Comptes Rendus
no. 9
Partial Differential Equations
Remarks on global approximate controllability for the 2-D Navier–Stokes system with Dirichlet boundary conditions
[Remarques sur la contrôlabilité globale approchée du système 2-D de Navier–Stokes avec conditions au bord de type Dirichlet]

Présenté par : Pierre-Louis Lions

Sergio Guerrero1 ; Oleg Yurievich Imanuvilov2 ; Jean-Pierre Puel3
1 Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, boîte corrier, 187, 75035 Paris cedex 05, France
2 Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80523-1874, USA
3 Laboratoire de mathématiques de Versailles, université de Versailles-St-Quentin, 45, avenue des États Unis, 78035 Versailles, France
Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 573-577.

Cette Note concerne le système de Navier–Stokes en dimension 2. Nous montrons un résultat de contrôlabilité approchée globale à l'aide de contrôles frontière.

This Note deals with the two-dimensional Navier–Stokes system. In this context, we prove a result concerning its global approximate controllability by means of boundary controls.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.09.023
Sergio Guerrero 1 ; Oleg Yurievich Imanuvilov 2 ; Jean-Pierre Puel 3

1 Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, boîte corrier, 187, 75035 Paris cedex 05, France
2 Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80523-1874, USA
3 Laboratoire de mathématiques de Versailles, université de Versailles-St-Quentin, 45, avenue des États Unis, 78035 Versailles, France 
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     title = {Remarks on global approximate controllability for the {2-D} {Navier{\textendash}Stokes} system with {Dirichlet} boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {573--577},
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     volume = {343},
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     year = {2006},
     doi = {10.1016/j.crma.2006.09.023},
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Sergio Guerrero; Oleg Yurievich Imanuvilov; Jean-Pierre Puel. Remarks on global approximate controllability for the 2-D Navier–Stokes system with Dirichlet boundary conditions. Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 573-577. doi : 10.1016/j.crma.2006.09.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.023/

[1] J.-M. Coron Global asymptotic stabilization for the controllable systems without drift, Math. Control Signals Systems, Volume 5 (1992) no. 3, pp. 295-312

[2] J.-M. Coron On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESIAN COCV, Volume 1 (1995/96), pp. 35-75

[3] C. Fabre; G. Lebeau Régularité et unicité pour le problème de Stokes, Comm. Partial Differential Equations, Volume 27 (2002) no. 3–4, pp. 437-475

[4] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, J.-P. Puel, Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl. 83 (12) 1501–1542

[5] O.Yu. Imanuvilov Boundary controllability of parabolic equations, Sb. Math., Volume 186 (1995), pp. 879-900

[6] Y. Giga; H. Sohr Abstract Lp estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal., Volume 102 (1991), pp. 72-94

[7] S. Guerrero, O. Yu. Imanuvilov, J.-P. Puel, Global approximate controllability of the 2-D Navier–Stokes equations with Dirichlet boundary conditions, Preprint, 2006

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