# 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:
 [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

show all references

##### 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
 [1] Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169 [2] Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021 [3] Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1 [4] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [5] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [6] A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289 [7] Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495 [8] Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273 [9] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [10] Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 [11] Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349 [12] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [13] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [14] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [15] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [16] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [17] Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269 [18] Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255 [19] Hammadi Abidi, Taoufik Hmidi, Sahbi Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 737-756. doi: 10.3934/dcds.2011.29.737 [20] Rafael Vázquez, Emmanuel Trélat, Jean-Michel Coron. Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 925-956. doi: 10.3934/dcdsb.2008.10.925

2018 Impact Factor: 1.292