FEEDBACK STABILIZATION OF A 3D FLUID FLOW BY SHAPE DEFORMATIONS OF AN OBSTACLE∗

Abstract. We consider a fluid flow in a time dependent domain Ωf (t) = Ω \Ωs(t) ⊂ R, surrounding a deformable obstacle Ωs(t). We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in Ωf (t), t > 0. We prove that, for any arbitrary exponential decay rate ω > 0, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary ∂Ωs(t) can be chosen so that the fluid flow is stabilized to rest, and the obstacle to its initial shape and its initial location, with the exponential decay rate ω > 0.


Setting of the problem
We consider a fluid occupying a time dependent bounded domain Ω f (t) in R 3 , for t ≥ 0. We consider the case where Ω f (t) = Ω \ Ω s (t), Ω is a regular bounded domain in R 3 , and Ω s (t) is another regular domain such that Ω s (t) ⊂ Ω, for t ≥ 0. Thus, the obstacle Ω s (t) is surrounded by the fluid domain Ω f (t) as in Figure 1, but other situations can be considered as well. We denote by Γ e the boundary of Ω, and by Γ s (t) the boundary of Ω s (t). Therefore Γ e ∪ Γ s (t) = ∂Ω f (t), Γ s (t) = Ω f (t) ∩ Ω s (t), and Γ s (t) ∩ Γ e = ∅. For simplicity, we assume that Γ s (t) is connected.
When w satisfies the Lamé system of linear elasticity in Ω s × (0, ∞), coupling system (1.2) with the elasticity system (including the equality of the Cauchy stress forces at the interface Γ s ) corresponds to a fluid-structure interaction system (see, e.g., [25]). Thus there is a close connection between the stabilization of fluid-structure interaction systems and the stabilization problem that we consider. Here, for simplicity, we first treat the stabilization of the fluid flow by shape deformations of the surrounded obstable Ω s (t). The stabilization of the system coupling the Navier-Stokes equations with the Lamé system will be studied in a forthcoming paper, based on the results obtained here.
There are a few existence results of strong solutions, local in time, for system (1.2), either with a flat initial fluid-solid interface [14,25], or in the case of the geometrical configuration considered here, see e.g. [3,4]. The results in [16] and in [12] provide regularity results for strong solutions (assuming the existence of these strong solutions). Several papers are concerned with the case where the solid is rigid, we refer to [5,11] and the references therein. There are other results dealing with models in which the fluid-solid interface is assumed to be fixed [14].
From the point of view of control theory a very few results are known for fluid-structure interaction models. We refer to the recent paper [28] where a stabilization result is obtained for a fluid-solid structure model, with a control acting in the solid equation. In [32,33], the authors study the asymptotic behavior of fluid-structure models coupling heat and wave equations, but the systems are linear and the fluid-structure interface is not moving. In [18], the authors study some stabilization problems, but still in that case the fluid-structure interface is fixed. The case of a moving interface, with a stabilizing feedback located at the fluid-structure interface is considered in [13]. In [21], the second author has studied the stabilization of a fluid-structure model in two dimensions, where the structure is a damped beam equation located at the boundary of the fluid domain. These results have been recently extended to more complex systems, see [7]. We also mention [9,10], where the shape of an elastic structure is used to reduce the drag of a fluid flow surrounding this an elastic structure.
We prove stabilization results for system (1.2) by rewriting it in Lagrangian variables. For that, we introduce u(y, t) = u(X(y, t), t) and p(y, t) = p(X(y, t), t), We notice that if u satisfies (1.2) 3 , then equation (1.3) restricted to Γ s reduces to ∂X ∂t (y, t) = ∂w ∂t (y, t) for all y ∈ Γ s .
Since w(·, 0) = 0, this equation reduces to (1.2) 4 . We look for w of the form where the functions (ξ i ) 1≤i≤Nc are regular and satisfiy To ensure that the functions (F i ) 1≤i≤Nc are sufficiently regular, we are going to look for (F i ) 1≤i≤Nc satisfying We set F = (F 1 , . . . , F Nc ), f = (f 1 , . . . , f Nc ), h = (h 1 , . . . , h Nc ). The system satisfied by ( u, p, F, f ) is of the form , where the nonlinear terms F, G, and g are defined by (1.5) In order to satisfy (1.1), we are going to look for a control h ∈ L 2 (0, ∞; R Nc ), in feedback form, such that the system (1.4) admits a unique solution ( u, p, F, f ) satisfying In order to impose the exponential decay rate ω > 0, we introduce the following change of unknowns u(y, t) = e ωt u(y, t), p(y, t) = e ωt p(y, t), F (y, t) = e ωt F (y, t), f (y, t) = e ωt f (y, t), h(y, t) = e ωt h(y, t).
The system satisfied by ( u, p, F , f ) is , where the nonlinear terms F, G, and g are defined by G( u) = e ωt G(e −ωt u) and g( u) = e ωt g(e −ωt u). (1.8) We notice that X(y, t) = X(y, t), and therefore Y (x, t) = Y (x, t).
Let us notice that, if the functions (ξ i ) 1≤i≤Nc are chosen so that Γs ξ i · n dx = 0, for all 1 ≤ i ≤ N c , then Γs g( u)(t) · n dx = 0. Indeed, The first equality in (1.9) is a consequence of Nanson's formula and the last one is a consequence of the fact that divu(t) = 0 in Ω f (t).
In (1.7), h = ( h 1 , · · · , h Nc ) T is the control, and the function ξ = (ξ i ) 1≤i≤Nc has to be suitably chosen in such a way that the linearized system associated with (1.7) is stabilizable by controls h ∈ L 2 (0, ∞; R Nc ). The equation F = ω F + f , F (0) = 0, is added to guarantee that X(·, t)| Γs tends to the identity with the exponential decay rate ω > 0, when t tends to infinity. Thus the fluid domain Ω f (t) tends to the initial configuration Ω f , with the exponential decay rate ω > 0, as t tends to infinity. And the equation f = ω f + h, f (0) = 0, is added to improve the regularity of f . Let us notice that if h is a control stabilizing system (1.7), then system (1.2) 1−3 will be stabilized, with the exponential decay rate ω > 0, by the boundary deformation X satisfying the differential equation

∂X ∂t
(y, t) = u(X(y, t), t), X(y, 0) = y for all y ∈ Γ s . (1.10) This is why we shall speak of stabilization by shape deformations of the fluid-obstacle interface. Before stating the main result of the paper, we recall that, for s ≥ 0, H s (Ω f ) denotes the usual Sobolev spaces. We would like to underline that we shall make the abuse of notation by writing H s (Ω f ) for H s (Ω f ; R 3 ) or H s (Ω f ; R 3×3 ) for vector valued or matrix valued functions. The distinction between these three spaces should be clear from the context. We may also use the notation H s (Ω f ) for H s (Ω f ; C 3 ), for eigenvalue problems in which eigenvalues and eigenfunctions may be with complex values.
For s > 0, the space H s,s/2 (Q ∞ f ) is nothing but L 2 (0, ∞; H s (Ω f )) ∩ H s/2 (0, ∞; L 2 (Ω f )). We shall also make the abuse of notation by writing H s, ). Throughout the paper, we assume that Ω f is of class C 5 . (1.11) The main result of the paper concerning the stabilization of the Navier-Stokes equations by boundary deformations is stated below. Let ω > 0 be given fixed, and let 3 2 , and div u 0 = 0. If in addition u 0 H 1+ (Ω f ) is small enough, then there exists a control h belonging to H 1+ /2 (0, ∞; R Nc ), in feedback form, such that the system (1.7) admits a solution ( u, F , f , X) belonging to The outline of the paper is as follows. We prove new regularity results for the Stokes system in Section 2. The stabilizability and the feedback stabilization of the linearized Navier-Stokes system controlled by boundary deformations is studied in Section 3. The associated closed-loop nonhomogeneous linear system is studied in Section 4. Section 5 is devoted to estimates of nonlinear terms of the Navier-Stokes system controlled by boundary deformations. Theorem 1.1 is proved in Section 6 by a fixed point method. In Section A, we have collected estimates on products of functions needed in Section 5.
As the existence of solution to the nonlinear closed loop system (1.7) is proved by the Banach fixed point Theorem, we only prove the uniqueness of the fixed point. This does not mean that the solution is unique. However, we think it is the case, and that the uniqueness can be proved as in [19] by showing that the system (6.1) admits a unique maximal solution. But due to the length of the proof in [19], we shall explore that issue in a forthcoming paper.

The Stokes equations
In this section, we are going to state results very similar to those in [25], except that in [25] all the regularity results are stated for equations defined over a bounded time interval (0, T ), while, here, we are going to prove results for equations defined over the time interval (0, ∞).

Some function spaces
We introduce the following function spaces We also introduce the dual spaces of n (Ω f ) as pivot space:

The Leray projector
Let us recall that where ∇H 1 (Ω f ) is the subspace of functions belonging to L 2 (Ω f ; R 3 ) and which are the gradients of functions belonging to H 1 (Ω f ). The Leray projector P is the orthogonal projector from where ζ is the solution to the elliptic equation and π satisfies The operator P may be continuously extended as an operator belonging to L( Similarly P may also be continuously extended as an operator belonging to L(( With the extrapolation method, the Stokes operator may be extended to an unbounded operator in V −1 (Ω f ) and in V −2 (Ω f ) as follows. The operator (A, ), due to elliptic regularity results and the fact that Ω f is of class C 5 (see, e.g., [31], Thm. 2.24), the solution ζ to (2.2) and the solution π to (2.3) belong to H 5 (Ω f ) (resp. H 3 (Ω f ; R 3 )) (the nonhomogeneous boundary condition for the solution π to (2.3) can be treated by a lifting).

Regularity results for the Stokes equations
We study the following Stokes system with Dirichlet boundary conditions and a nonhomogeneous divergence condition with the compatibility condition We recall that ∈ ( 1 2 , 3 2 ). We introduce the spaces The norm in E will be denoted by · E , and it is defined by The usual norms in F , G , and G will be respectively denoted by · F , · G , and · G . The norm in P is p → ∇p F .
Since we need to consider velocity fields u whose trace on Σ ∞ s belongs to H 1+ /2 (0, ∞; H 3/2+ (Γ s )), this additional condition has to be included in the definition of E .
We want to prove the following theorem.
and if in addition the following compatibility conditions are satisfied Remark 2.4. A similar theorem is proved in Theorem 3.2 of [1] in the case where 1 < < 3/2, and 0 < T < ∞. Another version of Theorem 2.3 is stated in Theorem 1.1 of [27], still in the case where 0 < T < ∞.

Equation (2.8)
Proposition 2.5. If g, g, and ξ obey the assumptions of Theorem 2.3, then the solution to system Proof. For g = div g and ξ satisfying the compatibility condition (2.6), we define L(g, ξ) and L p (g, ξ) by where (u, ρ) is the solution to the following stationary problem When ξ ∈ H 1+ /2 (0, ∞; H 3/2+ (Γ s )) and g ∈ G , since ξ and g depend on the time variable t, we shall also emphasize the dependence of u and ρ with respect to t by setting L(g(t), ξ(t)) = u(t) and L p (g(t), ξ(t)) = ρ(t). From Corollary 8.4 of [23], it follows that u(t) = L(g(t), ξ(t)) satisfies Thus, we obtain We also have (see [8]) Therefore the solutions u to (2.11) belongs to E , and ∇ρ ∈ F .
We look for the solution ( v, q) to (2.8) (2.12) Since u belongs to E , ∂u ∂t belongs to F , and, due to Theorem 5.3 of [8] (see also the arguments in the proof of Prop. 2.6), we have the estimate From the above estimates for (u, ρ) and ( w, ρ), it follows that (2.14)

Equation (2.9)
Proposition 2.6. If F and u 0 obey the assumptions of Theorem 2.3, then the solution to system (2 Proof. This is a classical regularity result for the Stokes equation. However, since we need estimates over the time interval (0, ∞) we give the main arguments of the proof. (See [8] for estimates over a bounded time interval.) Let us denote by v v0,F the solution to (2.9), and let us To study the regularity of v F , we consider the case where F ∈ H 1 (0, ∞; L 2 (Ω f ; R 3 )), and we are going to prove that where v F is the extension of v F by zero to (−1, 0), and F is the extension of F by zero to (−1, 0). Thus the from which we deduce (2.15) 2 . With (2.16) and the equation Next using Av F = P F − ∂ t v F , we deduce an estimate of v F in L 2 (0, ∞; V 2+ (Ω f )). The estimate for the pressure is derived from elliptic regularity results for the velocity and the pressure in the Stokes equation. The proof is complete.
End of proof of Theorem 2.3. The estimates of Theorem 2.3 clearly follow from Propositions 2.5 and 2.6.

The extended system
Let {ξ 1 , · · · , ξ Nc } be a family satisfying (3.3), and consider the extended system where the lifting operator L is introduced in (2.10). We can rewrite the system (3.4) in the form d dt where the control operator B is defined by We have also to extend the operator (A ω , D(A ω )) to an unbounded operator in V −2 (Ω f ) × R Nc × R Nc . This extension, defined by the extrapolation method, is denoted by where P is the operator introduced in Section 2.2, and It is why after studying in Proposition 3.5 the stabilizability of the pair (A ω , B) in V 0 n (Ω f ) × R Nc × R Nc , we construct a feedback K in Proposition 3.6 able to stabilize the pair (A ω , B) both in V 0 n (Ω f ) × R Nc × R Nc and in V −2 (Ω f ) × R Nc × R Nc . Thus the stabilizability of the pair (A ω , B) in V −2 (Ω f ) × R Nc × R Nc is obtained as a byproduct of that proposition.
, with a compact resolvent.

Proof. We can set
and we can proceed as in Proof of Theorem 3.6 of [21] to prove the analiticity of the semigroup both on V 0 n (Ω f ) × R Nc × R Nc and on V −2 (Ω f ) × R Nc × R Nc ). The compacteness of the resolvent can be easily checked.

Hautus criterion Proposition 3.4. The adjoint of the unbounded operator
where (σ(φ, N ψ (φ))n, ξ i ) Γs stands for Γs σ(φ, N ψ (φ))n · ξ i dx, and, for any Proof. The proof can be done in a classical way and is left to the reader.
Step 2. If (3.6) is not satisfied, that is if −ω ∈ σ(A), we choose − ω < −ω, close enough to −ω, satisfying is a pair of eigenfunction-eigenvalue of the Stokes operator with λ ≥ 0} is a pair of eigenfunction-eigenvalue of the Stokes operator with λ ≥ 0}.
The proof is complete.

the infinitesimal generator of an analytic and exponentially stable semigroup on
Proof. It is sufficient to apply the strategy developed in [20] for the Oseen equations. In [20], feedbacks are obtained by looking for feedback controls stabilizing the Oseen system projected onto the finite dimensional unstable subspace of the Oseen operator. Here, we have to find feedback controls stabilizing the extended system (3.4) projected onto the finite dimensional unstable subspace of A ω , that we denote by Z u . If P u is the projector in V 0 n (Ω f ) × R Nc × R Nc onto Z u satisfying P u A ω = A ω P u , this projector can be extended as a projector from V −2 (Ω f ) × R Nc × R Nc onto Z u (see [24], Lem. 3.17). Because of that property, if K u belongs to L(Z u , R Nc ), the operator K = K u P u belongs to L(V 0 n (Ω f ) × R Nc × R Nc , R Nc ) and also L(V −2 (Ω f ) × R Nc × R Nc , R Nc ). If in addition, K u is a feedback stabilizing exponentially the pair (P u A ω , P u B) in Z u , that is if e t(PuAω+PuBKu) t≥0 is exponentially stable on Z u , then e t(Aω+BK) t≥0 is exponentially stable both on V 0 n (Ω f ) × R Nc × R Nc and on V −2 (Ω f ) × R Nc × R Nc . The fact that (A ω + BK, D(A ω )) is the infinitesimal generator of an analytic semigroup on V 0 n (Ω f ) × R Nc × R Nc follows from the fact that BK is a bounded perturbation of (A ω , D(A ω )). The same argument is valid for the extension of the operator This approach, used in [20], has been first introduced in [24]. Another approach for constructing such feedbacks is introduced in [22] for the 3D Oseen system.

Closed-loop nonhomogeneous linear system
We want to prove regularity results for the system where K is the feedback operator introduced in Proposition 3.6, and the function δ defined by is added to satisfy the compatibility condition Γs v · n dx = Γs g · n dx.
As in [23], we can prove that (v, F, f ) is solution to (4.1) if and only if it satisfies f i (t) ξ i (y) + n δ.

5.
Estimates of the nonlinear terms over the time interval (0, ∞)

Estimates of X and Y
In addition to the spaces E , F , and P already introduced before, we also need The spaces S , D , and I (defined in Thm. 1.1) will be equipped with the norms and Notice that if u belongs to E , then X(y, t) = y + t 0 e −ωs u(y, s) ds belongs to I , and that We are now going to estimate J X (y, t) = I + t 0 e −ωs ∇ u(y, s) ds.
Lemma 5.1. Let 0 < γ < 1 and ω > 0 be chosen. Let us assume that Proof. We have Lemma 5.2. If u belongs to E , then the mapping J X − I belongs to S . And there exists a constant C(ω) such that Proof. We estimate J X − I in L ∞ (0, ∞; H 1+ (Ω f )) as follows: We have ∂ t J X (y, t) = e −ωt ∇ u(y, t). Since the proof is complete.

Lemma 5.3. If u belongs to E , and if
For all µ satisfying we set We denote by P + 0 the set of polynomials f with nonnegative coefficients and satisfying f (0) = 0. Lemma 5.4. There exists a polynomial K 1 ∈ P + 0 such that, for all µ > 0 satisfying (5.1), all u ∈ E ,µ , we have Proof. The proof follows from Lemmas 5.2, 5.3, A.1, A.5, and from the following identities We can also establish the following Lipschitz estimates.

Lipschitz estimate on F
For all µ satisfying (5.1), we set Lemma 5.6. There exists a polynomial C F ∈ P + 0 such that, for all µ > 0 satisfying (5.1), all u ∈ E ,µ , all p ∈ P ,µ , all u 1 ∈ E ,µ , all p 1 ∈ P ,µ , all u 2 ∈ E ,µ , and all p 2 ∈ P ,µ , we have Proof. To estimate F 1 , with Lemma 5.6, we notice that The Lipschitz properties can be proved in the same way.
Lemma 5.7. There exists a polynomial C G ∈ P + 0 such that, for all µ > 0 satisfying (5.1), all u ∈ E ,µ , all u 1 ∈ E ,µ , and all u 2 ∈ E ,µ , we have Proof. The proof is based on Lemma A.4.
Step 1. We have Step 2. Since u belongs to E , ∇ u belongs to H 1+ ,1/2+ /2 (Q ∞ ). We know that I − det J X J T Y belongs to S . The estimate of G( u) in H /2 (0, ∞; H 1 (Ω f )) follows from Lemma A.4. The Lipschitz estimate for G( u) can be proved in the same way.
Lemma 5.8. There exists a polynomial C g ∈ P + 0 such that, for all µ > 0 satisfying (5.1), all u ∈ E ,µ , all u 1 ∈ E ,µ , and all u 2 ∈ E ,µ , we have To obtain the estimate of g( u) in H 1+ /2 (0, ∞; L 2 (Ω f )), it is sufficient to apply Lemma A.5. The estimate of δ( u) follows from that of g( u). The Lipschitz estimate can be easily obtained in the same way.

Stabilization of the nonlinear fluid system
The goal of this section is to prove Theorem 1.1. Our goal is to show that the system e −ωs u(y, s)ds, for all y ∈ Ω f and all t ∈ [0, ∞), is a solution of (6.1), then it is also a solution of (1.7). Indeed, from the condition div u = div g( u) and from the definition of g( u), it follows that δ( u) = 0. Thus, to prove Theorem 1.1, it is sufficient to prove the existence of solution to system (6.1).
Our goal is to prove that, for µ > 0 small enough, N is a contraction in E ,µ × P ,µ . From Theorem 4.1, and Lemmas 5.7 to 5.9, it follows that for all (v, q) ∈ E ,µ × P ,µ , and for all (v 1 , q 1 ) ∈ E ,µ × P ,µ and all (v 2 , q 2 ) ∈ E ,µ × P ,µ . We choose µ > 0 such that and we choose u 0 such that Since (0, 0) ∈ E ,µ × P ,µ , the space E ,µ × P ,µ is nonempty. From (6.5), (6.7), and (6.8), it follows that N is a mapping from E ,µ × P ,µ into itself, and with (6.6) and (6.7), it follows that N is a contraction in E ,µ × P ,µ . Therefore N admits a fixed point which is a solution to (6.1). Due to Step 1, the proof of Theorem 1.1 is complete.

Appendix A. Product estimates
Lemma A.1. If f ∈ S and g ∈ S , then f g ∈ S .
The proof is complete.
To prove that f g belongs to H /2 (0, ∞; L 2 (Ω f )), we treat only the case when 1 < < 3 2 . The cases when 1 2 < < 1 and = 1 can be treated with obvious modifications as we did in the proof of Lemma A.1. To estimate the semi-norm |f g| H /2 (0,∞;L 2 (Ω f )) , we use difference quotients. We have |f g| 2 To estimate the first integral, we write |s−τ |<1 (f (s) − f (τ ))g(τ ) 2 For the second term, we can write The proof is complete.
To estimate f g in H /2 (0, ∞; H 1 (Ω f )), we can use difference quotients and adapt the proof of Lemma A.2.
For the second term, we have ∂ t g ∈ H , /2 (Q ∞ f ), and f ∈ S . We apply Lemma A.2.