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 & 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, (2010).   Google Scholar

[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., ().   Google Scholar

[3]

Mehdi Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control Optim., 48 (2009), 1797.  doi: 10.1137/070682630.  Google Scholar

[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.   Google Scholar

[5]

O. Yu. Èmanuilov, Boundary controllability of parabolic equations,, Uspekhi Mat. Nauk, 48 (1993), 211.   Google Scholar

[6]

O. Yu. Èmanuilov, Controllability of parabolic equations,, Mat. Sb., 186 (1995), 109.   Google Scholar

[7]

E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids,, J. Math. Fluid Mech., 9 (2007), 419.  doi: 10.1007/s00021-005-0207-1.  Google Scholar

[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.   Google Scholar

[9]

Enrique Fernández-Cara and Sergio Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability,, SIAM J. Control Optim., 45 (2006), 1399.   Google Scholar

[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.   Google Scholar

[11]

A. V. Fursikov and O. Yu. Èmanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations,, Uspekhi Mat. Nauk, 54 (1999), 93.   Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34,, Seoul National University Research Institute of Mathematics, (1996).   Google Scholar

[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, (1986).   Google Scholar

[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.   Google Scholar

[15]

P. Grisvard, Caractérisation de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40.  doi: 10.1007/BF00281421.  Google Scholar

[16]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29.   Google Scholar

[17]

S. Guerrero and F. Guillén-González, On the controllability of the hydrostatic Stokes equations,, J. Math. Fluid Mech., 10 (2008), 402.  doi: 10.1007/s00021-006-0237-3.  Google Scholar

[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.   Google Scholar

[19]

Oleg Imanuvilov and Takéo Takahashi, Exact controllability of a fluid-rigid body system,, J. Math. Pures Appl. (9), 87 (2007), 408.   Google Scholar

[20]

Oleg Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 6 (2001), 39.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

[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.  doi: 10.2977/prims/1145476103.  Google Scholar

[24]

Jean-Pierre Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790.  doi: 10.1137/050628726.  Google Scholar

[25]

Roger Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes,, Bull. Soc. Math. France, 96 (1968), 115.   Google Scholar

show all references

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, (2010).   Google Scholar

[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., ().   Google Scholar

[3]

Mehdi Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control Optim., 48 (2009), 1797.  doi: 10.1137/070682630.  Google Scholar

[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.   Google Scholar

[5]

O. Yu. Èmanuilov, Boundary controllability of parabolic equations,, Uspekhi Mat. Nauk, 48 (1993), 211.   Google Scholar

[6]

O. Yu. Èmanuilov, Controllability of parabolic equations,, Mat. Sb., 186 (1995), 109.   Google Scholar

[7]

E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids,, J. Math. Fluid Mech., 9 (2007), 419.  doi: 10.1007/s00021-005-0207-1.  Google Scholar

[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.   Google Scholar

[9]

Enrique Fernández-Cara and Sergio Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability,, SIAM J. Control Optim., 45 (2006), 1399.   Google Scholar

[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.   Google Scholar

[11]

A. V. Fursikov and O. Yu. Èmanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations,, Uspekhi Mat. Nauk, 54 (1999), 93.   Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34,, Seoul National University Research Institute of Mathematics, (1996).   Google Scholar

[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, (1986).   Google Scholar

[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.   Google Scholar

[15]

P. Grisvard, Caractérisation de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40.  doi: 10.1007/BF00281421.  Google Scholar

[16]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29.   Google Scholar

[17]

S. Guerrero and F. Guillén-González, On the controllability of the hydrostatic Stokes equations,, J. Math. Fluid Mech., 10 (2008), 402.  doi: 10.1007/s00021-006-0237-3.  Google Scholar

[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.   Google Scholar

[19]

Oleg Imanuvilov and Takéo Takahashi, Exact controllability of a fluid-rigid body system,, J. Math. Pures Appl. (9), 87 (2007), 408.   Google Scholar

[20]

Oleg Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 6 (2001), 39.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

[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.  doi: 10.2977/prims/1145476103.  Google Scholar

[24]

Jean-Pierre Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790.  doi: 10.1137/050628726.  Google Scholar

[25]

Roger Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes,, Bull. Soc. Math. France, 96 (1968), 115.   Google Scholar

[1]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441

[2]

Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687

[3]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

[4]

Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations & Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022

[5]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020

[6]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019031

[7]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159

[8]

Irina F. Sivergina, Michael P. Polis. About global null controllability of a quasi-static thermoelastic contact system. Conference Publications, 2005, 2005 (Special) : 816-823. doi: 10.3934/proc.2005.2005.816

[9]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437

[10]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183

[11]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699

[12]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[13]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639

[14]

Šárka Nečasová. Stokes and Oseen flow with Coriolis force in the exterior domain. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 339-351. doi: 10.3934/dcdss.2008.1.339

[15]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557

[16]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[17]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695

[18]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[19]

Debayan Maity. On the null controllability of the Lotka-Mckendrick system. Mathematical Control & Related Fields, 2019, 9 (4) : 719-728. doi: 10.3934/mcrf.2019048

[20]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]