Partial Differential Equations/Optimal Control
Remarks on exact controllability for Stokes and Navier–Stokes systems
Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 375-380.

This Note deals with the controllability of Stokes and Navier–Stokes systems with distributed controls with support in possibly small subdomains. We first present a new global Carleman inequality for the solutions to Stokes-like systems that leads to the null controllability at any time T>0. Then, we present a local result concerning exact controllability to trajectories of the Navier–Stokes system.

Cette Note concerne la contrôlabilité des systèmes de Stokes et Navier–Stokes avec contrôles distribués pour lesquels les supports sont éventuellement petits. On présente d'abord une nouvelle inégalité globale de Carleman pour les solutions d'un problème de type Stokes. On en déduit la contrôlabilité exacte à zéro en tout temps T>0. Ensuite, on présente un résultat local de contrôlabilité exacte sur les trajectoires pour le système de Navier–Stokes.

Published online:
DOI: 10.1016/j.crma.2003.12.016
Fernández-Cara, Enrique 1; Guerrero, Sergio 1; Imanuvilov, Oleg Yurievich 2; Puel, Jean-Pierre 3

1 Dpto. 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 États Unis, 78035 Versailles, France
     author = {Fern\'andez-Cara, Enrique and Guerrero, Sergio and Imanuvilov, Oleg Yurievich and Puel, Jean-Pierre},
     title = {Remarks on exact controllability for {Stokes} and {Navier{\textendash}Stokes} systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {375--380},
     publisher = {Elsevier},
     volume = {338},
     number = {5},
     year = {2004},
     doi = {10.1016/j.crma.2003.12.016},
     language = {en},
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Fernández-Cara, Enrique; Guerrero, Sergio; Imanuvilov, Oleg Yurievich; Puel, Jean-Pierre. Remarks on exact controllability for Stokes and Navier–Stokes systems. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 375-380. doi : 10.1016/j.crma.2003.12.016.

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[2] E. Fernández-Cara, S. Guerrero, Global Carleman inequalities for parabolic systems and applications to null controllability, in press

[3] Fursikov, A.; Imanuvilov, O.Yu. Controllability of Evolution Equations, Lecture Notes, vol. 34, Seoul National University, Korea, 1996

[4] Giga, Y.; Sohr, H. 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

[5] Imanuvilov, O.Yu. Controllability of parabolic equations, Math. Sb., Volume 186 (1995) no. 6, pp. 109-132 (in Russian)

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

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

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