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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

Enrique Fernández-Cara 1, and  Diego A. Souza 2,

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2. 

Dpto. de Matemática, Universidade Federal da Paraba, 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 $c$.
Keywords: Navier-Stokes and Boussinesq-like systems, control reduction., controllability.
Mathematics Subject Classification: Primary: 35B37, 93B05; Secondary: 35Q3.
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:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control,'' Translated from Russina by V. M. Volosov, Contemp. Soviet Math., Consultants Bureau, New York, 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), 2908-2921. 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), 1501-1542.  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), 146-173. doi: 10.1137/04061965X.  Google Scholar

[5]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolutions Equations,'' Lectures Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 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), 565-618. 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, Analyse Non Linéaire, 23 (2006), 29-61.  Google Scholar

[8]

O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Cal. Var., 6 (2001), 39-72.  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), 33-38. doi: 10.1016/S1631-073X(02)02389-0.  Google Scholar

show all references

References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control,'' Translated from Russina by V. M. Volosov, Contemp. Soviet Math., Consultants Bureau, New York, 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), 2908-2921. 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), 1501-1542.  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), 146-173. doi: 10.1137/04061965X.  Google Scholar

[5]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolutions Equations,'' Lectures Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 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), 565-618. 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, Analyse Non Linéaire, 23 (2006), 29-61.  Google Scholar

[8]

O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Cal. Var., 6 (2001), 39-72.  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), 33-38. doi: 10.1016/S1631-073X(02)02389-0.  Google Scholar

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