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.
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Enrique Fernández-Cara 1; Sergio Guerrero 1; Oleg Yurievich Imanuvilov 2; Jean-Pierre Puel 3
@article{CRMATH_2004__338_5_375_0, author = {Enrique Fern\'andez-Cara and Sergio Guerrero and Oleg Yurievich Imanuvilov and Jean-Pierre Puel}, 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}, }
TY - JOUR AU - Enrique Fernández-Cara AU - Sergio Guerrero AU - Oleg Yurievich Imanuvilov AU - Jean-Pierre Puel TI - Remarks on exact controllability for Stokes and Navier–Stokes systems JO - Comptes Rendus. Mathématique PY - 2004 SP - 375 EP - 380 VL - 338 IS - 5 PB - Elsevier DO - 10.1016/j.crma.2003.12.016 LA - en ID - CRMATH_2004__338_5_375_0 ER -
%0 Journal Article %A Enrique Fernández-Cara %A Sergio Guerrero %A Oleg Yurievich Imanuvilov %A Jean-Pierre Puel %T Remarks on exact controllability for Stokes and Navier–Stokes systems %J Comptes Rendus. Mathématique %D 2004 %P 375-380 %V 338 %N 5 %I Elsevier %R 10.1016/j.crma.2003.12.016 %G en %F CRMATH_2004__338_5_375_0
Enrique Fernández-Cara; Sergio Guerrero; Oleg Yurievich Imanuvilov; Jean-Pierre Puel. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.016/
[1] Optimal Control, Consultants Bureau, New York, 1987 (Translated from the Russian by V.M. Volosov)
[2] E. Fernández-Cara, S. Guerrero, Global Carleman inequalities for parabolic systems and applications to null controllability, in press
[3] Controllability of Evolution Equations, Lecture Notes, vol. 34, Seoul National University, Korea, 1996
[4] 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] Controllability of parabolic equations, Math. Sb., Volume 186 (1995) no. 6, pp. 109-132 (in Russian)
[6] Remarks on exact controllability for the Navier–Stokes equations, ESAIM: Control Optim. Calc. Var., Volume 6 (2001), pp. 39-72
[7] Global Carleman estimates for weak elliptic nonhomogeneous Dirichlet problem, Int. Math. Res. Notices, Volume 16 (2003), pp. 883-913
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