≤ C(L, ν) L 0 b(x)w 2 0 (x)dx ≤ C(T, L, ν) T 0 ν 2 0 w 2 x dxdt + T 0 L 0 w 2 dxdt . ( 4
} ⊂ L 2 xdx such that (4.47) 1 = ||w n 0 || 2 L 2 xdx > n T 0 ν 2 0 |w n x | 2 dxdt,
where w n denotes the solution of (4.44) with w 0 replaced by w n 0 . By (4.29) and (4.47), {w n } is bounded in L 2 (0, T, H 1 (0, L)), hence also in H 1 (0, T, H -2 (0, L)) by (4.44). Extracting a subsequence, we have by Aubin-Lions' lemma that w n converges strongly in L 2 (0, T, L 2 (0, L)). Thus, using (4.45) and (4.47), we see that w n 0 is a Cauchy sequence in L 2 xdx , and hence it converges strongly in this space. Let w 0 denote its limit in L 2 xdx , and let w denote the corresponding solution of (4.44). Then

||w 0 || L 2 xdx = 1, w n → w in L 2 (0, T, H 1 (0, L)).
But w n x → 0 in L 2 (0, T, L 2 (0, ν/2)) by (4.47). Thus w x ≡ 0 in (0, T ) × (0, ν/2), and hence w(t, x) = g(t) (for some function g) in (0, T ) × (0, ν/2). Since w satisfies (4.44), we infer from w(t, 0) = 0 that w ≡ 0 in (0, T ) × (0, ν/2), and also in (0, T ) × (0, L) by Holmgren's theorem. This would imply that w(0, x) = 0, in contradiction with ||w 0 || L 2 xdx = 1. Therefore (4.46) is proved, and (4.43) follows at once. We are in a position to apply H.U.M. Let Λ(v T ) = (Lx) -1 u(T, .) ∈ L 2 (L-x)dx , where u solves (4.36)-(4.38) with h = -ρ(x)v x . Then Λ : L 2 (L-x)dx → L 2 (L-x)dx is clearly continuous. On the other hand, from (4.42)

Λ(v T ), v T L 2 (L-x)dx = u(T, .), v T L 2 1 L-x dx ,L 2 (L-x)dx = T 0 ||ρ(x)v x || 2 L 2 dt ≥ C||v T || 2 L 2 (L-x)dx
, and it follows that the map v

T → Λ(v T ) is invertible in L 2 (L-x)dx . Define the map Γ : L 2 1 L-x dx → L 2 (0, T, L 2 (0, L)) by Γ(u 1 ) = h := -ρ(x)v x ,
where v is the solution of (4.39)-(4.41) with v T = Λ -1 ((Lx) -1 u 1 ). Γ is continuous from L 2 1 L-x dx to L 2 (0, T, L 2 (0, L)), and the solution u of (4.36)-(4.38) with u 0 = 0 and h = Γ(u 1 ) satisfies u(T, .) = u 1 . To prove that Γ is also continuous from L 2

T 0 ||w|| 2 H 1 0 (0,L) dt ≤ C||w 0 || 2 L 2 xdx .
This yields for w 0 ∈ L 2 (0, L) Assume now that w 0 ∈ D(A), and let u 0 = Aw 0 = -w 0,xxxw 0,x . Denote by w (resp. u) the solution of (4.44) issuing from w 0 (resp. u 0 ). Then Aw = -w xxxw x = u ∈ L 2 (0, T, H 1 0 (0, L)), and we infer that w ∈ L 2 (0, T, H 4 (0, L)). By interpolation, this gives that w ∈ L 2 (0, T, H 2 (0, L)) if w 0 ∈ H 1 0 (0, L), with an estimate of the form This completes the proof of (4.48) and of Theorem 4.7.

4.6. Exact controllability of the nonlinear system. Our aim is to prove the local exact controllability in L 2 1 L-x dx of system (4.2). Note that the solutions of (4.2) can be written as

u = u L + u 1 + u 2 ,
where u L is the solution of (4.3) with initial data u 0 ∈ L 2 1 L-x dx , u 1 is solution of (4.52)    u 1,t + u 1,x + u 1,xxx = f = (ρ(x)h) x in (0, T ) × (0, L), u 1 (t, 0) = u 1 (t, L) = u 1,x (t, L) = 0 in (0, T ), u 1 (0, x) = 0 in (0, L) with h = h(t, x) ∈ L 2 (0, T ; L 2 (0, L)), and u 2 is solution of (4.53)    u 2,t + u 2,x + u 2,xxx = g(t, x) in (0, T ) × (0, L), u 2 (t, 0) = u 2 (t, L) = u 2,x (t, L) = 0 in (0, T ), u 2 (0, x) = 0 in (0, L), with g = g(t, x) = -uu x .

The following result is concerned with the solutions of the non-homogeneous system (4.53).

Proposition 4.8. (i) Let H and V be as in (4.15)-(4.16) If u, v ∈ L 2 (0, T ; V ), then uv x ∈ L 1 (0, T ; H). Furthermore, the map (u, v) ∈ L 2 (0, T ; V ) 2 → uv x ∈ L 1 (0, T ; H) is continuous and there exists a constant c > 0 such that (4.54) uv x L 1 (0,T ;H) ≤ c u L 2 (0,T ;V ) v L 2 (0,T ;V ) .

(ii) For g ∈ L 1 (0, T ; H), the mild solution u of (4.53) given by Duhamel formula satisfies Proof. For u, v ∈ V , we have

||uv x || L 2 1 L-x dx ≤ ||u|| L ∞ || v x √ L -x || L 2 ≤ C||u|| V ||v|| V .
This gives (i). For (ii), we first assume that g ∈ C 1 ([0, T ], H), so that u 2 ∈ C 1 ([0, T ], H) ∩ C 0 ([0, T ], D(A 2 )).

Taking the inner product of u ) H where C, C ′ denote some positive constants. Integrating over (0, T ) and using the classical estimate ||u 2 || L ∞ (0,T,H) ≤ C||g|| L 1 (0,T,H) coming from semigroup theory, we obtain (ii) when g ∈ C 1 ([0, T ], H). The general case (g ∈ L 1 (0, T, H)) follows by density.

Let Θ 1 (h) := u 1 and Θ 2 (g) := u 2 , where u 1 (resp. u 2 ) denotes the solution of (4.52) (resp. (4.53)). Then Θ 1 : L 2 (0, T ; L 2 (0, L)) → G and Θ 2 : L 1 (0, T ; L 2 Using Proposition 4.8 and the contraction mapping principle, one can prove as in [START_REF] Goubet | On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval[END_REF][START_REF] Perla-Menzala | Stabilization of the Korteweg-de Vries equation with localized damping[END_REF][START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF] the existence and uniqueness of a solution u ∈ G of (4.2) when the initial data u 0 and the forcing term h are small enough. As the proof is similar to those of Theorem 4.9, it will be omitted.

We are in a position to prove the main result of Section 4, namely the (local) exact controllability of system (4.2). Theorem 4.9. Let T > 0. Then there exists δ > 0 such that for any u 0 , u 1 ∈ L 2 As in the linear case, the forcing term f = (ρ(x)h) x is actually a function in L 2 (T -t)dt (0, T, L 2 (0, L)) supported in (0, T ) × (Lν, L).

Proof. To prove this result, we apply the contraction mapping principle, following closely [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]. Let F denote the nonlinear map F : L 2 (0, T ; V ) → G, defined by

F (u) = u L + Θ 1 • Γ(u T -u L (T, •) + Θ 2 (uu x )(T, •)) -Θ 2 (uu x ),
where u L is the solution of (4.3) with initial data u 0 ∈ L 2 1 L-x dx , Θ 1 and Θ 2 are defined as above, and Γ is as in Theorem 4.7.

Remark that if u is a fixed point of F , then u is a solution of (4.2) with the control h = Γ(u Tu L (T, •) + Θ 2 (uu x )(T, •)), and it satisfies u(T, •) = u T , as desired. In order to prove the existence of a fixed point of F , we apply the Banach fixed-point theorem to the restriction of F to some closed ball B(0, R) in L 2 (0, T ; V ). (i) F is contractive. Pick any u, ũ ∈ B(0, R). Using (4.34) and (4.54)-(4.55), we deduce that for some constant C, independent of u, ũ, and R, we have (4.57) F (u) -F (ũ) L 2 (0,T ;V ) ≤ 2CR uũ L 2 (0,T ;V ) .

Hence

  (3.26) Combining (3.25) with (3.26), picking s ≫ 1, and replacing again u by e -sϕ v(Tt, Lx) yields (3.13).

2 3 2 3 2 3

 222 (0, L)). Since H (0, L) is an algebra, we conclude that ξv 2x ∈ L 2 (0, T, H (0, L)). Therefore (3.47) f 3 L 2 (0,T ;H 2/3 (0,L)) ≤ Cs v 2 L 2 (0,T ;H 7/3 (0,L))∩L ∞ (0,T ;H 4/3 (0,L)) .

  7) and satisfies (1.8).

Proposition 4 . 4 .

 44 Let H and V be as in (4.15)-(4.16), and let T > 0 be given. Then there exists some constant C = C(L, T ) such that for any u 0 ∈ H, the solution u(t) = e tA2 u 0 of (4.3) satisfies(4.25) ||u|| L ∞ (0,T,H) + ||u|| L 2 (0,T,V ) ≤ C||u 0 || H .

Proposition 4 . 5 . 4

 454 Let u 0 ∈ L 2 xdx and f ∈ L 2 (0, T ; H -1 (0, L)). Then there exists a unique solution u ∈ C([0, T ], L 2 xdx ) ∩ L 2 (0, T, H 1 (0, L)) to (4.26)-(4.28). Furthermore, there is some constantC > 0 such that (4.29) ||u|| L ∞ (0,T,L 2 xdx ) + ||u|| L 2 (0,T,H 1 (0,L)) ≤ C ||u 0 || L 2 xdx + ||f || L 2 (0,T,H -1 (0,L) .Proof. Assume first that u 0 ∈ D(A 1 ) and f ∈ C 0 ([0, T ], D(A 1 )) to legitimate the following computations. Multiplying each term in (4.26) by xu and integrating over (0, τ ) × (0, L) where 0 < τ < T yields (

  one can find a sequence {w n 0

1 0

 1 (0,L) dt ≤ C||w 0 || 2 L 2 .

2 H 2 2 H 1 0 2 H 1 0

 222121 (0,L) dt ≤ C||w 0 || (0,L) .The different constants C in (4.49)-(4.51) may be taken independent of T for 0 < T < T 0 . Thus, using Fubini's theorem, we obtain (0,L) dt ≤ C||w 0 || 2 L 2 xdx .

u 2 ∈

 2 C([0, T ] ; H) ∩ L 2 (0, T ; V ) =: G and we have the estimate(4.55) ||u 2 || L ∞ (0,T,H) + ||u 2 || L 2 (0,T,V ) ≤ C||g|| L 1 (0,T,H) .

1 L

 1 -x dx ) → G are well-defined continuous operators, by Propositions 4.6 and 4.8.

1 L

 1 one can find a control function h ∈ L 2 (0, T ; L 2 (0, L)) such that the solution u ∈ G of (4.2) satisfies u(T, •) = u 1 in (0, L).

  5 |q| 2 ]e -2sϕ dxdt

	Actually, we shall need a Carleman estimate for (3.2) with the potential ξ ∈ Y 1 4 . Let
	φ(t, x) = ϕ(t, L -x).
	Corollary 3.2. Let ξ ∈ Y 1 4 . Then there exist some positive constants s0 = s0 (T, ||ξ|| Y 1 4

1

  L-x dx into L 2 (T -t)dt (0, T, H 1 (0, L)), it is sufficient to prove the following estimate

	T
	0	||v(t)|| 2 H 2 (T -t)dt ≤ C||v T || 2 L 2 (L-x)dx	,
	for the solutions of (4.39)-(4.41) or, alternatively, the estimate
		T
	(4.48)	0	||w|| 2 H 2 tdt ≤ C||w 0 || 2 L 2 xdx
	for the solutions of (4.44). By Proposition 4.5,
	(4.49)

  2,t = A 2 u 2 + g with u 2 in H yields (4.56) (u 2,t , u 2 ) H ≤ -C||u 2 || 2 V + C ′ ||u 2 || 2 H + (g, u 2

  , F is contractive if R satisfies Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530 -Cidade Universitária -Ilha do Fundão, 21941-909 Rio de Janeiro (RJ), Brazil and Institut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, BP 70239, 54506 Vandoeuvre-les-Nancy Cedex, France E-mail address: capistrano@im.ufrj.br Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530 -Cidade Universitária -Ilha do Fundão, 21941-909 Rio de Janeiro (RJ), Brazil E-mail address: ademir@im.ufrj.br Institut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, BP 70239, 54506 Vandoeuvre-les-Nancy Cedex, France E-mail address: rosier@iecn.u-nancy.fr

	(4.58)	R <	1 4C	,

where C is the constant in (4.57).

Acknowledgments: RC was supported by CNPq and Capes (Brazil) via a Fellowship, Project " Ciência sem Fronteiras" and Agence Nationale de la Recherche (ANR), Project CISIFS, grant ANR-09-BLAN-0213-02. LR was partially supported by the Agence Nationale de la Recherche (ANR), Project CISIFS, grant ANR-09-BLAN-0213-02. AFP was partially supported by CNPq (Brazil) and the Cooperation Agreement Brazil-France.

(ii) F maps B(0, R) into itself. Using Proposition 4.4 and the continuity of the operators Γ, Θ 1 , and Θ 2 , we infer the existence of a constant C ′ > 0 such that for any u ∈ B(0, R), we have

Thus, taking R satisfying (4.58) and R < 1/(2C ′ ) and assuming that u

are small enough, we obtain that the operator F maps B(0, R) into itself. Therefore the map F has a fixed point in B(0, R) by the Banach fixed-point Theorem. The proof of Theorem 4.9 is complete.