Local exact controllability to trajectories of the magneto-micropolar fluid equations

In this paper we prove the exact controllability to trajectories of the magneto-micropolar fluid equations with distributed controls. We first establish new Carleman inequalities for the associated linearized system which lead to its null controllability. Then, combining the null controllability of the linearized system with an inverse mapping theorem, we deduce the local exact controllability to trajactories of the nonlinear problem.


Introduction and statement of main results.
Let Ω be a bounded connected domain in R d , d ∈ {2, 3}, whose boundary ∂Ω is regular enough. Let T > 0 and we will use the notations Q = Ω × (0, T ), Σ = ∂Ω × (0, T ), and we denote by n(x) the outward unit normal to ∂Ω at the point x ∈ ∂Ω.
Here we have used the following notations: In the case d = 2, we denote curla = ∂ x1 a 2 − ∂ x2 a 1 for a vector function a = (a 1 , a 2 ), and curlb = (∂ x2 b, −∂ x1 b) for a scalar function b.
In the case d = 3, we denote for a vector function a = (a 1 , a 2 , a 3 ). In this work, the control function acting on the equations satisfied by the magnetic B is assumed to have the form P (v1 O ) = v1 O + ∇χ, for some χ ∈ L 2 (0, T ; H 1 (Ω)). ( This form of the control v has been also considered in recent works on the local exact controllability of the MHD system [4,5,18,19]. There is only a recent result on the controllability of MHD system [3] in which the control acting on the magnetic field has support in an arbitrarily small open subset of the spatial domain, i.e., the control has the form 1 O P O (v1 O ), where P O is the classical Helmholtz projector related to O (i.e., the orthogonal projection operator from L 2 (O) d onto the completion of the set {v ∈ C ∞ 0 (O) d | ∇ · v = 0 in O} in the norm of L 2 (O) d . However, since the boundary conditions on the magnetic field in our system is different from that in [3], so here we cannot use ideas in [3] to establish our Carleman estimate for the component C of the adjoint system respectively to the magnetic field. Hence, we are not able to get an estimate of the right-hand side of the component C having the form O e −2sα ξ 3 |P O C| 2 dxdt as in [3]. So we only obtain the controllability of (1) with the control function acting on the magneto field has the form (2). The controllability of (1) with the control function acting on the magneto field has the form 1 O P O (v1 O ) remains an open question.
The magneto-micropolar fluid is a model of fluids in which micro-structures of the fluid and its electronic-magnetic properties are taken into account. In the past years, there have been a number of works devoted to studying mathematical questions related to the magneto-micropolar fluid equations. The existence and uniqueness of weak/strong solutions to (1) were studied in [8,14,25,27,28]. The regularity and blow-up criterion of solutions were studied in [13,23,33,35]. Besides, the long-time behavior of solutions was investigated in [1,6,21,22,24,29]. However, to the best of our knowledge, there is no work on the controllability of the magneto-micropolar fluid equations. This is the motivation of the present paper. Because here we focus on the controllability, we have omitted some physical constants in this model.
It is noticed that the magneto-micropolar fluid equations contain the micropolar equations (when B = 0), the MHD equations (when ω = 0), the Navier-Stokes equations (when B = 0 and ω = 0) as particular cases. The local exact controllability of the Navier-Stokes equations has been studied extensively in many works, see e.g. [10,12,26] and references therein. In recent years, the local exact controllability of the MHD system was also studied by a number of authors in [3,4,5,18,19], and that of the micropolar fluid equation was studied in [9,17].
To study system (1), we use the following function spaces H = y ∈ L 2 (Ω) d | ∇ · y = 0 and y · n = 0 on ∂Ω LOCAL CONTROLLABILITY TO TRAJECTORIES   359 with the norm The main question considered in this paper is that whether (1) is locally exactly controllable to the trajectories.
As long as the initial conditions are concerned, we will assume that We are now ready to formulate the main results in the present paper. First, the result in the case of two dimensions is given in the following theorem.
The following theorem is the result in the case of three dimensions.
Remark 1. From the above theorems, by taking ω = 0 and B = 0 we recover the local exact controllability result in [26] for Navier-Stokes equations, which improved the previous results in [10] and references therein. Moreover, by taking B = 0 only, we improved the previous result on local exact controllability to trajectories of the micropolar fluids in [9] in the sense that a weaker regularity of the given trajectory and initial data is required.
To do this, we will follow the strategy introduced by Fursikov and Imanuvilov [12] in the context of Navier-Stokes equations. Let us consider the linearized system around (y, ω, B): where f 1 , f 2 and f 3 are functions that decay exponentially to zero as t → T − . We will prove that, under appropriate assumptions for f 1 , f 2 and f 3 , these above linear system (8) is null controllable. After that, combining the null controllability of (8) with an inverse mapping theorem, it will lead to the local null exact controllability of (7).
A basic tool for proving the null controllability of (8) is a global Carleman inequality for solutions to the following associated adjoint system Here we have used the notations D s := ∇ + t ∇ and D a := ∇ − t ∇. In (9), the pressure functions are π, r.
To obtain the above main results, which particularly improve some recent related results, we have to establish new necessary Carleman inequalities. This is in fact the main contribution of our paper.
Let us explain the method used to construct our Carleman inequality. Firstly, using the Carleman estimate in [20,Theorem 4.1] (see also in [26,Theorem 3.4]) for the Stokes system with suitable f , we get the global integral estimates for the component ϕ in both cases d = 2 and d = 3. Since the magneto field has the homogeneous Dirichlet condition and the equation satisfying the magneto field has an addition pressure, then the global integral estimates for the component C can be established as same as the estimates for the component ϕ. The global integral estimate for the component ψ is obtained separately in two cases d = 2 and d = 3. In the case d = 2, we can use the Carleman inequality directly for the heat equation to the component ψ to get the estimate for ψ. However, in the case d = 3, we cannot use the Carleman inequality directly for the heat equations to the component ψ since the equation satisfying by ψ has the term ∇(∇ · ψ). To overcome this difficulty, we exploit some ideas in [17] by using the Carleman inequality [20, Theorem 2.2] for the nonhomogeneous heat equations with suitable powers of the weight functions. Then, we can establish our new Carleman estimates with slightly weaker requirement of the regularity of the trajectory as that in the case of micropolar fluid equations [9,Propositon 4].
The paper is organized as follows. In Section 2, we establish new Carleman inequalities for the solutions to the adjoint linearized system. Section 3 is devoted to proving Theorem 1.1 and Theorem 1.2. We first use the new Carleman inequality to prove the null controllability of the linearized system, then the conclusion of the proof of the main results is obtained by combining the null controllability of the linearized system and an inverse mapping theorem. In the Appendix we recall some well-known Carleman inequalities which are used in the proof.

2.2.
Proof of Carleman inequalities. We will prove Theorem 2.1 and Theorem 2.2 in several steps.
Step 1. Estimation of global terms ϕ and C: Notice that the system for the components ϕ (and C) in the adjoint system (9) can be viewed as the Stokes system (43) in the Appendix with t replaced by T − t and f = (D s ϕ)y − (D a C)B + curlψ + ( t ∇ψ)ω + g 1 (and f = −(D s ϕ)B + (D a C)y + g 3 ). So, applying Lemma 4.3 in the Appendix to components ϕ (and C) in (9) we get some positive constants s 0 ≥ 1, λ 0 ≥ 1 and C > 0 such that for any s ≥ s 0 and λ ≥ λ 0 , where we have used the fact that |curlϕ| 2 ≤ C|∇ϕ| 2 and |curlψ| 2 ≤ C|∇ψ| 2 . Therefore, taking λ ≥ max{λ 0 , C( y ∞ + B ∞ )}, we have from (14) that Step 2. Estimation of global term ψ: We will consider two cases: Case d = 2. Using the Carleman estimate (40) in the Appendix for ψ in (9) with d = 2, we deduce that for s ≥ C(T 3 + T 4 ) and λ ≥ C.
Case d = 3. We apply the divergence operator to the equation satisfied by ψ in (9) with d = 3 to deduce that Thus, we apply the Carleman estimate (42) in the Appendix for the equation (17) with different powers of ξ. More precisely, we apply that Carleman inequality to s 1/2 ξ 1/2 ∇ · ψ and we get that On the other hand, since ψ satisfies the system then using the Carleman (40) in the Appendix for ψ in (19), we deduce that for s ≥ C(T 3 + T 4 ) and λ ≥ C. Combining (18) and (20) for s ≥ max{s 0 , C(T 3 + T 4 )} and λ ≥ max{λ 0 , C(1 + y ∞ )}. Furthermore, integrating by parts and using the Cauchy inequality, we get for any ε > 0. Hence, choosing ε sufficiently small, one infers from (21) that We now estimate the trace terms. From the definition of · , we have where σ 1 := s 1/4 (ξ * ) 1/4 e −sα * .
We see that σ 1 ψ satisfies in Ω.

Conclusion of Theorem 2.2. Combining
3. Proof of the main results. In this section, we will give the proof of Theorem 1.2, i.e. the result in the case of three dimensions. The proof of Theorem 1.1 (the result in the case d = 2) is very similar to that in the case d = 3, so it is omitted here.
3.1. Null controllability for the linear system (8). We now prove the null controllability for the system (8) and this will be crucial when proving the local controllability of (1) in the next subsection.
We can rewrite problem (8) as follows where L( y, ω, B) = (L 1 ( y, ω, B), L 2 ( y, ω), L 3 ( y, B)) with We would like to find the controls (u, w, v) such that the solution ( y, ω, B) to (26) satisfies We first deduce the Carleman inequality with weight functions that do not vanish at t = 0. More precisely, let us consider the functioñ and we define new weight functions We will prove the following lemma.
Lemma 3.1. Let s and λ be like in Theorem 2.2. Then there exists a positive constant C 0 depending on T, s and λ, such that every solution (ϕ, ψ, C) of (9) satisfies Proof. The proof of this lemma is similar to those in some recent works on the controllability of the fluid models (see for instance [16]). More precisely, this lemma is a consequence of (13) and energy estimates satisfied by solutions of (9). In what follows, we only give the sketch of the proof. We introduce a function ϑ ∈ C 1 ([0, T ]) such that Then (ϑϕ, ϑψ, ϑC) satisfies in Ω. (29) Multiplying (29) 1 by ϑϕ, (29) 2 by ϑψ, (29) 3 by ϑC, then integrating over Ω and using the Cauchy inequality, there exists a positive constant C depending on (30) So, from inequality (30) we get the energy estimate This implies that From the last inequality and the fact that we have
We will prove the following result.

3.2.
Local controllability of the semilinear problem. In this subsection we give the proof of Theorem 1.2 by using similar arguments as in pioneering works [10,16]. We will use the following inverse mapping theorem (see [2]).
Theorem 3.2. Let B 1 and B 2 be two Banach spaces and let A : B 1 → B 2 satisfy A ∈ C 1 (B 1 ; B 2 ). Assume that b 1 ∈ B 1 , A(b 1 ) = b 2 and that A (b 1 ) : B 1 → B 2 is surjective. Then, there exists ε > 0 such that, for every b ∈ B 2 satisfying b − b 2 B2 < ε, there exists a solution of the equation In our setting, we use this theorem with the spaces 3 . Then, we consider the operator A( y, p, ω, B, u, w, v) = A 1 ( y, p, ω, B, u), A 2 ( y, ω, w), To apply Theorem 3.2, we first check that the operator A is of class C 1 (B 1 , B 2 ). Indeed, all terms arising in the definition of A are linear (and consequently C 1 ), , ( y · ∇) ω, and ( y · ∇) B − ( B · ∇) y. However, the operators So, it follows from (39) that ( y · ∇) y − ( B · ∇) B + 1 2 ∇( B · B) belongs to the class of C 1 .

CUNG THE ANH AND VU MANH TOI
So, from (39) we have that ( y · ∇) ω belongs to the class of C 1 .
The nonlinear term: ( y · ∇) B − ( B · ∇) y: We have the same estimates as in term ( y · ∇) y − ( B · ∇) B. Hence, this term belongs to the class of C 1 . Therefore, we have proved that A ∈ C 1 (B 1 , B 2 ) with In view of the null controllability result for the linearized system (8) given in Proposition 1, we can see that A (0, 0, 0, 0, 0, 0, 0) is surjective.
4. Appendix: Some well-known Carleman estimates. With the weight functions α and ξ defined in (11), we now recall some well-known Carleman estimates, which have been used in our proofs above. for any s ≥ C(T 3 + T 4 ) and any λ ≥ C.
Consider the equation where F 0 , F 1 , F 2 , F 3 ∈ L 2 (Q). Then we have the following result.
Recall here that Let us now consider the following Stokes system with z 0 ∈ V and f ∈ L 2 (0, T ; L 2 (Ω) d ). Then we have the following result for solutions to (43).