Comptes Rendus
no. 4
Partial Differential Equations/Optimal Control
On the controllability of the N-dimensional Navier–Stokes and Boussinesq systems with N−1 scalar controls
[Sur la contrôlabilité des systèmes de Navier–Stokes et Boussinesq N-dimensionnels avec N−1 contrôles scalaires]

Présenté par : Pierre-Louis Lions

Enrique Fernández-Cara1 ; Sergio Guerrero1 ; Oleg Yurievich Imanuvilov2 ; Jean-Pierre Puel3
1 Departamento E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064, USA
3 Laboratoire de mathématiques appliquées, université de Versailles – St Quentin, 45, avenue des Etats Unis, 78035 Versailles, France
Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 275-280.

Dans cette Note on présente quelques résultats de contrôlabilité pour des systèmes non linéaires du type Navier–Stokes et Boussinesq. On analyse l'existence de contrôles particuliers avec un nombre petit de degrés de liberté.

In this Note we present several controllability results for nonlinear systems of the Navier–Stokes and Boussinesq kind. We discuss the existence of particular controls with a small number of degrees of freedom.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.12.013
Enrique Fernández-Cara 1 ; Sergio Guerrero 1 ; Oleg Yurievich Imanuvilov 2 ; Jean-Pierre Puel 3

1 Departamento E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064, USA
3 Laboratoire de mathématiques appliquées, université de Versailles – St Quentin, 45, avenue des Etats Unis, 78035 Versailles, France 
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     title = {On the controllability of the {\protect\emph{N}-dimensional} {Navier{\textendash}Stokes} and {Boussinesq} systems with $ N-1$ scalar controls},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {275--280},
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Enrique Fernández-Cara; Sergio Guerrero; Oleg Yurievich Imanuvilov; Jean-Pierre Puel. On the controllability of the N-dimensional Navier–Stokes and Boussinesq systems with $ N-1$ scalar controls. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 275-280. doi : 10.1016/j.crma.2004.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.013/

[1] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov, J.P. Puel, Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl., in press

[2] O.Yu. Imanuvilov Remarks on exact controllability for the Navier–Stokes equations, ESAIM Control Optim. Calc. Var., Volume 6 (2001), pp. 39-72

[3] O.Yu. Imanuvilov; J.P. Puel Global Carleman estimates for weak elliptic non homogeneous Dirichlet problem, Int. Math. Res. Notices, Volume 16 (2003), pp. 883-913

[4] O.Yu. Imanuvilov; M. Yamamoto Carleman Estimate for a Parabolic Equation in a Sobolev Space of Negative Order and its Applications, Lecture Notes in Pure and Appl. Math., vol. 218, Dekker, New York, 2001

[5] J.-L. Lions; E. Zuazua A generic uniqueness result for the Stokes system and its control theoretical consequences (P. Marcellini; G. Talenti; E. Visentini, eds.), Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., vol. 177, Dekker, New York, 1996, pp. 221-235

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