Local null controllability of a fluid-rigid body interaction problem with Navier slip boundary conditions

The aim of this work is to show the local null controllability of a fluid-solid interaction system by using a distributed control located in the fluid. The fluid is modeled by the incompressible Navier-Stokes system with Navier slip boundary conditions and the rigid body is governed by the Newton laws. Our main result yields that we can drive the velocities of the fluid and of the structure to 0 and we can control exactly the position of the rigid body, provided that its shape is not a disk. One important ingredient consists in a new Carleman estimate for a linear fluid-rigid body system with Navier boundary conditions.


Introduction
Let Ω be a bounded, non empty open subset of R 2 with a regular boundary. We assume that Ω contains a rigid body and an incompressible viscous fluid. At each time t > 0, the domain of the rigid body is denoted by S(t) ⊂ Ω that is assumed to be compact with non empty interior and regular. The fluid domain is denoted by F(t) = Ω\S(t), and is assumed to be connected.
We consider the following system describing the evolution of the fluid which is governed by the incompressible Navier-Stokes system ∇ · U = 0 in (0, T ) × F(t).
where ν is the viscosity of the fluid. We denote for each time t, the position of the structure by h(t) ∈ R 2 and by R θ(t) the rotation matrix of angle θ of the solid defined by R θ(t) = cos θ(t) − sin θ(t) sin θ(t) cos θ(t) .
Then, the flow of the structure is given by X S (t, ·) : S −→ S(t) where X S (t, y) = h(t) + R θ(t) y, t ∈ (0, T ), y ∈ S, (1.2) where S is a fixed subset of R 2 , non empty, compact with a regular boundary. We notice that X S (t, ·) is invertible and a C ∞ -diffeomorphism, we denote its inverse by Y S (t, ·) : Thus, the Eulerian velocity of the structure is given by We denote by a ⊥ , the vector −a 2 a 1 , for any a = a 1 a 2 ∈ R 2 . We notice that R θ(t) R −1 θ(t) is a skew-symmetric matrix, then the Eulerian velocity of the structure writes where ω(t) = θ (t) represents the angular velocity of the rigid body.
We denote by S h,θ the set S h,θ = h + R θ S, and we define the corresponding fluid domain for any h ∈ R 2 , θ ∈ R. Then, with these notations, we have We point out that the fluid domain is depending on the displacement of the solid structure, consequently, it depends on time.
The motion of the structure is governed by the balance equations for linear and angular momenta mh (t) = − ∂S(t) T(U, P ) n dΓ t ∈ (0, T ), Jω (t) = − ∂S(t) (x − h(t)) ⊥ · T(U, P ) n dΓ t ∈ (0, T ). (1.3) We complete (1.1) and (1.3) by the Navier slip boundary conditions. In order to write these boundary conditions, we need to introduce some notations. Let τ be a tangent vector to ∂F(t). We denote by a n and a τ the normal and the tangential parts of a ∈ R 2 : a n = (a · n) n, a τ = a − a n .
Then, the boundary conditions write as follows where β Ω 0 and β S 0 are the friction coefficients.
Let h 0 , 0 ∈ R 2 , θ 0 , ω 0 ∈ R and u 0 ∈ [H 1 (F h 0 ,θ 0 )] 2 . We furnish the following initial conditions such that the following compatibility conditions are satisfied where u 0 S (x) = 0 + ω 0 (x − h 0 ) ⊥ . Without loss of generality, we assume that the center of gravity of S is at the origin. Then, h(t) will be the position of the center of mass of the rigid body S(t).
Our main objective in this paper is to look for a control v * acting on O such that for any (h T , θ T ) ∈ R 2 × R with we get that h(T ) = h T , θ(T ) = θ T and the velocities of the fluid and of the rigid body are equal to 0 at time T . The main result of this paper is stated below: Theorem 1.1. Assume that β S > 0 and let (h T , θ T ) that satisfies (1.7). Then, there exists ε > 0 such that for any (u 0 , h 0 , 0 , ω 0 , θ 0 ) that satisfies (1.6) and there exists a control v * ∈ L 2 (0, T ; [L 2 (O)] 2 ) such that U (T, ·) = 0 in F h T ,θ T , h(T ) = h T , h (T ) = 0, ω(T ) = 0, θ(T ) = θ T .
Without loss of generality, we can always assume that h T = 0, θ T = 0, and thus S h T ,θ T = S, F h T ,θ T = F.
In fact: in general, we have X S (t, y) = h(t) + R θ(t)−θ T (y − h T ), t ∈ (0, T ), y ∈ S. and in this case, we set Then, we notice that we are reduced to the case (1.2). Thus, by translation of vector −h T and rotation of angle −θ T , one can reduce the controllability problem to the case h T = 0 and θ T = 0. In what follows, the vectors n and τ stand respectively for the outer unit normal and the unit tangent vector to ∂F. Several works were devoted to the study of fluid-rigid body interaction systems, in particular, when the fluid is governed by the Navier-Stokes system. Existence results concerning this kind of systems with Dirichlet boundary conditions were considered in [9,12,13,20,24,[28][29][30] etc. For the case of the Navier slip boundary conditions (1.4), the existence of weak solutions is proved in [18] and the existence of strong solutions is obtained in [31]. In [19,31], the authors proved that collisions can occur in final time between the rigid body and the domain cavity with some assumptions on the solid geometry.
Concerning the controllability, let us mention [16,25], where the authors obtained the local exact controllability of the 2D or 3D Navier-Stokes equations with Dirichlet boundary conditions considering distributed controls. The local exact controllability of the Navier Stokes system with nonlinear Navier boundary conditions with distributed controls was studied in [22]. Moreover, in [23], the authors established the local controllability with N − 1 scalar controls. With Navier-slip conditions on the fluid equations, global null controllability is obtained for the weak solution in [11] such that the controls are only located on a small part of the domain boundary. Concerning controllability results of fluid-structure systems with Dirichlet boundary conditions, in dimension 2, we mention the paper [7], where the authors proved the null controllability in velocity and the exact controllability for the position of the rigid body assuming some geometric properties for the solid and provided that the initial conditions are small enough, more precisely a condition of smallness on the H 3 norm of the initial fluid velocity is needed. The authors used the Kakutani's fixed point theorem to deduce the null controllability of the nonlinear system. We have also the paper [26] where the authors considered the structure of a rigid ball, their result relies on semigroup theory. In the latest paper, only an assumption on the H 1 norm of the initial fluid velocity is needed. In dimension 3, we mention [6], the same result was proved without any assumptions on the solid geometry while a condition of smallness on the H 2 norm of the initial fluid velocity is needed. We also mention [27], where the authors considered the interaction between a viscous and incompressible fluid modeled by the Boussinesq system and a rigid body with arbitrary shape, they proved null controllability of the associated system. In the case of the stabilization of fluid-solid ineraction systems, we have [2,3].
In this paper, we prove the local null controllability of the system (1.1), (1.3), (1.4), (1.5), that is the case of the Navier slip boundary conditions in the presence of a rigid structure of arbitrary shape. We follow the same method as [26]: we use a change of variables to write our system in a fixed domain and use a fixed point argument to reduce our problem to the null controllability of a linear fluid-rigid body system, that is coupling the Stokes system with ODE for the structure velocity. To do this we derive a Carleman estimates for the corresponding system.
One of the main difficulties to obtain such an estimate is to manage the boundary conditions and more precisely to obtain estimates of the rigid velocity with the good weights. An important step for this calculation is a Carleman estimates for the Laplacian equation with divergence free condition and Navier slip boundary conditions, which is given in Section 4. We emphasize that this is the first result concerning the null controllability of a fluid-structure interaction system with boundary conditions different from the standard no-slip ones. Note that with the Navier boundary conditions considered here, one of the additional difficulties with respect to the Dirichlet boundary conditions lies on the fact that in the Carleman estimate, it is more complictaed to estimate the structure velocities from the fluid velocity. There are several possible extensions to this work. First let us recall that in [11], the authors obtain the global exact controllability of the Navier-Stokes system with Navier boundary conditions. One of their ingredients is to use the local exact null controllability of [22]. Here, one can also consider the global exact controllability but the arguments of [11] may be difficult to adapt due to the presence of the structure. Second, one can also consider a heat conducting fluid and remplace the Navier-Stokes system by the Boussinesq system. This has been done for instance in [27] with a rigid body and Dirichlet boundary conditions. Our method here should be adapted to this case and we would obtain a similar result. Finally, one can try to reduce the number of controls as it is done in [23] for the Navier-Stokes system with Navier boundary conditions. However, let us note that due to the presence of the structure velocities in the boundary conditions, some parts of the proof in [23], might be difficult to adapt, mainly the manipulation of the curl of the fluid velocity on the boundary.
The outline of this paper is as follows: in Section 2, we give some preliminaries. We emphasize that one of the main difficulties in this problem is that we are dealing with a coupled system set on a non cylindrical domain. Then, in Section 3, we remap the problem into an equivalent system given in a fixed geometry. In Section 5, we establish a new carleman inequality. In Section 6, we prove the null controllability of the linearized system. Finally, in Section 7, we prove Theorem 1.1 and deduce the null controllability of the system by applying a fixed-point argument.

Preliminaries
In this section, we prove some regularity results of an associated linearized problem. We consider the following linear system where w S (y) = w + k w y ⊥ , completed with the initial conditions We have the following regularity result for the system (2.1), (2.2) and (2.3) which is proved in [31].
Then, there exists a unique solution to problem (2.1), (2.2) and (2.3) such that Moreover, it satisfies the following estimate Proof. The proof of the above theorem is based on semigroup theory. For the sake of completeness, we just recall the main ideas of the proof. We note that w 0 and w are extended by 0 w + k 0 w y ⊥ and w + k w y ⊥ on S respectively. Let define the following Hilbert spaces We notice that the condition D(w) = 0 on S is equivalent to w = w S ∈ R on S where For w, v ∈ H, we define the inner product on H by Let define also the orthogonal projector P : [L 2 (Ω)] 2 −→ H.
The system (2.1), (2.2) and (2.3) can be reduced to the following form where the operator A is defined by In Lemma 3.1 of [31], it is proved that the operator A is self-adjoint and it generates a semigroup of contractions on H. Thus, we deduce Theorem 2.1 (see [31], Prop. 3.3).
We note here that since A is a self-adjoint operator, then for any w ∈ D(A), we have We also need some regularity results on the linear system (2.1), (2.2), (2.3).

Change of variables
To treat the free boundary problem (1.1), (1.3), (1.4), (1.5), we consider an equivalent system written in a fixed domain using a change of variables that was already introduced in [29]. In fact, we construct an extension of the structure flow (1.2) over Ω by a regular and incompressible flow. First, we need to control the distance between the structure and the boundary ∂(Ω\O).
The condition (1.7) implies that there exists d > 0 such that Then, we get Thus, we obtain In other words, we only assume that no collision occurs between the structure and the boundary ∂(Ω\O) at time T . In fact, if the initial data are small enough, then the displacement of the structure remains small, then (3.2) is satisfied. Thus, no contact can occur between the solid and the boundary for any t ∈ [0, T ]. Following [29], we can construct a change of variables X and Y with the following properties -For any t ∈ [0, T ], X and Y are C ∞ diffeomorphisms from Ω into itself, -The function X is invertible of inverse Y , -In a neighborhood of S = S(T ), X(t, y) = X S (t, y) = h(t) + R θ(t) y, -In a neighborhood of ∂Ω and of O, X(t, y) = y, -det ∇X(t, y) = 1, for all y ∈ Ω, -In a neighborhood of S, ∇X(t, y) = R θ(t) and ∇Y (t, X(t, y)) = R −1 θ(t) . Moreover, we have where C depends on T . Now, we set u(t, y) = Cof(∇X(t, y)) * U (t, X(t, y)), P (t, y) = p(t, X(t, y)).
Then, we have where n and τ respectively stand for the normal and the tangential vectors on ∂F, with Finally, we set the initial conditions for y ∈ F

Carleman estimate for the Laplacian problem with Navier slip boundary conditions
We prove first, a Carleman inequality for the Laplacian problem with non-homogeneous Navier boundary conditions. From Lemma 1.1 of [8], we can construct a function η ∈ C 2 (F) such that Let λ > 0 and let take α = e λη . We have the following proposition.
satisfies the inequality for any s s 1 and λ λ for a complete proof.
Proof. The proof is inspired from [22] where in our case, we need to take into account the non homogeneous Navier slip boundary conditions and thus, one need to manipulate carefully the surface integrals that appear.
We notice that Using the inequality in Theorem II.4.1 of [17] with r = 2, q = 2, we obtain Applying the same arguments, we get for I 3 Then, combining all these inequalities, we get We recall that Since ψ is divergence free, we have that We have used the fact that for any scalar function a : We recall the Green formula Thus, the last term in the right hand side of the inequality (4.32) gives To adsorb the second term of the right hand side, we proceed like ( [15], inequality (1.62)) which shows that the integral of e 2sα α 2 |∇ψ| 2 over O η can be estimated by e 2sα α 4 |ψ| 2 over a larger set O. Indeed, we define θ ∈ C 2 0 (O) such that θ ≡ 1 in O η and 0 θ 1. We obtain

Carleman estimate for the linearized system
We consider the following adjoint system Let η ∈ C 2 (F) which verifies (4.1) with O η ⊂⊂ O a non empty open set. Let λ > 0 and with N > 0 an integer number to be defined later on.
Using Theorem 2.1, we have Step 2: In this part, we are going to obtain a Carleman estimate for the system (5.8) by following the proof in [6]. However, we need to deal with the Navier boundary conditions (5.9). We apply the curl operator to the first equation of (5.8) in order to eliminate the pressure, to get We obtain a one dimensional heat equation. We recall that We apply Proposition A.1 replacing ψ by ∇ × ϕ. We get Arguing as pages 7-8 of [6], we treat the local terms appearing in the right hand side of (5.13), we obtain for λ C and s C(T N + T 2N ). We notice that ϕ satisfies the following problem where we have used that a = ϕ S 1 ∂S and b = β S (ϕ S ) τ 1 ∂S . We replace s in (5.16) by se 2N λ η L ∞ (Ω) and integrating over (0, T ), we get Applying the estimates obtained in Theorem 2.2 of [1], we get Then, we multiply (5.18) by s 3 λ 4 e −2s β (ξ * ) 3 , we get Adding (5.19), (5.17) and (5.14), we deduce Taking (s, λ) large enough, the fifth term in the right hand side of (5.20) can be transported to the left side. Indeed, since ϕ S is rigid, from Lemma 2.2 of [27], we have F |ϕ(t, ·)| 2 dy C ϕ S (t) · n 2 H 3/2 (∂S) , for any shape of the body S. Moreover, we have the following relation ∇ × ϕ = (∇ϕn) · τ − ((∇ϕ) * n) · τ, on ∂F, (5.22) where τ = −n 2 n 1 . In the other hand, we have Using the boundary conditions (5.9), we can write β S (ϕ S ) τ = ν (∇ϕn + (∇ϕ) * n) τ + β S ϕ τ , on ∂S.
Step 3: Now, it remains to treat the two terms Using (5.22), (5.23), and the fact that we get, Then, we have It implies Then The second and the third term in the right hand side of the above inequality can be absorbed using (5.27) by the left side of the inequality (5.32). To absorb the first term in the right hand side of the inequality (5.36), we use the elliptic estimate of the system (5.15), we obtain The terms in the right hand side of (5.37) can be absorbed by the left hand side of (5.32), moreover the last term in the right side of (5.36) can be manipulated as (5.27) and thus, it can be absorbed by the left side of (5.32). To estimate the second term in (5.33), observe that from (5.34) and (5.35), we have We take ζ 2 (t) = s −1/2 λ −1/2 e −sβ(t) (ξ * ) −1/2 (t) and let consider the system with the boundary condition We notice that in the above system all final conditions are equal to zero, then all the compatibility conditions mentioned in Proposition 2.2 are satisfied. To absorb the last two terms in the right hand side of (5.38), we use L 2 regularity results of the system satisfied by ζ 2 ϕ. In fact, we have We note that |ζ 2 | Cs 1/2 λ −1/2 (ξ * ) 1/2+1/N e −s β , |ζ 2 ρ | Cs 1/2 λ −1/2 (ξ * ) 1/2+1/N e −s β ρ.
Using the trace theorem, we have Let estimate the terms in the right hand side of (5.44). We have The terms appearing in the right hand side of (5.45) can be absorbed by the left hand side of the Carleman inequality (5.32). In the other hand, we have and By an interpolation argument, we get We rewrite the right hand side of the inequality (5.46) to obtain Applying again Young's inequality, we get for N 4 The first term in the right hand side of (5.48) can be absorbed by the left hand side of the Carleman inequality (5.32) while the second term is absorbed by the left hand side of (5.44).
Using again an interpolation argument, we obtain similarly, The left hand side of (5.49) can be rewritten as Then, for N 4 we get The first term in the right hand side of (5.51) can be absorbed by the left hand side of the Carleman inequality (5.32) while the second term is absorbed by the left hand side of (5.44).
In the other hand, that can be rewritten as Then, for N 4, we get The first term in the right hand side of (5.54) can be absorbed by the left hand side of the Carleman inequality (5.32) while the second term is absorbed by the left hand side of (5.44). On the other hand, we notice that |ζ 2 ρ | Cs 1/2 λ −1/2 e −s β (ξ * ) 1/2+1/N ρ and we get as for (5.38) Using the decomposition (5.7) and the regularity estimate (5.11), we deduce from the above inequality The first and the second term in the right hand side of (5.55) can be absorbed by the left hand side of the Carleman inequality (5.32). Since ρ = − 3 2 s( β) ρ, we have where we have used that Using interpolation arguments and the Young inequality, we find as for ζ 2 ϕ We get also and Combining (5.32), (5.59) and (5.7), we get finally (5.6) for N 4, λ C and s C(T N + T 2N ).
Therefore, the couple (v, q) satisfies the system Moreover, from (6.19), we have v = 0 in (0, T − ε) × O. Then, using the unique continuity property of the Stokes system (see for instance [14]), we get The boundary conditions read to Since β S > 0, we get that v = 0 and k v = 0 in (0, T − ε). Then, we obtain in particular that γ 2 = ( , k) = 0 from the equations of the structure motion which contradicts (6.18).

Fixed point
In this section, we prove Theorem 1.1 by applying a fixed-point argument. For this purpose, we follow the same steps as [26]. First, we give some estimates on the terms appearing in the system (3.4), (3.5) and (3.6). We have the following lemma that is proved in [29].
Lemma 7.1. Let X and Y satisfying the properties given in Section 3. We obtain for all (u, π) ∈ [H 2 (F)] 2 × H 1 (F), the following estimates, for all t ∈ [0, T ] We have also Lemma 7.2. Let X and Y satisfying the properties given in Section 3. We obtain for all (u, π) ∈ [H 2 (F)] 2 × H 1 (F) the following estimates, for all t ∈ [0, T ] Now, we are in position to prove Theorem 1.1.
Proof of Theorem 1.1. For all r > 0, let us set Let F ∈ K r , and assume that From Proposition 6.1, the solution (u, π, h, θ) of the linear system (6.1), (6.2), (6.4) with v * = E T (Z 0 , a 0 , F ) satisfies h(T ) = 0, θ(T ) = 0 and Using ( Using the condition (7.2), we can construct the change of variables defined in Section 3. We can thus, define the mapping Φ : K r −→ K r , that associates F ∈ K r , we set where (u, π, h, θ) is the solution of the linear system (6.1), (6.2) and (6.3). Combining Lemma 7.1, the estimate (7.2) and Then, for r small enough, we get Φ(K r ) ⊂ K r . Similarly, using Lemma 7.2, we get that .