# American Institute of Mathematical Sciences

June  2012, 2(2): 121-140. doi: 10.3934/mcrf.2012.2.121

## On the control of some coupled systems of the Boussinesq kind with few controls

 1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain 2 Dpto. de Matemática, Universidade Federal da Paraba, 58051-900, João Pessoa, Brazil

Received  February 2011 Revised  September 2011 Published  May 2012

This paper is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are the velocity field and pressure of the fluid $(\mathbf{y},p)$, the temperature $\theta$ and an additional variable $c$ that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by $\theta$ and $c$.
Citation: Enrique Fernández-Cara, Diego A. Souza. On the control of some coupled systems of the Boussinesq kind with few controls. Mathematical Control & Related Fields, 2012, 2 (2) : 121-140. doi: 10.3934/mcrf.2012.2.121
##### References:
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##### References:
 [1] V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control,'', Translated from Russina by V. M. Volosov, (1987).   Google Scholar [2] J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls,, Journal of Differrential Equations, 246 (2009), 2908.  doi: 10.1016/j.jde.2008.10.019.  Google Scholar [3] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system,, J. Math. Pures Appl. (9), 83 (2004), 1501.   Google Scholar [4] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier-Stokes and Boussinesq systems with $N - 1$ scalar controls,, SIAM J. Control Optim., 45 (2006), 146.  doi: 10.1137/04061965X.  Google Scholar [5] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolutions Equations,'', Lectures Notes Series, 34 (1996).   Google Scholar [6] A. V. Fursikov and O. Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equation,, Russian Math. Surveys, 54 (1999), 565.  doi: 10.1070/RM1999v054n03ABEH000153.  Google Scholar [7] S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Annales de l'Institut Henri Poincaré, 23 (2006), 29.   Google Scholar [8] O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations,, ESAIM Control Optim. Cal. Var., 6 (2001), 39.   Google Scholar [9] O. Yu. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems,, C. R. Math. Acad. Sci. Paris, 335 (2002), 33.  doi: 10.1016/S1631-073X(02)02389-0.  Google Scholar
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