Stabilization of the non-homogeneous Navier-Stokes equations in a 2d channel

In this article we study the local stabilization of the non-homogeneous Navier- Stokes equations in a 2d channel around Poiseuille flow. We design a feedback control of the velocity which acts on the inflow boundary of the domain such that both the fluid velocity and density are stabilized around Poiseuille flow provided the initial density is given by a constant added with a perturbation, such that the perturbation is supported away from the lateral boundary of the channel. Moreover the feedback control operator we construct has finite dimensional range.

1. Introduction 1.1. Settings of the problem. We are interested in stabilizing the density dependent Navier-Stokes equations around some stationary state (ρ s , v s ) (where (ρ s , v s , p s ) is a stationary solution) in a two dimensional channel Ω. For that we will use an appropriate boundary control u c acting on the velocity in the inflow part of the boundary ∂Ω. Let d be a positive constant. Throughout this article we will use the following notations (see (1.1) The unit outward normal to the boundary Γ is denoted by n. The velocity, density and pressure of the fluid are denoted respectively by v, ρ and p. The viscosity ν > 0 of the fluid is a positive constant. We consider the following control system in Ω, in Ω, (1.2) where u c χ Γc is a control function for the velocity v with χ Γc denoting the characteristics function of a set Γ c which is compactly supported on Γ. The set Γ c will be precisely defined shortly afterwards. The equation (1.2) 1 is the mass balance equation and (1.2) 4 is the momentum balance equation. The triplet (ρ s , v s , p s ) is the Poiseuille profile defined as follows Observe that (ρ s , v s , p s ) (given by (1.3)) is a stationary solution of the Navier-Stokes equations (1.2). We remark that in the definition (1.3) of the Poiseuille profile we can choose ρ s to be any positive constant in place of one up to modifying p s accordingly. Also in the definition (1.3) one can consider v s = (αx 2 (1 − x 2 ), 0), for a positive constant α > 0. The strategy and results of our analysis apply for any constant ρ s > 0 and α > 0. The aim of this article is to determine feedback boundary control u c (the control of the velocity) such that the solution (ρ, v) of the controlled system is exponentially stable around the stationary solution (ρ s , v s ) provided the perturbation (ρ 0 , v 0 ) of the steady state (ρ s , v s ) is sufficiently small (in some suitable norm).
In view of the stationary profile (1.3), it is natural to control the inflow part of the boundary, i.e. we will consider the control function u c supported on In fact we do slightly more and control on some open subset Γ c of Γ in . We consider Γ c of the following form for some fixed 0 < L < 1 2 . Remark 1.1. We consider the control zone of the form (1.5) to simplify the notations. In fact our analysis allows to consider any subset {0} × (A, B) (0 < A < B < 1) of Γ in as the control zone.
To state our results precisely, we introduce some appropriate functional spaces.
The spaces V s (Ω) and V s (Γ) are respectively equipped with the usual norms of H s (Ω) and H s (Γ), which will be denoted by · V s (Ω) and · V s (Γ) . From now onwards we will identify the space V 0 n (Ω) with its dual. For 0 < T ∞ let us introduce the following functional spaces adapted to deal with functions of the time and space variables.
We also fix the convention that for any two Banach spaces X and Y, the product space X × Y is endowed with the norm ∀ (x, y) ∈ X × Y, (x, y) X ×Y = x X + y Y , where . X and . Y denotes the norms in the corresponding spaces.
1.3. The main result. We now precisely state our main result in form of the following theorem. Theorem 1.2. Let β > 0, A 1 ∈ (0, 1 2 ). There exist a constant δ > 0 such that for all there exists a control u c ∈ H 1 (0, ∞; C ∞ (Γ c )), for which the system (1.2) admits a solution satisfying the following stabilization requirement
We now make precise the structure of the control function u c we are going to construct. We will show the existence of a natural number N c , and a family of smooth functions supported on Γ c such that the control u c acting on the velocity is given as follows u c (x, t) = e −βt Nc j=1 w j (t)g j (x), (1.8) where w c (t) = (w 1 (t), ...., w Nc (t)) is the control variable and is given in terms of a feedback operator K. More precisely, w c = (w 1 , ..., w Nc ) satisfies the following ODE where γ is a positive constant, P is the Leray projector from L 2 (Ω) to V 0 n (Ω) ([44, Section 1.4]) and K ∈ L(V 0 n (Ω) × R Nc , R Nc ) (the feedback operator K is determined in Section 2.2.2). The boundary control (1.8) we construct has a finite dimensional range and resembles with the control designed in [42]. The construction of our control basis {g j | 1 j N c } is different from the one done in [42]. In [42] it is constructed using generalized eigenvectors of the adjoint of Oseen operator while we construct it only by using eigenvectors of adjoint of Oseen operator relying on the construction of [36]. We will not consider any control on the transport equation modeling the density and as for the homogeneous Navier-Stokes equations, we show that considering a control u c of the velocity is enough to stabilize the whole system (1.2). The stabilizability of the constant density (or homogeneous) incompressible Navier-Stokes equation (with Dirichlet or mixed boundary condition) by a finite dimensional feedback Dirichlet boundary control has already been studied in the literature. For instance in [42] it is proved that in a C 4 domain the velocity profile v, solution to system (1.2) 4 -(1.2) 7 with ρ = 1 is locally stabilizable around a steady state v s (v s ∈ H 3 (Ω; R 2 )) by a finite dimensional Dirichlet boundary control localized in a portion of the boundary and moreover the control u c is given as a feedback of the velocity field. Unlike the constant density incompressible Navier-Stokes equations (which is of parabolic nature), the system (1.2) obeys a coupled parabolic-hyperbolic dynamics. Local exact controllability to trajectories of the system (1.2) was studied in [3]. In the present article we answer the question posed in [3] on the stabilizability of the system (1.2) around the Poiseuille profile. In proving the controllability results one of the main geometric assumptions of [3] is that Ω = Ω T out = {x ∈ Ω | ∃t ∈ (0, T ), s.t X(t, 0, x) ∈ R d \ Ω}, (1.9) where X is the flow corresponding to the target velocity trajectory v s defined as In the article [3] the assumption (1.9) plays the key role in controlling the density of the fluid. In our case since the target velocity trajectory is v s (defined in (1.3)) the assumption (1.9) is not satisfied because v s vanishes at the lateral boundary of the domain Ω. Hence to control the density we make a parallel assumption (1.6). Indeed, the assumption (1.6) implies that supp(ρ 0 ) ⋐ Ω T out .
The assumption (1.6) exploits the hyperbolic nature of the continuity equation (1.2) 1 in order to control the coupled system (1.2). The condition (1.6) in fact guarantees that the density exactly equals ρ s = 1, after some time vs (will be detailed in Section 3) so that the non-homogeneous Navier-Stokes equations become homogeneous after some finite time. In [3] the authors uses two control functions (one for the density and one for velocity) for the purpose of controlling the non-homogeneous fluid. Contrary to that we use only one control acting on the velocity to stabilize the coupled system (1.2).
1.4. Decomposition of the boundary Γ and comment on the support of control. Based on the velocity profile v s (as defined in (1.3)) we can rewrite the boundary of Ω as follows Figure 1).
(1.10)  . From now onwards we will use the notation Γ in to denote the inflow boundary of both the vector fields v s and v. This is a slight abuse of notation but we will prove the existence of the controlled trajectory v in a small neighborhood (in a suitable norm) of v s provided the perturbation v 0 is small. This will guarantee that Γ in and the inflow boundary of the vector field v s are identical. For the details we refer the reader to the Corollary 2.17.
We will look for a control function u c of the form (1.8) which is compactly supported in Γ c . More particularly we will construct the finite dimensional basis {g j | 1 j N c } of the control space in such a way that g j (∀ 1 j N c ) is smooth and supported in Γ c .
To solve a nonlinear stabilization problem the usual method is to first solve the stabilization problem for the linearized system and then use a fixed point method to conclude the stabilizability of the original nonlinear problem (1.13). In this article due to regularity issues of the transport equation we avoid linearizing the whole system. Instead, we only linearize the equation (1.13) 4 satisfied by y i.e. we replace the nonlinear terms appearing in the equation (1.13) 4 by a non homogeneous source term f and we leave the equation of the density (1.13) 1 unchanged. Hence we start by analyzing the stabilizability of the system y(x, 0) = y 0 in Ω. (1.14) (ii) Section 2 is devoted to study the stabilization of the linearized Oseen equations (1.14) 4 -(1.14) 8 . In that direction we first write (1.14) 4 -(1.14) 8 using operator notations. This is done in the spirit of [40] but with suitable modifications which are necessary since our domain is Lipschitz. To prove the stabilizability of this system we look for a control of the form (1.8). We will choose the functions {g j | 1 j N c }, supported on Γ c , so that we can prove some unique continuation property equivalent to the stabilizability of the system under consideration. This is inspired from [36]. Using the fact that g j (for all 1 j N c ) is supported on a smooth subset of Γ we further show that g j is in C ∞ (Γ). This in particular implies that the control u c , of the form (1.8), is smooth in the space variable. (iii) Next our aim is to find a boundary control which is given in terms of a feedback law. At the same time we have to design the control such that the velocity y belongs to the space V 2,1 (Q ∞ ). Indeed the H 2 (Ω) regularity of the velocity field will be used later to prove the stabilization of the continuity equation. This creates another difficulty because to prove the V 2,1 (Q ∞ ) regularity of y solution of (1.14) 4 -(1.14) 8 , one must have a compatibility between the initial velocity y 0 , assumed to be in V 1 0 (Ω) and the boundary condition (i.e. the control u). We deal with this issue by adding a system of ordinary differential equations satisfied by w c . The corresponding extended system satisfied by (y, w c ) reads as follows in Ω, (1.15) where γ > 0 is a positive constant and ϕ c (∈ R Nc ) is a new control variable which will be determined later as a feedback of the pair (y, w c ). Since y(., 0) = 0, imposing w c (0) = 0 furnishes the desired compatibility between the initial and boundary conditions of y which is necessary to obtain the V 2,1 (Q ∞ ) regularity of y. First we will construct the control ϕ c given in terms of a feedback operator which is able to stabilize the homogeneous (i.e. when f = 0) extended system (1.15) by solving a Riccati equation. Then we show that the same control stabilizes the entire non-homogeneous (i.e. with the non-homogeneous source term f ) extended system (1.15) by assuming that the non-homogeneous term f belongs to some appropriate space.
(iv) In Section 3, we study the stability of the continuity equation (1.14) 1 -(1.14) 3 . We assume the velocity field in V 2,1 (Q ∞ ) and σ 0 ∈ L ∞ (Ω) such that (1.6) (recall from (1.12) that σ 0 = ρ 0 ) holds. Since σ 0 ∈ L ∞ (Ω) and the transport equation has no regularizing effect we expect that σ ∈ L ∞ loc (Q ∞ ). The Cauchy problem for the continuity equation in the presence of an inflow boundary is rather delicate. In our case we use results from [9] for the existence of a unique renormalized weak solution of the problem (1.14) 1 -(1.14) 3 in the space L ∞ (Q ∞ ). Our proof of the stabilization of the transport equation satisfied by the density relies on the fact that the characteristics equation corresponding to the velocity field is well posed. As we are dealing with velocity fields in L 2 (0, ∞, H 2 (Ω)), which is not embedded in L 1 loc (0, ∞, W 1,∞ (Ω)) in dimension two, our analysis relies on [47] (see also [4,Theorem 3.7]), stating the well-posedness of the equation of the flow as a consequence of Osgood condition. Then considering the velocity field (v s + e −βt y) as a small perturbation of v s (see (1.3) for the definition) we prove that the characteristic curves corresponding to the perturbed velocity field stay close to that of v s in a suitable norm. Using the fact that the characteristics corresponding to the velocity fields v s and (v s + e −βt y) are close we show that the particles initially lying in the support of σ 0 are transported out of the domain in some finite time T > T A1 = d A1(1−A1) along the flow corresponding to the perturbed velocity field. Consequently, the solution ρ of the equation (1.2) 1 -(1.2) 3 reaches exactly the target density ρ s = 1 after the time T.
(v) Finally in Section 4, we will use Schauder's fixed point theorem to conclude that the control designed in step (iii) locally stabilizes the non linear coupled system (1.14) and consequently Theorem 1.2 follows.
1.6. Bibliographical comments. In the literature many works have been dedicated to the study of incompressible Navier-Stokes equations. For the classical results concerning the existence-uniqueness and regularity issues of the constant density incompressible Navier-Stokes equations we refer the reader to [44]. The reader can also look into [25] for a thorough analysis of the subject. Intricate situations may arise due to the lack of regularity when special geometric assumptions are imposed on the boundary ∂Ω. For example, the domain can have corners or edges of prescribed geometric shape. For the analysis of these situations the interested reader may look into [34] and [14]. In the present article the functional settings for the incompressible Navier-Stokes equations is motivated from [40]. The results of [40] are stated in a domain with smooth boundary. Thus to adapt the functional framework from [40] in the case of a rectangular domain we have used some results from [26] and [28]. Regarding the Cauchy problem of the non-homogeneous Navier-Stokes equations, the existence of classical solution for the non-homogeneous Navier-Stokes equations with homogeneous Dirichlet boundary condition for velocity in space dimension three is studied in [1]. Results concerning the existence-uniqueness of global in time strong solution (with small initial data and small volume force) in space dimension three can be found in [30]. In dimension two the existence and uniqueness of global in time solution (without any smallness restriction on the data) is also proved in [30]. In both of these references the velocity field is Lipschitz and the initial condition of the density is smooth enough, hence the transport equation satisfied by the density can be classically solved using the method of characteristics. To deal with less regular velocity field the concept of renormalized solution was initially developed in [15] and later suitably adapted in several contexts. For instance, one can find an application of a suitable variation of the Di-Perna-Lions theory to prove an existence and uniqueness result for the inhomogeneous Navier-Stokes equation in [13]. All of these articles assume that the velocity field satisfies v · n = 0. In the present article we are dealing with the target velocity v s , which is inflow on a part of the boundary ∂Ω. For a velocity field with inflow, one must assume a suitable boundary condition for the density so that the transport equation satisfied by the density is well posed. This problem is analyzed in the articles [9, Chapter VI] and [7], where the authors suitably define the trace for the weak solution of the transport equation. They also prove that these traces enjoy the renormalization property. In the present article we use the existence, uniqueness and stability results for the transport equation from [9] and [7]. For a more intricate case involving nonlinear outflow boundary condition, similar results can be found in [8].
There is a rich literature where the question of the feedback boundary stabilization of the constant density incompressible Navier-Stokes equation is investigated. For the feedback boundary stabilization of a general semilinear parabolic equation one can look into the article [22]. The feedback stabilization of the 2D and 3D constant density Navier-Stokes equations can be found in the articles [23] and [24] respectively. Concerning the stabilization of homogeneous Navier-Stokes equations one can also consult [42] and [39] where the feedback boundary controls are achieved by solving optimal control problems. We would also like to mention the articles [35] and [5] where the authors prove the feedback stabilization of the same model around the Poiseuille profile by using normal velocity controllers. The idea of constructing a finite dimensional boundary feedback control to stabilize a linear parabolic equation dates back to the work [45]. In our case we adapt the ideas from the articles [36] and [42] in order to construct a feedback boundary control with finite dimensional range to stabilize the linear Oseen equations. Actually for constant density fluids, the article [36] deals with a more intricate case involving mixed boundary conditions. Control properties of the variable density Navier-Stokes equations have been studied in the article [21], which proves several optimal control results in the context of various cost functionals. We also refer to the article [3] where the authors prove the local exact controllability to a smooth trajectory of the non-homogeneous incompressible Navier-Stokes equation. The study of the controllability and stabilizability issues of a system coupling equations of parabolic and hyperbolic nature is relatively new in the literature. We would like to quote a few articles in that direction. Null-controllability of a system of linear thermoelasticity (coupling wave and heat equations) in a n− dimensional, compact, connected C ∞ Riemannain manifold is studied in [31]. Controllability and stabilizability issues of compressible Navier-Stokes equations are investigated in [11], [10], [18] (in dim 1) and [17] (in dim 2 and 3). The compressible Navier-Stokes equations are also modeled by a coupled system of momentum balance and mass balance equations but the coupling is different from the one we consider in system (1.2). Let us emphasize that in the system (1.2) the control acts only on the velocity of the fluid and not on the density. In the literature there are articles dealing with controllability issues of a system of PDEs in which the controls act only on some components of the system. We would like to quote a few of them. We refer to [12] where the authors prove local null-controllability of the three dimensional incompressible Navier-Stokes equations using distributed control with two vanishing components. A related result concerning the stabilizability of 2−d incompressible Navier-Stokes equations using a control acting on the normal component of the upper boundary is proved in [16]. In [31] to prove the null-controllability of a system of linear thermoelasticity the authors consider the control on the wave equation i.e. on the hyperbolic part and not on the parabolic equation modeling the temperature. On the other hand controllability and stabilizability issues of one dimensional compressible Navier-Stokes equations have been studied in [11] and [10] by using only a control acting on the velocity. In the present article we also consider the control on the velocity and not on the density but our approach exploits more directly and in a more intuitive manner the geometry of the flow of the target velocity in order to control the hyperbolic transport equation modeling the density. 1.7. Outline. In section 2 we study the feedback stabilization of the velocity. Section 3 is devoted to the stabilization of the density. In Section 4 we use a fixed point argument to prove the stabilizability of the coupled system (1.2). Finally in Section 5 we briefly comment on how to adapt our analysis if one wishes to control the outflow boundary Γ out or the lateral boundary Γ 0 of the channel Ω.

Stabilization of the Oseen equations
The goal of this section is to discuss the stabilization of the Oseen equations (1.14) 4 -(1.14) 8 . We will first design a localized boundary control with finite dimensional range to stabilize the linear Oseen equation (1.14) 4 -(1.14) 8 . We will then construct the control as a feedback of (y, w c ), where the pair (y, w c ) solves the extended system (1.15). The plan of this section is as follows (i) In Section 2.1, we study the stabilization of the homogeneous linear system (with f = 0) (1.14) 4 -(1.14) 8 , using a finite dimensional boundary control.
(ii) We will analyze the feedback stabilization of the extended system (1.15) in Section 2.2. Moreover with this feedback control we will prove the V 2,1 (Q ∞ ) regularity of the solution of linear Oseen equations (1.14) 4 -(1.14) 8 . Using a further regularity regularity estimate (see (2.59)) of the control u we show that (e −βt y + v s ) has the same inflow and outflow as that of v s , provided the initial condition y 0 and the non-homogeneous source term f (appearing in (1.14) 4 -(1.14) 8 ) are suitably small (see Corollary 2.17 ).

2.1.
Stabilization of the linear Oseen equations. In the following section we will define some operators and present some of their properties which helps in studying the linearized Oseen equations (1.14) 4 -(1.14) 8 .

2.1.1.
Writing the equations with operators. The following results are taken from [40] where they are stated in a C 2 domain. It is necessary to make suitable changes to adapt those results in our case since the domain Ω in our case is Lipschitz. Without going into the details of the proofs we will just comment on how to adapt those results in our case. Let P be the orthogonal projection operator from L 2 (Ω) onto V 0 n (Ω) known as Helmholtz or Leray projector (see [44,Section 1.4]). We denote by (A, D(A)) (the Oseen operator) and (A * , D(A * )) the unbounded operators in V 0 n (Ω), defined by For the H 2 (Ω) regularity of the solutions of the homogeneous Dirichlet boundary value problems corresponding to the operators A and A * in a rectangular domain Ω, one can apply [26, Since v s is smooth with div(v s ) = 0, we can prove the following lemma.
There exists λ 0 > 0 in the resolvent set of A such that the following hold for all y ∈ D(A), for all y ∈ D(A * ).

(2.2)
In Lemma 2.1 we can always choose λ 0 > β, taking λ 0 larger if necessary. Throughout this article we will stick to this assumption. Now, Lemma 2.1 can be used to prove the following. Now we want to find a suitable operator B to write down the Oseen equation as a boundary control system. Consider the following system of equations 3) admits a unique solution (y, q) ∈ V 2 (Ω) × H 1 (Ω)/R and moreover the following inequality holds
A , the adjoint of D A computed as a bounded operator from V 0 (Γ) to V 0 (Ω) is a bounded linear operator from V 0 (Ω) to V 0 (Γ) and is given as follows

10)
Here |Γ| is the one dimensional Lebesgue measure of Γ. , mapping g to (z, π) (where g, z and π are as in (2.10)) satisfies the following In particular one has the following Now let us assume g ∈ H −1 (Ω), where H −1 (Ω) denotes the dual of H 1 0 (Ω) with L 2 (Ω) as the pivot space. Using (2.13) and the fact that (2.14) where Now [9, Theorem IV.5.2] furnishes the following regularity Hence we are done with the proof of Lemma 2.5.
Remark 2.7. In part (ii) of Lemma 2.5, the operator D * A is defined on the space of divergence free functions but in part (iii) we extended this definition by removing the divergence free constraint on the elements of the domain of D * A . This is possible since it is not necessary to have a divergence free function g in order to solve (2.10).
In order to localize the control of the velocity on Γ c (defined in (1.5)), we introduce the operator M, which is defined as follows So the operator M localizes the support of the control on Γ in and also guarantees that M g ∈ V 0 (Γ) for any g ∈ L 2 (Γ). Sometimes we might use the notation to denote the Cauchy stress tensor corresponding to a vector field v and a pressure p.
We now define the operator where (D(A * )) ′ denotes the dual of the space D(A * ) with V 0 n (Ω) as the pivot space. Proposition 2.9. (i) The adjoint of the operator B, computed for the duality structure ·, · (D(A * ) ′ ,D(A * )) , that we will denote by B * in the following, satisfies B * ∈ L(D(A * ), V 0 (Γ)) and for all Φ ∈ D(A * ),

23)
and T denotes the stress tensor as defined in (2.19).
(ii) There exists a positive constant ω > 0 such that the operator B * can be extended as a bounded linear Proof. (i) From Lemma 2.5, we know that (Ω) (for details on the characterization of domains of fractional powers we refer to [32]), one observes the following Now one can use the expression of B * as given by (2.25) and part (iii) of the Lemma 2.5 to prove (2.24).
Now following [40] the Oseen equations on Ω, (2.28) can be written in the following evolution equation form (2.29) In the following section we discuss some spectral properties of the Oseen operator A and then we define a suitable control space in order to construct a control function which stabilizes the Oseen equations.
2.1.2. Spectral properties of A and the stabilizability criterion. Since the resolvent of A is compact (see Lemma 2.2), the spectrum spec(A) of the operator A is discrete. Moreover since A is the generator of an analytic semi group (see Lemma 2.2), spec(A) is contained in a sector. Also the eigenvalues are of finite multiplicity and appear in conjugate pairs when they are not real. We denote by (λ k ) k∈N the eigenvalues of A. Without loss of generality we can always assume that there is no eigenvalue of A with zero real part by fixing a slightly larger β, if necessary. So we choose N u ∈ N such that ...Reλ Nu+1 < 0 < Reλ Nu ... Reλ 1 . (2.30) Following [36], we now choose the control space as follows The choice (2.31) of the control space plays an important role in proving a unique continuation property which implies the stabilizability of the pair (A, B). Let us choose the functions g j in (1.8) such that For later use we now prove an additional regularity result for the elements of the control space U 0 . The following regularity result is true only because the elements of U 0 are supported on a smooth subset of Γ.
Proof. The function m is supported on Γ c , which is C ∞ . In view of the representation (2.21) of the operator B * , we observe that to prove Lemma 2.10 it is enough to show that for each 1 k N u , any solution (φ, ψ) to the system (2.33) is C ∞ in some open set Ω Γc (⊂ Ω) such that ∂Ω Γc contains Γ c . Let us consider k ∈ {1, ..., N u } and (φ, ψ) solves the following We will work in a neighborhood of Γ c in order to avoid the singularities due to the presence of the corners (0, 0) and (0, 1). First consider a neighborhood N b Γc of Γ c such that neither of the points (0, 0) and (0, 1) belong to N b Γc . Now we consider an open set Ω Γc such that Ω Γc ⊂ Ω, ∂Ω Γc (the boundary of Ω Γc ) is Γc . One can check that the function (Θφ, Θψ) satisfies the following and also Θφ = 0 on ∂Ω Γc , which implies Now we apply [9, Theorem IV.5.8] to obtain, (Θφ, Θψ) ∈ H 3 (Ω Γc ) × H 2 (Ω Γc ). We can use a bootstrap argument to conclude that, (Θφ, Θψ) ∈ C ∞ (Ω Γc ). Hence we finally have g j ∈ C ∞ (Γ), for all 1 j N c .
We are looking for a control u taking values in U 0 . We write Before going into the proof of Theorem 2.11, let us recall that the pair (A, B) is stabilizable in V 0 n (Ω) iff for all y 0 ∈ V 0 n (Ω), there exists a control w c ∈ L 2 (0, ∞; R Nc ) such that the controlled system P y ′ = AP y + Bw c in (0, ∞), P y(0) = y 0 , The proof of Theorem 2.11 in a more intricate situation involving mixed boundary condition can be found in [36]. In [36] the localization operator M, localizing the control, is simply the cutoff function m whereas in our case M is as defined in (2.17). For the sake of completeness, we present the proof of Theorem 2.11 below, which follows step by step the one of [36] up to minor modifications. Let φ ∈ ker(λ k I − A * ). Also suppose that ψ is the pressure associated with φ, i.e. the pair (φ, ψ) solves (2.33). Now one can use (2.37) and Proposition 2.9 in order to verify that One can notice that M ReT(φ, ψ)n ∈ U 0 and M ImT(φ, ψ)n ∈ U 0 . On the other hand we know that {g j } 1 j Nc forms a basis of U 0 . Hence B * φ = 0 implies that This implies that T(φ, ψ)n = C 0 n on supp (m), (2.42) where C 0 is a constant given by Now recall that φ = 0 on Γ and the unit outward normal on Γ + c is (−1, 0). Also since φ ∈ V 2 (Ω), one can consider the trace of divφ on Γ to obtain that divφ = 0 on Γ. Using these facts one can at once deduce from (2.42) that ∂φ ∂n = 0 and ψ = C 0 on Γ + c . Now consider the domain Ω ex which is an extension of the domain Ω (see Figure 2). Extend the function φ into Ω ex by defining it zero outside Ω, denote the extension also by φ. Extend ψ into Ω ex by the constant C 0 outside Ω. We denote the extension of ψ by ψ itself. It is not hard to verify that the extended pair (φ, ψ) ∈ V 2 (Ω ex ) × H 1 (Ω ex )/R, solves the eigenvalue problem (2.33) in the extended domain Ω ex . Finally the unique continuation property from [19] shows that φ = 0 in Ω ex , thus in particular on Ω. Hence we are done with the proof of the Hautus test (2.39).
From Theorem 2.11 we know that the pair (A, B) is stabilizable by a control w c ∈ L 2 (0, ∞; R Nc ). Hence there exists a control u (of the form (2.36)) which belongs to the finite dimensional space U 0 (see (2.31)) and stabilizes the pair (A, B). Now our aim is to construct w c such that it is given in terms of a feedback control law. For that we will study the stabilization of the extended system (1.15) in the following section.

2.2.
Stabilization of the extended system (1.15) by a feedback control.

2.2.1.
Evolution equation associated with the extended system (1.15). We set Depending on the context the notation I denotes the identity operator for all of the spaces V 0 n (Ω), R Nc and Z. We equip the space Z with the inner product We fix a positive constant γ (where γ is the constant appearing in the extended system (1.15)). Now let us recall the representation (2.29) of the system (2.28). In the same note it follows that y = (P y, w c ) is a solution to equation (1.15) iff (P y, w c ) solves the following set of equations where f = (P f, 0) and recall the definition of B from (2.37). Now we define the operator ( A, D( A)) in Z as follows As we have identified V 0 n (Ω) with its dual, the space Z and Z * are also identified. We define the adjoint of ( A, D( A)) in Z as follows Proof. We will prove that ( A, D( A)) generates an analytic semigroup on Z by proving that ( A * , D( A * )) generates an analytic semigroup on Z. This is enough since one has the following by using [ where R(λ, ·) denotes the resolvent of the respective operator (see [27,Section 2.16] for details on resolvent) and hence ( A, D( A)) generates an analytic semigroup on Z follows from the fact that ( A * , D( A * )) generates an analytic semigroup on Z as a consequence of [46,p. 163,Def. 5.4.5].
Let us notice that the operator A * can be decomposed as follows Since This implies the following From the definition (2.45) of the operator A one can easily observe that the spectrum of A is discrete and is explicitly given as follows spec( A) = spec(A) ∪ {−γ}.

2.2.2.
Existence of a feedback control law. We introduce the notation J = (0, I). Let us notice that J belongs to L(R Nc , Z). This section is devoted to the construction of a feedback control ϕ c which is able to stabilize the linear equation Before going into the proof of Proposition 2.14, let us recall that the pair ( A, J) is stabilizable in Z iff for all y 0 ∈ Z, there exists a control ϕ c ∈ L 2 (0, ∞; R Nc ) such that the controlled system Proof of Proposition 2.14.
We is the generator of an exponentially stable analytic semigroup on Z.
From now onwards we will not use the explicit expression of the feedback controller K which was constructed in the proof of Proposition 2.14, in fact we will only use that K ∈ L( Z, R Nc ) and D( A+ JK) = D( A).
The following result justifies our choice of denoting the inflow and outflow boundary of v s and a perturbation of v s using the same notation.
Proof. The proof is a direct consequence of Corollary 2.16, in particular the estimate (2.59).

Stability of the continuity equation
This section is devoted to the study of the transport equation satisfied by density which is modeled by (1.14) 1 together with (1.14) 2 and (1.14) 3 . This equation is linear in σ but nonlinear in (σ, y). First let us briefly discuss the stabilization of the linearized transport equation modeling the density with zero inflow boundary condition. This will give us an idea about how to obtain analogous results for its nonlinear counterpart.

Comments on the linear transport equation at velocity v s . The linearized continuity equation with the zero inflow boundary condition is given by
in Ω. (3.1) We can explicitly solve (3.1) to obtain for all (x 1 , x 2 ) ∈ Ω. In particular if we assume that σ 0 satisfies the condition (1.6), the solution σ to (3.1) vanishes after some finite time T A1 = d A1(1−A1) . Hence we see that with zero inflow boundary condition the solution of the linearized transport equation is automatically stabilized (in fact controlled) after some finite time. The equation (3.1) is just a prototype of the transport equation (1.14) 1,2,3 exhibiting similar property and we will discuss this in the following section.

3.2.
Stability of the transport equation (1.14) satisfied by density. We consider the transport equation satisfied by the density with the nonlinearity (y · ∇)σ. We assume that y V 2,1 (Q∞) is small enough and the following holds (e −βt y + v s ) · n < 0 on Γ in , (e −βt y + v s ) · n = 0 on Γ 0 , and (e −βt y + v s ) · n > 0 on Γ out . Here the transport equation satisfied by the density is given by in Ω, (3.4) where y is in V 2,1 (Q ∞ ), (3.3) holds, σ 0 ∈ L ∞ (Ω) and satisfies the condition (1.6) (recall from (1.12) that σ 0 = ρ 0 ). Provided y is suitably small in the norm V 2,1 (Q ∞ ), (3.1) can be seen as an approximation of (3.4), and as we will see in Theorem 3.5, solutions of (3.1) and of (3.4) share some similar behavior. We are in search of a unique solution of (3.4) in the space L ∞ (Q ∞ ). In the following discussion we will borrow several results from [9] on the existence, uniqueness and stability of the continuity equation. For later use, we shall consider a general transport equation of the form in Ω, (3.5) where v is a divergence free vector field in L 2 (0, ∞; V 2 (Ω)), and First let us define the notion of weak solution for the transport equation (3.5) 1 .
One can interpret the boundary trace of a weak solution (as defined in Definition 3.1) of (3.5) 1 in a weak sense. Following [9] we introduce some notations which will be used to define the trace of a weak solution of (3.5) 1 . Let m denote the boundary Lebesgue measure on Γ. Now for any T > 0, associated to the vector field v, we introduce the measure dµ v = (v · n)dmdt on Σ T and denote by dµ + v (respectively dµ − v ) its positive (resp. negative) part in such a way that is the outflow (resp. inflow) part of Σ T corresponding to the vector field v. The following two theorems, Theorem 3.2 and Theorem 3.3, are stated in [9] for a weaker assumption on the velocity field v. Here we state the results with v ∈ L 2 (0, T ; V 2 (Ω)) for the particular equation (3.5). (ii) There exists a unique function γ σ ∈ L ∞ (Σ T , |dµ v |) such that for any test function φ ∈ C 0,1 (Q T ) and for any [t 0 , iii) The renormalization property: For any function ξ : R → R of class C 1 , for any φ ∈ C 0,1 (Q T ) and for any [t 0 , The following theorem states some results on the well posedness of the weak solution σ of the Cauchy-Dirichlet transport problem (3.5). (ii) The trace γ σ of σ satisfies the inflow boundary condition, γ σ = 0, dµ − v almost everywhere on Σ in,v,T and σ satisfies the initial condition σ(x, 0) = σ 0 in Ω. In the following, we call this function σ satisfying (i) and (ii), the solution of (3.5).
(iii) Moreover for 0 < t < T, the solution σ of (3.5) satisfies Let us also recall, for later purpose, the following stability result for the transport equation with respect to its velocity field: Lemma 3.4. [9, Theorem VI.1.9] Let T > 0. Suppose that σ 0 ∈ L ∞ (Ω) and let {v m } m be a sequence of functions in L 2 (0, T ; V 2 (Ω)) such that there exists v ∈ L 2 (0, T ; Now suppose that σ m ∈ L ∞ (Q T ) is the unique weak solution (in sense of Definition 3.1.) of the following initial and boundary value problem in Ω. Now we state the main theorem of this section: (1−A1) be fixed. Our approach will be based on the flow X corresponding to the vector field v s + e −βt y. In order to introduce it in a more convenient manner, we first extend the domain into R 2 . Observe that the definition of v s can be naturally extended to R 2 into a Lipschitz function by setting v s (x 1 , x 2 ) = v s (x 2 ) if x 2 ∈ (0, 1) and 0 if x 2 ∈ R \ (0, 1). We denote this extension by v s itself. For the following analysis we use the functional space (this is consistent with the notations defined in Section 1.2). Now we introduce an extension operator E from Ω to R 2 .
such that: • for every y ∈ L 2 (Ω), Ey | Ω = y, • the restriction of E to H 2 (Ω) defines a linear operator from H 2 (Ω) to H 2 (R 2 ), • the restriction of E to H 2 (Ω) ∩ W 1,∞ (Ω) defines a linear operator from H 2 (Ω) ∩ W 1,∞ (Ω) to The existence of such an extension operator is a direct consequence of [33,Theorem 2.2]. We now introduce the flow X(x, t, s) defined for x ∈ R 2 and (t, s) ∈ [0, ∞) 2 , by the following differential equation:    ∂X(x, t, s) ∂t = (v s + e −βt Ey)(X(x, t, s), t), (3.13) The integral formulation of (3.13) can be written as follows (3.14) As the vector field due to the Osgood condition (see [47] and [4, Theorem 3.7]) we know that equation (3.14) has a unique continuous solution. Similarly, we introduce the flow X 0 corresponding to the vector field v s as the solution of the following differential equation: As v s is Lipschitz, the flow, which can also be seen as the solution of is well defined in classical sense. Proof. The proof of Lemma 3.6 can be performed by using arguments which are very standard in the literature. For the convenience of the reader we include the proof. 1. As H 2 (R 2 ) is embedded in L ∞ (R 2 ), using Hölder's inequality we can at once obtain the following estimate for all (t, s) ∈ [0, T ] 2 and x ∈ R 2 , t s e −βθ Ey(X(x, θ, s), θ)dθ K Ey H 2,1 (R 2 ×(0,∞)) , for some constant K > 0.
Since E is a bounded operator from L 2 (Ω) to L 2 (R 2 ) and from H 2 (Ω) to H 2 (R 2 ), there exists a constant K > 0 such that Now we can use Grönwall's inequality to obtain (3.17).
Recall that the solution of (3.1) vanishes after some finite time T A1 = d A1(1−A1) . At the same time Lemma 3.6 suggests that for any finite time T > 0, the flow X 0 (x, t, s) stays uniformly close to X(x, t, s) in R 2 × (0, T ) provided y V 2,1 (Q∞) is small enough. In view of these observations, in the following we design a Lyapunov functional corresponding to a localized energy, to prove that σ vanishes after the time T 1 > T A1 when y V 2,1 (Q∞) is small enough, which will prove Theorem 3.5. Let ε be a fixed positive constant in (0, A 1 ) such that Our primary goal is to prove that, for a velocity field y satisfying (3.3) and such that y V 2,1 (QT 1 ) is small enough and an initial condition σ 0 ∈ L ∞ (Ω) satisfying (1.6), the solution σ of (3.4) satisfies σ(x, T 1 ) = 0 for all x ∈ Ω. (3.20) In fact, the condition (3.3) does not play any role. We shall thus prove a slightly more general result: there exists K 3 > 0, such that for any velocity field y such that y V 2,1 (QT 1 ) ≤ K 3 and any initial condition σ(x, t) = 0 on Σ in,y,∞ , σ(x, 0) = σ 0 in Ω, satisfies (3.20).
We will achieve this goal using two steps. In the first one, we shall consider smooth (∈ V 2,1 (Q T1 ) ∩ L 2 (0, T 1 ; W 1,∞ (Ω))) vector field y. In the second one, we will explain how the same result can be obtained for all vector fields y ∈ V 2,1 (Q T1 ).
The general case y ∈ V 2,1 (Q T1 ). We now discuss the case in which y does not satisfy the regularity (3.22) and y only belongs to V 2,1 (Q ∞ ) as stated in Theorem 3.5. In order to deal with this case, we use the density of V 2,1 (Q T1 ) ∩ L 2 (0, T 1 ; W 1,∞ (Ω)) in V 2,1 (Q T1 ). In particular, if y belongs to V 2,1 (Q ∞ ) and satisfies (3.31), we can find a sequence y n of functions of V 2,1 (Q T1 ) ∩ L 2 (0, T 1 ; W 1,∞ (Ω)) such that y n strongly converges to y in V 2,1 (Q T ) and for all n, y n V 2,1 (QT 1 ) < K 3 . Using then the previous arguments, we can show that for all n, σ n (x, T 1 ) = 0 for all x ∈ Ω, where σ n denotes the solution of (3.9) on the time interval (0, T 1 ). The strong convergence of (y n ) to y in V 2,1 (Q T1 ), hence of y n to y in L 1 (Q T1 ) and of y n · n to y · n in L 1 (Σ T1 ), and Lemma 3.4 then imply (3.20).
End of the proof of Theorem 3.5. We shall then show that, when y ∈ V 2,1 (Q ∞ ) satisfies the condition (3.31), the solution σ of (3.4) stays zero for times larger than T 1 . This is obvious, as one can replace (3.4) 3 by σ(x, T 1 ) = 0 on Ω and solve the Cauchy problem (3.4) in the time interval [T 1 , ∞) to obtain that σ is the trivial solution This concludes the proof of Theorem 3.5.
Remark 3.7. In the above proof, we have handled separately the case y ∈ V 2,1 (Q T1 )∩L 2 (0, T 1 ; W 1,∞ (Ω)) from the case of a general vector field y ∈ V 2,1 (Q T1 ), because the solution Ψ of (3.24) for a vector field y ∈ V 2,1 (Q T1 ) has a priori only Hölder regularity (see in particular [4,Theorem 3.7]), and thus cannot be used directly as a test function in the weak formulation (3.7) to obtain (3.27).
Remark 3.8. In general to prove the stabilizability of a non-linear problem it is usual to first study the stabilizability of the corresponding linear problem and then consider the non-linear term as a source term to obtain analogous stabilizability result corresponding to the complete non-linear system. But the reader may notice that contrary to the usual method we did not consider the non-linear term (y · ∇)σ (nonlinear in (σ, y) but linear in σ) as a source term while dealing with the system (3.4). This is because the transport equation has no regularizing effect on its solution, hence it is not possible to consider the non-linear term in (3.4) as a source term and to recover the solution in L ∞ (Q ∞ ).

Stabilization of the two dimensional Navier-Stokes equations.
Proof of Theorem 1.2. We will prove Theorem 1.2 using the Schauder fixed point theorem. We now discuss the strategy of the proof. (i) First we define an appropriate fixed point map. This will be done in Section 4.1.
(ii) Then we fix a suitable ball which is stable by the map defined in step (i). This is done in the Section 4.2.
(iii) In Section 4.3 we show that the ball defined in step (ii), is compact in some appropriate topology. We then prove that the fixed point map from step (i) in that topology is continuous. (iv) At the end we draw the final conclusion to prove Theorem 1.2.

4.1.
Definition of a fixed point map. Let us recall the fully non linear system (including the boundary controls) under consideration: in Ω, in Ω, where F (y, σ) = −e −βt σ ∂y ∂t − e −βt (y · ∇)y − e −βt σ(v s · ∇)y − e −βt σ(y · ∇)v s − e −2βt σ(y · ∇)y + βe −βt σy, and w c = (w 1 , ..., w Nc ). To prove the existence of a solution of the system (4.1) we are going to define a suitable fixed point map. Now assume that σ 0 ∈ L ∞ (Ω) and satisfies (1.6). Recall the definition of g j 's from (2.32). Let us suppose that y ∈ V 2,1 (Q ∞ ) satisfies (3.11) and on the boundary it is given in the following form where w c = ( w 1 , ..., w Nc ) ∈ H 1 (0, ∞; R Nc ). In addition the coefficients w c are assumed to be such that y satisfies the following boundary condition where the constant L was fixed in (1.5). We further assume that y 0 ∈ V 1 0 (Ω). We consider the following set of equations in Ω, in Ω, (4.4) where We also fix the constant K 3 appearing in Theorem 3.5. Let 0 < µ < K 3 . We define a convex set D µ as follows Notice that (0, 0) belongs to D µ , hence D µ is non-empty. Let ( σ, y, w c ) ∈ L ∞ (Q ∞ ) × V 2,1 (Q ∞ ) × H 1 (0, ∞; R Nc ) be the solution of system (4.4) corresponding to ( y, w c ) ∈ D µ . We consider the following map In the sequel we will choose the constant µ ∈ (0, K 3 ), small enough such that χ maps D µ into itself. We will then look for a fixed point of the map χ. Indeed if (y f , w f,c ) is a fixed point of the map χ, by construction, there exists a function σ f such that the triplet (σ f , y f , w f,c ) solves (4.1). Hence in order to prove Theorem 1.2 it is enough to show that the map χ has a fixed point in D µ .
Proof. We divide the proof in two steps.
As the trace operator is linear and bounded from H 2 (Ω) onto H 3/2 (Γ), y n | Σ∞ converges weakly to y | Σ∞ in L 2 (0, ∞; H 3/2 (Γ)). On the other hand as y n ∈ D µ , for each n where w n,c = (w n,1 , ..., w n,Nc ). Now since w n,c converges weakly to w c in H 1 (0, ∞; R Nc ) we have the following convergence in the sense of distribution Since the distributional limit and the weak limit (in the space L 2 (0, ∞; H 3/2 (Γ))) of y n | Σ∞ coincides, one at once obtains the expression (4.24) of y | Σ∞ . Also using the continuous embedding Hence one has the following by lower semi continuity of norm with respect to the above weak type convergence Hence y | Σ∞ satisfies (4.3). This finishes the proof ofȳ ∈ D µ .
This implies that for any T > 0 for all n ∈ N. Let ǫ > 0. Choose T ǫ > 0 such that So using (4.25) we have for all m, n ∈ N.
We know from Rellich's compactness theorem and Aubin-Lions lemma ( [2]) that the embedding of V 2,1 (Q Tǫ ) × H 1 (0, T ǫ ; R Nc ) into L 2 (0, T ǫ , L 2 (Ω) × R Nc ) is compact. Hence up to a subsequence (denoted by the same notation) {z n } n is Cauchy in L 2 (0, T ǫ , L 2 (Ω) × R Nc ). So it follows that there exists N 0 ∈ N such that for all natural numbers m, n N 0 , Now combining (4.26), (4.27) and a diagonal extraction argument, we can construct a subsequence {z n } n which is a Cauchy sequence in the Banach space L 2 (0, ∞, (1 + t) −1 dt; L 2 (Ω) × R Nc ). The proof is complete. Proof. The proof follows from the arguments used in proving Lemma 4.4 and is left to the reader.
Proof. Let { y n } n where y n = ( y n , w n,c ) be a sequence in D µ and assume that this sequence { y n } n strongly converges to y where y = ( y, w c ) in the norm L 2 (0, ∞, (1 + t) −1 dt; L 2 (Ω) × R Nc ).
As for all n ∈ N, y n V 2,1 (Q∞)×H 1 (0,∞;R Nc ) µ, up to a subsequence we have the following weak convergence Now corresponding to the vector field y n , let us denote by σ n the solutions to (4.4) 1 -(4.4) 3 . Similarly σ is the solution to (4.4) 1 -(4.4) 3 which corresponds to the vector field y. As y n converges strongly to y in the norm L 2 (0, ∞, (1 + t) −1 dt, L 2 (Ω)), for any T > 0, y n converges to y in particular in the norm L 1 (Q T ). Besides, the strong L 1 (Σ T ) convergence of y n · n towards y · n is obvious in view of the identities (4.2) and the strong convergence of w n to w in L 1 (0, T ), which immediately follows from the weak convergence of w n to w in H 1 (0, ∞). Hence from Lemma 3.4, we obtain that σ n strongly converges to σ in C 0 ([0, T ], L q (Ω)) for all 1 q < +∞. Due to the suitable choice of µ in Lemma 4.3, we can conclude from Theorem 3.5 (in particular from (3.12)) that each of σ n and σ vanishes for t T 1 . So σ n − −−− → n→∞ σ strongly in L ∞ (0, ∞; L q (Ω)) ∀ 1 q < +∞, ∀n ∈ N, σ n (t) = σ(t) = 0 for all t T 1 . Also from (4.7) and (4.18) we know that the L ∞ (Q ∞ ) norm of the sequence σ n is uniformly bounded.
We will now check that F ( y n , σ n ) converges weakly in L 2 (Q ∞ ) to F ( y, σ). As ( y n , w n,c ) ∈ D µ , from the estimate (4.8) we obtain a uniform bound for F ( y n , σ n ) L 2 (Q∞) . So there exists a subsequence of F ( y n , σ n ) which weakly converges in L 2 (0, ∞; L 2 (Ω)). This is therefore enough to show that the sequence F ( y n , σ n ) converges to F ( y, σ) weakly in D ′ (Q ∞ ) (i.e. in the sense of distribution). Let us first check the weak convergence of the term −e −βt σ n ∂ yn ∂t . From (4.30) we know that σ n strongly converges to σ in L 2 (Q ∞ ) and each of σ n and σ vanishes for all t T 1 (see (4.30)). Also from (4.29) we have that ∂ yn ∂t converges weakly to ∂ yn ∂t in L 2 (Q ∞ ). Hence their product σ n ∂ yn ∂t converges weakly to σ ∂ y ∂t in L 1 (Q ∞ ). So it is now easy to verify that e −βt σ n ∂ yn ∂t converges to e −βt σ ∂ y ∂t weakly in L 1 (Q ∞ ). Now we consider e −2βt ( y n · ∇) y n . As y n is bounded and weakly convergent to y in V 2,1 (Q ∞ ), using Lemma 4.5, we have e −2βt y n − −−− → n→∞ e −2βt y strongly in L 2 (Q ∞ ), (4.31) and ∇ y n ⇀ ∇ y in L 2 (Q ∞ ) as n → ∞. Therefore e −2βt ( y n · ∇) y converges to e −2βt ( y · ∇) y weakly in L 1 (Q ∞ ). Since y n converges weakly to y in V 2,1 (Q ∞ ), one has the following ∇ y n ⇀ ∇ y in L ∞ (0, ∞; L 2 (Ω)) ∩ L 2 (0, ∞; H 1 (Ω)) as n → ∞.
One observes that all the assumptions of Schauder fixed point theorem are satisfied by the map χ on D µ , endowed with the norm L 2 (0, ∞, (1 + t) −1 dt; L 2 (Ω) × R Nc ). Therefore, Schauder fixed point theorem yields a fixed point (y f , w f,c ) of the map χ in D µ . Hence the trajectory (σ f , y f , w f,c ) solves the non linear problem (4.1). Moreover, as a consequence of Theorem 3.5 the following holds σ f (., t) = 0 in Ω for t T 1 . for some positive constant C. Now in view of the change of unknowns (1.11), we obtain the existence of a trajectory (ρ, v) ∈ L ∞ (Q ∞ ) × V 2,1 (Q ∞ ) which solves (1.2) and satisfies the decay estimate (1.7). The proof of Theorem 1.2 is complete.

Further comments
Our result considers that the control u c is supported on Γ c , which is an open subset of the inflow part Γ in (see (1.5)) of the boundary. This is in fact natural to control the inflow boundary of the channel. At the same time we remark that our analysis applies if one wants to control the outflow boundary Γ out or the lateral boundary Γ 0 of the channel Ω. In what follows we briefly discuss these cases. (i) Controlling the outflow boundary. In this case the control zone Γ c is an open subset of Γ out . After the change of unknowns (1.11), one can imitate the linearization procedure (as done while transforming (1.13) into (1.14)). In this linearized system the transport equation modeling the density (1.14) 1 -(1.14) 3 will remain unchanged but the boundary conditions on the velocity equations (1.14) 4 -(1.14) 8 should be replaced by y = 0 on (Γ 0 ∪ Γ in ) × (0, ∞) and y = Nc j=1 w j (t)g j (x) on Γ out × (0, ∞). Still the proof of the boundary controllability of the Oseen equations can be carried in a similar way as done in Section 2 and in the same spirit of Corollary 2.17, one can prove that if the initial condition y 0 and the nonhomogeneous term f are suitably small then the inflow and the outflow boundaries of the perturbed vector field (v s + e −βt y) coincide with that of v s . Since the transport equation (1.14) 1 -(1.14) 3 remains unchanged in this case, the analysis done in Section 3 applies without any change. The fixed point argument done in Section 4 to prove the stabilization of the coupled system (1.2) also applies without change.
(ii) Controlling the lateral boundary. In this case the control zone Γ c is an open subset of Γ 0 . In particular we assume that Γ c ⊂ Γ b (where Γ b = (0, d)×{0} ⊂ Γ 0 ). Now the inflow and outflow boundaries of the velocity vector (e −βt y + v s ) cannot be characterized by using the notations Γ in and Γ out (as defined in (1.10)), since Γ c can contain an inflow part and an outflow part and one can not prove a result similar to Corollary 2.17. More precisely here we can use the following notations for time t > 0, In a similar way as we have obtained (1.14) from (1.2), one gets the following system One can use arguments similar to the ones in Section 2 in order to stabilize y solving (5.2) 4 -(5.2) 8 . The functions g j can be constructed with compact support in Γ b (imitating the construction (2.32)), and we can recover the C ∞ regularity of the boundary control and V 2,1 (Q ∞ ) regularity of y. Hence the flow corresponding to the vector field (e −βt y + v s ) is well defined in classical sense, consequently one can adapt the arguments used in Section 3 to prove that σ, the solution of (5.2) 1 -(5.2) 3 belongs to L ∞ (Q ∞ ) and vanishes after some finite time provided the initial condition σ 0 is supported away from the lateral boundaries and y is small enough. The use of a fixed point argument to prove the stabilizability of the solution of (1.2) is again a straightforward adaptation of the arguments used in Section 4.