Remarks on local controllability for the Boussinesq system with Navier boundary condition

TL;DR

This paper investigates the local controllability of the Boussinesq system with Navier boundary conditions, showing that in 2D only one control is needed, while in 3D two controls are necessary.

Contribution

It establishes local exact controllability results for the Boussinesq system with nonlinear Navier-slip boundary conditions, highlighting differences between 2D and 3D cases.

Findings

In 2D, local controllability achieved with a single control.

In 3D, two scalar controls are required for controllability.

Results extend controllability theory to systems with Navier boundary conditions.

Abstract

This note deals with the local exact controllability to a particular class of trajectories for the Boussinesq system with nonlinear Navier-slip boundary conditions and internal controls having vanishing components. Briefly speaking, in two dimensions, the local exact controllability property is obtained using only one control in the heat equation, meanwhile two scalar controls are required in three dimensions.

Equations

$$\left\{\begin{array}[]{llll}y_{t}-\nabla\cdot(Dy)+(y,\nabla)y+\nabla p=u\chi_{\omega}+\theta e_{N},\quad\nabla\cdot y=0&\text{ in }&Q,\\ \theta_{t}-\Delta\theta+y\cdot\nabla\theta=v1_{\omega}&\text{ in }&Q,\\ y\cdot n=0,\,(\sigma(y,p)\cdot n)_{tg}+f(y)_{tg}=0,\quad\nabla\theta\cdot n=0&\text{ on }&\Sigma,\\ y(\cdot,0)=y_{0}(\cdot),\,\,\theta(\cdot,0)=\theta_{0}(\cdot)&\text{ in }&\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{llll}y_{t}-\nabla\cdot(Dy)+\nabla p=h_{1}+u\chi_{\omega}+\theta e_{N},\quad\nabla\cdot y=0&\text{ in }&Q,\\ \theta_{t}-\Delta\theta+y\cdot\nabla\overline{\theta}=h_{2}+v1_{\omega}&\text{ in }&Q,\\ y\cdot n=0,\,(\sigma(y,p)\cdot n)_{tg}+(A(x,t)y)_{tg}=0,\quad\nabla\theta\cdot n=0&\text{ on }&\Sigma,\\ y(\cdot,0)=y_{0}(\cdot),\,\,\theta(\cdot,0)=\theta_{0}(\cdot)&\text{ in }&\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{llll}-\varphi_{t}-\nabla\cdot(D\varphi)+\nabla\pi=g-\psi\nabla\overline{\theta},\quad\nabla\cdot\varphi=0&\text{ in }&Q,\\ -\psi_{t}-\Delta\psi=g_{0}+\varphi\cdot e_{N}&\text{ in }&Q,\\ \varphi\cdot n=0,\,(\sigma(\varphi,\pi)\cdot n)_{tg}+(A^{t}(x,t)\varphi)_{tg}=0,\quad\nabla\psi\cdot n=0&\text{ on }&\Sigma,\\ \varphi(\cdot,T)=\varphi^{T}(\cdot),\,\,\psi(\cdot,T)=\psi^{T}(\cdot)&\text{ in }&\Omega,\end{array}\right.$$

$$\begin{array}[]{ll}W&=\{u\in H^{1}(\Omega)^{N}:\nabla\cdot u=0\,\,\text{in}\,\,\Omega,\,\,\,u\cdot n=0\,\,\text{on}\,\,\partial\Omega\},\\ H&=\{u\in L^{2}(\Omega)^{N}:\nabla\cdot u=0,\,\,\text{in}\,\,\Omega\,\,\,u\cdot n=0\,\,\text{on}\,\,\partial\Omega\},\\ P^{1}_{\varepsilon}&=H^{5/4+\varepsilon}{(0,T;L^{2}(\partial\Omega)^{N\times N}}),\quad P^{2}=L^{2}(0,T;H^{5/2}(\partial\Omega)^{N\times N}),\quad\forall\varepsilon>0,\\ Y_{m}&:=L^{2}(0,T;H^{2m}(\Omega)^{N})\cap H^{m}(0,T;L^{2}(\Omega)^{N}),\quad m=1,2.\end{array}$$

$$\overline{\theta }\in L^{\infty }(0,T;W^{3,\infty }(\Omega )),\nabla {\overline{\theta }}_t\in L^{\infty }(Q{)}^N.$$

$$\left\{\begin{array}[]{llll}\nabla\overline{p}=\overline{\theta}e_{N},\quad\overline{\theta}_{t}-\Delta\overline{\theta}=0&\text{ in }&Q,\\ \nabla\overline{\theta}\cdot n=0&\text{ on }&\Sigma,\\ \overline{\theta}(\cdot,0)=\overline{\theta}_{0}(\cdot)&\text{ in }&\Omega.\end{array}\right.$$

$$\begin{array}[]{llllll}\alpha(x,t)&=\dfrac{e^{2\lambda\|\eta\|_{\infty}}-e^{\lambda\eta(x)}}{(t(T-t))^{11}},&\xi(x,t)&=\dfrac{e^{\lambda\eta(x)}}{(t(T-t))^{11}},&\alpha^{*}(t)&=\max_{x\in\overline{\Omega}}\alpha(x,t),\\ \xi^{*}(t)&=\min_{x\in\overline{\Omega}}\xi(x,t),&\widehat{\alpha}(t)&=\min_{x\in\overline{\Omega}}\alpha(x,t),&\widehat{\xi}(t)&=\max_{x\in\overline{\Omega}}\xi(x,t).\end{array}$$

$$|\nabla \eta |>0\text{mbox}in\overline{\Omega }\setminus {\omega }_0,\eta >0\text{mbox}in\Omega \text{mbox}and\eta \equiv 0\text{mbox}on\partial \Omega ,$$

$$s^3$$

$$\displaystyle\leq C\Bigl{(}\iint\limits_{Q}e^{-2as\alpha^{*}}(|g|^{2}+|g_{0}|^{2})dxdt+(N-2)s^{7}\int\limits_{0}^{T}\int\limits_{\omega^{\prime}}e^{-4s\hat{\alpha}+2(1-a)s\alpha^{*}}(\hat{\xi})^{12}|\varphi_{j}|^{2}dxdt$$

$$\displaystyle\hskip 56.9055pt+s^{13}\int\limits_{0}^{T}\int\limits_{\omega}e^{-8s\hat{\alpha}+(6-2a)s\alpha^{*}}(\hat{\xi})^{24}|\psi|^{2}dxdt\Bigr{)}$$

$$(D{y}_0\cdot n{)}_{tg}+(f(y_0){)}_{tg}=0\text{mbox}on\partial \Omega \text{mbox}and|(y_0,{\theta }_0)-(0,{\overline{\theta }}_0){\Vert }_{[H^3(\Omega {)}^N\cap W]\times H^1(\Omega )}\le \delta ,$$

$$y(\cdot ,T)=\overline{y}(\cdot ,T)\text{mbox}and\theta (\cdot ,T)=\overline{\theta }(\cdot ,T)\text{mbox}in\Omega .$$

$$\left\{\begin{array}[]{lllll}-w_{t}-\nabla\cdot(Dw)+\nabla\pi_{w}=\rho g;&\quad-z_{t}-\nabla\cdot(Dz)+\nabla\pi_{z}=-\rho^{\prime}\varphi-\tilde{\psi}\nabla\overline{\theta}&\text{ in }&Q,\\ \nabla\cdot w=0;&\quad\nabla\cdot z=0&\text{ in }&Q,\\ w\cdot n=0,\,(\sigma(w,\pi_{w})\cdot n)_{tg}+(A^{t}(x,t)w)_{tg}=0;&\quad z\cdot n=0,\,(\sigma(z,\pi_{z})\cdot n)_{tg}+(A^{t}(x,t)z)_{tg}=0&\text{ on }&\Sigma,\\ w(\cdot,T)=0;&\quad z(\cdot,T)=0&\text{ in }&\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{llll}-\tilde{\psi}_{t}-\Delta\tilde{\psi}=\rho g_{0}+\rho\varphi_{3}-\rho^{\prime}\psi&\text{ in }&Q,\\ \nabla\tilde{\psi}\cdot n=0&\text{ on }&\Sigma,\\ \tilde{\psi}(\cdot,T)=0&\text{ in }&\Omega,\end{array}\right.$$

$$\underset{Q}{\iint }$$

$$\le$$

$$\displaystyle I(s,z)+J(s,\tilde{\psi})\leq C\Biggl{(}\|\rho g\|^{2}_{L^{2}(Q)^{3}}+\|\rho g_{0}\|^{2}_{L^{2}(Q)}+s^{5}\int\limits_{0}^{T}\int\limits_{\omega_{1}}e^{-2s\alpha}\xi^{5}|\tilde{\psi}|^{2}dxdt$$

$$\displaystyle\quad+\sum\limits_{k=1,3}\Biggl{[}\int\limits_{0}^{T}\int\limits_{\omega_{1}}e^{-2s\alpha}(s^{5}\xi^{5}|z_{k}|^{2}+s^{3}\xi^{3}|\nabla z_{k}|^{2})dxdt+\int\limits_{0}^{T}\int\limits_{\omega_{2}}e^{-2s\alpha}\xi^{2}|\nabla\partial_{k}\pi_{z}|^{2}dxdt\Biggr{]}\Biggr{)},$$

$$I(s,z):=\sum _{k=1,3}$$

$$+\Vert s^{1/2}{e}^{-s{\alpha }^{\ast }}({\xi }^{\ast }{)}^{9/22}z{\Vert }_{Y_1}^2+\Vert s^{-1/2}{e}^{-s{\alpha }^{\ast }}({\xi }^{\ast }{)}^{-15/22}z{\Vert }_{L^2(0,T;H^4(\Omega {)}^3)\cap H^2(0,T;L^2(\Omega {)}^3)}^2$$

$$+\Vert s^{1/2}{e}^{-s{\alpha }^{\ast }}({\xi }^{\ast }{)}^{9/22}{\pi }_z{\Vert }_{L^2(0,T;H^1(\Omega ))}^2.$$

$$L_{1}w:=w_t-\nabla \cdot Dw\text{mbox}and{L}_{2}w:=w_t-\Delta w$$

$$\{(y,p,u_1,\theta ,v):e^{as{\beta }^{\ast }}y,e^{2s\hat{\beta }-(1-a)s{\beta }^{\ast }}(\hat{\gamma }{)}^{-6}(u_1,0,0){\chi }_{\omega },\tilde{\rho }({\partial }_t{u}_1,0,0)\in L^2(Q{)}^3,e^{as{\beta }^{\ast }}\theta \in L^2(Q),$$

$$e^{4s\hat{\beta }-(3-a)s{\beta }^{\ast }}(\hat{\gamma }{)}^{-12}v{1}_{\omega }\in L^2(Q),\tilde{\rho }{u}_1\in L^2(0,T;H^2(\Omega )),supp{u}_1\subset \omega \times (0,T),$$

$$e^{as{\beta }^{\ast }}({\gamma }^{\ast }{)}^{-12/11}y\in Y_1,e^{as{\beta }^{\ast }}({\gamma }^{\ast }{)}^{-12/11}\theta \in L^2(0,T;H^2(\Omega ))\cap H^1(0,T;L^2(\Omega )),$$

$$e^{(a+1)s{\beta }^{\ast }}({\gamma }^{\ast }{)}^{-3/2}(L_{1}y+\nabla p-(u_1,0,0){\chi }_{\omega }-\theta e_3)\in L^2(Q{)}^3,$$

$$e^{(a+1)s{\beta }^{\ast }}({\gamma }^{\ast }{)}^{-5/2}(L_2\theta +y\cdot \nabla \overline{\theta }-v{1}_{\omega })\in L^2(Q)\}=:E,$$

$$\beta (x,t)=\frac{e^{2\lambda \Vert \eta {\Vert }_{\infty }}-e^{\lambda \eta (x)}}{{\ell }^{11}(t)},$$

$${\gamma }^{\ast }(t)=\underset{x\in \overline{\Omega }}{\min }\gamma (x,t),$$

$$y_0\in W,{\theta }_0\in H^1(\Omega ),e^{(a+1)s{\beta }^{\ast }}({\gamma }^{\ast }{)}^{-3/2}{h}_1\in L^2(Q{)}^3\text{mbox}and{e}^{(a+1)s{\beta }^{\ast }}({\gamma }^{\ast }{)}^{-5/2}{h}_2\in L^2(Q).$$

$$\left\{\begin{array}[]{llll}y_{t}-\nabla\cdot(Dy)+\nabla p=h_{1}+(u_{1},0,0)\chi_{\omega}+\theta e_{3},\quad\nabla\cdot y=0&\mbox{ in }&Q,\\ \theta_{t}-\Delta\theta+y\cdot\nabla\overline{\theta}=h_{2}+v1_{\omega}&\mbox{ in }&Q,\\ y\cdot n=0,\,(\sigma(y,p)\cdot n)_{tg}+(f(y))_{tg}=0,\quad\nabla\theta\cdot n=0&\mbox{ on }&\Sigma,\\ y(\cdot,0)=y_{0}(\cdot),\,\,\theta(\cdot,0)=\theta_{0}(\cdot)&\mbox{ in }&\Omega.\end{array}\right.$$

$$\Vert h_1{\Vert }_{Y_1}+\Vert h_2{\Vert }_{L^2(Q)}+\Vert y_0{\Vert }_{H^3(\Omega {)}^3\cap W}+\Vert {\theta }_0{\Vert }_{H^1(\Omega )}\le \delta$$

$$\mathcal{A}:{\mathcal{B}}_1\to {\mathcal{B}}_2$$

$$\mathcal{A}(b_1^0)=b_2^0$$

$$\Vert b_2^0-b_2{\Vert }_{{\mathcal{B}}_2}<\delta$$

Full text

Résumé Remarque sur la contrôlabilité locale du système de Boussinesq avec la condition de frontière de Navier. Cette note concerne la contrôlabilité locale d’une classe particulière de trajectoires, ceci pour le système de Boussinesq avec la condition de Navier non linéaire et certains contrôles internes. Brièvement, la propiété de contrôlabilité exacte locale s’obtient en dimension deux n’utilisant que le contrôl associé à l’équation de la chaleur. Tandis que, deux contrôles scalaires sont nécessaires pour obtenir nôtre résultat dans le cas de dimension trois.

1 Introduction

The interaction of incompressible fluids with a diffusion process can be modeled by a coupled system between the Navier–Stokes and heat equations, usually called Boussinesq system. On bounded domains, both heat and the velocity field can show a different behaviour on its boundary. In this paper, nonlinear Navier–type boundary conditions for the fluid flow and homogeneous Neumann conditions for the diffusion equation are considered in order to study the local exact controllability for the Boussinesq system with few scalar controls.

Henceforth, let $\Omega$ be a nonempty bounded connected open subset of $\mathbb{R}^{N}$ ($N=2$ or $N=3$) of class $C^{\infty}$. Let $T>0$ and let $\omega\subset\Omega$ be a (small) nonempty open subset which is the control domain. Here, we will use the notation $Q:=\Omega\times(0,T)$,  $\Sigma:=\partial\Omega\times(0,T)$ and $n$ the outward unit normal vector to $\Omega$. Moreover, $C$ denotes a generic positive constant which may depend on $\Omega$ and $\omega$.

In this Note, we will consider the Boussinesq system with Navier–slip and Neumann conditions

[TABLE]

as well as the linearized Boussinesq system (around a target flow of the form ($(0,\overline{p},\overline{\theta})$)

[TABLE]

where $y=y(x,t)$ is the velocity field of the fluid, $\theta=\theta(x,t)$ their temperature, $v$ and $u=(u_{1},\dots,u_{N})$ stands for the controls, which are acting in a arbitrary fixed domain $\omega\times(0,T)$, where $\chi_{\omega}$ is a smooth positive function such that $\chi_{\omega}=1$ in $\omega^{\prime}$, $\omega^{\prime}\Subset\omega$, and $1_{\omega}$ is the indicator function. Here, the gravity vector field is given by $e_{N}=(0,1)$ for $N=2$, or $e_{N}=(0,0,1)$ for $N=3$. Moreover, $f:\mathbf{R}^{N}\to\mathbf{R}^{N}$ is a nonlinear regular function given, $\sigma(y,p):=-pId+Dy$ is the stress tensor, $A$ is a $N\times N$ matrix–valued function in a suitable space, and tg stands for the tangential component of the corresponding vector field, i.e., $y_{tg}=y-(y\cdot n)n$.

In the context of controllability, the first results for the Boussinesq system were made by Fursikov and Imanuvilov in [9] and [10]. The work by S. Guerrero [12] shows the local exact controllability to the trajectories of the Boussinesq system with Dirichlet boundary conditions and $N+1$ distributed scalar controls supported in small sets.

Additionally, recent works have been developed for controllability problems with reduced number of controls. For instance, N. Carreño and S. Guerrero in [1] have proven the local null controllability for the Navier–Stokes with Dirichlet conditions and $N-1$ scalar controls. The recent work made by S. Guerrero and C. Montoya shows that the local null controllability property is achieved for the $N$–dimensional Navier–Stokes system with Navier–slip conditions and $N-1$ scalar controls [13]. The methodology in the previous articles are Carleman estimates. In the three dimensional case of the Navier–Stokes system with Dirichlet conditions, J-M. Coron and P. Lissy developed in [4] a new strategy to prove the local null controllability using only one scalar control.

Concerning the $N$-dimensional Boussinesq system with Dirichlet conditions, in [7] the authors proved that the local exact controllability to the trajectories can be achieved with $N-1$ scalar controls, under certain geometric assumption on the control domain. N. Carreño showed the local controllability of the $N$–Boussinesq system using $N-1$ scalar controls, without conditions on the control domain [2]. Finally, this Note improves the results of [1] and [13].

Our results below extend the results of [1] and [13]. Taking into account the relation between the observability and controllability property, it will be appropriate to consider the following adjoint system related to (2):

[TABLE]

where $g,\varphi^{T},g_{0}$ and $\psi^{T}$ satisfying adequate regularity assumptions. We will introduce several spaces and hypotheses over $\overline{\theta}$ which will be needed in order to have suitable Carleman estimates for the solution of (3):

[TABLE]

and

[TABLE]

Here, the target flow $(0,\overline{p},\overline{\theta})$ satisfies the problem

[TABLE]

Our first main result is a new Carleman estimate for the solution of (3). Several weight functions are needed:

[TABLE]

Here, $\eta\in C^{2}(\overline{\Omega})$ and satisfies that

[TABLE]

where $\omega_{0}\Subset\omega_{1}\Subset\omega^{\prime}\Subset{\omega}$ is a nonempty open set. The existence of such a function $\eta$ is proved in [8].

Theorem 1.1

Assume $A\in P^{1}_{\varepsilon}\cap P^{2}$ and $(0,\overline{p},\overline{\theta})$ satisfying (4)–(5). There exists a constant $\lambda_{0}$, such that for any $\lambda\geq\lambda_{0}$ there exist two constants $C(\lambda)>0$ increasing on $\|A\|_{P^{1}_{\varepsilon}\cap P^{2}}$ and $s_{0}(\lambda)>0$ such that for any $j\in\{1,2\}$, any $a>0$, any $g\in L^{2}(Q)^{3}$, any $g_{0}\in L^{2}(Q)$, any $\varphi^{T}\in H$ and any $\psi^{T}\in L^{2}(\Omega)$, the solution of (3) satisfies

[TABLE]

for every $s\geq s_{0}$.

The second main result in this Note concerns the local controllability to a particular class of trajectories of (1). This result is presented as follows:

Theorem 1.2

Assume $f\in C^{4}(\mathbb{R}^{N};\mathbb{R}^{N})$ with $f(0)=0$ and $i\in\{1,\dots,N-1\}$ fixed. Let $(0,\overline{p},\overline{\theta})$ be a solution to (5) satisfying (4). Then, for every $T>0$ and $\omega\subset\Omega$, there exists $\delta>0$ such that, for every $(y_{0},\theta_{0})\in[H^{3}(\Omega)^{N}\cap W]\times H^{1}(\Omega)$ satisfying

[TABLE]

we can find controls $v\in L^{2}(\omega\times(0,T))$ and $u\in L^{2}(0,T;H^{2}(\omega)^{N})\cap H^{1}(0,T;L^{2}(\omega)^{N})$ with $u_{i}\equiv 0$ and $u_{N}\equiv 0$ such that the corresponding solution $(y,p,\theta)$ to (1) satisfies

[TABLE]

In the following sections, we will indicate the main ideas of the proof of Theorem 1.1 and Theorem 1.2.

2 A new Carleman inequality

In this section, we give the proof of Theorem 1.1. Our arguments are based in [1, 3, 7, 13]. From (3) and using the decomposition $\rho\varphi=w+z,\quad\rho\pi=\pi_{z}+\pi_{w}$ and $\rho\psi=\tilde{\psi}$, where $\rho(t)=e^{-as\alpha^{*}}$ and $a>0$, it is very easy to verify that $(w,\pi_{w}),(z,\pi_{z})$ and $\tilde{\psi}$ are solutions to the systems

[TABLE]

and

[TABLE]

We will use the Carleman inequality for parabolic equations with Neumann conditions [8] for the system (10) in order to estimate the global terms associated to $\tilde{\psi}$. Thus, there exists $\tilde{\lambda}>0$ such that for any $\lambda>\tilde{\lambda}$ there exists a positive constant $C$ depending on $\lambda,\Omega,\omega_{2},\|\overline{\theta}\|_{L^{\infty}(0,T;W^{3,\infty}(\Omega))}$ such that

[TABLE]

for every $s\geq C$.

The arguments below are given for the case $N=3$. For $k=1,3$, we can deduce the inequality

[TABLE]

where $J(s,\tilde{\psi})$ denotes the left–hand side of (11), and for $k=1,3$, $I(s,z)$ is defined by

[TABLE]

Here, $\omega_{1}$ and $\omega_{2}$ are open sets such that $\omega_{1}\Subset\omega_{2}\Subset\omega$. The rest of the proof is oriented towards the absorption of the local pressure term in (12). However, we have omitted these details since analogous arguments can be found in [13], Section 3. Let us remark that the regularity over $\overline{\theta}$ given in (4) is used in several estimates associated to the pressure term. The other local terms can be estimated in an easier way. Therefore, those local estimates lead to the desired Carleman inequality (7).

3 Local controllability for the Boussinesq system

The proof of Theorem 1.2 follows the ideas in [1] and [13]. Thus, in a first step a null controllability result for (2) with an appropriate right–hand side $h_{1},\,h_{2}$. Here, the idea is to look for a solution in an appropriate weighted functional space. Let us

[TABLE]

and let us define the space $E$ as follows:

[TABLE]

where $\tilde{\rho}:=e^{4s\hat{\beta}+2(1-a)s\beta^{*}}(\hat{\gamma})^{-12}e^{-(1+a)s\beta^{*}}(\gamma^{*})^{9/22}$ and whose weight functions are given by

[TABLE]

In this case, $\ell\in C^{2}([0,T])$ is a positive function in $[0,T)$ such that $\ell(t)>t(T-t)$ for all $t\in[0,T/4]$ and $\ell(t)=t(T-t)$ for all $t\in[T/2,T]$.

Proposition 3.1

Let $s$ and $\lambda$ be like in Theorem 1.1 and $(0,\overline{p},\overline{\theta})$ satisfy (5). Assume that

[TABLE]

Then, there exists controls $u_{1}$ and $v$ such that, if $(y,p,\theta)$ is the associated solution to (2), we have $(y,p,u_{1},\theta,v)\in E$. In particular $y(\cdot,T)=0$ and $\theta(\cdot,T)=0$ in $\Omega$.

The rest of the proof of Theorem 1.2 relies on two fixed point theorems, namely, one for the nonlinearity posed on the boundary condition, and another one, for the convective term in (1). We will mention only these results since the methodology given in [13] can be adapted to (1). Thus, for $N=3$, we consider the nonlinear system

[TABLE]

Theorem 3.2

Let us assume that $f\in C^{4}(\mathbf{R}^{3};\mathbf{R}^{3})$ with $f(0)=0$. Then, for every $T>0$ and $\omega\subset\Omega$, there exists $\delta>0$ such that, for every $a>0$ and for every $(y_{0},\theta_{0})\in H^{3}(\Omega)^{3}\cap W\times H^{1}(\Omega)$,  $h_{1}\in Y_{1},h_{2}\in L^{2}(Q)$ satisfying $e^{(a+1)s\beta^{*}}(\gamma^{*})^{-3/2}h_{1}\in L^{2}(Q)^{3}$  and  $e^{(a+1)s\beta^{*}}(\gamma^{*})^{-5/2}h_{2}\in L^{2}(Q),$

[TABLE]

and (8), there exists controls $v\in L^{2}(0,T;L^{2}(\omega))$ and $u_{1}\in L^{2}(0,T;H^{2}(\omega))\cap H^{1}(0,T;L^{2}(\omega))$ and an associated solution $(y,p,\theta)$ of (15) satisfying $(y,\theta)\in Y_{2}\times L^{2}(0,T;H^{2}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$ and such that $(y,p,u_{1},\theta,v)\in E$.

Theorem 3.3

Suppose that $\mathcal{B}_{1},\mathcal{B}_{2}$ are Banach spaces and

[TABLE]

is a continuously differentiable map. We assume that for $b_{1}^{0}\in\mathcal{B}_{1},b_{2}^{0}\in\mathcal{B}_{2}$ the equality

[TABLE]

holds and $\mathcal{A}^{\prime}(b_{1}^{0}):\mathcal{B}_{1}\to\mathcal{B}_{2}$ is an epimorphism. Then there exists $\delta>0$ such that for any $b_{2}\in\mathcal{B}_{2}$ which satisfies the condition

[TABLE]

there exists a solution $b_{1}\in\mathcal{B}_{1}$ of the equation

[TABLE]

Let us set

[TABLE]

For $a=2>1$, we apply Theorem 3.3 with the spaces

[TABLE]

[TABLE]

and where

[TABLE]

By defining the operator $\mathcal{A}:\mathcal{B}_{1}\to\mathcal{B}_{2}$ by

[TABLE]

for every $(\tilde{y},\tilde{p},u_{1},\tilde{\theta},v)\in\mathcal{B}_{1}$, one can easily check the conditions for $\mathcal{A}$ in order to complete the proof of Theorem 1.2.

Some open problems. It would be interesting to know if the local controllability to the trajectories with $N-1$ scalar controls holds for $\overline{y}\neq 0$ and $\omega$ like in Theorem 1.2. However, is not clear at all and therefore is an open problem even for the Navier–Stokes system.

On the other side, could be reasonable to expect results of the same kind whether one considers nonlinear conditions such as $\nabla\theta\cdot n+g(\theta)=0$, where $g$ is a suitable function to study.

Recently, Coron et al. have proved a global exact controllability result for the Navier–Stokes and Navier–type conditions (for small time), see [5]. A challenging problem would be to use the Boussinesq system proposed in this Note in order to apply and prove analogous results to [5].

Acknowledgements

The author would like to express his gratitude to Sergio Guerrero for his suggestions, which have contributed to a better presentation of this paper.

This work has been supported by FONDECYT grant 3180100.