LOCAL NULL CONTROLLABILITY OF A RIGID BODY MOVING INTO A BOUSSINESQ FLOW

. In this paper, we study the controllability of a ﬂuid-structure interaction system. We consider a viscous and incompressible ﬂuid modeled by the Boussinesq system and the structure is a rigid body with arbitrary shape which satisﬁes Newton’s laws of motion. We assume that the motion of this system is bidimensional in space. We prove the local null controllability for the velocity and temperature of the ﬂuid and for the position and velocity of rigid body for a control acting only on the temperature equation on a ﬁxed subset of the ﬂuid domain.


Introduction and main result.
Let Ω be a bounded, nonempty, open subset of R 2 with C 2 boundary that contains a rigid body and a viscous incompressible fluid. The domain of the rigid body is denoted by Sptq Ă Ω and it is assumed to be of class C 2 , compact, simply connected and with non-empty interior. The fluid domain is denoted by Fptq " ΩzSptq and it is assumed to be connected. Since, we assume that the structure is a rigid solid, we can describe Sptq with two functions t Þ Ñ hptq P R 2 and t Þ Ñ βptq P R through the formulas Sptq " S hptq,βptq , Fptq " F hptq,βptq . (1) In the above relations and in what follows, we write for any h P R 2 and for any β P R, S h,β " h`R β S and F h,β " ΩzS h,β , where S is a fixed subset of R 2 of class C 2 , compact, simply connected and with non-empty interior. In (2), R β is the rotation matrix, defined by R β "ˆc os β´sin β sin β cos β˙.
We assume that there exist h 0 P R 2 , β 0 P R such that S h0,β0 Ă Ω.
Without loss of generality, we can assume that the center of gravity of S is at the origin. In that case, hptq is the position of the centre of mass of the rigid body.
Let O be an open subset with O Ă Ω. The fluid-rigid body system is controlled by a force field supported in O and we suppose that O Ă Fptq.
In the above system, p upt, yq is the velocity field of the fluid, p ppt, yq denotes the pressure of the fluid and p θpt, yq is the temperature. Here ν ą 0 is the kinematic viscosity and µ ą 0 is the thermal diffusivity. For all x "ˆx 1 x 2˙P R 2 , we denote by x K , the vectorˆ´x 2 x 1˙. Moreover the boundaries of the rigid body and fluid domain are denoted by BSptq and BFptq respectively. The outward unit normal to BFptq is denoted by p npt, xq. The constants M and J are the mass and the moment of inertia of the rigid body. For the sake of convenience, we will assume that the rigid body is homogeneous with a constant density ρ S P R˚and thus we have M " ρ S |S|, J " ρ S ż S |y| 2 dy.
The Cauchy stress tensor is defined as: σpp u, p pq "´p pI 2`2 νDpp uq, where Dpp uq is the symmetric gradient:

LOCAL NULL CONTROLLABILITY OF A BODY-BOUSSINESQ FLOW SYSTEM 795
The state of system (4)- (13) is pp u, p p, p θ, h, βq and we want to emphasize the fact that the domains Fptq and Sptq are depending on the state and thus evolve through the dynamics induced by the system (10)- (11). This is one of the main difficulties in this problem: we are working on a non cylindrical domain and the spatial domain is unknown. A standard tool to handle this difficulty consists in using a change of variables in order to rewrite the system in a cylindrical domain. We need however to take care that such a change of variables is constructed from the state and this leads to some technical estimates on the coefficients coming from this transformation.
Several studies on the existence of weak solutions or strong solutions of fluidstructure interaction system have been published in recent years, usually without the equation on the temperature. The stationary problem was studied in Serre [39] and in Galdi [23]. An existence result of strong solutions in two or three dimension was proved in Grandmont and Maday [26] under the assumption that the inertia of the rigid body is large enough with respect to the inertia of the fluid. The existence and uniqueness of strong solutions in the case of a bounded domain has been proved in [40] without the hypothesis of [26] about the inertia of the rigid body. In the case of whole space, existence and uniqueness of strong solutions in two dimensions have been proved by Takahashi and Tucsnak [41] for an infinite cylinder and a similar result has been proved in three dimension by Silvestre and Galdi [24] for a rigid body having an arbitrary form. The question of existence of weak solutions has been investigated by many authors: [11], [7], [38], [15], [14], [28] etc. We can also mention a result on existence of weak solutions of the case where the fluid motion is modeled by the Boussinesq system: in [35], Nečasová proved the existence of weak solutions in three dimension for the problem of motion of one or several rigid bodies immersed in an incompressible non-Newtonian and heat-conducting fluid.
The controllability of the Navier-Stokes system has been the objective of considerable work over the last years. In the case of the two dimensional incompressible Navier-Stokes equations with the Navier slip boundary conditions, an approximate controllability result for boundary or distributed controls was proved by Coron in [8] and local exact controllability was established by Imanuvilov in [30]. In [18] and [31] the authors obtained the local exact controllability of the 2D or 3D Navier-Stokes equations with Dirichlet boundary condition with distributed controls supported in a small subset. They established a new Carleman inequality for the linearized Navier-Stokes system, which leads to null controllability and then they deduced a local result concerning the exact controllability. Fursikov and Imanuvilov established the local exact boundary controllability to the trajectories of the N dimensional Boussinesq system with N`1 scalar controls acting over the whole boundary and the local exact controllability to the same trajectories with N`1 scalar distributed controls when Ω is a torus in [20], [21], [22] by deducing a global Carleman estimate for the adjoint system. The techniques in [18] have been adapted in [27] to obtain the local exact controllability to the trajectories of the N dimensional Boussinesq systems with N`1 distributed scalar controls supported in subsets of the domain. In [25], the authors also establish same result as in [27] but via a method based on applying fictitious control on the divergence equation.
Here we want to emphasize that there have been many works in the literature where the authors deal with the controllability problem of Navier-Stokes type systems via reduced number of controls. In [19], the authors show that the N dimensional Navier-Stokes and Boussinesq systems can be controlled with only N´1 scalar controls under some geometrical assumptions on control domains. In [9], Coron and Guerrero established the null controllability of the N dimensional Stokes system with internal controls having one vanishing component with no condition imposed on the control domain. Local null controllability of the N dimensional Navier-Stokes and Boussinesq system with N´1 scalar controls in an arbitrary control domain has been obtained in [6], [5]. Here we want to mention that in [19], [5] for Boussinesq system, the authors obtained the local exact controllability result with two vanishing components of velocity control. Let us mention that in [33], Lions and Zuazua showed that three dimensional Stokes system is not necessarily null controllable with two vanishing components for the control even if the control is distributed on the entire domain. But in [10], local null controllability of the three dimensional Navier-Stokes system with a control distributed in an arbitrarily small nonempty open subset having two vanishing components has been proved by Coron and Lissy by using the return method and a Gromov method.
There are few articles in the last decade concerning the controllability results on fluid-structure interaction problem. In a paper of Raymond and Vanninathan [37], they considered a simplified model in 2D where the fluid equations are replaced by the Helmholtz equations and the motion of a solid represented by a harmonic oscillator. In that case, the domain is supposed to be fixed but one of the difficulties comes from the fact that there is no control in the solid part. They established exact controllability results for this model with an internal control only in the fluid part. In the work of Doubova and Fernández-Cara [12], they proved the local null controllability by boundary controls for a 1D model where point mass is immersed in a fluid which evolves in p´1, 1q. In that case, the domain is not fixed any more and the proof of the result is based on the global null controllability of the linearized system (by Carleman estimates) and on Kakutani's fixed point theorem. In [29], the authors established exact controllability of a 2D fluid-structure system where the body is a ball. In the paper of Boulakia and Osses [4], the authors dealt with the same problem as in [29], except that the body can have more general shape. In [3], Boulakia and Guerrero proved the local null controllability of a fluid-solid interaction problem in three dimension. Finally, in [34], the authors studied the local null controllability problem for the simplified one dimensional model considered in [12] and they managed to reduce the number of controls.
Our aim in this article is to control the fluid-structure system (4)- (13). More precisely, we want to control the position of the rigid body, the velocities of the fluid and of rigid body and the temperature of the fluid at a given time T ą 0. Our main result can be stated as follows: There exists ε ą 0 such that for every and we can find a control w 0 P L 2 p0, T ; L 2 pOqq such that the solution of (4)-(13) satisfies p upT,¨q " 0, h 1 pT q " 0, β 1 pT q " 0, and Observe that by using a translation and a rotation we can always assume that and thus Therefore in what follows, we assume (18).
Our main result consists of the local null controllability of a fluid-structure system in dimension two by applying a control only on the temperature equation. In our knowledge, there are no results on the controllability of fluid-structure interaction problems that deal with reduced number of controls (that is, the number of controls is less that the number of equations). We use the same change of variables and similar type fixed point argument as in [29]. But, unlike [29], we have considered the Boussinesq system and we are interested in the controllability via reduced number of controls. In [5], the author proved the local exact controllability of the N -dimensional Boussinesq system with internal controls having two vanishing components in velocity control and the main tool is to use a suitable Carleman inequality. We also prove the main result by showing a Carleman estimate. In our case, we have to incorporate some terms due to the presence of rigid body. This paper is organized as follows. In Section 2, we give the notation used in this paper and we recall some results. In Section 3, we introduce a change of variables to rewrite the problem (4)-(13) in a fixed spatial domain. In Section 4, we study the existence and regularity of a linearized problem in a fixed domain associated to our problem. Section 5 is devoted to establish a suitable Carleman inequality of the adjoint system of the linearized problem in a fixed domain. Then, in Section 6, we first give a link between controllability properties and Carleman estimates and then prove the controllability of an auxiliary linear system associated to (4)- (13). Finally, Section 7 is devoted to the proof of Theorem 1 where we use a fixed point procedure to obtain a solution of the nonlinear system.

Notation and preliminaries.
2.1. Notation. We set L 2 pΩq " L 2 pΩ; R 2 q, H 1 pΩq " H 1 pΩ; R 2 q and the same notation conventions will be used for trace spaces. We introduce the following spaces that we use frequently later on: }u} H 1,2 pp0,T qˆBF q "´}u} 2 H 1 p0,T ;L 2 pBF qq`} u} 2 We also define We recall that (see, for instance, [43, Lemma 1.1, p.18]) for any u P H 1 , there exist u P R 2 and ω u P R such that upyq " u`ωu y K , @ y P S.

Preliminaries.
Proof. Let us prove that is a norm of R 2 . It is enough to show the following implication: a`by 1 " 0 py 1 P BSq ùñ a " 0, b " 0.
We have BS Ă ty 1 P R | a`by 1 " 0u .
If b ‰ 0, then we obtain that BS is included in the line which is a contradiction. Thus b " 0, which implies a " 0 and consequently, (20) defines a norm of R 2 and we havë Lemma 2.2. Assume z P H 1 , with z " `ωy K in S. Then there exists a positive constant C independent of z, , ω such that If S is not a disk, we also have }z} L 2 pF q ě C|ω|.

LOCAL NULL CONTROLLABILITY OF A BODY-BOUSSINESQ FLOW SYSTEM 799
Proof. Using Theorem 1.2 p.9 in [42], there exists C such that First let us consider the case where S is a disk. Then, using that the center of S is 0, relation (21) writes Let us show that Þ Ñ } ¨n} H´1 {2 pBSq is a norm of R 2 . Indeed assume ¨n " 0 on BS.
If ‰ 0, there exists a point of BS such that n " {| | and thus, ¨n " | | ‰ 0. Thus we conclude from (22) that If S is not a disk, let us prove that is a norm of R 3 . We want to prove the following implication: p `ωy K q¨n " 0 py P BSq ùñ " 0, ω " 0. This is equivalent to show pa`byq¨τ " 0 py P BSq ùñ a " 0, b " 0.
Let us introduce f pyq :" a¨y`b |y| 2 2 . Then, Bf Bτ pyq " pa`byq¨τ for any y P BS. If pa`byq¨τ " 0 for any y P BS, then it implies that there exists c P R such that f pyq`c " 0 for any y P BS. This yields The set in the right-hand side is either empty, a point, a line, a circle or R 2 . The last case is the only one possible and it is equivalent to a " 0 and b " 0. Thus we conclude from (21) that }z} L 2 pF q ě Cp| |`|ω|q.
3. The change of variables.

3.1.
Construction of the change of variables. Assume Sptq is defined by (1) and S Ă Ω. We also take a control region O such that The above assumptions imply that distpS, Oq ě d 0 and distpS, BΩq ě d 0 for some d 0 ą 0. Then we can easily prove the following result: then distpS h,β , Oq ě d0 2 and distpS h,β , BΩq ě d0 2 .

ARNAB ROY AND TAKÉO TAKAHASHI
Taking ε ă c 0 in (15), we deduce that We want to construct change of variables X : Ω Ñ Ω that transforms F onto Fptq and S onto Sptq. Thus we can define X pt, yq " y`kpyqrhptq`R βptq y´ys, t P p0, T q, y P Ω.
Here k : Ω Ñ R is a smooth function such that kpyq " for c small enough.
With the above choices, ‚ in a neighborhood of S, X pt, yq " hptq`R βptq y, and thus X pt, Sq " Sptq. ‚ in a neighborhood of BΩ and of O, X pt, yq " y. Let the inverse of X pt,¨q is denoted by Ypt,¨q. Observe that, in a neighborhood of Sptq, we have Ypt, xq " R´β ptq px´hptqq.
Here CofpM q is the cofactor matrix of M , which satisfies M pCofpM qq˚" pCofpM qq˚M " detpM q Id .
We transform (4)-(13) by using this change of variables. Such a calculation is already done in [1] except for the temperature equation. We give here only the part of the calculation that corresponds to the temperature equation and we refer to [1] for the calculation of the other equations. From (29), we have: In order to transform the Neumann boundary condition (9), we also need to rewrite the exterior normal to BFptq. Let us denote by n the exterior normal to BF. Then, p n " n on BΩ, and p npt, xq " R βptq npR´β ptq px´hptqqq x P BSptq.
In a neighborhood of Sptq, Ypt, xq " pR´β ptq px´hptqqq and in a neighborhood of BΩ, Y " Id. Thus on BSptq, and on BΩ, B p θ Bn " Bθ Bn pYq.
Thus, we can rewrite the system (4)-(13) as: div u " 0, in p0, T qˆF, upt, yq " 0, t P p0, T q, y P BΩ, upt, yq " ptq`ωptqy K , t P p0, T q, y P BS, up0, yq " u 0 pyq and θp0, yq " θ 0 pyq, y P F, hp0q Here we want to underline the fact that the linear and nonlinear operators rK u s,rN u s, rL u s, rG u s, rN θ s, rL θ s depend on h and β and the operators rM u s, rM θ s depend on h, β, , ω through the change of variables X and its inverse Y. The definitions of the operators are given through the following formulas: We have set 4. Some linear systems. In this section we analyze two linear systems associated with (36)-(47): upt, yq " 0, t P p0, T q, y P BΩ, upt, yq " ptq`ωptqy K , t P p0, T q, y P BS, up0, yq " u 0 pyq and θp0, yq " θ 0 pyq, y P F, and upt, yq " 0, t P p0, T q, y P BΩ, upt, yq " ptq`ωptqy K , t P p0, T q, y P BS, For both systems, we extend u and r f to Ω by setting: upt, yq " ptq`ωptqy K , @pt, yq P p0, T qˆS, In particular, u is a rigid velocity in S, that is Dpuq " 0 in p0, T qˆS. We recall that H 1 is defined by (19). We set We consider the inner product on L 2 pΩqˆL 2 pFq defined by The corresponding norm is equivalent to the usual norm in In order to work with (58)-(70), we use an approach based on semigroups. We define: and DpAq " DpA 1 qˆDpA 2 q.
(82) For all u P DpA 1 q, we set where P is the orthogonal projector from L 2 pΩq onto H 1 .

ARNAB ROY AND TAKÉO TAKAHASHI
We also define for θ P DpA 2 q, It is shown in [41, Proposition 4.2] that A 1 is a self-adjoint, maximal dissipative operator. It is also well-known that A 2 is a self-adjoint, maximal dissipative operator. Thus, using a perturbation argument (see [36, Corollary 2.2, Chapter 3, page 81]), we deduce the following result: Proposition 4.1. The operator pA, DpAqq defined by (85) is the generator of an analytic semigroup on H. Its adjoint is given by DpA˚q " DpAq and Observe that As a consequence of Proposition 4.1, and by using the isomorphism theorem (see, for instance, [2, Theorem 3.1, p. 143]), we have the following result: Let T ą 0 and r f P L 2 p0, T ; L 2 pFqq, r g P L 2 p0, T ; L 2 pFqq, w 0 P L 2 p0, T ; L 2 pOqq r h p1q P L 2 p0, T ; R 2 q, r h p2q P L 2 p0, T ; Rq, u 0 P H 1 pFq, θ 0 P H 1 pFq be such that: div u 0 " 0 in F, u 0 pyq " 0 on BΩ, u 0 pyq " 0`ω0 y K for y P BS. Then the linear system (58)-(70) admits a unique solution pu, p, θ, , ωq with Moreover, the solution pu, p, θ, , ωq satisfies the following estimate: In what follows, we also need some properties of the linear system (71)-(78) that we can write as Corollary 4.3. Let T ą 0 and r f P L 2 p0, T ; L 2 pFqq, u 0 P H 1 pFq such that: Then the linear system (71)-(78) admits a unique solution pu, p, , ωq with Moreover, the solution pu, p, , ωq satisfies the following estimate: 5. The carleman inequality. Let us introduce the adjoint system of (58)-(70): φpT, yq " φ T pyq and ψpT, yq " ψ T pyq, y P F, In this section, our aim is to establish a suitable Carleman estimate for the adjoint system (93). Let us introduce the weight functions used for this estimate.
Let us consider η P C 2 pFq satisfying η " 0 on BF and Bη Bn where Such functions are standard for Carleman estimates. Let us give some properties that are used in what follows: ξ M ptq ď Cξ m ptq pt P p0, T qq, for some positive constants C depending on T and on λ. Now, we can state the following Carleman inequality: Theorem 5.1. Let T ą 0 and O be a nonempty open subset such that O Ă F. Then there exists a constant λ 0 ą 0 such that for any λ ě λ 0 there exist constants Cpλq ą 0 and s 0 pλq ą 0 such that for all f P L 2 p0, T ; L 2 pFqq, g P L 2 p0, T ; L 2 pFqq, h p1q P L 2 p0, T ; R 2 q, h p2q P L 2 p0, T ; Rq and for all φ T P H 1 , ψ T P L 2 pFq, T P R 2 , ω T P R satisfying φ T " T`ωT y K in S, the solution of (93) satisfies the inequality: for all s ě s 0 .
Proof. In this proof, we follow similar ideas as in [9] and [5]. Throughout the proof, C stands for a positive constant depending only on F, O and η.
First, the proof of the above estimate is done by density, for more regular solutions. More precisely, we can assume that where we have as usual extended f and φ T in S by respectively h p1q`hp2q y K and T`ωT y K . In that case, our solution satisfieŝ φ ψ˙P L 2 p0, T ; DppA˚q 2 qq X H 2 p0, T ; Hq.
Step 1. decomposition of the solution of (93). Let pφ, q, ψ, φ , ω φ q be the solution to (93). We set The function ρ is C 8 pr0, T sq and for any k P N, From (105) and (106), we deduce the following relations and We then consider the following decomposition

ARNAB ROY AND TAKÉO TAKAHASHI
where pv, p v , v , ω v q, pz, p z , z , ω z q and r ψ satisfy the following systems : and Note that since we have z P L 2 p0, T ; DpA 2 1 qq X H 2 p0, T ; H 1 q.
Step 2. Carleman estimates for heat equation, Laplace and Gradient operators First we apply the divergence operator to the first equation of (114) and we deduce that ∆q z " 0. Then we apply the operator ∇∆ " p B By1 ∆, B By2 ∆q to the first equation of (114) satisfied by z 2 and we obtaiń Bp∇∆z 2 q Bt´∆ p∇∆z 2 q " ∇p´ρ 1 ∆φ 2 q in p0, T qˆF.
This means that ∇∆z 2 satisfies a heat equation with nonhomogeneous boundary conditions. For such an equation, we have the following Carleman estimates, obtained in [32]: there exists C ą 0, λ 0 ą 0, s 0 ą 0 such that for any λ ě λ 0 , s ě s 0 Now by using a Carleman estimate on the gradient operator (see [9, Lemma 3]) on ∆z 2 there exist λ 1 , s 1 , C such that for λ ě λ 1 and s ě s 1 . Let Then we can use a Carleman estimate for the Laplace operator (see for instance [3]). We recall the proof of such an estimate in the appendix (Corollary A.2).
On BS, we have Using Lemma 2.1, we have

ARNAB ROY AND TAKÉO TAKAHASHI
On the other hand, there exists a constant depending only on BS such that Combining (101), (121), (122) and (123) we deduce for λ ě λ 3 and s ě s 3 . We set We recall a standard Carleman estimate for equation (115) (see, for instance [16] Let us introduce the following quantities and Gathering (119), (120), (124), (126) and the above definitions, we deduce Step 3. recovering z 1 and z¨e1 Using that z " 0 on p0, T qˆBΩ and that the domain Ω is bounded, we can apply the Poincaré inequality Combining the above estimate with the fact that div z " 0, we deduce Using Lemma 2.2, we have Step 4. estimate of B 3 Here (110) and (112) allow us to write By applying Corollary 4.3 on system (113), we have Using (108) and applying estimate (135), we deduce From the above estimate, (101) and (134), we obtain Similarly, by using (101), (110), (112) and (135) Adding (136) and (137), we deduce Step 5. estimate of B 1 We recall here a technical lemma that is obtained in [6, Step 3, Section 2.1]: There exist constants λ 5 , s 5 and C depending on F, O 0 , O 1 such that for every s ě s 5 , λ ě λ 5 , ε ą 0
Combining the above estimate and (141), we deduce We now estimate the right-hand side of (142). Let us write p z " e´s α M pξ m q´1 4 z, p q z " e´s α M pξ m q´1 4 q z ; (143) Since pz, q z , z , ω z q satisfies (114), pp z, p q z , p z , x ω z q is the solution of the following system p zpt, yq " 0, t P p0, T q, y P BΩ, p zpt, yq " p z ptq`x ω z y K , t P p0, T q, y P BS, where Note that if we extend F p4q by F p5q`F p6q y K for y P S, we have from (116) and (117) that F p4q P L 2 p0, T ; DpA 1 qq X H 1 p0, T ; H 1 q.
Hence by above estimate and (140), (142), we get Step 7. going back to φ, φ , ω φ By taking s large enough, from (139) we can conclude that: Again by using (108), (112), (135), (150), (156) and (161), for all λ ě λ 7 , s ě s 7 , we have Step 8. removing the local term in φ 2 We are going to estimate the last term of inequality (162) by following the same approach as in [5]: Consider a non-negative function χ P C 2 c pOq such that χ " 1 in O 1 . Now by using equation (115), we get Our main aim is to estimate the local integrals of r ψ and g. Then via integration by parts and Young's inequality, we obtain that for any ε ą 0, there exists C ą 0 such that 2 | 2 dy dt`r Ips, ρφ, ρ φ , ρω φ q¸(163) 820 ARNAB ROY AND TAKÉO TAKAHASHÌ C˜s 12 Thus finally from (162) and (163), we get We have finished the proof of Proposition 5.1.
6. Null controllability of the linearized system. In this section, we use the Carleman estimate obtained in Theorem 5.1 to deduce the null controllability of a linear system associated with (36)-(47). We recall that H is defined in (79) and the operator A is defined in (80)-(85). We define the control operator B P LpL 2 pOq, Hq as and the operator C P LpH, R 3 q is defined as Cpu, θq " p u , ω u q, if u " u`ωu y K in S.
Let us fix s ě s 0 , λ ě λ 0 as in Theorem 5.1 and consider ρ i for i P t1, 2, 3u and r ρ in the following way and r ρptq " Thus ρ i and r ρ are continuous functions such that ρ i pT q " 0 and ρ i ą 0 in r0, T q, r ρpT q " 0 and r ρ ą 0 in r0, T q.
We define the following spaces Our main result here is the following Theorem 6.1. There exists a linear bounded operator such that for any pZ 0 , d 0 , F q P HˆR 3ˆF , the control w 0 " E T ppZ 0 , d 0 , F qq is such that the solution pZ, dq to equation (165) satisfy Z P Z and dpT q " 0. Moreover, if we assume that Z 0 P Dpp´Aq 1 2 q, then we have Z r ρ P L 2 p0, T ; DpAqq X Cpr0, T s; Dpp´Aq and we have the following estimate:ˇˇˇˇˇˇˇZ r ρˇˇˇˇˇˇˇˇL 2 p0,T ;DpAqqXCpr0,T s;Dpp´Aq Proof. We use [29,Theorem 4.1]: the existence of E T is obtained from the following observability inequality for adjoint equation (166): We thus prove the above estimate and this gives us the existence of E T and the second part of the theorem. Indeed, using [29,Corollary 4.3], this second part comes from the following relations pr ρq 1 ρ 2 pr ρq 2 P L 8 p0, T q and that can be obtained from the definition of functions (167)-(170) and from the relations (101)-(108). It remains to prove (173). First, we notice that (166) can be written in the following form: φpt, yq " 0, t P p0, T q, y P BΩ, φpt, yq " φ ptq`ω φ ptqy K , t P p0, T q, y P BS, Jω 1 φ ptq "´ż BS y K¨σ pφ, qqn dΓ`Jω γ 1`J ω γ 2 , t P p0, T q, φpT, yq " 0 and ψpT, yq " 0, y P F, where γ 1 " pγ 1 1 , γ 1 2 q P H 1ˆL 2 pFq and γ 2 " p γ 2 , ω γ 2 q P R 3 . In particular, we have γ 1 pt, yq " γ 1 ptq`ω γ 1 ptqy K t P p0, T q, y P S.
With the above notation, the condition (173) can be rewritten as The proof of (176) is based on Theorem 5.1. We set and then, (109) implies that Then by following similar steps as in [5, Lemma 3.2] (using in particular the energy estimates), we can deduce from the above estimate In order to prove (176) from the above estimate, it is sufficient to show the following inequality: We argue by contradiction: assume that (179) is false. Then there exists a sequence pγ 2 n , γ 1 1,n , γ 1 2,n , φ n , ψ n q such that (175) holds and such that Writing Φ n " pφ n , ψ n q, we have "´9 Φ n ptq " A˚Φ n ptq`γ 1 n ptq`C˚γ 2 n , Φ n pT q " 0.
Thus we have established inequality (179) and combining this inequality with (178), we have proven (176). 7. The nonlinear problem. This section is devoted to the proof of the main result.
7.1. Estimates of the nonlinear terms. In this section, we give some estimates on the coefficients appearing in the system (36)-(47).
We assume here that h and β satisfy hpT q " 0, βpT q " 0, With our choice of r ρ (see (170) and (97)), we deduce in particular that Following the proofs of [1,Proposition 12] and [1,Lemma 31], we obtain the following estimates Lemma 7.1. Assume (26). Then, for any pu, p, θq P H 2 pFqˆH 1 pFqˆH 2 pFq, the following relations holds for a.e. t P p0, T q: Since we will use the Banach fixed point theorem, we also need to estimate the differences of coefficients. More precisely, let us consider, for i " 1, 2, h piq and β piq that satisfy h piq pT q " 0, β piq pT q " 0, pph piq q 1 , pβ piq q 1 q r ρ P L 2 p0, T q.

ARNAB ROY AND TAKÉO TAKAHASHI
With our choice of r ρ (see (170) and (97)), we deduce in particular that We assume that for all i, h piq and β piq satisfy (26). In particular we can define the change of variables X piq , Y piq , and the operators Following the proof of [1,Lemma 33], we obtain the following estimates of the difference of coefficients: Lemma 7.2. For any pu, p, θq P H 2 pFqˆH 1 pFqˆH 2 pFq, the following relations hold for a.e. t P p0, T q: 7.2. The fixed point argument. We are now in position to prove the main result.
From (14), (88) and the properties of X and Y (Section 3), we can check that where A is defined by (80)-(82).
The proof of Theorem 1 is based on a fixed point argument. If we set Z "ˆu θ˙, Zptq " AZptq`Bw 0 ptq`r F pZ, dq, 9 dptq " CZptq, with r F pZ, dq "ˆP f 2 rM θ θs´rN θ pu, θqs`µrpL θ´∆ qθs˙, where #´r pK u´I2 q Bu Bt s´rM u us´rN u us`νrpL u´∆ qus`rp∇´G u qps in F, Now from Theorem 6.1, we know there exists a control w 0 " E T pZ 0 , d 0 , F q such that the solution of Zptq " AZptq`Bw 0 ptq`F ptq, satisfies (171) and (172). Let us consider r ą 0 (that is fixed later) and let us set If }Z 0 } Dpp´Aq 1{2 q ď r and }d 0 } R 3 ď r, F P K r , We take r small enough so that (24) and (26) holds true and we can construct the change of variables X and Y as in Section 3. We can thus define where r F pZ, dq is given by (194)-(195). By using Lemma 7.1, (199) and r ρ 2 ρ1 P L 8 p0, T q, we can verify that T : and In particular for r small enough, T maps K r to K r . Similarly, by using Lemma 7.2 and (199), we deduce that and thus for r small enough, T admits a unique fixed point F in K r . The corresponding solution of (196) is the solution of (193) and satisfiešˇˇˇˇˇˇˇZ r ρˇˇˇˇˇˇˇˇL 2 p0,T ;DpAqqXCpr0,T s;Dpp´Aq and dpT q " 0. In particular, we obtain (16)- (17).
Appendix A. Carleman estimates for the Laplace operator. In this section, we recall a Carleman estimate for the Laplace equation. We give the proof of such an estimate for completeness.
Then there exist constants λ 1 , κ 1 , C depending only on F, O 0 , O 1 such that, for any λ ě λ 1 , κ ě κ 1 and u P H 2 pFq, the following inequality holds: Proof. We follow the same steps as [21] and [17] but here we incorporate the boundary terms. Let us set f "´∆u and σ " e κζ u and g " e κζ f.

ARNAB ROY AND TAKÉO TAKAHASHI
The next term of (208) that we estimate is: We first estimate the quantities C 2 and C 3 : and Since λ ą 1, We have the following estimate on C 4 : Finally, we estimate I 22 : dΓ .
Thus, we have obtained the estimate (205).
From the above proposition, we deduce the following result.
Now if we multiply the above inequality by exp´´2s e 2λ}η} L 8 Eptq 8¯a nd integrate from 0 to T , we obtain (234).