# American Institute of Mathematical Sciences

June  2011, 1(2): 149-175. doi: 10.3934/mcrf.2011.1.149

## Global Carleman inequalities for Stokes and penalized Stokes equations

 1 Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex, France

Received  November 2010 Revised  March 2011 Published  June 2011

In this note we use the result of [22] to prove a global Carleman inequality related to the null controllability of penalized Stokes kind systems. The constants of the obtained Carleman inequality are uniform in terms of the penalization parameter $\varepsilon$. It then provides a null control with a uniformly (in $\varepsilon$) bounded $L^2$ norm. With a limiting argument we also deduce a new Carleman inequality for Stokes type system. Thus, we apply theses results to obtain the null controllability of Oseen and Navier-Stokes system in the penalized and in the non penalized cases.
Citation: Mehdi Badra. Global Carleman inequalities for Stokes and penalized Stokes equations. Mathematical Control and Related Fields, 2011, 1 (2) : 149-175. doi: 10.3934/mcrf.2011.1.149
##### References:
 [1] M. Badra, J.-M. Buchot and L. Thevenet, Méthode de pénalisation pour le Contrôle Frontière des équations de Navier-Stokes, submitted to Journal Européen des Systèmes Automatisés, special issue "Méthodes Numériques et Applications des Systèmes à Paramètres Répartis," 2010. [2] 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, to appear in Discrete Contin. Dyn. Syst. Ser. A. [3] Mehdi Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control Optim., 48 (2009), 1797-1830. doi: 10.1137/070682630. [4] Jean-Michel Coron and Sergio Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921. [5] O. Yu. Èmanuilov, Boundary controllability of parabolic equations, Uspekhi Mat. Nauk, 48 (1993), 211-212. [6] O. Yu. Èmanuilov, Controllability of parabolic equations, Mat. Sb., 186 (1995), 109-132. [7] E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids, J. Math. Fluid Mech., 9 (2007), 419-453. doi: 10.1007/s00021-005-0207-1. [8] 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. [9] Enrique Fernández-Cara and Sergio Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446 (electronic). [10] Enrique Fernández-Cara, Sergio Guerrero, Oleg Yu. Imanuvilov and Jean-Pierre 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 (electronic). [11] A. V. Fursikov and O. Yu. Èmanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54 (1999), 93-146. [12] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [13] Vivette Girault and Pierre-Arnaud Raviart, "Finite Element Methods for Navier-Stokes Equations," Theory and Algorithms, Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. [14] Manuel González-Burgos, Sergio Guerrero and Jean-Pierre Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., 8 (2009), 311-333. [15] P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421. [16] S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61. [17] S. Guerrero and F. Guillén-González, On the controllability of the hydrostatic Stokes equations, J. Math. Fluid Mech., 10 (2008), 402-422. doi: 10.1007/s00021-006-0237-3. [18] Sergio Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029-1054. [19] Oleg Imanuvilov and Takéo Takahashi, Exact controllability of a fluid-rigid body system, J. Math. Pures Appl. (9), 87 (2007), 408-437. [20] Oleg Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72 (electronic). [21] Oleg Yu. Imanuvilov and Jean-Pierre Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., (2003), 883-913. [22] Oleg Yu. Imanuvilov, Jean Pierre Puel and Masahiro Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378. doi: 10.1007/s11401-008-0280-x. [23] Oleg Yu. Imanuvilov and Masahiro Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274. doi: 10.2977/prims/1145476103. [24] Jean-Pierre Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828 (electronic). doi: 10.1137/050628726. [25] Roger Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152.

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##### References:
 [1] M. Badra, J.-M. Buchot and L. Thevenet, Méthode de pénalisation pour le Contrôle Frontière des équations de Navier-Stokes, submitted to Journal Européen des Systèmes Automatisés, special issue "Méthodes Numériques et Applications des Systèmes à Paramètres Répartis," 2010. [2] 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, to appear in Discrete Contin. Dyn. Syst. Ser. A. [3] Mehdi Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control Optim., 48 (2009), 1797-1830. doi: 10.1137/070682630. [4] Jean-Michel Coron and Sergio Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921. [5] O. Yu. Èmanuilov, Boundary controllability of parabolic equations, Uspekhi Mat. Nauk, 48 (1993), 211-212. [6] O. Yu. Èmanuilov, Controllability of parabolic equations, Mat. Sb., 186 (1995), 109-132. [7] E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids, J. Math. Fluid Mech., 9 (2007), 419-453. doi: 10.1007/s00021-005-0207-1. [8] 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. [9] Enrique Fernández-Cara and Sergio Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446 (electronic). [10] Enrique Fernández-Cara, Sergio Guerrero, Oleg Yu. Imanuvilov and Jean-Pierre 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 (electronic). [11] A. V. Fursikov and O. Yu. Èmanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54 (1999), 93-146. [12] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [13] Vivette Girault and Pierre-Arnaud Raviart, "Finite Element Methods for Navier-Stokes Equations," Theory and Algorithms, Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. [14] Manuel González-Burgos, Sergio Guerrero and Jean-Pierre Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., 8 (2009), 311-333. [15] P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421. [16] S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61. [17] S. Guerrero and F. Guillén-González, On the controllability of the hydrostatic Stokes equations, J. Math. Fluid Mech., 10 (2008), 402-422. doi: 10.1007/s00021-006-0237-3. [18] Sergio Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029-1054. [19] Oleg Imanuvilov and Takéo Takahashi, Exact controllability of a fluid-rigid body system, J. Math. Pures Appl. (9), 87 (2007), 408-437. [20] Oleg Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72 (electronic). [21] Oleg Yu. Imanuvilov and Jean-Pierre Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., (2003), 883-913. [22] Oleg Yu. Imanuvilov, Jean Pierre Puel and Masahiro Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378. doi: 10.1007/s11401-008-0280-x. [23] Oleg Yu. Imanuvilov and Masahiro Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274. doi: 10.2977/prims/1145476103. [24] Jean-Pierre Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828 (electronic). doi: 10.1137/050628726. [25] Roger Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152.
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