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

Montoya, Cristhian

Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 169-175.

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. En bref, la propriété de contrôlabilité exacte locale s’obtient en dimension deux en n’utilisant que le contrôle associé à l’équation de la chaleur. En revanche, deux contrôles scalaires sont nécessaires pour obtenir notre résultat dans le cas de dimension trois

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, whereas two scalar controls are required in three dimensions.

[1] Carreño, Nicolás Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain, Math. Control Relat. Fields, Volume 2 (2012) no. 4, pp. 361-382 | DOI | MR | Zbl

[2] Carreño, Nicolás; Guerrero, Sergio Local null controllability of the $N$-dimensional Navier–Stokes system with $N-1$ scalar controls in an arbitrary control domain, J. Math. Fluid Mech., Volume 15 (2013) no. 1, pp. 139-153 | DOI | MR | Zbl

[3] Coron, Jean-Michel; Guerrero, Sergio Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differ. Equations, Volume 246 (2009) no. 7, pp. 2908-2921 | DOI | MR | Zbl

[4] Coron, Jean-Michel; Lissy, Pierre Local null controllability of the three-dimensional Navier–Stokes system with a distributed control having two vanishing components, Invent. Math., Volume 198 (2014) no. 3, pp. 833-880 | DOI | MR | Zbl

[5] Coron, Jean-Michel; Marbach, Frédéric; Sueur, Franck Small-time global exact controllability of the navier-stokes equation with navier slip-with-friction boundary conditions (2016) (https://arxiv.org/abs/1612.08087)

[6] Fernández-Cara, Enrique; Guerrero, Sergio; Imanuvilov, Oleg Yu.; Puel, Jean-Pierre Some controllability results for the $N$-dimensional Navier–Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optimization, Volume 45 (2006) no. 1, pp. 146-173 | DOI | MR | Zbl

[7] Fursikov, Andreĭ V.; Imanuvilov, Oleg Yu. Controllability of evolution equations, Lecture Notes Series, Seoul, 34, Seoul National Univ., 1996 | MR | Zbl

[8] Fursikov, Andreĭ V.; Imanuvilov, Oleg Yu. Local exact boundary controllability of the Boussinesq equation, SIAM J. Control Optimization, Volume 36 (1998) no. 2, pp. 391-421 | DOI | MR | Zbl

[9] Fursikov, Andreĭ V.; Imanuvilov, Oleg Yu. Exact controllability of the Navier–Stokes and Boussinesq equations, Usp. Mat. Nauk, Volume 54 (1999) no. 3, pp. 93-146 | MR | Zbl

[10] Guerrero, Sergio Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 23 (2006) no. 1, pp. 29-61 | DOI | Numdam | MR | Zbl

[11] Guerrero, Sergio Local exact controllability to the trajectories of the Navier–Stokes system with nonlinear Navier-slip boundary conditions, ESAIM, Control Optim. Calc. Var., Volume 12 (2006) no. 3, pp. 484-544 | DOI | Numdam | MR | Zbl

[12] Guerrero, Sergio; Montoya, Cristhian Local null controllability of the $N$-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and $N-1$ scalar controls, J. Math. Pures Appl., Volume 113 (2018), pp. 37-69 | DOI | MR | Zbl