Rémi Buffe

Takéo Takahashi

Controllability of a fluid-structure interaction system coupling the Navier-Stokes system and a damped beam equation

Keywords: Null controllability, Navier-Stokes systems, Carleman estimates, fluid-structure interaction systems 2010 Mathematics Subject Classification. 76D05, 35Q30, 74F10, 76D55, 76D27, 93B05, 93B07, 93C10

We show the local null-controllability of a fluid-structure interaction system coupling a viscous incompressible fluid with a damped beam located on a part of its boundary. The controls act on arbitrary small parts of the fluid domain and of the beam domain. In order to show the result, we first use a change of variables and a linearization to reduce the problem to the null-controllability of a Stokes-beam system in a cylindrical domain. We obtain this property by combining Carleman inequalities for the heat equation, for the damped beam equation and for the Laplace equation with high-frequency estimates. Then, the result on the nonlinear system is obtained by a fixed-point argument.

Introduction

We consider a fluid-structure interaction system composed by a viscous incompressible fluid, modeled by the Navier-Stokes system, and by an elastic structure located at a part of the boundary of the fluid domain. We assume that the structure displacement is governed by a damped beam equation. The corresponding model has been introduced in [START_REF] Quarteroni | Computational vascular fluid dynamics: problems, models and methods[END_REF] as a first model to study the blood flow in vessels. To simplify our work, we consider here a particular geometry in dimension 2 of space (see Figure 1). The fluid domain is confined into an infinite strip where the bottom boundary is fixed and where the top boundary corresponds to the beam. We also assume periodic condition in the x 1 variables. To be more precise, we set

I := R/(2πZ),
and for any deformation ζ : I → (-1, ∞), we consider the fluid domain associated with this deformation:

Ω ζ = {(x 1 , x 2 ) ∈ I × R ; x 2 ∈ (0, 1 + ζ(x 1 ))} .
(1.1)

Then the fluid-structure interaction system writes

               ∂ t w + (w • ∇)w -div T(w, π) = 1 ω f t > 0, x ∈ Ω ζ(t) , div w = 0 t > 0, x ∈ Ω ζ(t) , w(t, x 1 , 1 + ζ(t, x 1 )) = (∂ t ζ)(t, x 1 )e 2 , t > 0, x 1 ∈ I, w = 0 t > 0, x ∈ Γ 0 , ∂ tt ζ + α 1 ∂ 4 x1 ζ -α 2 ∂ 2 x1 ζ -α 3 ∂ t ∂ 2 x1 ζ = -H ζ (w, π) + 1 J g t > 0, x 1 ∈ I, w(0, •) = w 0 in Ω ζ 0 1 , ζ(0, •) = ζ 0 1 , ∂ t ζ(0, •) = ζ 0 2 in I, (1.2)
where α 1 > 0, α 2 ⩾ 0, α 3 > 0, and where Γ 0 = I × {0}.

In the above system, we have used the following notations: (e 1 , e 2 ) is the canonical basis of R 2 and T(w, π) = 2D(w) -πI 2 , D(w) = 1 2 (∇w + (∇w) * ) ,

H ζ (w, π)(t, x 1 ) = (1 + |∂ x1 ζ| 2 ) 1/2 [T(w, π)n] (t, x 1 , 1 + ζ(t, x 1 )) • e 2 . (1.3)
We have also denoted by n the unit exterior normal to Ω ζ(t) . In (1.2), w and π are respectively the velocity and the pressure of the fluid and they satisfy the Navier-Stokes system (two first lines), with no-slip boundary conditions (third and forth equations). The elastic displacement satisfies the damped beam equation written in the fifth line of (1.2). Finally, our aim is to control (1.2) by using two distributed controls f and g respectively localized in an arbitrary small nonempty open subset ω of Ω and in an arbitrary small nonempty open subset J of I.

Let us remark that the well-posedness and the stabilization of system (1.2) have been already studied in the literature. Let us quote some of the corresponding articles: [START_REF] Chambolle | Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate[END_REF] (existence of weak solutions), [START_REF] Veiga | On the existence of strong solutions to a coupled fluid-structure evolution problem[END_REF], [START_REF] Lequeurre | Existence of strong solutions to a fluid-structure system[END_REF], [START_REF] Grandmont | Existence of global strong solutions to a beam-fluid interaction system[END_REF] and [START_REF] Maity | L p theory for the interaction between the incompressible Navier-Stokes system and a damped plate[END_REF] (existence of strong solutions), [START_REF] Raymond | Feedback stabilization of a fluid-structure model[END_REF] (stabilization of strong solutions), [START_REF] Badra | Feedback boundary stabilization of 2D fluid-structure interaction systems[END_REF] (stabilization of weak solutions around a stationary state). We can also mention some works devoted to the case δ = 0 (undamped beam equation/wave equation): [START_REF] Grandmont | Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate[END_REF], [START_REF] Muha | Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls[END_REF], [START_REF] Čanić | Fluid-structure interaction in hemodynamics: modeling, analysis, and numerical simulation[END_REF] (weak solutions), [START_REF] Grandmont | Existence of local strong solutions to fluidbeam and fluid-rod interaction systems[END_REF], [START_REF] Badra | Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation[END_REF], [START_REF] Badra | Gevrey regularity for a system coupling the Navier-Stokes system with a beam: the non-flat case[END_REF], [START_REF] Badra | Maximal regularity for the Stokes system coupled with a wave equation: application to the system of interaction between a viscous incompressible fluid and an elastic wall[END_REF] (strong solutions). Some authors have tackled the study of more complex models: [START_REF] Lengeler | Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell[END_REF][START_REF] Lengeler | Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell[END_REF] (linear elastic Koiter shell), [START_REF] Muha | Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls[END_REF] (dynamic pressure

Ω ζ Γ 0 -- -- {x 2 = 1 + ζ(x 1 )} 0 2π 1 Ω Γ 0 -- -- Γ 1 (beam) 0 2π
1 Figure 1: Our geometry boundary conditions), [START_REF] Muha | A nonlinear, 3D fluid-structure interaction problem driven by the timedependent dynamic pressure data: a constructive existence proof[END_REF][START_REF] Muha | Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy[END_REF] (3D cylindrical domain with nonlinear elastic cylindrical Koiter shell), [START_REF] Trifunović | Existence of a weak solution to the fluid-structure interaction problem in 3D[END_REF] and [START_REF] Trifunović | Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D[END_REF] (nonlinear elastic and thermoelastic plate equations), [START_REF] Maity | Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation[END_REF], [START_REF] Maity | Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation[END_REF] (compressible fluids), etc.

A standard strategy to study this kind of systems consists in using a change of variables to write the fluid system into a cylindrical domain, and then in linearizing the system after this transformation. A large part of the work is thus devoted to the corresponding linear system, the results for the nonlinear system are deduced by estimating the coefficients coming from the change of variables and by using a fixed-point argument. We follow here this approach and after a change of variable and a linearization (see Section 6 for the details), we are reduced to work on the spatial domain Ω := Ω 0 = I × (0, 1) (see Figure 1) and to show the null controllability of the following linear system

               ∂ t w -∆w + ∇π = 1 ω f in (0, T ) × Ω, div w = 0 in (0, T ) × Ω, w = 0 on (0, T ) × Γ 0 , w = (∂ t ζ)e 2 on (0, T ) × Γ 1 , ∂ 2 t ζ + α 1 ∂ 4 x1 ζ -α 2 ∂ 2 x1 ζ -α 3 ∂ t ∂ 2 x1 ζ = -T(w, π)n • e 2 + 1 J g in (0, T ) × I, w(0, •) = w 0 in Ω, ζ(0, •) = ζ 0 1 , ∂ t ζ(0, •) = ζ 0 2 in I, (1.5)
where

Γ 1 = I × {1}.
In what follows, to simplify the notation, we take

α 1 = α 2 = α 3 = 1.
The values of these constants do not play any role in our study. As it is standard (see, for instance, [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Theorem 11.2.1, p.357]), the controllability of (1.5) is equivalent to an observability inequality for the adjoint system

               ∂ t u -∆u + ∇p 0 = 0 in (0, T ) × Ω, div u = 0 in (0, T ) × Ω, u = 0 on (0, T ) × Γ 0 , u = ∂ t ηe 2 on (0, T ) × Γ 1 , ∂ 2 t η + ∂ 4 x1 η -∂ 2 x1 η -∂ t ∂ 2 x1 η = -T(u, p 0 )n |Γ1 • e 2 in (0, T ) × I, u(0, •) = u 0 in Ω, η(0, •) = η 0 1 , ∂ t η(0, •) = η 0 1 in I.
(1.6)

Before writing the corresponding observability inequality, let us mention an important remark and introduce some notation. We set

L 2 0 (I) := f ∈ L 2 (I) ; 2π 0 f (x 1 ) dx 1 = 0 .
Remark 1.1. Using the particular geometry considered here, we can simplify the above adjoint system. First on Γ 1 , n = e 2 and using (1.3), we deduce

-T(u, p 0 )n • e 2 = -2∂ x2 u 2 + p 0 = 2∂ x1 u 1 + p 0 = p 0 on Γ 1 , (1.7)
since u 1 (x 1 , 1) = 0 for x 1 ∈ I. Moreover, using the incompressibility of the fluid and the boundary conditions, we deduce that

0 = Ω div u dx = d dt 2π 0 η dx 1 .
Assuming that η 0 1 ∈ L 2 0 (I) then, we deduce that for all t ⩾ 0, η(t, •) ∈ L 2 0 (I). Using this condition on the beam equation leads to the following condition on the pressure: 2π 0 p 0 (t, x 1 , 1) dx 1 = 0.

(1.8)

In particular, in contrast with the standard Stokes system, the pressure is not determined up to a constant.

We define the operators associated with the beam equation:

D(A 1 ) := H 4 (I) ∩ L 2 0 (I), A 1 η := ∂ 4 x1 η -∂ 2 x1 η, (1.9)
D(A 2 ) := H 2 (I) ∩ L 2 0 (I), A 2 η := -∂ 2 x1 η. (1.10)
We also define the Hilbert space of states for our system:

H := (u, η 1 , η 2 ) ∈ L 2 (Ω) × D(A 1/2 1 ) × L 2 0 (I) ; u 2 = η 2 on Γ 1 , u 2 = 0 on Γ 0 , div u = 0 in Ω , (1.11)
endowed with the canonical scalar product of

L 2 (Ω) × D(A 1/2 1 ) × L 2 (I).
With the above remark and notation, the adjoint system writes

               ∂ t u -∆u + ∇p 0 = 0 in (0, T ) × Ω, div u = 0 in (0, T ) × Ω, u = 0 on (0, T ) × Γ 0 , u = ∂ t ηe 2 on (0, T ) × Γ 1 , ∂ 2 t η + A 1 η + A 2 ∂ t η = p 0| Γ 1 in (0, T ) × I, u(0, •) = u 0 in Ω, η(0, •) = η 0 1 , ∂ t η(0, •) = η 0 2 in I, (1.12)
with the condition (1.8). Our main result stated below is an observability inequality for (1.12):

Theorem 1.2. Assume T > 0, ω ⋐ Ω and J ⋐ I are nonempty open sets. For any [u 0 , η 0 1 , η 0 2 ] ∈ H, the solution of (1.12) satisfies

∥u(T, •)∥ 2 L 2 (Ω) + ∥η(T, •)∥ 2 H 2 (I) + ∥∂ t η(T, •)∥ 2 L 2 (I) ⩽ k 2 T (0,T )×ω |u| 2 dx dt + (0,T )×J |∂ t η| 2 dx 1 dt , (1.13)
and we can choose k T in the form

k T = Ce C/T 2 , (1.14)
with a constant C > 0.

The controllability of fluid-structure interaction systems has already been tackled in the case where the structure is a rigid body in [START_REF] Boulakia | Local null controllability of a fluid-solid interaction problem in dimension 3[END_REF], [START_REF] Boulakia | Local null controllability of a two-dimensional fluid-structure interaction problem[END_REF], [START_REF] Doubova | Some control results for simplified one-dimensional models of fluid-solid interaction[END_REF], [START_REF] Imanuvilov | Exact controllability of a fluid-rigid body system[END_REF], [START_REF] Roy | Local null controllability of a rigid body moving into a Boussinesq flow[END_REF]. Up to our knowledge, the above theorem is the first result of controllability for the system (1.5). Let us mention also [START_REF] Sourav | Observability and unique continuation of the adjoint of a linearized simplified compressible fluid-structure model in a 2D channel[END_REF] where the author obtains an observability inequality for the adjoint of a linearized simplified compressible fluid-structure model similar to our system.

Let us point out that due to the structural damping in the beam equation (-∂ t ∂ 2 x1 ζ) the corresponding beam equation becomes a parabolic equation (see, for instance, [START_REF] Shu | Proof of extensions of two conjectures on structural damping for elastic systems[END_REF]). In a previous work [START_REF] Buffe | Controllability of a Stokes system with a diffusive boundary condition[END_REF], we have replaced the damped beam equation by a heat equation and we have shown the corresponding controllability result. The proof done here is inspired by our previous work, and in particular, in the proof of the observability, we first apply results on the heat equations to the fluid velocity by considering the pressure as a source term, (in the spirit of [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF]). Then, we estimate the pressure by using that it satisfies a Laplace equation. Since the boundary conditions of this Laplace equation are difficult to handle, our estimates on the pressure depend on the boundary value of the pressure and more precisely on the high frequencies of the pressure on the boundary of the fluid domain. To conclude, we apply some energy inequalities combined with a high frequency argument in the horizontal direction to estimate these high frequencies. Using the microlocal analysis near boundaries and interfaces to derive Carleman estimates and to show the controllability of coupled parabolic systems is quite standard and one can quote for instance [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems[END_REF][START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part II: Transmission problems[END_REF][START_REF] Buffe | Stabilization of the wave equation with Ventcel boundary condition[END_REF][START_REF] Jérôme | Controllability of a parabolic system with a diffuse interface[END_REF][START_REF] Rousseau | Carleman estimates for anisotropic elliptic operators with jumps at an interface[END_REF][START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF][START_REF] Rousseau | Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF] and the recent books [START_REF] Jérôme | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF][START_REF] Jérôme | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF] for elliptic counterparts).

One of the main differences with [START_REF] Buffe | Controllability of a Stokes system with a diffusive boundary condition[END_REF] is that we work here directly with the time variable whereas in the previous work we show a spectral inequality and then use an abstract method ( [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the nullcontrollability of parabolic systems[END_REF][START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]) to deduce the corresponding observability inequality. Here we do not follow the same approach since it uses that the main operator of our system is self-adjoint, and here our main operator is not self-adjoint or even a perturbation of a self-adjoint operator as in the framework considered in [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the nullcontrollability of parabolic systems[END_REF]. A consequence of working directly with the time variable is that the separation between low and high frequencies is done through a pseudo-differential operator, which symbol depends on time, and in particular we need some standard commutator estimates from these operators in order to handle the high frequencies.

Remark 1.3. With respect to [START_REF] Buffe | Controllability of a Stokes system with a diffusive boundary condition[END_REF] or to the stabilization result [START_REF] Badra | Feedback boundary stabilization of 2D fluid-structure interaction systems[END_REF], one should expect to obtain the controllability of (1.2) or of (1.5) without any control on the beam equation (g ≡ 0). However, with our present approach, it seems difficult to handle the elastic displacement without any observation on the beam equation. Even with the presence of two controls, a particular treatment of the coupling between the pressure and the elastic displacement in the proof of the observability is needed. Concerning the particular geometry, we are using it several times in order to simplify several proofs but the corresponding result in a general geometry should hold even if it is not a direct consequence of our work.

We deduce from Theorem 1.2 the local controllability of (1.2): Theorem 1.4. Assume T > 0 and that ω ⋐ Ω and J ⋐ I are nonempty open sets. There exists R 0 > 0 such that for any

ζ 0 1 ∈ D(A 3/4 1 ), ζ 0 2 ∈ D(A 1/4 1 ), w 0 ∈ H 1 (Ω ζ 0 1 ) satisfying div w 0 = 0 in Ω, w 0 = 0 on Γ 0 , w 0 (x 1 , 1 + ζ 0 1 (x 1 )) = ζ 0 2 (x 1 )e 2 (x 1 ∈ I), (1.15)
and

ζ 0 1 H 3 (I) + ζ 0 2 H 1 (I) + w 0 H 1 Ω ζ 0 1 ⩽ R 0 , (1.16
)

there exists a control (f, g) ∈ L 2 (0, T ; L 2 (ω)) × L 2 (0, T ; L 2 (J ))
such that the solution of (1.2) satisfies

ζ(T, •) = 0, ∂ t ζ(T, •) = 0 in I, w(T, •) = 0 in Ω.
The proof of Theorem 1.4 is quite standard from Theorem 1.2: we need to estimate the coefficients of the change of variables and use a fixed point argument. Similar procedure is done to show the well-posedness or the stabilization of the system. We only sketch the proof of Theorem 1.4, the details can be found for instance in [START_REF] Raymond | Feedback stabilization of a fluid-structure model[END_REF], [START_REF] Badra | Feedback boundary stabilization of 2D fluid-structure interaction systems[END_REF].

The outline of the article is as follows: in the next section, we complete the functional setting needed in this article, introduce the Carleman weights and some classical results on pseudodifferential operators. Section 3 is devoted to Carleman estimates: a Carleman estimate for the heat equation, a Carleman estimate for the damped beam equation and a Carleman estimate for the pressure. Gathering them yields an estimate of the fluid velocity and pressure and of the elastic displacement by terms localized in ω or in J and by high frequencies of the pressure on the boundary. To get rid of these last terms, we show in Section 4 high frequency estimates using the Stokes system. This allows us to show the observability inequality in Section 5. We give the sketch of the proof of Theorem 1.4 in Section 6. Finally, in Appendix A, we recall some technical results concerning the Carleman estimates of Section 3. Notation 1.5. In the whole paper, we use C as a generic positive constant that does not depend on the other terms of the inequality. The value of the constant C may change from one appearance to another. We also use the notation X ≲ Y if there exists a constant C > 0 such that we have the inequality X ⩽ CY .

Notation and preliminaries

Functional setting

We complete the notation introduced in the introduction: we consider the control operator for the beam equation:

B J g := P L 2 0 (I) (1 J g) ,
where P L 2 0 (I) : L 2 (I) → L 2 0 (I) is the orthogonal projection. With the above notation and (1.9), (1.10), the beam equation in (1.5) writes

∂ 2 t ζ + A 1 ζ + A 2 ∂ t ζ = P L 2 0 (I) π + B J g.
We also consider the orthogonal projection on the space H defined by (1.11):

P : L 2 (Ω) × D(A 1/2 1 ) × L 2 0 (I) → H.
We recall (see, for instance, [1, Proposition 3.1]) that the orthogonal of H in L 2 (Ω) × D(A

1/2 1 ) × L 2 0 (I) is given by H ⊥ = (∇p, 0, P L 2 0 (I) p |Γ1 ) ; p ∈ H 1 (Ω) . (2.1)
Then we define the space

V := (u, η 1 , η 2 ) ∈ H 1 (Ω) × D(A 3/4 1 ) × D(A 1/4 1 ) ; u = η 2 e 2 on Γ 1 , u = 0 on Γ 0 , div u = 0 in Ω ,
and the unbounded operator A associated with (1.5):

D(A) := V ∩ H 2 (Ω) × D(A 1 ) × D(A 1/2 1 ) , A   u η 1 η 2   := P   ∆u η 2 -A 1 η 1 -A 2 η 2   .
It is shown (see, for instance, [START_REF] Badra | Feedback boundary stabilization of 2D fluid-structure interaction systems[END_REF]Proposition 3.11]) that A is the infinitesimal generator of an analytic semigroup on H. We have in particular that if F ∈ L 2 (0, T ; H), Φ 0 ∈ V, then there exists a unique solution

Φ ∈ L 2 (0, T ; D(A)) ∩ C 0 ([0, T ]; V) ∩ H 1 (0, T ; H) to dΦ dt = AΦ + F in (0, T ), Φ(0) = Φ 0 (2.2)
and we have the estimate

∥Φ∥ L 2 (0,T ;H 2 (Ω)×D(A1)×D(A 1/2 1 )) + ∥Φ∥ H 1 (0,T ;L 2 (Ω)×D(A 1/2 1 )×L 2 (I)) ≲ ∥F ∥ L 2 (0,T ;L 2 (Ω)×D(A 1/2 1 )×L 2 (I)) + Φ 0 V . (2.3)
Finally, we consider the control operator:

B f g := P   1 ω f 0 B J g   .
Using the above notation and (2.1), we can write (1.5) as

d dt   w ζ ∂ t ζ   = A   w ζ ∂ t ζ   + B f g in (0, T ),   w ζ ∂ t ζ   (0) =   w 0 ζ 0 1 ζ 0 2   . (2.4)
We say that the above system is null-controllable in time T > 0 if for any w 0 , ζ 0 1 , ζ 0 2 ∈ H, there exists a control f, g ∈ L 2 (0, T ; L 2 (ω) × L 2 (J )) such that the solution of the above system satisfies

  w ζ ∂ t ζ   (T ) = 0.
A classical result (see, for instance, [52, Theorem 11.2.1, p.357]) states that the null-controllability is equivalent to the final-state observability of the adjoint system: there exists k T > 0 such that for any

  u 0 η 0 1 η 0 2   ∈ H, the solution of d dt   u η 1 η 2   = A *   u η 1 η 2   in (0, T ),   u η 1 η 2   (0) =   u 0 -η 0 1 η 0 2   (2.5) satisfies   u η 1 η 2   (T ) 2 H ⩽ k 2 T T 0 B *   u η 1 η 2   (t) 2 L 2 (ω)×L 2 (J )
dt.

(2.6)

One can show that

D(A * ) = D(A), A *   u η 1 η 2   = P   ∆u -η 2 A 1 η 1 -A 2 η 2   and B *   u η 1 η 2   = u |ω η 2|J
.

Setting η = -η 1 we see that (2.5) writes as (1.12) or in the following abstract form

d dt   u η ∂ t η   = A   u η ∂ t η   in (0, T ),   u η ∂ t η   (0) =   u 0 η 0 1 η 0 2   . (2.7)
The observability inequality (2.6) writes as (1.13).

Weight functions for the Carleman estimates

We consider nonempty open subsets J 0 ⋐ J and ω 0 ⋐ ω and (see, for instance, [18, Lemma 1.1], [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Theorem 9.4.3]) two smooth functions ψ I and ψ Ω satisfying

ψ I > 0 in I, ψ ′ I (x 1 ) = 0 ⇒ x 1 ∈ J 0 , (2.8)
ψ Ω > 0 in Ω, ψ Ω = 0 and ∂ n ψ Ω = -1 on ∂Ω, ∇ψ Ω (x) = 0 ⇒ x ∈ ω 0 , (2.9)
with J 0 ⋐ J , ω 0 ⋐ ω.

(2.10)

In fact, using our particular geometry, one can show directly the existence of such functions ψ I and ψ Ω . We set ψ 1 (x 1 ) := 2 + sin(x 1 ) and we consider

ψ 2 ∈ C ∞ ([0, 1]), odd, ψ 2 (x 2 ) = x 2 in a neighborhood of 0, and ψ ′ 2 (x 2 ) = 0 ⇔ x 2 = 1/2.
We also consider θ ∈ C ∞ (R) with compact support in (0, 1) and such that θ ≡ 1 in a neighborhood of 1/2. Then for ε > 0 small enough,

ψ Ω (x 1 , x 2 ) = ψ 2 (x 2 ) + εθ(x 2 )ψ 1 (x 1 )
satisfies ψ Ω > 0 in Ω, ψ Ω = 0 and ∂ n ψ Ω = -1 on ∂Ω and it has only two critical points: (π/2, 1/2) and (-π/2, 1/2). By a change of variables on ψ 1 and on ψ Ω (see, for instance, [52, Proposition 14.3.1]), we obtain functions ψ I and ψ Ω satisfying (2.8) and (2.9).

We also denote by ℓ the function defined by

ℓ(t) := t(T -t). (2.11) Let us consider Ψ := ∥ψ I ∥ L ∞ (I) + ∥ψ Ω ∥ L ∞ (Ω)
and for λ ⩾ µ > 0, let us define the following functions

φ(t, x 1 , x 2 ) := 1 ℓ(t) 2 (e λψΩ(x1,x2)+µψ I (x1)+8λΨ -e 10λΨ ), ξ(t, x 1 , x 2 ) := 1 ℓ(t) 2 e λψΩ(x1,x2)+µψ I (x1)+8λΨ , (2.12)
φ 0 (t, x 1 ) := 1 ℓ(t) 2 (e µψ I (x1)+8λΨ -e 10λΨ ), ξ 0 (t, x 1 ) := 1 ℓ(t) 2 e µψ I (x1)+8λΨ .

(2.13)

We also define for λ ⩾ µ > 0 the function

ψ(x 1 , x 2 ) := µ λ ψ I (x 1 ) + ψ Ω (x 1 , x 2 ).
(2.14)

Spatial truncation

In order to use pseudodifferential operators in the x 1 variables, we consider that our functions are 2π-periodic functions defined in the domains

Ω ∞ := R × (0, 1), Γ ∞ 0 := R × {0}, Γ ∞ 1 := R × {1}.
In the adjoint system (1.12), we also replace the pressure p 0 that satisfies (1.8) by a pressure p satisfying another condition. More precisely, we consider ω 1 an open set such that ω 0 ⋐ ω 1 ⋐ ω and we define

c p (t) := - ω1 p 0 (t, x) dx and p := p 0 + c p . (2.15)
Then the pressure p verifies the condition

ω1 p(t, x) dx = 0 in (0, T ). (2.16) We consider χ ∞ ∈ C ∞ (R; [0, 1]) with compact support and such that χ ∞ ≡ 1 in [0, 2π]. We set u ∞ := χ ∞ u, p ∞ := χ ∞ p, η ∞ := χ ∞ η.
(2.17)

Then we deduce from (1.12) that

       ∂ t u ∞ -∆u ∞ + ∇p ∞ = f (1) in (0, T ) × Ω ∞ , div u ∞ = f (2) in (0, T ) × Ω ∞ , u ∞ = 0 on (0, T ) × Γ ∞ 0 , u ∞ = ∂ t η ∞ e 2 on (0, T ) × Γ ∞ 1 , (2.18)
and

∆p ∞ = f (3) in (0, T ) × Ω ∞ , (2.19)
where

f (1) := -(χ ∞ ) ′′ u -2 (χ ∞ ) ′ ∂ x1 u + (χ ∞ ) ′ pe 1 , f (2) := (χ ∞ ) ′ u 1 , f (3) = (χ ∞ ) ′′ p + 2(χ ∞ ) ′ ∂ x1 p.
(2.20)

Pseudodifferential operators

We consider a parameter τ ⩾ 1 and an order function

Λ τ (k) := τ 2 + k 2 (k ∈ R), (2.21)
where k corresponds to the Fourier variable associated with x 1 . For m ∈ R, we denote by S m τ the space of complex smooth functions a

= a(x 1 , k, τ ) defined on R × R × [1, ∞) and such that for all α, β ∈ N there exists C α,β > 0 ∂ α x1 ∂ β k a(x 1 , k, τ ) ⩽ C α,β Λ m-β τ (k) ((x 1 , k, τ ) ∈ R × R × [1, ∞)). (2.22)
For instance, we have Λ m τ ∈ S m τ and for any C ∈ R, the function

(k, τ ) → τ 2 -Ck 2 τ 2 + k 2
is in S 0 τ . We also recall the following classical lemma (see, for instance, [25, Proposition 2.3] or [22, p.73, Lemma 18.1.10] in the classical setting)

Lemma 2.1. If a ∈ S 0 τ and χ 0 ∈ C ∞ (R). Then χ 0 (a) ∈ S 0 τ .
From a ∈ S m τ , we can define the following operator on the Schwartz space on R :

[Op(a)u] (x 1 ) := 1 2π R 2 e ik(x1-y1) a(x 1 , k, τ )u(y 1 ) dy 1 dk.
We can also extend this operator to the Schwartz space on [0, T ] × R × [0, 1] by a similar formula:

[Op(a)u] (t, x 1 , x 2 ) := 1 2π R 2 e ik(x1-y1) a(x 1 , k, τ )u(t, y 1 , x 2 ) dy 1 dk.
From symbolic calculus, we have the following results (see, for instance, [25, pp.27-28, Theorem 2.22 and Corollary 2.23])

Theorem 2.2. Let m, m ′ ∈ R and let a ∈ S m τ , b ∈ S m ′ τ . Then there exist c ∈ S m+m ′ τ and d ∈ S m+m ′ -1 τ such that Op(a) • Op(b) = Op(c), [Op(a), Op(b)] = Op(d).
We can extend the operator associated with a symbol of order m to Sobolev spaces. For instance we have the following result (see [25, p.29

H m+m ′ (R) → H m ′ (R) and if m, m ′ ∈ N, we have i+j⩽m ′ τ 2i ∂ j x1 Op(a)u 2 L 2 (R) ≲ i+j⩽m+m ′ τ 2i ∂ j x1 u 2 L 2 (R) .
In what follows, we assume that the parameter τ is related to functions defined in Section 2.2 through the formula

τ := τ (t) = sλe 8λΨ ℓ 2 (t) . (2.23)
In particular, τ is a function of time and there exist s 0 > 0 and λ 0 > 0 such that if s ⩾ s 0 T 4 and λ ⩾ λ 0 , then

τ ⩾ τ λ ⩾ 1. (2.24)
Remark 2.4. Due to (2.23), the symbols in S m τ depends on the time variable through the parameter τ . The continuity estimates of Theorem 2.3 are uniform with respect to τ , and thus with respect to the time variable if it only appears in the parameter τ . In what follows, some symbols may depend on time, but not as a function of τ , this occurs for instance when considering ∂ t τ . In that case, we always decompose such symbols in terms of the form b(t)a(x 1 , k, τ ) where b is a bounded function of time, and a ∈ S m τ .

An important example of symbol used in what follows is a function of the form

χ(τ, k) := χ 0 τ 2 -Ck 2 τ 2 + k 2 ,
where C is a constant and χ 0 ∈ C ∞ (R). From Lemma 2.1, we have that χ ∈ S 0 τ and one can check that

[∂ x1 , Op(χ)] = [∂ x2 , Op(χ)] = 0.
Moreover, we have the following result on the time derivative of χ:

Lemma 2.5. Let χ be defined as above. Then

∂ t χ ∈ ℓ ′ (λs) 1/2 τ 5 2 S -2 τ .
Proof. By standard computations and (2.23),

∂ t χ(τ, k) = χ ′ 0 τ 2 -Ck 2 τ 2 + k 2 2 (C + 1) k 2 τ ∂ t τ (τ 2 + k 2 ) 2 , ∂ t τ = -2ℓ ′ sλe 8λΨ ℓ 3 . (2.25)
We have

sλe 8λΨ ℓ 3 ⩽ τ 3/2 (sλ) 1/2
so that using Lemma 2.1 and Theorem 2.2, we deduce the result.

Carleman estimates

In this section, we show a Carleman estimate for the solutions of (1.6). Using the weights introduced in Section 2.2, we define the following weighted integrals:

I 1 (s, λ, η) := λ (0,T )×I e 2sφ0 s 10 ξ 10 0 |η| 2 + s 8 ξ 8 0 |∂ x1 η| 2 + s 6 ξ 6 0 |∂ 2 x1 η| 2 + |∂ t η| 2 dt dx 1 + λ (0,T )×I e 2sφ0 s 4 ξ 4 0 |∂ 3 x1 η| 2 + |∂ x1 ∂ t η| 2 dt dx 1 + λ (0,T )×I e 2sφ0 s 2 ξ 2 0 |∂ 4 x1 η| 2 + |∂ t ∂ 2 x1 η| 2 + |∂ 2 t η| 2 dt dx 1 + λ (0,T )×I e 2sφ0 ∂ 5 x1 η 2 + |∂ 2 t ∂ x1 η| 2 + |∂ t ∂ 3 x1 η| 2 dt dx 1 , (3.1)
I 2 (s, λ, u) := (0,T )×Ω λ 2 |∇ 2 u| 2 + (∂ t u) 2 e 2sφ dt dx + (0,T )×Ω s 2 λ 4 ξ 2 e 2sφ |∇u| 2 dt dx + (0,T )×Ω s 4 λ 6 ξ 4 e 2sφ |u| 2 dt dx, (3.2)
and

I 3 (s, λ, p ∞ ) := (0,T )×Ω ∞ s 3 λ 4 ξ 3 e 2sφ |p ∞ | 2 dt dx + (0,T )×Ω ∞ sλ 2 ξe 2sφ |∇p ∞ | 2 dt dx + (0,T )×∂Ω ∞ s 3 λ 3 ξ 3 0 e 2sφ0 |p ∞ | 2 dt dx 1 + (0,T )×∂Ω ∞ sλξ 0 e 2sφ0 |∂ x1 p ∞ | 2 dt dx 1 . (3.3) Remark 3.1.
The above quantities depend also on µ but since we will fix the value of µ = µ 0 after Section 3.1, we suppress reference to it in the notation.

For µ 0 > 1, we set K + := e µ0 max ψ I , K -:= e µ0 min ψ I .

(3.4)

In particular, with the definition (2.13) of ξ 0 and the definition (2.23) of τ , we have

K -τ ⩽ sλξ 0 ⩽ K + τ. (3.5)
Using Lemma 2.1, we can define the following symbol of order 0:

χ(τ, k) := χ 0   τ 2 -4K+ K 3 - k 2 τ 2 + k 2   ∈ S 0 τ , with χ 0 ∈ C ∞ (R; [0, 1]) such that χ 0 = 1 in [3/4, ∞) 0 in (-∞, 1/2] . (3.6)
The main result of this section is stated below:

Proposition 3.2. Assume J 0 ⋐ J 1 ⋐ J and ω 0 ⋐ ω 1 ⋐ ω. There exist µ 0 > 0, λ 0 > 0 and s 0 such that for µ = µ 0 , λ ⩾ λ 0 and s ⩾ s 0 (T 2 + T 4 ), any smooth solution [u, p 0 , η] of (1.6) satisfies I 1 (s, λ, η) + I 2 (s, λ, u) + I 3 (s, λ, p ∞ ) ≲ λ (0,T )×J1 e 2sφ0 s 10 ξ 10 0 |η| 2 + s 2 ξ 2 0 |∂ 2 t η| 2 dt dx 1 + (0,T )×ω1 e 2sφ s 4 λ 6 ξ 4 |u| 2 dt dx + s 3 λ 4 ξ 3 |p ∞ | 2 dt dx + (0,T )×∂Ω ∞ τ |∂ x1 Op(1 -χ) [e sφ0 p ∞ ]| 2 dt dx 1 , (3.7)
where p ∞ is given by (2.15) and (2.17).

In order to prove Proposition 3.2, we first combine a Carleman estimate for the fluid velocity and a Carleman estimate for the elastic deformation (see Section 3.1 and Section 3.2). Both estimates contain pressure terms in the right-hand side and to estimates them, we perform a Carleman estimate for the pressure in Section 3.3. In this last estimate, we need to put in the right-hand side the trace of the pressure at the boundary, microlocalized in the high frequency regime.

A Carleman estimate for the elastic deformation

In this section, we obtain a Carleman estimate for the elastic deformation, mainly based on the results in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF]. This is the only part of the work where µ ⩾ µ 0 , after this, we will fix µ = µ 0 in the weights φ, ξ, φ 0 , ξ 0 . To avoid introducing many notations, we keep the same notation µ 0 , s 0 , λ 0 during the proofs, but their values may change from one appearance to another.

First, we deduce from the definitions (2.13), the existence of µ 0 such that for λ ⩾ µ ⩾ µ 0 , t ∈ [0, T ] and x 1 ∈ I, and α ⩾ 0,

∂ α x1 φ 0 + ∂ α x1 ξ 0 ≲ µ α ξ 0 (k ⩾ 1), ∂ t ∂ α x1 φ 0 + ∂ t ∂ α x1 ξ 0 ≲ T µ α ξ 3/2 0 , ∂ 2 t ∂ α x1 φ 0 + ∂ 2 t ∂ α x1 ξ 0 ≲ T 2 µ α ξ 2 0 . (3.8)
Moreover, there exists µ 0 such that for λ ⩾ µ ⩾ µ 0 , for t ∈ [0, T ] and for x 1 ∈ I \ J 0 ,

µξ 0 ≲ |∂ x1 φ 0 | , µ 2 ξ 0 ≲ ∂ 2 x1 φ 0 . (3.9)
With these properties, we can obtain the following result which is proven in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF]. More precisely, the Carleman estimate below is obtained in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF] with slightly different weights but the author only uses the above properties in his proof. For sake of completeness, we give in Appendix A.1 a sketch of the corresponding proof.

Theorem 3.3. Assume r ∈ R and J 0 ⋐ J 1 ⋐ J . There exist constants s 0 > 0 and µ 0 > 0 such that for any smooth function η, for any s ⩾ s 0 (T 2 + T 4 ), and for any λ ⩾ µ ⩾ µ 0 , we have

(0,T )×I e 2sφ0 s 2r+7 µ 2r+8 ξ 2r+7 0 |η| 2 + s 2r+5 µ 2r+6 ξ 2r+5 0 |∂ x1 η| 2 dt dx 1 + (0,T )×I e 2sφ0 s 2r+3 µ 2r+4 ξ 2r+3 0 |∂ 2 x1 η| 2 + |∂ t η| 2 dt dx 1 + (0,T )×I e 2sφ0 s 2r+1 µ 2r+2 ξ 2r+1 0 |∂ 3 x1 η| 2 + |∂ t ∂ x1 η| 2 dt dx 1 + (0,T )×I e 2sφ0 s 2r-1 µ 2r ξ 2r-1 0 ∂ 4 x1 η 2 + |∂ 2 t η| 2 + |∂ t ∂ 2 x1 η| 2 dt dx 1 ≲ (0,T )×I e 2sφ0 s 2r µ 2r ξ 2r 0 (∂ 2 t + ∂ 4 x1 -∂ 2 x1 -∂ t ∂ 2 x1 )η 2 dt dx 1 + (0,T )×J1 s 2r+7 µ 2r+8 ξ 2r+7 0 e 2sφ0 |η| 2 dt dx 1 . (3.10)
As a corollary, we have the following result Corollary 3.4. Assume J 0 ⋐ J 1 ⋐ J . There exist constants s 0 > 0 and µ 0 > 0 such that for any smooth function η, for any s ⩾ s 0 (T 2 + T 4 ), and for any λ ⩾ µ ⩾ µ 0 , we have

+ (0,T )×I e 2sφ0 s 2 µ 3 ξ 2 0 |∂ 4 x1 η| 2 + |∂ t ∂ 2 x1 η| 2 + |∂ 2 t η| 2 + µ ∂ 5 x1 η 2 + |∂ 2 t ∂ x1 η| 2 + |∂ t ∂ 3 x1 η| 2 dt dx 1 ≲ (0,T )×I e 2sφ0 sµξ 0 |∂ x1 (∂ 2 t -∂ 2 x1 + ∂ 4 x1 -∂ t ∂ 2 x1 )η| 2 dt dx 1 + (0,T )×J1 e 2sφ0 s 10 µ 11 ξ 10 0 |η| 2 + s 2 µ 3 ξ 2 0 |∂ 2 t η| 2 dt dx 1 . (3.11)
Proof. We first apply Theorem 3.3 to ∂ x1 η with r = 1/2 and with an open set J 2 such that J 0 ⋐ J 2 ⋐ J 1 :

(0,T )×I e 2sφ0 s 8 µ 9 ξ 8 0 |∂ x1 η| 2 + s 6 µ 7 ξ 6 0 |∂ 2 x1 η| 2 + s 4 µ 5 ξ 4 0 |∂ 3 x1 η| 2 + |∂ t ∂ x1 η| 2 dt dx 1 + (0,T )×I e 2sφ0 s 2 µ 3 ξ 2 0 |∂ 4 x1 η| 2 + |∂ t ∂ 2 x1 η| 2 + µ ∂ 5 x1 η 2 + |∂ 2 t ∂ x1 η| 2 + |∂ t ∂ 3 x1 η| 2 dt dx 1 ≲ (0,T )×I e 2sφ0 sµξ 0 ∂ x1 (∂ 2 t + ∂ 4 x1 -∂ 2 x1 -∂ t ∂ 2 x1 )η 2 dt dx 1 + (0,T )×J2 s 8 µ 9 ξ 8 0 e 2sφ0 |∂ x1 η| 2 dt dx 1 . (3.12)
Then, we use a Carleman estimate for the gradient operator (see, for instance, [14, Lemma 3]): there exists s 0 > 0 such that for any smooth function ζ, and for any s ⩾ s 0 T 4 , (0,T )×I

s r+2 µ r+3 ξ r+2 0 e 2sφ0 ζ 2 dt dx 1 ≲ (0,T )×J2 s r+2 µ r+3 ξ r+2 0 e 2sφ0 ζ 2 dt dx 1 + (0,T )×I s r µ r+1 ξ r 0 e 2sφ0 (∂ x1 ζ) 2 dt dx 1 .
This Carleman estimate, combined with (3.12), yields that for s ⩾ s 0 (T 2 + T 4 ),

(0,T )×I e 2sφ0 s 10 µ 11 ξ 10 0 |η| 2 + s 8 µ 9 ξ 8 0 |∂ x1 η| 2 + s 6 µ 7 ξ 6 0 |∂ 2 x1 η| 2 + |∂ t η| 2 dt dx 1 + (0,T )×I e 2sφ0 s 4 µ 5 ξ 4 0 |∂ 3 x1 η| 2 + |∂ x1 ∂ t η| 2 dt dx 1 + (0,T )×I e 2sφ0 s 2 µ 3 ξ 2 0 |∂ 4 x1 η| 2 + |∂ t ∂ 2 x1 η| 2 + |∂ 2 t η| 2 + µ ∂ 5 x1 η 2 + |∂ 2 t ∂ x1 η| 2 + |∂ t ∂ 3 x1 η| 2 dt dx 1 ≲ (0,T )×I e 2sφ0 sµξ 0 |∂ x1 (∂ 2 t -∂ 2 x1 + ∂ 4 x1 -∂ t ∂ 2 x1 )η| 2 dt dx 1 + (0,T )×J2 e 2sφ0 s 10 µ 11 ξ 10 0 |η| 2 + s 8 µ 9 ξ 8 0 |∂ x1 η| 2 + s 6 µ 7 ξ 6 0 |∂ t η| 2 + s 2 µ 3 ξ 2 0 |∂ 2 t η| 2 dt dx 1 . (3.13)
Then proceeding as in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF], one can absorb the local terms in ∂ x1 η and in ∂ t η by using a cut-off function and integrations by parts and we deduce the result.

A Carleman estimate for the velocity

From now on, we take µ = µ 0 as in Theorem 3.3 and take λ ⩾ µ 0 . The constants that follow in the article may depend on µ 0 . We have the following standard Carleman estimate for the heat equation (see, for instance, [START_REF] Fursikov | Controllability of evolution equations[END_REF] or [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF]). For sake of completeness, we also give a sketch of the proof of the following result in Appendix A.2.

Theorem 3.5. Assume µ = µ 0 and ω 0 ⋐ ω 1 ⋐ ω. There exist s 0 > 0 and λ 0 > 0 such that for any λ ⩾ λ 0 s ⩾ s 0 (T 2 + T 4 ), and for any smooth function u such that

u = 0 on (0, T ) × Γ 0 , u 1 = 0 on (0, T ) × Γ 1 , ∂u 2 ∂n = 0 on (0, T ) × Γ 1 ,
we have

(0,T )×Ω |∇ 2 u| 2 + (∂ t u)

A Carleman estimate for the pressure

In order to obtain a Carleman estimate for the pressure, we use that from (1.12), the pressure p 0 is harmonic in Ω. We recall that p ∞ is defined from p 0 by (2.15) and (2.17). In particular, it satisfies the Laplace equation(2.18) but without any explicit boundary condition. Thus in our Carleman estimate, we keep in the right-hand side a boundary term microlocalized in a high frequency regime (represented by supp(1 -χ), with χ defined by (3.6)).

We recall that τ is defined in (2.23).

Proposition 3.6. Assume µ = µ 0 and ω 0 ⋐ ω 1 ⋐ ω. There exist s 0 > 0 λ 0 > 0 and C > 0 such that for any s ⩾ s 0 (T 2 + T 4 ), λ ⩾ λ 0 and for any smooth function p, the function

p ∞ := pχ ∞ satisfies (0,T )×Ω ∞ s 3 λ 4 ξ 3 e 2sφ |p ∞ | 2 dt dx + (0,T )×Ω ∞ sλ 2 ξe 2sφ |∇p ∞ | 2 dt dx + (0,T )×∂Ω ∞ s 3 λ 3 ξ 3 0 e 2sφ0 |p ∞ | 2 dt dx 1 + (0,T )×∂Ω ∞ sλξ 0 e 2sφ0 |∂ x1 p ∞ | 2 dt dx 1 ⩽ C (0,T )×Ω ∞ e 2sφ |∆p ∞ | 2 dt dx + (0,T )×ω1 s 3 λ 4 ξ 3 e 2sφ |p ∞ | 2 dt dx + (0,T )×∂Ω ∞ τ |∂ x1 Op(1 -χ) [e sφ0 p ∞ ]| 2 dt dx 1 . (3.15)
Proof. We start by a standard Carleman estimate for p ∞ in Ω ∞ , using that χ ∞ has a compact support. First, we set q = e sφ p ∞ and we perform standard computations (see, for instance, [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF], [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], [25, pp.106-117]), to obtain the existence of positive constants c, C, s 0 such that for s ⩾ s 0 (T 2 + T 4 ),

c (0,T )×Ω ∞ s 3 λ 4 ξ 3 q 2 + sλ 2 ξ |∇q| 2 + 1 sξ |∆q| 2 dt dx + (0,T )×∂Ω ∞ -s 3 λ 3 ξ 3 |∇ψ| 2 ∂ψ ∂n q 2 -2sλ 2 ξ |∇ψ| 2 ∂q ∂n q -2sλξ∇ψ • ∇q ∂q ∂n + sλξ ∂ψ ∂n |∇q| 2 dt dx 1 ⩽ C (0,T )×Ω ∞ (-∆p) 2 e 2sφ dt dx + (0,T )×ω1 s 3 λ 4 ξ 3 q 2 dx . (3.16)
From (2.14) and (2.9), we have

∂ψ ∂x 1 = µ 0 λ ψ ′ I , ∂ψ ∂n = -1 on ∂Ω ∞ .
Thus there exist λ 0 > 0 and s 0 > 0 such that for λ ⩾ λ 0 , s ⩾ s 0 (T 2 + T 4 ), we have on (0, T ) × ∂Ω,

-s 3 λ 3 ξ 3 |∇ψ| 2 ∂ψ ∂n q 2 -2sλ 2 ξ |∇ψ| 2 ∂q ∂n q -2sλξ∇ψ • ∇q ∂q ∂n + sλξ ∂ψ ∂n |∇q| 2 = s 3 λ 3 ξ 3 |∇ψ| 2 q 2 + sλξ ∂q ∂n 2 -sλξ ∂q ∂x 1 2 -2sλ 2 ξ |∇ψ| 2 ∂q ∂n q -2sµ 0 ξψ ′ I ∂q ∂x 1 ∂q ∂n ⩾ 1 2 s 3 λ 3 ξ 3 0 q 2 + 1 2 sλξ 0 ∂q ∂n 2 -2sλξ 0 ∂q ∂x 1 2 . (3.17)
Let us denote by q the Fourier transform of q in the x 1 direction. Then by using the Plancherel theorem, there exists c > 0 such that

(0,T )×∂Ω ∞ 1 2 s 3 λ 3 ξ 3 0 q 2 -2sλξ 0 ∂q ∂x 1 2 dx 1 dt ⩾ c (0,T )×∂Ω ∞ τ K 3 -τ 2 -4K + k 2 | q| 2 dk dt
and thus, there exist two constant c, C > 0 such that

(0,T )×∂Ω ∞ 1 2 s 3 λ 3 ξ 3 0 q 2 -2sλξ 0 ∂q ∂x 1 2 dx 1 dt + C (0,T )×∂Ω ∞ (1 -χ) 2 τ k 2 | q| 2 dk dt ⩾ c (0,T )×∂Ω ∞ τ τ 2 + k 2 | q| 2 dk dt. (3.18)
Using again the Plancherel theorem, and combining the above relation with (3.17) and with (3.16), we deduce the result.

Gathering the Carleman estimates

We are now in a position to prove Proposition 3.2

Proof. Assume that (u, p 0 , η) is the solution of (1.12). We consider p defined from p 0 by (2.15) and p ∞ defined from p by (2.17). We apply Corollary 3.4, Theorem 3.5 and Proposition 3.6 and using that ∇p 0 = ∇p, we obtain the following relations for I 1 , I 2 and I 3 (defined by (3.1)-(3.3)):

I 1 (s, λ, η) ≲ (0,T )×I e 2sφ0 sλξ 0 |∂ x1 p| 2 dt dx 1 + λ (0,T )×J1 e 2sφ0 s 10 ξ 10 0 |η| 2 + s 2 ξ 2 0 |∂ 2 t η| 2 dt dx 1 , (3.19)
I 2 (s, λ, u) ≲ (0,T )×Ω sλ 2 ξe 2sφ |∇p| 2 dt dx + (0,T )×ω1 s 4 λ 6 ξ 4 e 2sφ |u| 2 dt dx, (3.20)
and

I 3 (s, λ, p ∞ ) ≲ (0,T )×Ω ∞ e 2sφ |f (3) | 2 dt dx + (0,T )×ω1 s 3 λ 4 ξ 3 e 2sφ |p ∞ | 2 dt dx + (0,T )×∂Ω ∞ τ |∂ x1 Op(1 -χ) [e sφ0 p ∞ ]| 2 dt dx 1 . (3.21)
Then, we can estimate f (3) by using (2.20) and we deduce that

(0,T )×Ω ∞ e 2sφ |f (3) | 2 dt dx ⩽ Cλ -2 I 3 (s, λ, p ∞ ).
Using that χ ∞ ≡ 1 in (0, 2π) × (0, 1), and taking λ ⩾ λ 0 with λ 0 > 0 sufficiently large, we can combine the three Carleman estimates (3.19)-(3.21) and the above relation to obtain (3.7).

High frequency estimates

In this section, we eliminate the last term in (3.7) by showing high frequency estimates for u and p. The method used here is the same as the one used in [START_REF] Buffe | Controllability of a Stokes system with a diffusive boundary condition[END_REF]. We conjugate the system (2.18) with e sφ0 , using that the spatial derivatives of φ 0 involve only powers of µ 0 that is fixed, instead of powers of λ for the spatial derivatives of φ. This allows us to perform energy estimates of the Stokes system, by considering all the terms coming from the conjugaison as lower order terms in the high frequency regime.

Estimates from the Stokes system

We recall that u ∞ , p ∞ and η ∞ are defined in (2.17) by using the function χ ∞ . We introduce

χ ∞ ∈ C ∞ 0 (R, [0, 1]), χ ∞ ≡ 1 in supp χ ∞ . (4.1)
Then, we set ǔ := e sφ0 u ∞ , p := e sφ0 p ∞ , η := e sφ0 η ∞ , (4.2)

u := Op(1 -χ)ǔ, p := Op(1 -χ)p, η := Op(1 -χ)η, (4.3)
u ∞ := χ ∞ Op(1 -χ)ǔ, p ∞ := χ ∞ Op(1 -χ)p, η ∞ := χ ∞ Op(1 -χ)η. (4.4)
Our aim is to estimate p (see (3.7)) but we need to use χ ∞ to work on a bounded domain and to apply the elliptic regularity of the Stokes system. In order to estimate p, we use that, with our choice of truncation functions, we have the relations

p = p ∞ + [1 -χ ∞ , Op(1 -χ)]p.
Then, using the commutator property in Theorem 2.2, we can estimate p from p ∞ and p. Using (3.5), (2.12) and (2.13), we have

τ ≲ sλξ 0 , τ ≲ sλξ.
This leads us to define (see (3.1)-(3.3))

I 4 (s, λ, η) := (0,T )×I λ -9 τ 10 |η| 2 + λ -7 τ 8 |∂ x1 η| 2 + λ -5 τ 6 |∂ 2 x1 η| 2 + |∂ t η| 2 dt dx 1 + (0,T )×I λ -3 τ 4 |∂ 3 x1 η| 2 + |∂ x1 ∂ t η| 2 dt dx 1 + (0,T )×I λ -1 τ 2 |∂ 4 x1 η| 2 + |∂ t ∂ 2 x1 η| 2 + |∂ 2 t η| 2 + λ ∂ 5 x1 η 2 + |∂ 2 t ∂ x1 η| 2 + |∂ t ∂ 3 x1 η| 2 dt dx 1 , (4.5)
I 5 (s, λ, ǔ) := λ 2 (0,T )×Ω |∇ 2 ǔ| 2 + (∂ t ǔ) 2 + τ 2 |∇ǔ| 2 + τ 4 |ǔ| 2 dt dx, (4.6)
and

I 6 (s, λ, p) := λ (0,T )×Ω ∞ τ 3 |p| 2 + τ |∇p| 2 dt dx + (0,T )×∂Ω ∞ τ 3 |p| 2 + τ |∂ x1 p| 2 dt dx 1 . (4.7)
Noting that

I 4 (s, λ, η) ≲ I 1 (s, λ, η), I 5 (s, λ, ǔ) ≲ I 2 (s, λ, u), I 6 (s, λ, p) ≲ I 3 (s, λ, p ∞ ),
we deduce from (3.7) that

I 4 (s, λ, η) + I 5 (s, λ, ǔ) + I 6 (s, λ, p) ≲ λ (0,T )×J1 e 2sφ0 s 10 ξ 10 0 |η| 2 + s 2 ξ 2 0 |∂ 2 t η| 2 dt dx 1 + (0,T )×ω1 e 2sφ s 4 λ 6 ξ 4 |u| 2 dt dx + s 3 λ 4 ξ 3 |p ∞ | 2 dt dx + (0,T )×∂Ω ∞ τ |∂ x1 p| 2 dt dx 1 . (4.8)
The aim of this section is to show the following result:

Proposition 4.1. There exist λ 0 > 0 and s 0 > 0 such that for λ ⩾ λ 0 and s ⩾ s 0 (T This yields the following semi-classical trace inequality:

Lemma 4.2. There exists s 0 > 0 such that for any s ⩾ s 0 T 4 and for any f ∈ H 1 (Ω ∞ ),

τ 1/2 Op(1 -χ)f | ∂Ω ∞ L 2 (∂Ω ∞ ) ≲ ∥∇ Op(1 -χ)f ∥ L 2 (Ω ∞ ) .
Proof. We write g := Op(1 -χ)f and

g 2 (x 1 , 1) = g 2 (x 1 , x 2 ) + 2 1 x2 g(x 1 , y 2 )∂ x2 g(x 1 , y 2 ) dy 2 so that τ Γ ∞ 1 g(x 1 , 1) 2 dx 1 ⩽ τ ∥g∥ 2 L 2 (Ω ∞ ) + 2τ ∥g∥ L 2 (Ω ∞ ) ∥∂ x2 g∥ L 2 (Ω ∞ ) ⩽ (τ + 4τ 2 ) ∥g∥ 2 L 2 (Ω ∞ ) + ∥∂ x2 g∥ 2 L 2 (Ω ∞ )
and we conclude by using (4.10).

In particular, using Lemma 4.2, we can estimate the last term of (4.8) as follows:

(0,T )×∂Ω ∞ τ ∂ x1 p | ∂Ω ∞ 2 dt dx 1 ≲ ∥∇∂ x1 p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) . (4.11)
We have also the following result that will allows us to estimate boundary terms:

Lemma 4.3. There exists s 0 > 0 such that for any s ⩾ s 0 T 4 and for any f ∈ H 2 (R),

∥Op(1 -χ)f ∥ H 3/2 (R) ≲ τ -1/2 Op(1 -χ)∂ 2 x1 f L 2 (R) .
Proof. Denoting by f the Fourier transform of f , we have

∥Op(1 -χ)f ∥ 2 H 3/2 (R) = R (1 + k 2 ) 3/2 (1 -χ(τ, k)) 2 f (k) 2 dk
and by using (4.10), we deduce the result.

In order to estimate p (and prove Proposition 4.1), we consider the system verified by u ∞ and p ∞ : from (2.18), we have

           ∂ t u ∞ -∆ u ∞ + ∇ p ∞ = f (1) in (0, T ) × Ω ∞ , div u ∞ = f (2) in (0, T ) × Ω ∞ , u ∞ = 0 on (0, T ) × Γ ∞ 0 , u ∞ = he 2 on (0, T ) × Γ ∞ 1 , u ∞ (0, •) = u ∞ (T, •) = 0 in Ω ∞ , (4.12)
where

f (1) = χ ∞ Op(1 -χ)e sφ0 f (1) + (s∂ t φ 0 ) u ∞ -s ∂ 2 x1 φ 0 u ∞ + s 2 (∂ x1 φ 0 ) 2 u ∞ -2s∂ x1 φ 0 ∂ x1 u ∞ + s∇φ 0 p ∞ + -(s∂ t φ 0 ) + s ∂ 2 x1 φ 0 -s 2 (∂ x1 φ 0 ) 2 + 2s∂ x1 φ 0 ∂ x1 , χ ∞ Op(1 -χ) ǔ + [-s∇φ 0 , χ ∞ Op(1 -χ)]p -χ ∞ Op(∂ t χ)ǔ -( χ ∞ ) ′′ u -2 ( χ ∞ ) ′ ∂ x1 u + ( χ ∞ ) ′ pe 1 , (4.13) f (2) = χ ∞ Op(1 -χ)(e sφ0 f (2) ) + s∂ x1 φ 0 u ∞ 1 -[s∂ x1 φ 0 , χ ∞ Op(1 -χ)]ǔ 1 + ( χ ∞ ) ′ u 1 , (4.14)
h := χ ∞ Op(1 -χ)(∂ t η -s (∂ t φ 0 ) η). (4.15)
We also define

I( u, p) := ∥∂ t ∂ x1 u∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥∂ x1 u∥ 2 L 2 (0,T ;H 2 (Ω ∞ )) + ∥∇∂ x1 p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥τ ∂ t u∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ 3 u 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ 2 ∂ x1 u 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ ∂ 2 x1 u 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥τ ∂ x1 ∂ x2 u∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ ∂ 2 x2 u 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ 2 ∂ x2 u 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ 2 p 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥τ ∇ p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) . (4.16)
From relation (4.10), we have

I( u, p) ≲ ∥∂ t ∂ x1 u∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥∂ x1 u∥ 2 L 2 (0,T ;H 2 (Ω ∞ )) + ∥∇∂ x1 p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) . (4.17)
We have the following a priori estimate on (4.12).

Proposition 4.4. There exist λ 0 > 0 and s 0 > 0 such that for λ ⩾ λ 0 and s ⩾ s 0 (T 2 + T 4 ), any smooth solutions [u, p 0 , η] of (1.6) satisfies

I( u, p) ≲ ∂ x1 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∂ x1 f (2) 2 L 2 (0,T ;H 1 (Ω ∞ )) + ∂ t f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∂ x1 h 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 )) + τ -1/2 ∂ t ∂ x1 h 2 L 2 (0,T ;L 2 (Γ ∞ 1 ))
+ λ -1 (I 5 (s, λ, ǔ) + I 6 (s, λ, p)) . (4.18)

Proof. First, we differentiate (4.12) with respect to x 1 :

           ∂ t ∂ x1 u ∞ -∆∂ x1 u ∞ + ∇∂ x1 p ∞ = ∂ x1 f (1) in (0, T ) × Ω ∞ , div ∂ x1 u ∞ = ∂ x1 f (2) in (0, T ) × Ω ∞ , ∂ x1 u ∞ = 0 on (0, T ) × Γ ∞ 0 , ∂ x1 u ∞ = ∂ x1 he 2 on (0, T ) × Γ ∞ 1 , ∂ x1 u ∞ (0, •) = ∂ x1 u ∞ (T, •) = 0 in Ω ∞ . (4.19)
Let us consider a bounded smooth domain Ω ♮ ⊂ Ω ∞ containing supp χ ∞ × (0, 1). Let us also write

h ♮ = ∂ x1 u ∞ = 0 on (0, T ) × ∂Ω ♮ \ Γ ∞ 1 , ∂ x1 u ∞ = ∂ x1 he 2 on (0, T ) × ∂Ω ♮ ∩ Γ ∞ 1 .
Using [35, p.33, Theorem 7.5], there exists

H ∈ H 2 ( (x 1 , x 2 ) ∈ R 2 ; x 2 < 1 ) such that H = ∂ x1 h on Γ ∞ 1 .
Multiplying H by an adequate cut-off function we deduce the existence of H ♮ ∈ H 2 (Ω ♮ ) such that H ♮ = h ♮ on ∂Ω ♮ . Therefore h ♮ ∈ H 3/2 (∂Ω ♮ ) and we have the estimate

h ♮ H 3/2 (∂Ω ♮ ) ≲ ∂ x1 h H 3/2 (Γ ∞ 1 )
.

With the above notation, we deduce from (4.19

) that (∂ x1 u ∞ , ∂ x1 p ∞ ) satisfies a Stokes system in Ω ♮ :    -∆∂ x1 u ∞ + ∇∂ x1 p ∞ = ∂ x1 f (1) -∂ t ∂ x1 u ∞ in (0, T ) × Ω ♮ , div ∂ x1 u ∞ = ∂ x1 f (2) in (0, T ) × Ω ♮ , ∂ x1 u ∞ = h ♮ on (0, T ) × ∂Ω ♮ . (4.20)
Using the elliptic regularity of the Stokes system (see, for instance, [49, Proposition 2.2 p.33]) we obtain

∥∂ x1 u ∞ ∥ 2 L 2 (0,T ;H 2 (Ω ∞ )) + ∥∇∂ x1 p ∞ ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ ∂ x1 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥∂ t ∂ x1 u ∞ ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∂ x1 f (2) 2 L 2 (0,T ;H 1 (Ω ∞ )) + ∂ x1 h 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 ))
. (4.21)

On the other hand, by multiplying the first equation of (4.19) by ∂ t ∂ x1 u and integrating by parts, we deduce

T 0 ∥∂ t ∂ x1 u ∞ ∥ 2 L 2 (Ω ∞ ) dt + (0,T )×Γ ∞ 1 ∂ x1 p ∞ | Γ ∞ 1 ∂ t ∂ x1 h dx 1 dt + (0,T )×Ω ∞ ∂ 2 x1 p ∞ ∂ t f (2) dtdx = (0,T )×Ω ∞ ∂ x1 f (1) • ∂ t ∂ x1 u ∞ dxdt.
The above relation yields that for any ε > 0,

∥∂ t ∂ x1 u ∞ ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ ε ∥∇∂ x1 p ∞ ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ε τ 1/2 ∂ x1 p ∞ 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + 1 ε ∂ t f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) + 1 ε τ -1/2 ∂ t ∂ x1 h 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + ∂ x1 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ ))
.

We deduce from the above relation, from (4.4) and from Lemma 4.2 that

∥∂ t ∂ x1 u ∞ ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ ε ∥∇∂ x1 p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ε ∥ p∥ 2 L 2 (0,T ;H 1 (Ω ∞ )) + 1 ε ∂ t f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) + 1 ε τ -1/2 ∂ t ∂ x1 h 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + ∂ x1 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ ))
. (4.22)

Now using (4.1) and (2.17), we have

∇ 2 ∂ x1 u = ∇ 2 ∂ x1 u ∞ + ∇ 2 ∂ x1 [(1 -χ ∞ ), Op(1 -χ)] ǔ.
Since, (1 -χ ∞ ), 1 -χ ∈ S 0 τ , we deduce from Theorem 2.2 and Theorem 2.3 that

∇ 2 ∂ x1 u 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ ∇ 2 ∂ x1 u ∞ 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ 2 ǔ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥τ ∇ǔ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∇ 2 ǔ 2 L 2 (0,T ;L 2 (Ω ∞ )) .
Using the compact support of χ ∞ and the periodicity of ue sφ0 , we deduce from the above relation and from

(4.6) that ∇ 2 ∂ x1 u 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ ∇ 2 ∂ x1 u ∞ 2 L 2 (0,T ;L 2 (Ω ∞ )) + λ -2 I 5 (s, λ, ǔ). (4.23)
Similarly, with (4.7), we have

∥∇∂ x1 p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ ∥∇∂ x1 p ∞ ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + λ -1 I 6 (s, λ, p). (4.24)
Finally,

∂ t ∂ x1 u = ∂ t ∂ x1 u ∞ -∂ x1 [1 -χ ∞ , Op(∂ t χ)]ǔ + ∂ x1 [1 -χ ∞ , Op(1 -χ)]∂ t ǔ,
and from Lemma 2.5,

∂ t χ ∈ ℓ ′ (λs) 1/2 τ 1 2 S 0 τ .
In particular, if s ⩾ T 2 , we can combine the two previous relations with Theorem 2.2 and Theorem 2.3 to obtain Combining (4.18), (4.11) and (4.8), we deduce the existence of λ 0 > 0 and s 0 > 0 such that for λ ⩾ λ 0 and s ⩾ s 0 (T 2 + T 4 ),

∥∂ t ∂ x1 u∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ ∥∂ t ∂ x1 u ∞ ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + λ -2 I 5 (s, λ, ǔ). ( 4
I 4 (s, λ, η) + I 5 (s, λ, ǔ) + I 6 (s, λ, p) + I( u, p) ≲ λ (0,T )×J1 e 2sφ0 s 10 ξ 10 0 |η| 2 + s 2 ξ 2 0 |∂ 2 t η| 2 dt dx 1 + (0,T )×ω1 e 2sφ s 4 λ 6 ξ 4 |u| 2 dt dx + s 3 λ 4 ξ 3 |p ∞ | 2 dt dx + ∂ x1 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∂ x1 f (2) 2 L 2 (0,T ;H 1 (Ω ∞ )) + ∂ t f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) + τ -1/2 ∂ t ∂ x1 h 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + ∂ x1 h 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 ))
. (4.26)

4.2 Estimates of f (1) , f (2) and h

To obtain Proposition 4.1, it remains to estimate the right-hand side of (4.26). We recall that f (1) , f (2) and h are given by (4.13), (4.14) and (4.15). Combining (3.8), (3.5), (2.13) and (2.22), we deduce that for α ⩾ 0,

∂ α x1 φ 0 ≲ τ λs (k ⩾ 1), ∂ t ∂ α x1 φ 0 ≲ T τ λs 3/2 , ∂ 2 t ∂ α x1 φ 0 ≲ T 2 τ λs 2 (4.27) ∂ α x1 φ 0 ∈ τ λs S 0 τ (k ⩾ 1), ∂ t ∂ α x1 φ 0 ∈ ℓ ′ τ λs 3/2 S 0 τ , ∂ 2 t ∂ α x1 φ 0 ∈ 2ℓ + 3(ℓ ′ ) 2 τ λs 2 S 0 τ . (4.28)
Proposition 4.5. There exist s 0 > 0 and λ 0 > 0 such that the function f (1) defined by (4.13) satisfies for s ⩾ s 0 (T 2 + T 4 ) and for λ ⩾ λ 0 ,

∂ x1 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ ))
⩽ λ -1 I 4 (s, λ, η) + I 5 (s, λ, ǔ) + I 6 (s, λ, p) + I( u, p) .

Proof. Differentiating (4.13) yields,

∂ x1 f (1) = 6 i=1 F (i) (4.29)
where 1) , (4.30)

F (1) := ∂ x1 χ ∞ Op(1 -χ)e sφ0 f (
F (2) := ∂ x1 (s∂ t φ 0 ) u ∞ -s ∂ 2 x1 φ 0 u ∞ + s 2 (∂ x1 φ 0 ) 2 u ∞ -2s∂ x1 φ 0 ∂ x1 u ∞ + s∇φ 0 p ∞ , (4.31)
F (3) := ∂ x1 -(s∂ t φ 0 ) + s ∂ 2 x1 φ 0 -s 2 (∂ x1 φ 0 ) 2 + 2s∂ x1 φ 0 ∂ x1 , χ ∞ Op(1 -χ) ǔ, (4.32)
F (4) := ∂ x1 [-s∇φ 0 , χ ∞ Op(1 -χ)]p, F (5) := -∂ x1 ( χ ∞ Op(∂ t χ)ǔ) , (4.33)
F (6) := ∂ x1 -( χ ∞ ) ′′ u -2 ( χ ∞ ) ′ ∂ x1 u + ( χ ∞ ) ′ pe 1 . (4.34)
From (4.30), we have (1) .

F (1) = χ ∞ Op(1 -χ) s∂ x1 φ 0 e sφ0 f (1) + e sφ0 ∂ x1 f (1) + ( χ ∞ ) ′ Op(1 -χ)e sφ0 f
From (2.20), (4.27), the properties of χ ∞ and the periodicity of u and p in the x 1 variable,

s∂ x1 φ 0 e sφ0 f (1) + e sφ0 ∂ x1 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ )) + e sφ0 f (1) 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ (0,T )×(0,2π)×(0,1) e 2sφ0 τ 2 λ 2 |u| 2 + |∂ x1 u| 2 + |p| 2 + ∂ 2 x1 u 2 + |∂ x1 p| 2 dx dt ≲ (0,T )×Ω ∞ ∂ 2 x1 ǔ 2 + τ λ 2 |∂ x1 ǔ| 2 + τ λ 4 |ǔ| 2 + |∂ x1 p| 2 + τ λ 2 |p| 2 dx dt.
Since 1 -χ ∈ S 0 τ (see Lemma 2.1), we deduce from the above estimate, from Theorem 2.3, and from (4.6)-(4.7), that

F (1) 2 L 2 (0,T ;L 2 (Ω ∞ )) ⩽ λ -1 (I 5 (s, λ, ǔ) + I 6 (s, λ, p)) . (4.35)
From (2.24), (4.31) and (4.27), we have for s ⩾ s 0 T 2 + T 4 and λ ⩾ λ 0 ,

F (2) ≲ τ 2 λ 2 (| u| + |∂ x1 u|) + τ λ ∂ 2 x1 u + | p| + |∂ x1 p| , (4.36)
and thus with (4.16),

F (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) ⩽ λ -2 I( u, p). (4.37)
On the other hand, from (4.32), (4.33),

F (3) = -(s∂ t φ 0 ) + s ∂ 2 x1 φ 0 -s 2 (∂ x1 φ 0 ) 2 + 2s∂ x1 φ 0 ∂ x1 , χ ∞ Op(1 -χ) ∂ x1 ǔ + -(s∂ t ∂ x1 φ 0 ) + s ∂ 3 x1 φ 0 -2s 2 ∂ x1 φ 0 ∂ 2 x1 φ 0 + 2s∂ 2 x1 φ 0 ∂ x1 + 2s∂ x1 φ 0 ∂ 2 x1 , χ ∞ Op(1 -χ) ǔ, (4.38)
F (4) = [-s∇φ 0 , χ ∞ Op(1 -χ)]∂ x1 p + [-s∇∂ x1 φ 0 , χ ∞ Op(1 -χ)]p. (4.39)
From (4.28),

s ∂ 2 x1 φ 0 -s 2 (∂ x1 φ 0 ) 2 -2s∂ x1 φ 0 ik ∈ 1 λ S 2 τ , s∂ t φ 0 ∈ ℓ ′ s 1/2 λ S 3/2 τ s ∂ 3 x1 φ 0 -2s 2 ∂ x1 φ 0 ∂ 2 x1 φ 0 -2s∂ 2 x1 φ 0 ik -2s∂ x1 φ 0 k 2 ∈ 1 λ S 3 τ , s∂ t ∂ x1 φ 0 ∈ ℓ ′ s 1/2 λ S 3/2 τ , -s∇φ 0 , -s∇∂ x1 φ 0 ∈ 1 λ S 1 τ ,
so that, from Theorem 2.2, Theorem 2.3 and (4.16),

F (3) 2 L 2 (0,T ;L 2 (Ω ∞ )) + F (4) 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ 1 λ 2 τ 2 ǔ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥τ ∂ x1 ǔ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∂ 2 x1 ǔ 2 L 2 (0,T ;L 2 (Ω ∞ )) + 1 λ 2 ∥p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥∂ x1 p∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ 1 λ 2 (I 5 (s, λ, ǔ) + I 6 (s, λ, p)) . (4.40)
From (4.33),

F (5) = -χ ∞ Op(∂ t χ)∂ x1 ǔ -( χ ∞ ) ′ Op(∂ t χ)ǔ. (4.41)
From Lemma 2.5,

∂ t χ ∈ ℓ ′ (λs) 1/2 S 1/2
τ so that from Theorem 2.3 and (4.6),

F (5) 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ λ -1 ∥τ ∂ x1 ǔ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∥τ ǔ∥ 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∂ 2 x1 ǔ 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ λ -2 I 5 (s, λ, ǔ). (4.42)
Finally, from (4.34), (4.16) and (2.24),

F (6) 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ λ -2 I( u, p).
Gathering (4.29), (4.35), (4.37), (4.40), (4.42) and the above estimate, we deduce the result.

Proposition 4.6. There exist s 0 > 0 and λ 0 > 0 such that the function f (2) defined by (4.14) satisfies for s ⩾ s 0 (T 2 + T 4 ) and for λ ⩾ λ 0 ,

∂ x1 f (2) 2 L 2 (0,T ;H 1 (Ω ∞ )) + ∂ t f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) ⩽ λ -1 I 4 (s, λ, η) + I 5 (s, λ, ǔ) + I 6 (s, λ, p) + I( u, p) .
Proof. From (4.14)

∂ x1 f (2) = G (1) + G (2) + G (3) + G (4) , (4.43)
with

G (1) := ∂ x1 χ ∞ Op(1 -χ) e sφ0 f (2) , G (2) := ∂ x1 (s∂ x1 φ 0 u ∞ 1 ) , (4.44)
G (3) := -∂ x1 [s∂ x1 φ 0 , χ ∞ Op(1 -χ)]ǔ 1 , G (4) := ∂ x1 ( χ ∞ ) ′ u 1 . (4.45)
From (4.44), we have

G (1) = ( χ ∞ ) ′ Op(1 -χ) e sφ0 f (2) + χ ∞ Op(1 -χ)∂ x1 e sφ0 f (2) , ∂ x1 G (1) = ( χ ∞ ) ′′ Op(1 -χ) e sφ0 f (2) + 2 ( χ ∞ ) ′ Op(1 -χ)∂ x1 e sφ0 f (2) + χ ∞ Op(1 -χ)∂ 2 x1 e sφ0 f (2) , ∂ x2 G (1) = ( χ ∞ ) ′ Op(1 -χ) e sφ0 ∂ x2 f (2) + χ ∞ Op(1 -χ)∂ x1 e sφ0 ∂ x2 f (2) , with ∂ x1 e sφ0 f (2) = s∂ x1 φ 0 e sφ0 f (2) + e sφ0 ∂ x1 f (2) , ∂ 2 x1 e sφ0 f (2) = s∂ 2 x1 φ 0 e sφ0 f (2) + s 2 (∂ x1 φ 0 ) 2 e sφ0 f (2) + 2s∂ x1 φ 0 e sφ0 ∂ x1 f (2) + e sφ0 ∂ 2 x1 f (2) , ∂ x1 e sφ0 ∂ x2 f (2) = s∂ x1 φ 0 e sφ0 ∂ x2 f (2) + e sφ0 ∂ x1 ∂ x2 f (2) .
The above relations, combined with (2.20), (4.27), with the properties of χ ∞ and with the periodicity of u and p in the x 1 variable, imply

e sφ0 f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∇ e sφ0 f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) + ∇∂ x1 e sφ0 f (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ (0,T )×(0,2π)×(0,1) e 2sφ0 τ λ 4 |u| 2 + τ λ 2 |∇u| 2 + ∂ 2 x1 u 2 + |∂ x1 ∂ x2 u| 2 dx dt ≲ (0,T )×Ω ∞ ∇ 2 ǔ 2 + τ λ 2 |∇ǔ| 2 + τ λ 4 |ǔ| 2 dx dt.
Thus, using 1 -χ ∈ S 0 τ (Lemma 2.1) along with Theorem 2.3, the property of χ ∞ and (4.6), we deduce that

G (1) 2 L 2 (0,T ;H 1 (Ω ∞ )) ⩽ λ -1 I 5 (s, λ, ǔ). (4.46)
Moreover, using (4.27), we deduce

G (2) + ∇G (2) ≲ τ λ | u| + |∂ x1 u| + ∂ 2 x1 u + |∂ x2 u| + |∂ x1 ∂ x2 u| . (4.47)
We deduce from the above relation and (4.16) that

G (2) 2 L 2 (0,T ;H 1 (Ω ∞ )) ⩽ λ -2 I( u, p). (4.48)
From (4.45),

∂ x1 G (3) = -∂ 2 x1 [s∂ x1 φ 0 , χ ∞ Op(1 -χ)]ǔ 1 , ∂ x2 G (3) = -∂ x1 [s∂ x1 φ 0 , χ ∞ Op(1 -χ)]∂ x2 ǔ1
. Thus from (4.28), (4.6), Theorem 2.2 and Theorem 2.3,

G (3) 2 L 2 (0,T ;H 1 (Ω ∞ )) ≲ λ -4 I 5 (s, λ, ǔ). (4.49)
From (4.45),

∂ x1 G (4) = ∂ 2 x1 ( χ ∞ ) ′ u 1 , ∂ x2 G (4) = ∂ x1 ( χ ∞ ) ′ ∂ x2 u 1 ,
and thus, from (4.16) and (2.24), G

L 2 (0,T ;H 1 (Ω ∞ )) ≲ λ -2 I( (4) 2
and the above relation, we deduce the estimate for ∂ x1 f (2) . To estimate ∂ t f (2) , we derive (4.14) with respect to time:

∂ t f (2) = H (1) + H (2) + H (3) , (4.50)
with

H (1) := -χ ∞ Op(-∂ t χ) e sφ0 f (2) + χ ∞ Op(1 -χ) s∂ t φ 0 f (2) + ∂ t f (2) e sφ0 , (4.51)
H (2) := ∂ t s∂ x1 φ 0 u ∞ 1 + ( χ ∞ ) ′ u 1 , (4.52)
H (3) := -∂ t [s∂ x1 φ 0 , χ ∞ Op(1 -χ)]ǔ 1 . (4.53)
Combining (2.20), Lemma 2.5, (4.27), Theorem 2.3, the property of χ ∞ and (4.6), we deduce that

H (1) 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ λ -1 I 5 (s, λ, ǔ). (4.54)
Using (2.20), (4.27) and (4.16), we also find

H (2) 2 L 2 (0,T ;L 2 (Ω ∞ )) ⩽ λ -2 I( u, p). (4.55)
For the last term, we write

H (3) = -[s∂ x1 ∂ t φ 0 , χ ∞ Op(1 -χ)]ǔ 1 + [s∂ x1 φ 0 , χ ∞ Op(∂ t χ)]ǔ 1 -[s∂ x1 φ 0 , χ ∞ Op(1 -χ)]∂ t ǔ1 .
Combining (4.28), Lemma 2.5, Theorem 2.2 and Theorem 2.3, we deduce

H (3) 2 L 2 (0,T ;L 2 (Ω ∞ )) ≲ λ -1 I 5 (s, λ, ǔ).
Gathering the above estimate, (4.54) and (4.55) yields the result.

Proposition 4.7. There exist s 0 > 0 and λ 0 > 0 such that the function h defined by (4.15) satisfies for s ⩾ s 0 (T 2 + T 4 ) and for λ ⩾ λ 0 ,

τ -1/2 ∂ t ∂ x1 h 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + ∂ x1 h 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 )) ≲ λ -1 I 4 (s, λ, η).
Proof. From (4.15)

∂ x1 h = ( χ ∞ ) ′ Op(1 -χ) h (1) + χ ∞ Op(1 -χ) h (2) , (4.56)
with h (1)

:= ∂ t η -s∂ t φ 0 η, h (2) := ∂ x1 h (1) = ∂ t ∂ x1 η -s (∂ t ∂ x1 φ 0 ) η -s (∂ t φ 0 ) ∂ x1 η.
Applying Lemma 4.3 and using (4.56), we deduce that

∂ x1 h 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 )) ≲ Op(1 -χ) h (1) 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 )) + Op(1 -χ) h (2) 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 )) ≲ τ -1/2 Op(1 -χ)∂ 2 x1 h (1) 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + τ -1/2 Op(1 -χ)∂ 2 x1 h (2) 2 L 2 (0,T ;L 2 (Γ ∞ 1 ))
.

Then, using that 1 -χ ∈ S 0 τ , (4.27) and Theorem 2.3, we find

∂ x1 h 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 )) ≲ τ -1/2 ∂ t ∂ 3 x1 η 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + τ -1/2 ∂ t ∂ 2 x1 η 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + 3 j=0 τ 3/2 λ 3/2 ∂ j x1 η 2 L 2 (0,T ;L 2 (Γ ∞ 1 ))
.

From (4.5), we deduce from the above relation that

∂ x1 h 2 L 2 (0,T ;H 3/2 (Γ ∞ 1 )) ≲ λ -1 I 4 (s, λ, η). (4.57)
By differentiating (4.56) with respect to t, we obtain

∂ t ∂ x1 h = -( χ ∞ ) ′ Op(∂ t χ) h (1) + ( χ ∞ ) ′ Op(1 -χ)∂ t h (1) -χ ∞ Op(∂ t χ) h (2) + χ ∞ Op(1 -χ)∂ t h (2) . (4.58)
Applying Lemma 2.5, Theorem 2.3 and (4.27), we have for s ⩾ s 0 (T 2 + T 4 ) and λ ⩾ λ 0 ,

τ -1/2 ∂ t ∂ x1 h 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) ≲ τ 3/2 λ 3/2 ∂ x1 η 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + τ 3/2 λ 3/2 η 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + τ -1/2 ∂ 2 t η 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + τ -1/2 ∂ 2 t ∂ x1 η 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + τ λ 3/2 ∂ x1 ∂ t η 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) + τ λ 3/2 ∂ t η 2 L 2 (0,T ;L 2 (Γ ∞ 1 ))
. (4.59) From (4.5), we deduce from the above relation that

τ -1/2 ∂ t ∂ x1 h 2 L 2 (0,T ;L 2 (Γ ∞ 1 )) ≲ λ -1 I 4 (s, λ, η). (4.60)
The proof of Proposition 4.1 consists now in combining (4.26) with Proposition 4.5, Proposition 4.6 and Proposition 4.7. In the next section, we show the observability result from the above relation.

Proof of the observability

This section is devoted to the proof of Theorem 1.2. We first remove in (4.9) the local terms in p and in ∂ 2 t η. Setting φ 1 (t) := 1 ℓ(t) 2 (e 8λΨ -e 10λΨ ), ξ 1 (t) := 1 ℓ(t) 2 e 8λΨ , (5.1)

φ 2 (t) := 1 ℓ(t) 2 (e (9λ+µ0)Ψ -e 10λΨ ), ξ 2 (t) := 1 ℓ(t) 2 e (9λ+µ0)Ψ , (5.2)
we have (see (2.12))

φ 1 (t) ⩽ φ(t, •) ⩽ φ 2 (t), ξ 1 (t) ⩽ ξ(t, •) ⩽ ξ 2 (t) (t ∈ (0, T )).
Let us set ρ 0 := λτ 2 e sφ1 = λ 3 e 16λΨ s 2 ℓ 4 e sφ1 , (5.3)

ρ 1 := s 11/2 λ -7 ξ 11/2 2
e 4sφ2-3sφ1 , ρ 2 := s 9 λ -1 ξ 9 2 e 4sφ2-3sφ1 .

(5.4)

Then we have the following result.

Proposition 5.1. There exist s 0 > 0 and λ 0 > 0 such that for any s ⩾ s 0 (T 2 + T 4 ) and for any λ ⩾ λ 0 , any smooth solution of (1.6) satisfies Now we set ρ 3 (t) := λ 3 e 6λΨ s 1/2 ℓ(t) e sφ1(t) , ρ 4 (t) := λ 3 e -4λΨ s -1 ℓ(t) 2 e sφ1(t) .

T 0 ρ 2 0 ∥[u, η, ∂ t η]∥ 2 H dt ≲ (0,T )×J ρ 2
(5.9)

We have ρ 3 , ρ 4 ∈ C 1 ([0, T ]), ρ 3 (0) = ρ 4 (0) = 0 and

|ρ ′ 3 | ≲ ρ 0 and |ρ ′ 4 | ≲ ρ 3 .
(5.10)

We recall that (1.12) can be written as (2.7). Then, we deduce

d dt ρ 3   u η ∂ t η   = Aρ 3   u η ∂ t η   + ρ ′ 3   u η ∂ t η   in (0, T ),   ρ 3 u ρ 3 η ρ 3 ∂ t η   (0) = 0.
(5.11)

From (5.10), (5.7) and (2.3),

∥ρ 3 ∂ t u∥ L 2 (0,T ;L 2 (Ω)) + ∥ρ 3 u∥ L 2 (0,T ;H 2 (Ω)) + ρ 3 ∂ 2 t η L 2 (0,T ;L 2 (Γ1)) + ∥ρ 3 ∂ t η∥ L 2 (0,T ;H 2 (Γ1)) + ∥ρ 3 η∥ L 2 (0,T ;H 4 (Γ1)) ≲ ∥ρ 0 u∥ 2 L 2 (0,T ;L 2 (Ω)) + ∥ρ 0 ∂ t η∥ 2 L 2 (0,T ;L 2 (Γ1)) + ∥ρ 0 η∥ 2 L 2 (0,T ;H 2 (Γ1)) .
(5.12) Then, we deduce from (2.7) that

d dt   ρ 4 d dt   u η ∂ t η     = A   ρ 4 d dt   u η ∂ t η     + ρ ′ 4 d dt   u η ∂ t η   in (0, T ),   ρ 4 d dt   u η ∂ t η     (0) = 0.
(5.13) From (5.10), (5.7), (5.12) and (2.3),

ρ 4 ∂ 2 t u L 2 (0,T ;L 2 (Ω)) + ∥ρ 4 ∂ t u∥ L 2 (0,T ;H 2 (Ω)) + ρ 4 ∂ 3 t η L 2 (0,T ;L 2 (Γ1)) + ρ 4 ∂ 2 t η L 2 (0,T ;H 2 (Γ1)) + ∥ρ 4 ∂ t η∥ L 2 (0,T ;H 4 (Γ1)) ≲ ∥ρ 0 u∥ 2 L 2 (0,T ;L 2 (Ω)) + ∥ρ 0 ∂ t η∥ 2 L 2 (0,T ;L 2 (Γ1)) + ∥ρ 0 η∥ 2 L 2 (
+ ε ρ 4 ∂ 3 t η 2 L 2 (0,T ;L 2 (Γ1)) + C (0,T )×J1 s 6 λ -4 ξ 6 0 e 4sφ0-2sφ1 |∂ t η| 2 dt dx 1 ⩽ ε ρ 4 ∂ 3 t η 2 L 2 (0,T ;L 2 (Γ1)) + C (0,T )×J1 s 6 λ -4 ξ 6 0 e 4sφ0-2sφ1 |∂ t η| 2 dt dx 1 . (5.16)
Then, we integrate by parts the last term and we obtain that for any ε > 0, there exists C > 0 such that (0,T )×J1

s 6 λ -4 ξ 6 0 e 4sφ0-2sφ1 |∂ t η| 2 dt dx 1 ⩽ ε ρ 3 ∂ 2 t η 2 L 2 (0,T ;L 2 (Γ1)) + C (0,T )×J1
s 11 λ -14 ξ 11 0 e 8sφ0-6sφ1 |η| 2 dt dx 1 . (5.17)

Similarly, for any ε > 0, there exists C > 0 such that (0,T )×ω1 On the other hand, using (2.7) and the fact that 0 ∈ ρ(A) (see, for instance, [1, Proposition 3.5]),

∥∂ t [u, η, ∂ t η]∥ H = ∥A[u, η, ∂ t η]∥ H ⩾ c ∥[u, η, ∂ t η]∥ H .
Combining the above estimate and (5.22) implies

T 0 ρ 2 0 ∥[u, η, ∂ t η]∥ 2 H dt ≲ (0,T )×J ρ 2 1 |∂ t η| 2 dt dx 1 + (0,T )×ω ρ 2 5 |∂ t u| 2 dt dx. (5.23)
We integrate by parts the last term: recalling (5.21), we obtain that for any ε > 0, there exists C > 0 such that (0,T )×ω1

ρ 2 5 |∂ t u| 2 dt dx ⩽ C (0,T )×ω1 s 11 λ 2 ξ 11 2 e 4sφ2-2sφ1 |u| 2 dt dx + ε ρ 4 ∂ 2 t u 2 L 2 (0,T ;L 2 (Ω)) + C (0,T )×ω1
s 18 λ -2 ξ 18 2 e 8sφ2-6sφ1 |u| 2 dt dx. (5.24)

We deduce (5.5) by combining (5.23), (5.24) and (5.15).

Using Proposition 5.1, one can deduce Theorem 1.2:

Proof of Theorem 1.2. We fix λ = λ 0 and s = s 0 (T 2 + T 4 ) in (5.3) and (5.4). In particular the constants that follows may depend on λ 0 and s 0 . Then we deduce from (5.5) that

3T /4 T /4 ρ 2 0 ∥[u, η, ∂ t η]∥ 2 H dt ≲ (0,T )×J ρ 1 (t) 2 |∂ t η| 2 dt dx 1 + (0,T )×ω ρ 2 (t) 2 |u| 2 dt dx. (5.25)
From (5.3) and (5.4), there exists C > 0 such that

C T e -C/T 2 ⩽ ρ 0 in T 4 , 3T 4
,

and ρ 1 ≲ 1, ρ 2 ≲ 1.
Since A is the generator of a semigroup of contractions (see, for instance, [1, Proposition 3.4]), we deduce the result from the above relations.

6 Proof of Theorem 1.4

We give here a sketch of the proof of Theorem 1.4. First we construct a change of variables to write (1.2) in a cylindrical domain, then we use the "source term method" and Theorem 1.2 to show Theorem 1.4 by a fixed-point argument.

Change of variables

We can assume that for some δ > 0 ω ⊂ I × (0, 1 -δ).

Let us consider a smooth function θ ∈ C ∞ ([0, 1]; [0, 1]) with compact support in (1 -δ, 1] and such that θ ≡ 1 in [1 -δ/2, 1]. We consider the change of variables

X(t, •) : Ω → Ω ζ(t) , (y 1 , y 2 ) → (y 1 , y 2 + θ(y 2 )ζ(t, y 1 )) (6.1) that is a diffeomorphism if ∥θ ′ ∥ L ∞ (0,1) ∥ζ∥ L ∞ (0,T ;L ∞ (I)) ⩽ 1 2 . ( 6.2)
We denote by Y (t, •) the inverse of X(t, •). We write W (t, y) := Cof(∇X) * (t, y)w(t, X(t, y)), Π(t, y) := π(t, X(t, y))

X 0 := X(0, •), W 0 := Cof(∇X 0 ) * w 0 • X 0 .
We also write a := Cof(∇Y ) * . (6.3)

After some standard calculations (see, for instance,

[2]) (1.2) is transformed into            ∂ t W -div T(W, Π) = 1 ω f + F ζ (W, Π) t > 0, x ∈ Ω, div W = 0 t > 0, x ∈ Ω, W = ∂ t ζe 2 t > 0, x ∈ Γ 1 , W = 0 t > 0, x ∈ Γ 0 , ∂ tt ζ + A 1 ζ + A 2 ∂ t ζ = P L 2 0 (I) (Π + 1 J g + G ζ (W )) t > 0, x 1 ∈ I, (6.4)
W (0, •) = W 0 in Ω, ζ(0, •) = ζ 0 1 , ∂ t ζ(0, •) = ζ 0 2 in I, (6.5)
with

[F ζ (W, Π)] i := - k ∂ t a i,k (X)W k + (a i,k (X) -δ i,k ) ∂ t W k - k,l a i,k (X) ∂W k ∂y l ∂Y l ∂t (X) + k,j ∂ 2 a i,k ∂x 2 j (X)W k + 2 k,j,l ∂a i,k ∂x j (X) ∂W k ∂y l ∂Y l ∂x j (X) + k,j,l a i,k (X) ∂W k ∂y l ∂ 2 Y l ∂x 2 j (X) + k,j,l,m a i,k (X) ∂Y l ∂x j (X) ∂Y m ∂x j (X) -δ i,k δ l,j δ m,j ∂W k ∂y l ∂y m - l ∂Y l ∂x i (X) -δ l,i ∂Π ∂y l - j,m,k a j,m (X) ∂a i,k ∂x j (X)W m W k - 1 det(∇X) m,k a i,k (X)W m ∂W k ∂y m , (6.6)
G ζ (W ) := -∂ x1 ζ + (∂ x1 ζ) 2 ∂W 1 ∂x 2 x2=1 . (6.7)
Then, we can write Theorem 1.4 as follows:

Theorem 6.1. Assume T > 0, ω ⋐ Ω and J ⋐ I are nonempty open sets. There exists R 0 > 0 such that for any

[W 0 , ζ 0 1 , ζ 0 2 ] ∈ V with [W 0 , ζ 0 1 , ζ 0 2 ] V ⩽ R 0 , (6.8
)

there exists a control (f, g) ∈ L 2 (0, T ; L 2 (ω)) × L 2 (0, T ; L 2 (J ))
such that the solution of (6.4), (6.5), (6.6) and (6.7) satisfies

ζ(T, •) = 0, ∂ t ζ(T, •) = 0 in I, W (T, •) = 0 in Ω.

The fixed point argument

Using the notation of Section 2.1, the result of Theorem 1.2 states the existence of k T satisfying (1.14) such that for any u 0 , η 0 1 , η 0 2 ,

e T A *   u 0 η 0 1 η 0 2   2 H ⩽ k 2 T T 0 B * e tA *   u 0 η 0 1 η 0 2   2 L 2 (ω)×L 2 (J )
dt. (6.9)

From standard results (see, for instance, [52, Theorem 11.2.1, p.357]), this yields the null-controllability of (1.5).

Using the "source term method" (see, [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF]), one can improve this result. Let us consider the following weight functions

σ 1 (t) := e - C 1 (T -t) 2 , σ 2 (t) := e - C 2 (T -t) 2 , σ 3 (t) := e - C 3 (T -t) 2 (6.10)
and the corresponding spaces (for σ = σ 1 , σ 2 or σ 3 )

L p σ (0, T ; X ) := {f /σ ∈ L p (0, T ; X )} , C α σ ([0, T ]; X ) := {f /σ ∈ C α ([0, T ]; X )} , H s σ (0, T ; X ) := {f /σ ∈ H s (0, T ; X )} , for p ⩾ 1, k ∈ N, s ∈ R +
and X a Banach space. The abstract result proved in [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF] yields the following result:

A Technical results

A.1 A Carleman estimates for the damped beam equation

The proof of Theorem 3.3 follows directly from the proof done in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF]. The differences with respect to this article is the weight in time and the powers of s, µ, and ξ 0 . For sake of completeness, we give here a brief sketch of the proof of Theorem 3.3 by using what is already done in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF].

We recall that φ 0 and ξ 0 are given by (2.13). We set

f η := ∂ 2 t η + ∂ 4 x1 η -∂ 2 x1 η -∂ t ∂ 2 x1 η, (A.1) ζ := e sφ0 ξ r 0 η. (A.2)
We say that a function g is l. o. t (lower order term) if it satisfies for some

ε 1 , ε 2 ⩾ 0, ε 1 ε 2 ̸ = 0, |g| ≲ s -ε1 λ -ε2 ξ -ε1 0 s 7/2 µ 4 ξ 7/2 0 |ζ| + s 5/2 µ 3 ξ 5/2 0 |∂ x1 ζ| + s 3/2 µ 2 ξ 3/2 0 ∂ 2 x1 ζ + |∂ t ζ| +s 1/2 µξ 1/2 0 ∂ 3 x1 ζ + |∂ t ∂ x1 ζ| + s -1/2 ξ -1/2 0 ∂ 4 x1 ζ + ∂ t ∂ 2 x1 ζ + ∂ 2 t ζ .
From the Leibniz formula

e sφ0 ξ r 0 ∂ 4 ∂x 4 1 e -sφ0 ξ -r 0 ζ = 4 α=0 4 α e sφ0 ∂ α ∂x α 1 e -sφ0 ζ ξ r 0 ∂ 4-α ∂x 4-α 1 ξ -r 0 .
From (3.8), we obtain

ξ r 0 ∂ 4-α ∂x 4-α 1 ξ -r 0 ≲ µ 4-α
and thus a direct computation and (3.8) yield that for s ⩾ s 0 (T 2 + T 4 ), There exists s 0 > 0 such that for s ⩾ s 0 (T 2 + T 4 ),

1 ⩽ sξ, |∇ α ξ| + |∇ α φ| ≲ λ α ξ (k ⩾ 1), |∂ t ∇ α ξ| + |∂ t ∇ α φ| ≲ λ α T ξ 3/2 , ∂ 2 t ∇ α ξ + ∂ 2 t ∇ α φ ≲ λ α T 2 ξ 2 , 1 ⩽ s ξ, ∇ α ξ + |∇ α φ| ≲ λ α ξ (k ⩾ 1), ∂ t ∇ α ξ + |∂ t ∇ α φ| ≲ λ α T ξ 3/2 , ∂ 2 t ∇ α ξ + ∂ 2 t ∇ α φ ≲ λ α T 2 ξ 2 ,
Using the above relations and following standard calculations (see, for instance, [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF]), we obtain the existence of s 0 , c, C, c, C > 0 such that for s ⩾ s 0 (T We deduce Theorem 3.5 by taking r = 1/2.

. 25 )

 25 Combining (4.17), (4.21), (4.22), (4.23), (4.24) and (4.25), we deduce the result by taking ε > 0 small enough.

∂ α ∂x α 1 e -sφ0 ζ ξ r 0 ∂ 4 1 eand thus e sφ0 ξ r 0 ∂ 4 ∂x 4 1 esφ0 ξ r 0 ∂ 2 ∂x 2 1 e∂t 2 e.e sφ0 ξ r 0 ∂ 2 ∂t 2 e. 1 e 1 es 7 µ 8 ξ 7 0 |ζ| 2 + s 5 µ 6 ξ 5 0 |∂ x1 ζ| 2 + s 3 µ 4 ξ 3 0 |∂ 2 x1 ζ| 2 + |∂ t ζ| 2 + sµ 2 ξ 0 |∂ 3 x1 ζ| 2 + 1 0 ∂ 4 x1 ζ 2 + |∂ 2 t ζ| 2 + |∂ t ∂ 2 x1 ζ| 2 dt dx 1 (A. 7 )Then, we deduce ∥M 1 ζ∥ 2 L 2 2 L 2 2 (M 1 ζ • M 2 ζ dx 1 dt = ∥e sφ0 ξ r 0 f η ζ∥ and using that ∂u 2

 1041010120221125222212172222212 -sφ0 ζ = -4s 3 (∂ x1 φ 0 ) 3 ∂ x1 ζ + 6s 2 (∂ x1 φ 0 ) 2 ∂ 2 x1 ζ -4s∂ x1 φ 0 ∂ 3 x1 ζ + ∂ 4 x1 ζ + s 4 (∂ x1 φ 0 ) 4 ζ -12s 3 (∂ x1 φ 0 ) 2 ∂ 2 x1 φ 0 ζ + l. o. t -sφ0 ξ -r 0 ζ = -4s 3 (∂ x1 φ 0 ) 3 ∂ x1 ζ + 6s 2 (∂ x1 φ 0 ) 2 ∂ 2 x1 ζ -4s∂ x1 φ 0 ∂ 3 x1 ζ + ∂ 4 x1 ζ + s 4 (∂ x1 φ 0 ) 4 ζ -12s 3 (∂ x1 φ 0 ) 2 ∂ 2 x1 φ 0 ζ + l. o. t . (A.3) Note that -12s 3 (∂ x1 φ 0 ) 2 ∂ 2 x1 φ 0 ζ = l. o.t, but we follow the trick of[START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF] to keep this term in order to show the Carleman estimate.We can show similarly thate -sφ0 ξ -r 0 ζ = l. o. t . -sφ0 ξ -r 0 ζ = e sφ0 ∂ 2 ∂t 2 e -sφ0 ζ + 2e sφ0 ∂ ∂t e -sφ0 ζ ξ r Thus, using (3.8), for s ⩾ s 0 (T 2 + T 4 ), -sφ0 ξ -r 0 ζ = ∂ 2 t ζ + l. o. t .From (3.8), for s ⩾ s 0 (T 2 + T 4 ), and for p = 0, 1, 2, -sφ0 ζ + l. o. t .Consequently, using (3.8), we deduce that for s ⩾ s 0 (T 2 + T 4 ), -sφ0 ξ -r0 ζ = s 2 (∂ x1 φ 0 ) 2 ∂ t ζ -2s∂ x1 φ 0 ∂ t ∂ x1 ζ + ∂ t ∂ 2 x1 ζ + l. o. t .Gathering (A.3), (A.4), (A.5), and the above relation and combining them with (A.1) and (A.2), we deduceM 1 ζ + M 2 ζ = e sφ0 ξ r 0 f η + l. o. t, (A.6) with M 1 ζ := s 4 (∂ x1 φ 0 ) 4 ζ + 6s 2 (∂ x1 φ 0 ) 2 ∂ 2 x1 ζ + ∂ 4 x1 ζ + 2s (∂ x1 φ 0 ) ∂ t ∂ x1 ζ + ∂ 2 t ζ, and M 2 ζ := -4s 3 (∂ x1 φ 0 ) 3 ∂ x1 ζ -4s∂ x1 φ 0 ∂ 3 x1 ζ -∂ t ∂ 2 x1 ζ -s 2 (∂ x1 φ 0 ) 2 ∂ t ζ -12s 3 (∂ x1 φ 0 ) 2 ∂ 2 x1 φ 0 ζ.In what follows, we say that a term G is a L. O. T (Lower Order Term) if there existε 1 ⩾ 0, ε 2 ⩾ 0, ε 1 ε 2 ̸ = 0, such that |G| ≲ (0,T )×I s -ε1 λ -ε2 ξ -ε1 0 |∂ t ∂ x1 ζ| 2 + s -1 ξ -(0,T ;L 2 (Ω)) + ∥M 2 ζ∥ (0,T ;L 2 (Ω)) + ∂n = 0 on (0, T ) × Γ 1 , we deduce that ∂v 2 ∂n = -∂ v 2 ∂n on (0, T ) × Γ 1 .(A.12)Since µ = µ 0 , taking λ 0 ⩾ µ 0 and λ ⩾ λ 0 , we deduce that |ψ| + |∇ψ| + ∇ 2 ψ + ψ + ∇ψ + ∇ 2 ψ ≲ 1.

  ) and with standard computations, we can come back to u:(0,T )×Ω e 2sφ s 2r+3 λ 2r+4 ξ 2r+3 |u| 2 + s 2r+1 λ 2r+2 ξ 2r+1 |∇u| 2 +s 2r-1 λ 2r ξ 2r-1 ∇ 2 u 2 + s 2r-1 λ 2r ξ 2r-1 |∂ t u| 2 dxdt ≲ (0,T )×Ω |∂ t u -∆u| 2 (sλξ) 2r e 2sφ dxdt + (0,T )×ω1 s 2r+3 λ 2r+4 ξ 2r+3 e 2sφ |u| 2 dxdt. (A.18)

  , Theorem 2.26]) Theorem 2.3. Let m, m ′ ∈ R, and let a ∈ S m τ . Then, Op(a) :

  2sφ0 s 10 µ 11 ξ 10 0 |η| 2 + s 8 µ 9 ξ 8 0 |∂ x1 η| 2 + s 6 µ 7 ξ 6 0 |∂ 2 x1 η| 2 + |∂ t η| 2 dt dx 1 |∂ x1 ∂ t η| 2 dt dx 1

	(0,T )×I
	+	e 2sφ0 s 4 µ 5 ξ 4 0 |∂ 3 x1 η| 2 +
	(0,T )×I

e

  2 e 2sφ dt dx + (0,T )×Ω s 2 λ 2 ξ 2 e 2sφ |∇u| 2 dt dx + 4 λ 4 ξ 4 e 2sφ |u| 2 dt dx. (3.14)

		s 4 λ 4 ξ 4 e 2sφ |u| 2 dt dx
		(0,T )×Ω
	≲	sξe 2sφ |(∂ t -∆)u| 2 dt dx +
	(0,T )×Ω	(0,T )×ω1

s

  2 + T 4 ), any smooth solutions [u, p 0 , η] of (1.6) satisfiesI 4 (s, λ, η) + I 5 (s, λ, ǔ) ≲ λ 2sφ s 4 λ 6 ξ 4 |u| 2 dt dx + s 3 λ 4 ξ 3 |p| 2 dt dx, (4.9)where η and ǔ are defined by (4.2) and p is defined by (2.15).

	e 2sφ0 s 10 ξ 10 0 |η| 2 + s 2 ξ 2 0 |∂ 2 t η| 2 dt dx 1
	(0,T )×J1
	+
	(0,T )×ω1
	Before proving Proposition 4.1, let us first introduce some preliminary results and notation. Recalling that
	χ is defined in (3.6), we deduce that if χ ̸ = 1 then
	τ ≲ |k|.	(4.10)

e

  1 |∂ t η| 2 dt dx 1 + 3 λ 4 ξ 3 e 2sφ |p| 2 dt dx ≲ 2sφ2 |∂ t u| 2 + |∆u| 2 dt dx.(5.8)

						ρ 2 2 |u| 2 dt dx.	(5.5)
						(0,T )×ω
	Proof. Using (2.16) and applying the Poincaré-Wirtinger inequality, we deduce that
						s 3 λ 4 ξ 3 2 e 2sφ2 |∇p| 2 dt dx
			(0,T )×ω1		(0,T )×ω1
	and with (1.12),
			s 3 λ 4 ξ 3 e 2sφ |p| 2 dt dx ≲	s 3 λ 4 ξ 3 2 e 2sφ2 |∂ t u| 2 + |∆u| 2 dt dx.	(5.6)
			(0,T )×ω1			(0,T )×ω1
	From (5.3), (4.5) and (4.6), we have
	0	T	ρ 0 (t) 2 ∥u(t)∥ 2 L 2 (Ω) + ∥η(t)∥ 2 H 2 (I) + ∥∂ t η(t)∥ 2 L 2 (I)	dt ≲ I 4 (s, λ, η) + I 5 (s, λ, ǔ).	(5.7)
	Combining (4.9), (5.7) and (5.6), we deduce
	∥ρ 0 u∥ 2 L 2 (0,T ;L 2 (Ω)) + ∥ρ 0 ∂ t η∥	2 L 2 (0,T ;L 2 (Γ1)) + ∥ρ 0 η∥	2 L 2 (0,T ;H 2 (Γ1))
			≲ λ	e 2sφ0 s 10 ξ 10 0 |η| 2 + s 2 ξ 2 0 |∂ 2 t η| 2 dt dx 1
				(0,T )×J1
			+	s 4 λ 6 ξ 4 e 2sφ |u| 2 dt dx +	s 3 λ 4 ξ 3 2 e
			(0,T )×ω1		(0,T )×ω1

s

  0,T ;H 2 (Γ1)) . (5.14) Then, from the standard elliptic regularities for the stationary Stokes system ([49, Proposition 2.2 p.33]) and for A 1 , we have moreover∥ρ 4 u∥ L 2 (0,T ;H 4 (Ω)) + ∥ρ 4 η∥ L 2 (0,T ;H 6 (Γ1))Using (3.8), we deduce that for s ⩾ s 0 (T 2 + T 4 ), for any ε > 0, there exists C > 0 such that

		≲ ∥ρ 0 u∥ 2 L 2 (0,T ;L 2 (Ω)) + ∥ρ 0 ∂ t η∥	2 L 2 (0,T ;L 2 (Γ1)) + ∥ρ 0 η∥ 2 L 2 (0,T ;H 2 (Γ1)) . (5.15)
	By integration by parts, we obtain
	(0,T )×J1	s 2 λξ 2 0 e 2sφ0 |∂ 2 t η| 2 dt dx 1 =	1 2	(0,T )×J1	∂ 2 t s 2 λξ 2 0 e 2sφ0 |∂ t η| 2 dt dx 1
						-	s 2 λξ 2 0 e 2sφ0 ∂ 3 t η∂ t η dt dx 1 .
						(0,T )×J1
		s 2 λξ 2 0 e 2sφ0 |∂ 2 t η| 2 dt dx 1 ⩽ C		s 5 λξ 5 0 e 2sφ0 |∂ t η| 2 dt dx 1
	(0,T )×J1			(0,T )×J1

  s 3 λ 4 ξ 3 2 e 2sφ2 |∂ t u| 2 dt dx ⩽ CFinally, we consider a nonnegative smooth function χ 1 with compact support in ω and such that χ 1 ≡ 1 in ω 1 .

							s 6 λ 4 ξ 6 2 e 2sφ2 |u| 2 dt dx
						(0,T )×ω1
				+ ε ρ 4 ∂ 2 t u	2 L 2 (0,T ;L 2 (Ω)) + C	(0,T )×ω1	s 8 λ 2 ξ 8 2 e 4sφ2-2sφ1 |u| 2 dt dx
						⩽ ε ρ 4 ∂ 2 t u	2 L 2 (0,T ;L 2 (Ω)) + C	(0,T )×ω1	s 8 λ 2 ξ 8 2 e 4sφ2-2sφ1 |u| 2 dt dx. (5.18)
	Then by integrating by parts,
			s 3 λ 4 ξ 3 2 e 2sφ2 |∆u| 2 dt dx
		(0,T )×ω1
			⩽		χ 1 s 3 λ 4 ξ 3 2 e 2sφ2 |∆u| 2 dt dx =	s 3 λ 4 ξ 3 2 e 2sφ2 ∆ (χ 1 ∆u) u dt dx
				(0,T )×ω		(0,T )×ω
						⩽ ε ∥ρ 4 u∥	2 L 2 (0,T ;H 4 (Ω)) + C	(0,T )×ω	s 8 λ 2 ξ 8 2 e 4sφ2-2sφ1 |u| 2 dt dx. (5.19)
	Gathering (5.8), (5.16), (5.17), (5.18), and (5.19), and using (5.12), (5.14) and (5.15) we deduce
	0	T	ρ 0 (t) 2 ∥u(t)∥	2 L 2 (Ω) + ∥η(t)∥ 2 H 2 (I) + ∥∂ t η(t)∥	2 L 2 (I)	dt
							≲	ρ 2 1 |η| 2 dt dx 1 +	ρ 2 5 |u| 2 dt dx, (5.20)
							(0,T )×J1	(0,T )×ω
	with ρ 1 defined by (5.4) and with
						ρ 5 := s 4 λξ 4 2 e 2sφ2-sφ1 .	(5.21)
	To end the proof of Proposition 5.1, we need to replace in the above estimate the observation by η with an
	observation by ∂ t η. This is done by using the smoothing effet of the parabolic system (2.7). More precisely, we
	apply (5.20) to (∂ t u, ∂ t η, ∂ 2 t η) and we deduce
	0	T	ρ 0 (t) 2 ∥∂ t u(t)∥	2 L 2 (Ω) + ∥∂ t η(t)∥ 2 H 2 (I) + ∂ 2 t η(t)	2 L 2 (I)	dt
							≲	ρ 2 1 |∂ t η| 2 dt dx 1 +	ρ 2 5 |∂
							(0,T )×J	(0,T )×ω

t u| 2 dt dx.

(5.22)

  2 + T 4 ), |∂ t u -∆u| 2 ξ 2r e 2s φ dxdt + (0,T )×ω1s 3 λ 4 ξ 3 | v| 2 dxdt . (A.14)34Summing (A.13) and (A.14) and using (A.12), (A.12), we deduce that c (0,T )×Ωs 3 λ 4 ξ 3 |v| 2 + sλ 2 ξ |∇v| |∂ t u -∆u| 2 ξ 2r e 2s φ dxdt + (0,T )×ω1 s 3 λ 4 ξ 3 | v| 2 dxdt . (A.15)Then, using that φ ⩽ φ and ξ ⩽ ξ, we deduce that (0,T )×Ωs 3 λ 4 ξ 3 |v| 2 + sλ 2 ξ |∇v| |∂ t u -∆u| 2 ξ 2r e 2sφ dxdt + (0,T )×ω1 s 3 λ 4 ξ 3 |v| 2 dxdt. (A.16)From the elliptical regularity of the Laplace operator, we deduce (0,T )×Ωs 3 λ 4 ξ 3 |v| 2 + sλ 2 ξ |∇v|

									2 +	1 sξ	|∆v| 2 +	1 sξ	|∂ t v|	2	dxdt
				+ c	(0,T )×Ω	s 3 λ 4 ξ 3 | v| 2 + sλ 2 ξ |∇ v| 2 +	1 s ξ	|∆ v| 2 +	1 s ξ	|∂ t v| 2	dxdt
				⩽ C					|∂ t u -∆u| 2 ξ 2r e 2sφ dxdt +	s 3 λ 4 ξ 3 |v| 2 dxdt
							(0,T )×Ω		(0,T )×ω0
							+ C
									(0,T )×Ω
									2 +	1 sξ	|∆v|	2 +	1 sξ	|∂ t v| 2	dxdt
	c	(0,T )×Ω	s 3 λ 4 ξ 3 |v| 2 + sλ 2 ξ |∇v| 2 + ≲ (0,T )×Ω 1 sξ |∆v| 2 +	1 sξ	|∂ t v|	2	dxdt
			-	(0,T )×∂Ω -(0,T )×∂Ω 2s 3 λ 3 ξ 3 |∇ψ| 4sλξ (∇v∇ψ) • 2 ∂ψ ∂n |v| 2 dx 1 dt -∂v ∂n dx 1 dt + 2 + 1 sξ ∇ 2 v 2 + 1 sξ |∂ t v|	(0,T )×∂Ω (0,T )×∂Ω	4sλ 2 ξ |∇ψ| 2 ∂v ∂n 2sλξ ∂ψ ∂n |∇v| 2 dx 1 dt • v dx 1 dt
				-	(0,T )×∂Ω	2∂ t v •	∂v ∂n	dx 1 dt -	(0,T )×∂Ω	2s 2 λξ	∂ψ ∂n	∂ t φ |v| 2 dx 1 dt
							⩽ C		|∂ t u -∆u|	2 ξ 2r e 2sφ dxdt +	s 3 λ 4 ξ 3 |v|	2 dxdt	(A.13)
									(0,T )×Ω	(0,T )×ω1
	and
	c	(0,T )×Ω	s 3 λ 4 ξ 3 | v| 2 + sλ 2 ξ |∇ v| 2 +	1 s ξ	|∆ v| 2 +	1 s ξ	|∂ t v|	2	dxdt
			-	(0,T )×∂Ω	2s 3 λ 3 ξ 3 ∇ ψ	2 ∂ ψ ∂n	| v| 2 dx 1 dt -	(0,T )×∂Ω	4sλ 2 ξ ∇ ψ	2 ∂ v ∂n	• v dx 1 dt
				-	(0,T )×∂Ω	4sλ ξ ∇ v∇ ψ •	∂ v ∂n	dx 1 dt +	(0,T )×∂Ω	2sλ ξ	∂ ψ ∂n	|∇ v| 2 dx 1 dt
				-	(0,T )×∂Ω	2∂ t v •	∂ v ∂n	dx 1 dt -	(0,T )×∂Ω	2s 2 λ ξ	∂ ψ ∂n	∂ t φ | v| 2 dx 1 dt
							⩽ C
									(0,T )×Ω

2 dxdt ≲ (0,T )×Ω |∂ t u -∆u| 2 ξ 2r e 2sφ dxdt + (0,T )×ω1

s 3 λ 4 ξ 3 |v| 2 dxdt (A.17

L 2 (0,T ;L 2 (Ω)) + L. O. T . (A.8)

Acknowledgements

The second author was partially supported by the Agence Nationale de la Recherche, Project TRECOS, ANR-20-CE40-0009.

Proposition 6.2. Assume (6.9) with (1.14). Then there exist σ 1 , σ 2 , σ 3 as in (6.10) and a bounded map

σ1 (0, T ; L 2 (Ω) × L 2 (I)) → L 2 σ2 (0, T ; L 2 (ω) × L 2 (J )) such that for any [W 0 , ζ 0 1 , ζ 0 2 ] ∈ V and for any (F, G) ∈ L 2 σ1 (0, T ; L 2 (Ω) × L 2 (I)), the solution of

with the control

(0,T ;L 2 (Ω)×L 2 (I)) . (6.13) Moreover, we can assume

We are now in a position to prove Theorem 6.1 and thus Theorem 1.4.

Proof of Theorem 6.1. Assume that [W 0 , ζ 0 1 , ζ 0 2 ] satisfies (6.8) for some R 0 and let us assume that

) such that the corresponding solution of (6.11), (6.12) satisfies

([0,T ];L 2 (I)) ⩽ CR 0 (6.15) for some constant C > 0. Using the Sobolev embeddings, we have in particular that

for some constant C > 0. This yields that for R 0 small enough, (6.2) holds and we can consider the change of variables of Section 6.1. We thus define X, a, F and G by respectively, (6.1), (6.3), (6.6) and (6.7). Moreover, following the arguments in [START_REF] Raymond | Feedback stabilization of a fluid-structure model[END_REF], [START_REF] Badra | Feedback boundary stabilization of 2D fluid-structure interaction systems[END_REF] and using (6.14), one can show that

and in particular for R 0 small enough, the closed set

One can also show that for R 0 > 0 small enough, the above map is a strict contraction on B R0 . Using the Banach fixed point we deduce the existence of fixed point (F, G) for Z. One can notice that the corresponding solution (W, Π, ζ) of (6.11)-(6.12) verifies the conclusion of Theorem 6.1.

Writing I i,j for the product term of the i-th term of M 1 ζ with the j-th term if M 2 ζ, we have

and we have to estimate all the terms I i,j . This is done in a precise way in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF] using (3.8). For instance, by integration by parts,

and using (3.8), we deduce, as in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF] that

Then, following the computations in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF], we find

Then by using standard techniques as in [START_REF] Sourav | Carleman estimate for an adjoint of a damped beam equation and an application to null controllability[END_REF], we deduce the result.

A.2 A Carleman estimate for the heat equation

We give here a sketch of the proof of Theorem 3.5. We recall that φ and ξ are defined by (2.12) and we define

We also recall that ψ is defined by (2.14) and we define ψ(x 1 , x 2 ) := µ λ ψ I (x 1 ) -ψ Ω (x 1 , x 2 ). (A.10)

We have in particular φ = 1 ℓ 2 (e λ(ψ+8Ψ) -e 10λΨ ), ξ := 1 ℓ 2 e λ(ψ+8Ψ) , φ = 1 ℓ 2 (e λ( ψ+8Ψ) -e 10λΨ ), ξ := 1 ℓ 2 e λ( ψ+8Ψ) .

We set v := e sφ ξ r u, v := e s φ ξ r u.

Using (2.9), we have


Morgane Bourhis
email: morgane.bourhis@grenoble-inp.fr

Pascal Perrier

Christophe Savariaux

Takayuki Ito

Compensatory movement of the tongue for speech production with or without masking noise

Introduction Results

Somatosensory and auditory inputs play an important role in speech production [START_REF] Guenther | Role of the auditory system in speech production[END_REF].

The tongue showed quick compensatory response working for stabilizing the tongue contour and maintaining the resultant produced sound [START_REF] Ito | Quick compensatory mechanisms for tongue posture stabilization during speech production[END_REF].

This compensatory response may be driven by somatosensory feedback alone.

Objective Methods

Subjects: 12 native French speakers Does a perturbation of the tongue during speech production induce compensatory response in the absence of auditory feedback?

Auditory masking and no masking conditions were tested in random order. The current compensatory response can be not of auditory-basis, but of somatosensory-basis.

Further analysis of produced speech sound and kinematic movement are required.

We will further investigate the current hypothesis by studying on hearing impaired participants, which is equivalent with the current experimental condition, and deafferented subjects, who lost somatosensory feedback.

The responses were similar to the ones in [START_REF] Ito | Quick compensatory mechanisms for tongue posture stabilization during speech production[END_REF]. Difference between onset of compensatory movement and peak of compensatory movement were significant (F(1,10)=10.279, p<0,05) indicating that The compensatory responses were induced.

Data were aligned at perturbation onset and were averaged across trials. Averaged movement and standard error has been calculated across subjects. A two way ANOVA on the TB position was applied in which auditory conditions and onset and peak of compensatory movement were considered as main factor. A one way ANOVA was also applied on the displacement before the perturbation.

Sensor locations: Tongue Tip (TT), Blade (TB), and Dorsum (TD), Upper Lip (UL), Lower Lip(LL), and Jaw (J) Speech Task: whispering the vowel /e/ for 3.5 sec after the presentation of green circle.

Produced sound and articulatory movement were recorded through the Wave system

Sensory feedback for speech production

Compensatory movement of the tongue

Trials : 3 sessions of 50 trials (+ 5 control trials)

The responses were similar in with and without auditory masking conditions.

Main effect of the masking condition F(1,10) = 0,605, p>0.1

Tongue Perturbation:

Stepwise pattern (1 N for 1 s) at TP in randomly selected 1/5 of the trials.

Speech somatosensory goal can be achieved even in the absence of auditory error [START_REF] Tremblay | Somatosensory basis of speech production[END_REF].

Magnified view between 100 and 300 ms

Amplitude of the compensatory responses

The position at TB was not significantly different between the no masking and auditory masking condition before the perturbation (F(1,10)=0,05, p>0,1).

  [START_REF] Ito | Quick compensatory mechanisms for tongue posture stabilization during speech production[END_REF]

	Auditory feedback
	Somatosensory feedback

Funding

This work was supported by Agence Nationale de la Recherche (ANR-21-CE28-0022) and National Institute on Deafness and Other Communication Disorders (R01-