Local null controllability of a two-dimensional fluid-structure interaction problem

Vol. 14, N 1, 2008, pp. 1–42 www.esaim-cocv.org

DOI: 10.1051/cocv:2007031

LOCAL NULL CONTROLLABILITY OF A TWO-DIMENSIONAL FLUID-STRUCTURE INTERACTION PROBLEM

Muriel Boulakia

and Axel Osses

Abstract. In this paper, we prove a controllability result for a fluid-structure interaction problem.

In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given T > 0, the system can be driven at rest and the structure to its reference configuration at time T. To show this result, we first consider a linearized system. Thanks to an observability inequality obtained from a Carleman inequality, we prove an optimal controllability result with a regular control. Next, with the help of Kakutani’s fixed point theorem and a regularity result, we pass to the nonlinear problem.

Mathematics Subject Classification. 35Q30, 93C20.

Received October 27, 2005. Revised April 5, 2006.

Published online July 20, 2007.

1. Introduction and main result 1.1. Introduction

We consider a rigid structure immersed in a viscous incompressible fluid. At timet, the structure occupies the smooth connected domain Ω_S(t). The structure and the fluid are contained in a fixed bounded connected open set Ω⊂R²with a regular boundary. We suppose that Ω_S(0) and Ω have a smooth boundary (for instanceC²).

The time evolution of the fluid eulerian velocity uis governed by the incompressible Navier-Stokes equations (for simplicity, we assume that the fluid density is constant and equal to 1):

∂_tu+ (u· ∇)u

(t, x)−divσ(u, p)(t, x) =f(t, x)1_ω(x),∀x∈Ω_F(t),∀t∈(0, T),

divu(t, x) = 0, ∀x∈Ω_F(t),∀t∈(0, T). (1.1)

For anyt∈(0, T), these equations are satisfied on Ω_F(t) = Ω\Ω_S(t), the fluid domain. The tensorσ(u, p) is the Cauchy tensor given by

σ(u, p) = 2(u)−pId,

Keywords and phrases. Controllability, fluid-solid interaction, Navier-Stokes equations, Carleman estimates.

1 Laboratoire de Math´ematiques Appliqu´ees, Universit´e de Versailles-St-Quentin, 45 avenue des ´Etats-Unis, 78035 Versailles Cedex, France;[email protected]

2Departamento de Ingener´ıa Matem´atica and Centro de Modelamiento Matem´atico UMI 2807 CNRS, Facultad de Ciencias de F´ısicas y Matem´aticas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile;[email protected]

c EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.esaim-cocv.org or http://dx.doi.org/10.1051/cocv:2007031

where(u) =¹₂(∇u+∇u^t) is the symmetric part of the gradient. Here, pis the pressure of the fluid. Without lost of generality, we have supposed that the viscosity is equal to 1. Finallyf is the control function which acts over a fixed small nonempty open subsetω of the fluid domain Ω_F(t) and 1_ω is the characteristic function of the domainω.

The motion of the structure is given by the translation velocity which is the velocity of the center of mass of the structure a(t)∈R² and by the instantaneous rotation velocity denoted r(t)∈R. The equations of the structure motion are given by the balance of linear and angular momentum. So, without the action of external forces, we have, for all t∈(0, T)

m¨a(t) =

∂ΩS(t)σ(u, p)ndσ(x), (1.2)

Jr(t) =˙

∂ΩS(t)(σ(u, p)n)·(x−a(t))^⊥dσ(x). (1.3) We have denoted bym >0 the mass of the rigid structure andJ >0 its moment of inertia. Moreover,x^⊥ is defined by

∀x= (x₁, x₂)∈R², x^⊥= (−x2, x₁).

At last, nis the outward unit normal to∂Ω_S(t). On the interface, we consider a non-slip boundary condition.

Therefore, we have, for allt∈(0, T)

u(t, x) = 0,∀x∈∂Ω, (1.4)

u(t, x) = ˙a(t) +r(t)(x−a(t))^⊥,∀x∈∂Ω_S(t). (1.5) We define up to a constant the angleθ associated to the rotation velocity

r= ˙θ.

The system is completed by the following initial conditions:

u(0,·) =u₀ in Ω_F(0), a(0) =a₀, a(0) =˙ a₁, θ(0) =θ₀, r(0) =r₀, (1.6) wherea₀∈R² the center of mass at initial time,θ₀∈R,u₀∈H³(Ω_F(0))²,a₁∈R² andr₀∈Rsatisfy

divu₀= 0 in Ω_F(0), u₀=a₁+r₀(x−a₀)^⊥ on∂Ω_S(0) andu₀= 0 on∂Ω. (1.7) At timet, the domain occupied by the structure Ω_S(t) is defined by

Ω_S(t) =X(t,Ω_S(0)), whereX denotes the flow associated to the motion of the structure:

X(t, y) =a(t) +R_θ(t)−θ0(y−a₀),∀y∈Ω_S(0),∀t∈(0, T). (1.8) Here,R_θ is the rotation matrix of angleθ. We have chosen to denote byy the lagrangian coordinate and byx the eulerian coordinate. We can also notice that equation (1.5) allows to extenduon the whole domain Ω. We still denoteuthe global velocity defined on the solid domain by

u(t, x) = ˙a(t) +r(t)(x−a(t))^⊥,∀x∈Ω_S(t),∀t∈(0, T).

We also extendu₀ on Ω in the same way. Thus, if we define

V ={v∈H₀¹(Ω)/divv= 0 in Ω},

then, fora.e. tin (0, T),u(t) belongs toV.

This problem satisfies ana priori estimate. Indeed, if we denoteE the global energy:

E(t) = 1 2

ΩF(t)|u(t, x)|²dx+m

2 |a(t)|˙ ²+J

2|r(t)|²+ _t

0

ΩF(t)|∇u(t, x)|²dxdt, we have

E(t)≤E(0) +C(T) _t

0

ω|f(t, x)|²dxdt.

Let us mention that [3] and [5] prove the existence of local solutions for this model (see also the references therein). In [18], a global existence result is proven: in particular, weak solutions of the fluid-structure problem are defined beyond collisions. Moreover, [19] obtains a regularity result valid as long as no collisions occur. In our study, we will need to keep this non-collision condition. We also want to avoid contact between the structure and the control domain. We consider an initial position such that

Ω_S(0)⊂Ω\ω,d

Ω_S(0), ∂

Ω\ω

>0,

∂ΩS(0)(y−a₀) dσ(y) = 0. (1.9) The last hypothesis will be necessary to obtain the Carleman inequality given in subsection 1.5. Indeed, thanks to this hypothesis, we will be able to deduce estimates for the structure velocity from estimates on the interface of the fluid velocity. It will come from the fact that, ifu= ˙a+r(x−a)^⊥ on∂Ω_S(t), we have

∂ΩS(t)|u|²=|a|˙ ²

∂ΩS(t)1 +|r|²

∂ΩS(t)|x−a|²=|a|˙ ²

∂ΩS(0)1 +|r|²

∂ΩS(0)|y−a₀|²,

thanks to the last hypothesis of (1.9). This hypothesis will be satisfied for a ball, an ellipse and more generally for any structure symmetric with respect to the center of mass.

In this paper, we will be concerned with the null controllability of the system presented above. In [6], the local null controllability is proved in dimension one for a particle evolving in a fluid modeled by Burgers equation.

This one-dimensional model has been analyzed in [21] and in [22]. Simplified problems for the interaction between an elastic structure and a fluid are studied in [16,17,23]. The controllability of Navier-Stokes equations is the subject of recent works. The methods used to deal with Navier-Stokes equations in our fluid-structure problem are essentially due to papers [10, 12].

Our article has been announced in a preprint [2]. Let us mention that a simultaneous and independent work has been achieved in [14]. Some differences can be emphasized. Indeed, in this paper, the geometry of the rigid solid is necessarily a ball while, in our paper, it only has to satisfy some symmetric hypothesis. The methods used in [14] and in our work are different even if, in the two works, the main tool is a Carleman inequality. In particular, in [14], the nonlinear problem is not proved with a compactness argument and thus initial conditions are not as regular as in our work.

Remark 1. In (1.9), we only assume that no contact occurs between the structure and the global boundary at initial time. As we will see, we keep this non-collision condition for allt∈(0, T). Indeed, if initial data are small, then the control function is also small (see Prop. 6) and thus the displacement of the structure stays small. Thus, if initial data are small enough, we then get that

d

Ω_S(t), ∂

Ω\ω

>0.

To conclude this subsection, we introduce function spaces on moving domains. In the following, for the sake of readability, we omit to indicate with respect to which variable we are integrating, except when this is not obvious.

Definition 1. We consider a domainS⊂Ω and, for eacht, the domainS(t) = Ψ(t, S)⊂Ω where Ψ : (t, y)∈(0, T)×Ω→Ω

belongs to H²(0, T;C²(Ω)) and is such that, for allt∈(0, T), Ψ(t,·) is aC²-diffeomorphism from Ω on Ω and fromS onS(t). For a functionu(t,·) :S(t)→R, we define

U(t, y) =u(t,Ψ(t, y)),∀t∈(0, T),∀y∈S.

Then, we define, for all 1≤p, q≤+∞, for allk∈N, L^p

0, T;W^k,q S(t)

=

u / U ∈L^p

0, T;W^k,q

S

, and, forl= 1,2,

W^l,p

0, T;W^k,q S(t)

=

u / U ∈W^l,p

0, T;W^k,q(S . In each space, we consider the associated norms

u _L^p_(0,T;W^k,q_(S(t)))= U _L^p_(0,T;W^k,q_(S)), u _W^l,p_(0,T;W^k,q_(S(t)))= U _W^l,p_(0,T;W^k,q_(S)). We give some useful properties satisfied by these spaces.

Proposition 1. We use the same notations and hypotheses as in Definition1.

• A functionubelongs toL^p

0, T;W^k,q S(t)

if and only if, for a.e. t∈(0, T), x→u(t, x)belongs toW^k,q

S(t) and

_T

0 u(t) ^p_Wk,qS(t))<∞. Moreover, the norm

T

0 · ^p_Wk,q(S(t))

_1/p

is equivalent to · _L^p_(0,T;W^k,q_(S(t))):

C₁ u _Lp(0,T;W^k,q(S(t)))≤ ^T

0 u(t) ^p_{Wk,q(S(t)) 1/p}

≤C₂ u _Lp(0,T;W^k,q(S(t))),

whereC₁>0 andC₂>0 depend on the norm ofΨandΨ⁻¹ inL^∞(0, T;C²(Ω)).

• If ubelongs toW^1,p

0, T;W^k,q S(t)

∩L^p

0, T;W^k+1,q S(t)

,∂_tudefined by

∂_tu(t, x) =∂_tU(t,Ψ⁻¹(t, x))−∂_tΨ(t,Ψ⁻¹(t, x))· ∇u(t, x) (1.10) belongs toL^p(0, T;W^k,q(S(t))).

1.2. Compatibility conditions on the initial data

With (1.7), we have already given compatibility conditions which have to be satisfied by our initial data. In particular, we want the velocity to be continuous on the interface at initial time. These compatibility conditions are necessary to obtain a first regularity result on the velocities of the fluid and the structure (the precise result is given below by Prop. 2). Our study will also require a second regularity result on the acceleration associated

to the fluid and structure motions (this result is given by Prop. 3). To obtain this result, we will need an additional compatibility condition expressing that the acceleration is continuous on the interface and on the global boundary at initial time. This kind of compatibility conditions appears for general classes of problems (we refer to [20] for a general theory).

First, we have to define the acceleration of the fluid and of the structure at time t = 0. They will be determined by the equations of the motion as explained in the following lemma. Since our control functionf will be null at initial time, the compatibility condition will not depend onf.

Lemma 1. Let u₀∈H³(Ω_F(0))²,a₀∈R²,a₁∈R² andr₀∈R be given. We consider the following problem

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

u₁+ (u₀· ∇)u₀−divσ(u₀, p₀) = 0 inΩ_F(0), ma₂=

∂ΩS(0)σ(u₀, p₀)n, J r₁=

∂ΩS(0)(σ(u₀, p₀)n)·(x−a₀)^⊥, divu₁= 0 inΩ_F(0),

u₁·n= 0on∂Ω, u₁·n=

a₂+r₁(x−a₀)^⊥−r²₀(x−a₀)− ∇u0

a₁+r₀(x−a₀)^⊥

·non∂Ω_S(0).

Then this problem admits a solution(u₁, p₀, a₂, r₁)∈H¹(Ω_F(0))²×H²(Ω_F(0))×R²×R. Moreover, this solution is unique (up to a constant for p₀).

Proof of Lemma 1. We define the solution (u_1,0, p_0,0) ∈ H¹(Ω_F(0))²×H²(Ω_F(0)) obtained by a Helmholtz decomposition

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

u_1,0+∇p_0,0=−(u₀· ∇)u₀+ ∆u₀ in Ω_F(0), divu_1,0= 0 in Ω_F(0),

u_1,0·n= 0 on∂Ω, u_1,0·n=

−r₀²(x−a₀)− ∇u₀

a₁+r₀(x−a₀)^⊥

·non∂Ω_S(0).

In the sequel of the proof, we will denote by x¹ and x² the coordinates of a vectorx∈R². We consider the following problems:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u_1,1+∇p0,1= 0 in Ω_F(0) divu_1,1= 0 in Ω_F(0) u_1,1·n= 0 on∂Ω u_1,1·n=n¹ on∂Ω_S(0)

,

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u_1,2+∇p0,2= 0 in Ω_F(0) divu_1,2= 0 in Ω_F(0) u_1,2·n= 0 on∂Ω u_1,2·n=n² on∂Ω_S(0)

and ⎧

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎩

u_1,3+∇p_0,3= 0 in Ω_F(0) divu_1,3= 0 in Ω_F(0) u_1,3·n= 0 on∂Ω

u_1,3·n= (x−a₀)^⊥·non∂Ω_S(0).

These three problems admit solutions in H¹(Ω_F(0))² ×H²(Ω_F(0)). We are looking for u₁, p₀, a₂ and r₁ satisfying, up to a constant forp₀,

u₁=u_1,0+a¹₂u_1,1+a²₂u_1,2+r₁u_1,3, p₀=p_0,0+a¹₂p_0,1+a²₂p_0,2+r₁p_0,3. (1.11)

Thus, the dependence ofu₁ andp₀ with respect toa¹₂,a²₂ andr₁ is affine. From this expression, we deduce the system which has to be satisfied bya₂ andr₁

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

ma¹₂=−a¹₂

∂ΩS(0)p_0,1n¹−a²₂

∂ΩS(0)p_0,2n¹−r₁

∂ΩS(0)p_0,3n¹+F₁¹, ma²₂=−a¹₂

∂ΩS(0)

p_0,1n²−a²₂

∂ΩS(0)

p_0,2n²−r₁

∂ΩS(0)

p_0,3n²+F₁², J r₁=−a¹₂

∂ΩS(0)p_0,1n·(x−a₀)^⊥−a²₂

∂ΩS(0)p_0,2n·(x−a₀)^⊥−r₁

∂ΩS(0)p_0,3n·(x−a₀)^⊥+F₂ with

F₁= 2

∂ΩS(0)(u₀)n−

∂ΩS(0)p_0,0n, F₂= 2

∂ΩS(0)((u₀)n)·(x−a₀)^⊥−

∂ΩS(0)(p_0,0n)·(x−a₀)^⊥. By noticing that, for instance,

∂ΩS(0)p_0,1n¹=

ΩF(0)|u_1,1|²,

∂ΩS(0)p_0,2n¹=

ΩF(0)u_1,1·u_1,2,

we can easily prove that, sincem >0 andJ >0, the matrix associated to this system is symmetric and definite positive. Thus, our system admits a unique solutiona¹₂,a²₂andr₁and then we deduceu₁andp₀from (1.11).

This lemma allows to define the accelerationu₁of the fluid at initial time and the acceleration of the center of massa₂ and of the angler₁ at initial time. It asserts the continuity of the normal trace of the acceleration.

In order to get the continuity of the whole trace of the acceleration, we make the following assumption on (u₁, a₂, r₀):

u₁= 0 on∂Ω, u₁=a₂+r₁(x−a₀)^⊥−r²₀(x−a₀)− ∇u₀

a₁+r₀(x−a₀)^⊥

on∂Ω_S(0). (1.12) Indeed, if we consider the expression (1.5) and we derive it with respect to time, we obtain this expression at initial time. To derive this expression, we have to be careful since the domain∂Ω_S(t) depends on time. Thus, we first have to express this equality on∂Ω_S(0) thanks to the flowX defined by (1.8). Condition (1.12) can be expressed in terms of initial data u₀,a₀,a₁ andr₀.

We make the following hypothesis foru₀,a₀,a₁,θ₀ andr₀:

u₀∈H³(Ω_F(0))², a₀∈R², a₁∈R², θ₀∈Randr₀∈R,

divu₀= 0 in Ω_F(0), u₀=a₁+r₀(x−a₀)^⊥ on∂Ω_S(0) and u₀= 0 on∂Ω, (u₁, a₂, r₁) defined by Lemma 1 satisfy (1.12).

⎫⎪

⎬

⎪⎭ (1.13)

1.3. Main result

We introduce the notion of controllability:

Definition 2. We will say that our problem isnull controllableat time T if there exists a control function f ∈L²((0, T)×ω)²such that

u(T,·) = 0 in ΩF(T), a(T) = 0,a(T˙ ) = 0, θ(T) = 0, r(T) = 0, (1.14) or, equivalently,

u(T,·) = 0 in Ω, a(T) = 0, θ(T) = 0,

where (u, a, θ) is the solution, together with a pressure p, of the problem defined by equations (1.1) to (1.6).

Thus, we want to drive the fluid and the structure at rest and we also want the structure to be located in the reference configurationR_−θ0(Ω_S(0)−a₀). The main result of this article is:

Theorem 1. We suppose that u₀, a₀, a₁, θ₀ and r₀ satisfy (1.13) and we consider an initial structure do- main Ω_S(0) such that(1.9) is satisfied. LetT >0 be a fixed final time. Then, there exists ε >0 depending on T and on the domains Ω,ω andΩ_S(0) such that, if

u_{0 H}³_(ΩF(0))²+|a₀|+|a₁|+|θ₀|+|r₀| ≤ε, the problem defined by equations (1.1)to(1.6)is null controllable at timeT.

Remark 2. We can also consider N structures occupying the domains Ωⁱ_S(t), 1 ≤ i ≤ N, immersed in the fluid. The two equations for the structure motion are replaced by 2N equations for the translationa_i and the rotation velocity r_i associated to the i-th solid. Each structure has to satisfy (1.9) and we also have to avoid contact between two different structuresi.e.

d

Ωⁱ_S(0),Ω^j_S(0)

>0,∀1≤i, j≤N.

Then we can prove that the same Carleman inequality (1.30) holds for the structure domain Ω_S(t) defined by Ω_S(t) =

1≤i≤N

Ωⁱ_S(t) and we can obtain the same local null controllability result.

Remark 3. By standard arguments in controllability, we can prove that this result also holds for a control domain located on the boundary of the cavity Ω.

To begin with, we will prove a controllability result on a linearized problem. Let (˜a,r) be given in˜ H²(0, T)²× H¹(0, T). We define ˜θ the angle associated to the rotation velocity ˜r defined up to a constant. Thus, for any t∈(0, T), the structure domainΩ_S(t) is defined by

Ω_S(t) =X(t, Ω_S(0)), (1.15)

whereX denotes the flow associated to the structure velocity and is defined by

X(t, y) = ˜a(t) +R_{θ(t)−θ˜ 0}(y−a₀),∀t∈(0, T),∀y∈Ω_S(0). (1.16)

We assume that ˜aand ˜θsatisfy

˜

a(0) =a₀,a(0) =˙˜ a₁,θ(0) =˜ θ₀, ˜r(0) =r₀,Ω_S(t)⊂Ω\ω,d

Ω_S(t), ∂

Ω\ω

≥α,∀t∈[0, T], (1.17) where α >0 is a fixed real number small enough. The last two properties are satisfied at time t= 0 because X(0,·) = Id in Ω_S(0) and we have supposed that Ω_S(0) satisfies (1.9). We can also define the corresponding fluid domain by

Ω_F(t) = Ω\Ω_S(t).

Next, let ˜ube given such that

˜

u∈L^∞(0, T;L^∞(Ω_F(t)))²∩W^1,4(0, T;L⁴(Ω_F(t)))²∩L^∞(0, T;H¹(Ω_F(t)))², (1.18) div ˜u= 0 inΩ_F(t),u˜= ˙˜a+ ˜r(x−a)˜ ^⊥ on∂Ω_S(t),u˜= 0 on∂Ω, (1.19)

˜

u(t= 0) =u₀ in Ω_F(0). (1.20)

As for the velocityu, we can extend ˜uonΩ_S(t) by the velocity of the structure.

We will say that (u, p, a, r) is a solution of the linearized problem around (˜u,˜a,˜r) if and only if, for allt∈(0, T),

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

∂_tu+ (˜u· ∇)u

(t, x)−divσ(u, p)(t, x) =f(t, x)1_ω(x),∀x∈Ω_F(t), m¨a(t) =

∂ΩS(t)σ(u, p)n, Jr(t) =˙

∂ΩS(t)(σ(u, p)n)·(x−a(t))˜ ^⊥, divu(t, x) = 0, ∀x∈Ω_F(t),

u(t, x) = 0,∀x∈∂Ω,

u(t, x) = ˙a(t) +r(t)(x−˜a(t))^⊥,∀x∈∂Ω_S(t),

u(0,·) =u₀ in Ω_F(0), a(0) =a₀,a(0) =˙ a₁, θ(0) =θ₀, r(0) =r₀.

(1.21)

We easily obtain an a priori energy estimate for this problem. Indeed denotingE(t) the global energy:

E(t) = 1 2

ΩF(t)|u(t, x)|²dx+m

2 |a(t)˙ |²+J

2|r(t)|²+ _t

0

ΩF(t)|∇u(t, x)|²dxdt, we have

E(t) ≤E(0) +C(T) _t

0

ω|f(t, x)|²dxdt.

It seems worth noting that, in order to have an energy estimate for the linearized problem, the given velocities ˜u, a˙˜and ˜rhave to satisfy continuity and divergence-free conditions (1.19). Since the trace of ˜uhas to be defined, we have taken ˜uinL^∞(0, T;H¹(Ω_F(t)))².

First of all, we will prove a controllability result for this linearized problem. The result is formulated as follows:

Theorem 2. We consider initial data u₀∈H¹(Ω_F(0))²,a₀∈R², a₁∈R², θ₀∈Randr₀∈R satisfying(1.7) and an initial structure domain Ω_S(0) such that(1.9) is satisfied.

Let T >0be a fixed final time. We suppose that(˜a,˜r)∈H²(0, T)²×H¹(0, T)are such that (1.17)holds for someα >0and thatu˜satisfies conditions(1.18) to(1.20). Then, problem(1.21)is null controllable at timeT. To prove the controllability result for the linearized problem, we need to introduce the homogeneous adjoint problem. It is defined by the following system, for allt∈(0, T)

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

−∂_tv−(˜u· ∇)v

(t, x)−divσ(v, q)(t, x) = 0,∀x∈Ω_F(t), m¨b(t) =−

∂ΩS(t)σ(v, q)n, Jγ(t) =˙ −

∂ΩS(t)(σ(v, q)n)·(x−˜a(t))^⊥, divv(t, x) = 0,∀x∈Ω_F(t),

v(t, x) = 0,∀x∈∂Ω,

v(t, x) = ˙b(t) +γ(t)(x−˜a(t))^⊥,∀x∈∂Ω_S(t),

v(T,·) =v₀^T in Ω_F(T), b(T) = 0,b(T˙ ) =b^T₁, γ(T) =γ₀^T.

(1.22)

The initial datav₀^T ∈H¹(Ω_F(T))², b^T₁ andγ₀^T satisfy

v₀^T =b^T₁ +γ₀^T(x−˜a(T))^⊥ on∂Ω_S(T), v^T₀ = 0 on∂Ω and divv^T₀ = 0 inΩ_F(T). (1.23)

1.4. Extension of the structure flow

We have already introduced the definition of the structure flow by (1.16). In the following, we will need to extend this flow up to the global boundary∂Ω by a regular and incompressible flow. To construct this extension, conditions of non-collision between the structure and the boundary of Ω have to be satisfied. According to condition (1.17), we have Ω_S(t)⊂(Ω\ω)_α, for each t ∈[0, T] where we have denoted, for a subset A ofR², A={x∈A/d (x, ∂A)≥}. We have the following result:

Lemma 2. Let (˜a,r)˜ ∈H²(0, T)²×H¹(0, T)be given. We define Ω_S(t) by(1.15) and we suppose that (1.17) is satisfied for someα >0. We can extend the velocity

˙˜

a+ ˜r(x−˜a)^⊥

defined on Ω_S(t)by a velocity u˜_S ∈H¹(0, T;C²(Ω))² satisfying, for allt∈(0, T) div ˜u_S = 0 inΩ,

˜

u_S = 0inΩ\(Ω\ω)_α/4,u˜_S = ˙˜a+ ˜r(x−a)˜ ^⊥ in(Ω\ω)_α/2, and such that

˜

u_{S H}1(0,T;C²(Ω))²≤C( a ˙˜ _H¹_(0,T)²+ ˜r _H¹_(0,T)), (1.24) whereC depends on T andα.

We do not detail how we obtain this incompressible velocity which extends the velocity defined on the structure: we refer to [19] for the proof of this result. We define the flow associated to ˜u_S. We still denote itX since it extends the flow defined on the structure by (1.16).

Lemma 3. Under the same hypotheses as in Lemma2, the flowX associated tou˜_S defined in Lemma2satisfies:

• for each t∈[0, T],X(t,·)is a C²-diffeomorphism from Ωon Ωand from Ω_F(0)on Ω_F(t). We denote by Y(t,·)the inverse of X(t, ·)defined on Ω;

• X andY belong toH²(0, T;C²(Ω))²;

• ∀(t, y)∈(0, T)×Ω,det∇X(t, y) = 1;

• ∀t∈(0, T),∀y∈Ω\(Ω\ω)_α/4,X(t, y) =y;

• ∀t∈(0, T),∀y∈Ω_S(0) +B(0, α/2),X(t, y) = ˜ a(t) +R_{θ(t)−θ˜ 0}(y−a₀), whereB(0, α/2) denotes the ball of center 0 and of radiusα/2. Moreover, we have

X _H²_(0,T;C²_(Ω))²+ Y _H2(0,T;C²(Ω))²≤C( ˜a _H²_(0,T)²+ ˜r _H¹_(0,T)), where the constant C depends onT andα.

Proof of Lemma3. Thanks to the regularity of ˜u_Sobtained in Lemma 2 and the properties of the flow associated to a velocity, we easily obtain the first three points of the lemma.

Now, on Ω\(Ω\ω)_α/4, since ˜u_S = 0, we have that X(t, ·) = Id. Moreover, for each t ∈ (0, T), for each y∈Ω_S(0) +B(0, α/2), we have

˜

a(t) +R_{θ(t)−θ˜ 0}(y−a₀)∈Ω_S(t) +B(0, α/2)⊂(Ω\ω)_α/2.

Consequently, by uniqueness of the flow, the last point of the lemma is satisfied.

Remark 4. If ˜a belongs to W^1,∞(0, T)² and ˜r belongs to L^∞(0, T), Lemmas 2 and 3 still hold with the appropriate changes (the flows belong to W^1,∞(0, T;C²(Ω))².

1.5. A Carleman inequality

To obtain our controllability result, we prove a Carleman inequality result for the adjoint system (1.22).

We consider a nonempty open set ω₀ such that ω₀ ⊂⊂ ω (i.e. ω₀ ⊂ ω). We will introduce time-dependent weight functions defined on the moving domainΩ_F(t). First of all, we consider a steady weight functionβ₀in C²

Ω_F(0)

depending on Ω,ω₀ and Ω_S(0) such that β₀= 0 on∂Ω∪∂Ω_S(0), β₀>0 in Ω_F(0),

∇β₀·n≤c₁<0 on∂Ω,∇β₀·n≥c₂>0 on∂Ω_S(0), |∇β₀|>0 in Ω_F(0)\ω₀.

On the boundary of Ω, the vector n is the outward unit normal to Ω and on the boundary of the structure domain,nis the outward unit normal to the structure domain (and thus the inward normal to the fluid domain).

For the proof of this result, we refer to [11]. We suppose that (1.17) holds for someα >0. Then, thanks toβ₀, we define the time-dependent weight functionβ which follows the displacement of the structure by

β(t, x) =β₀(Y(t, x)),∀x∈Ω_F(t),∀t∈(0, T), whereY is defined by Lemma 3.

Lemma 4. The functionβ belongs toL^∞(0, T;W^2,∞(Ω_F(t)))∩W^1,∞(0, T;W^1,∞(Ω_F(t))) and is such that:

β= 0 on∂Ω∪∂Ω_S(t), β >0 inΩ_F(t),

∇β·n≤c₁<0on ∂Ω,∇β·n≥c₂>0 on ∂Ω_S(t),|∇β|>0 inΩ_F(t)\ω₀. (1.25) Moreover, we have the following estimate:

β _L∞(0,T;W^2,∞(ΩF(t)))+ β _W1,∞(0,T;W^1,∞(ΩF(t)))≤C, (1.26) whereC depends on T andα.

To introduce the Carleman inequality satisfied by a solution of the adjoint linearized problem (1.22), we define, forλ≥1, the functionsV andϕby: ∀t∈(0, T),∀x∈Ω_F(t),

V(t, x) = e^10λM −e^{λ(8M+β(t,x))}

t⁴(T−t)⁴ , ϕ(t, x) = e^{λ(8M+β(t,x))}

t⁴(T−t)⁴ , (1.27)

where M = β_{0 L}∞(ΩF(0)). For this choice of M, we can already notice that V is a positive function since β _{L∞(0,T;L∞(Ω}

F(t)))= β_{0 L}∞(ΩF(0)). Moreover,V andϕhave the following properties:

∇V=−λϕ∇β, ∇ϕ=λϕ∇β.

We also define,∀t∈(0, T), Vˆ(t) = inf

x∈ΩF(t)V(t, x) =e^10λM−e^9λM

t⁴(T −t)⁴ ,V^∗(t) = sup

x∈ΩF(t)

V(t, x) = e^10λM −e^8λM

t⁴(T −t)⁴ , (1.28) ˆ

ϕ(t) = sup

x∈ΩF(t)

ϕ(t, x) = e^9λM

t⁴(T−t)⁴, ϕ^∗(t) = inf

x∈ΩF(t)ϕ(t, x) = e^8λM

t⁴(T−t)⁴· (1.29)

Then, the following global Carleman estimate for problem (1.22) holds:

Theorem 3. Let ˜a∈H²(0, T)², ˜r∈H¹(0, T)and u˜ be given such that (1.17) holds for some α >0 and such that conditions (1.18) to(1.20)are satisfied.

Then, there exists a constantC and two constantssˆandλˆ such that, for everyv^T₀ ∈ L²(Ω_F(T))²,b^T₁ ∈R², s^T₀ ∈R, the corresponding solution(v, q, b, γ)of (1.22)satisfies, for any s≥sˆandλ≥λ,ˆ

_T

0

ΩF(t)e^−2sV 1

sϕ

|∆v|²+|∂_tv|²

+sλ²ϕ|∇v|²+s³λ⁴ϕ³|v|²

+sλ _T

0 e^−2sV∗ϕ^∗¨b²+|γ|˙ ² +s³λ³

_T

0

∂ΩS(t)

e^−2sV∗(ϕ^∗)³|v|²+sλ _T

0

∂ΩS(t)

e^−2sV∗ϕ^∗|∇v n|²

≤Cs^19/2λ¹³ _T

0

ωe^{2sV∗−4sVˆ}ϕˆ¹⁰|v|². (1.30) The constantC only depends onT,αandβ₀, andsˆandλˆ depend onT,α,β₀ and the norm of˜ainH²(0, T)²,

˜

r inH¹(0, T)andu˜ inL^∞((0, T)×Ω)²∩W^1,4(0, T;L⁴(Ω_F(t)))².

Remark 5. In this work, we suppose that the viscosity µ is equal to 1. It can be interesting to wonder how the constant in our Carleman inequality depends onµifµis not fixed. It is known that the constant in global Carleman inequalities for parabolic equations behaves as exp(C/T), whereC >0 is a constant depending on the domain andT >0 is the length of the time interval (see for instance [9]). Let us consider the heat equation

u_t−µ∆u=f in (0, T),

whereµ >0 and make the change of variablesτ=µ tthen we retrieve a heat equation withµ= 1

u_τ−∆u=fin (0, µ T),

whereu(τ) =u(τ /µ),f(τ) =f(τ /µ)/µ, and therefore, with the classical computations, we find that the constant in the global Carleman inequality is of order exp(C/(µ T)).It has been also shown that this constant is optimal for the observability inequality at least in one dimension (see [4]). In our case, the situation is essentially the same.

The proof of this theorem will be given in Section 2, but before, we will study some regularity properties which will be useful in the sequel.

1.6. Regularity results on the linearized problem

We give regularity results for the following non-homogeneous linearized system associated to (1.21): for all

t∈(0, T), ⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

∂_tu+ (˜u· ∇)u

(t, x)−divσ(u, p)(t, x) =g_F(t, x),∀x∈Ω_F(t), m¨a(t) =

∂ΩS(t)σ(u, p)n+g_T(t), Jr(t) =˙

∂ΩS(t)(σ(u, p)n)·(x−˜a(t))^⊥+g_R(t), divu(t, x) = 0,∀x∈Ω_F(t),

u(t, x) = 0,∀x∈∂Ω,

u(t, x) = ˙a(t) +r(t)(x−˜a(t))^⊥, ∀x∈∂Ω_S(t),

u(0,·) =u₀in Ω_F(0), a(0) =a₀, a(0) =˙ a₁, θ(0) =θ₀, r(0) =r₀,

(1.31)

whereg_F is the force acting on the fluid andg_T andg_Rare the force and the torque acting on the structure.

Of course, these results are also true for the linear adjoint system and can be shown in the same way. In [19], the first regularity result is proved for the nonlinear fluid-structure direct problem. Thus, the proposition which follows is a result contained in [19]. We only give a sketch of the proof and we refer to [19] and the references therein for complementary explanations. Let us define

U(0, T; Ω) =L²(0, T;H²(Ω))∩H¹(0, T;L²(Ω))∩L^∞(0, T;H¹(Ω)).

Proposition 2. Let initial data u₀ ∈ H¹(Ω_F(0))², a₀ ∈ R², a₁ ∈ R², θ₀ ∈ R, r₀ ∈ R and forces g_F ∈ L²(0, T;L²(Ω_F(t)))², g_T ∈L²(0, T)²,g_R ∈L²(0, T) be given. We suppose that initial data satisfy (1.7), that

˜

u∈L^∞(0, T;L^∞(Ω_F(t)))²∩L^∞(0, T;H¹(Ω_F(t)))²,(˜a,˜r)∈W^1,∞(0, T)²×L^∞(0, T) satisfy (1.19) and (1.17) for someα >0. Then, the system (1.31)admits a unique solution

u∈ U(0, T;Ω_F(t))², p∈L²(0, T;H¹(Ω_F(t))), a∈H²(0, T)², r∈H¹(0, T).

Moreover, we have the estimate

u _{U(0,T;ΩF(t))}2+ p _L2(0,T;H¹(ΩF(t)))+ a _H²_(0,T)²+ r _H¹_(0,T)

≤C

(u0, a₁, r₀) _H¹_(ΩF(0))²_×R²_×R+ gF L²(0,T;L²(ΩF(t)))²+ gT L²(0,T)²+ gR L²(0,T) ,

where the constant C depends on T, α, the norm of u˜ in L^∞((0, T)×Ω)² (and thus on the norm of (˜a,r)˜ in W^1,∞(0, T)²×L^∞(0, T)).

Proof of Proposition 2. This result is obtained by doing a change of variables to come back to initial configu- rations Ω_F(0) and Ω_S(0). Thanks to Lemma 3 (see Rem. 4), we define the flowsX andY. Let us define the new variables

U(t, y) =∇Y(t,X(t, y))u(t,X(t, y)), P(t, y) =p(t,X(t, y)), A(t) = _t

0 R_θ0−θ(t˜ )a(t˙ ) dt. It can be proved (see [19]) that (u, p, a, r) is a solution of (1.31) if and only if (U, P, A, r) satisfies

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

∂_tU−[LU] + [M U] + [N U,U] + [GP] =G_F in (0, T)×Ω_F(0), mA¨=

∂ΩS(0)σ(U, P)n+G_T +m˜rA˙^⊥ in (0, T), Jr˙=

∂ΩS(0)(σ(U, P)n)·(y−a₀)^⊥+G_R in (0, T), divU = 0 in (0, T)×Ω_F(0),

U = 0 on (0, T)×∂Ω,

U = ˙A+r(y−a₀)^⊥ on (0, T)×∂Ω_S(0),

U(0,·) =u₀ in Ω_F(0), A(0) = 0,A(0) =˙ a₁, r(0) =r₀,

(1.32)

where we have defined

G_F(t, y) =∇Y(t,X(t, y)) g_F(t,X(t, y)), G_T(t) =R_θ

0−θ(t)˜ g_T(t), G_R(t) =g_R(t),U(t, y) =∇Y(t,X(t, y)) ˜u(t,X(t, y)).