Local controllability of the N-dimensional Boussinesq system with N-1 scalar controls in an arbitrary control domain

In this paper we deal with the local exact controllability to a particular class of trajectories of the N-dimensional Boussinesq system with internal controls having 2 vanishing components. The main novelty of this work is that no condition is imposed on the control domain.


Introduction
Let Ω be a nonempty bounded connected open subset of R N (N = 2 or 3) of class C ∞ . Let T > 0 and let ω ⊂ Ω be a (small) nonempty open subset which is the control domain. We will use the notation Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ).
In [5], the authors proved that local exact controllability can be achieved with N − 1 scalar controls acting in ω when ω intersects the boundary of Ω and (1.4) is satisfied. More precisely, we can find controls v 0 and v, with v N ≡ 0 and v k ≡ 0 for some k < N (k is determined by some geometric assumption on ω, see [5] for more details), such that the corresponding solution to (1.1) satisfies (1.3).
Remark 2. It would be interesting to know if the local controllability to the trajectories with N − 1 scalar controls holds forȳ = 0 and ω as in Theorem 1.1. However, up to our knowledge, this is an open problem even for the case of the Navier-Stokes system.
Remark 3. One could also try to just control the movement equation, that is, v 0 ≡ 0 in (1.1). However, this system does not seem to be controllable. To justify this, let us consider the control problem in Ω; where we have homogeneous Neumann boundary conditions for the temperature. Integrating in Q, integration by parts gives so we can not expect in general null controllability.
Some recent works have been developed in the controllability problem with reduced number of controls. For instance, in [3] the authors proved the null controllability for the Stokes system with N − 1 scalar controls, and in [2] the local null controllability was proved for the Navier-Stokes system with the same number of controls.
The present work can be viewed as an extension of [2]. To prove Theorem 1.1 we follow a standard approach introduced in [6] and [10] (see also [4]). We first deduce a null controllability result for the linear system in Ω, (1.8) where f and f 0 will be taken to decrease exponentially to zero in t = T . The main tool to prove this null controllability result for system (1.8) is a suitable Carleman estimate for the solutions of its adjoint system, namely, in Ω, (1.9) where g ∈ L 2 (Q) N , g 0 ∈ L 2 (Q), ϕ T ∈ H and ψ T ∈ L 2 (Ω). In fact, this inequality is of the form (1.10) if N = 3, and of the form if N = 2, where j = 1 or 2 and ρ k (t) are positive smooth weight functions (see inequalities (2.4) and (2.5) below). From these estimates, we can find a solution (y, θ, v, v 0 ) of (1.8) with the same decreasing properties as f and f 0 . In particular, (y(T ), θ(T )) = (0, 0) and v i = v N = 0. We conclude the controllability result for the nonlinear system by means of an inverse mapping theorem.
This paper is organized as follows. In section 2, we prove a Carleman inequality of the form (1.10) for system (1.9). In section 3, we deal with the null controllability of the linear system (1.8). Finally, in section 4 we give the proof of Theorem 1.1.
For the sake of completeness, let us also state this result for the 2-dimensional case.
To prove Proposition 1 we will follow the ideas of [3] and [5] (see also [2]). An important point in the proof of the Carleman inequality established in [3] is that the laplacian of the pressure in the adjoint system is zero. In [2], a decomposition of the solution was made, so that we can essentially concentrate in a solution where the laplacian of the pressure is zero. For system (1.9) this will not be possible because of the coupling term ψ∇θ. However, under hypothesis (1.6) we can follow the same ideas to obtain (2.4). All the details are given below.

Technical results
Let us present now the technical results needed to prove Carleman inequalities (2.4) and (2.5). The first of these results is a Carleman inequality for parabolic equations with nonhomogeneous boundary conditions proved in [11]. Consider the equation where F 0 , F 1 , . . . , F N ∈ L 2 (Q). We have the following result.
Lemma 2.1. There exists a constant λ 0 only depending on Ω, ω 0 , η and ℓ such that for any λ > λ 0 there exist two constants C(λ) > 0 and s(λ), such that for every s ≥ s and every Recall that The next technical result is a particular case of Lemma 3 in [3].
The last technical result concerns the regularity of the solutions to the Stokes system that can be found in [12] (see also [13]). Lemma 2.4. For every T > 0 and every F ∈ L 2 (Q) N , there exists a unique solution for some p ∈ L 2 (0, T ; H 1 (Ω)), and there exists a constant C > 0 depending only on Ω such that and there exists a constant C > 0 depending only on Ω such that (2.11) From now on, we set N = 3, i = 2 and j = 1, i.e., we consider a control for the movement equation in (1.1) (and (1.8)) of the form v = (v 1 , 0, 0). The arguments can be easily adapted to the general case by interchanging the roles of i and j.
Notice that from the identities in (2.15), the regularity estimate (2.10) for w and |ρ ′ | 2 ≤ Cs 2 ρ 2 (ξ) 9/4 we obtain for k = 1, 3 where we have also used the fact that s 2 e −2sα ξ 9/4 is bounded and 1 ≤ Cξ 3/4 in Q. Now, from z| Σ = 0 and the divergence free condition we readily have (notice that α * and ξ * do not depend on x) Using these two last estimates in (2.20), we get for every s ≥ C.
Combining inequalities (2.21) and (2.22), and taking into account that s 2 e −2sα ξ 2 ρ 2 is bounded, the identities in (2.15), estimate (2.10) for w and |ρ ′ | ≤ Cs(ξ * ) 9/8 ρ we have for every s ≥ C. It remains to treat the boundary terms of this inequality and to eliminate the local term in z 3 .
Estimate of the boundary terms. First, we treat the first boundary term in (2.23). Notice that, since α * and ξ * do not depend on x, we can readily get by integration by parts, for k = 1, 3, so e −sα * ∇∆z k 2 L 2 (Σ) 3 is bounded by I(s, z). On the other hand, we can bound the first boundary term as follows: Therefore, the first boundary terms can be absorbed by taking s large enough. Now we treat the second boundary term in the right-hand side of (2.23). We will use regularity estimates to prove that z 1 and z 3 multiplied by a certain weight function are regular enough. First, let us observe that from (2.15) and the regularity estimate (2.10) for w we readily have We define now z := se −sα * (ξ * ) 7/8 z, π z := se −sα * (ξ * ) 7/8 π z .
From (2.13), ( z, π z ) is the solution of the Stokes system: in Ω, By the same arguments as before, and thanks to (2.26), we can easily prove that R 2 ∈ L 2 (0, T ; H 2 (Ω) 3 ) ∩ H 1 (0, T ; L 2 (Ω) 3 ) (for the first term in R 2 , we use again (2.15) and (2.26)) and furthermore To end this part, we use a trace inequality to estimate the second boundary term in the right-hand side of (2.23): Thus, using (2.15) and (2.10) for w in the right-hand side of (2.23), we have for the moment for every s ≥ C. Furthermore, notice that using again (2.15), (2.10) for w and (2.26) we obtain from the previous inequality for every s ≥ C, where Estimate of ϕ 3 . We deal in this part with the last term in the right-hand side of (2.29). We introduce a function ζ 1 ∈ C 2 0 (ω) such that ζ 1 ≥ 0 and ζ 1 = 1 in ω 1 , and using equation (2.14) we have and we integrate by parts in this last term, in order to estimate it by local integrals of ψ, g 0 and ǫ I(s, ρ ϕ). This approach was already introduced in [5].

Null controllability of the linear system
Here we are concerned with the null controllability of the system in Ω, where y 0 ∈ V , θ 0 ∈ H 1 0 (Ω), f and f 0 are in appropriate weighted spaces, the controls v 0 and v 1 are in L 2 (ω × (0, T )) and Lq = q t − ∆q.
Before dealing with the null controllability of (3.1), we will deduce a Carleman inequality with weights not vanishing at t = 0. To this end, let us introduce the following weight functions:  6). Then, there exists a constant C > 0 (depending on s, λ andθ) such that every solution (ϕ, π, ψ) of (1.9) satisfies: Let us also state this result for N = 2.  6). Then, there exists a constant C > 0 (depending on s, λ andθ) such that every solution (ϕ, π, ψ) of (1.9) satisfies: Proof of Lemma 3.1: We start by an a priori estimate for system (1.9). To do this, we introduce a function ν ∈ C 1 ([0, T ]) such that We easily see that (νϕ, νπ, νψ) satisfies in Ω, thus we have the energy estimate ). Using the properties of the function ν, we readily obtain From this last inequality, and the fact that 0 Ω e −5sβ * (γ * ) 5 |ψ| 2 dx dt 0 Ω e −3sβ * (|g| 2 + |g 0 | 2 )dx dt Note that the last two terms in (3.5) are bounded by the left-hand side of the Carleman inequality (2.4). Since α = β in Ω × (T /2, T ), we have: Combining this with the Carleman inequality (2.4), we deduce we can readily get which, together with (3.5), yields (3.3). Now we will prove the null controllability of (3.1). Actually, we will prove the existence of a solution for this problem in an appropriate weighted space. Let us introduce the space It is clear that E is a Banach space for the following norm: Remark 4. Observe in particular that (y, p, v 1 , θ, v 0 ) ∈ E implies y(T ) = 0 and θ(T ) = 0 in Ω. Moreover, the functions belonging to this space posses the interesting following property: e 5/2sβ * (γ * ) −2 (y · ∇)y ∈ L 2 (Q) 3 and e 5/2sβ * (γ * ) −5/2 y · ∇θ ∈ L 2 (Q).
Sketch of the proof: The proof of this proposition is very similar to the one of Proposition 2 in [9] (see also Proposition 2 in [4] and Proposition 3.3 in [2]), so we will just give the main ideas.

Proof of Theorem 1.1
In this section we give the proof of Theorem 1.1 using similar arguments to those in [10] (see also [4], [5], [9] and [2]). The result of null controllability for the linear system (3.1) given by Proposition 3 will allow us to apply an inverse mapping theorem. Namely, we will use the following theorem (see [1]).
Theorem 4.1. 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 δ > 0 such that, for every b ′ ∈ B 2 satisfying b ′ − b 2 B2 < δ, there exists a solution of the equation Let us set y = y, p =p + p and θ =θ + θ.
This concludes the proof of Theorem 1.1.