no. 13-14
Optimal Control/Calculus of Variations
A least-squares formulation for the approximation of null controls for the Stokes system
[Une formulation moindres carrés de lʼapproximation de contrôles à zéro pour le système de Stokes]

Présenté par : Olivier Pironneau

Münch, Arnaud1 ; Pedregal, Pablo2
1 Laboratoire de mathématiques, Université Blaise-Pascal (Clermont-Ferrand-2), UMR CNRS 6620, campus des Cézeaux, 63177 Aubière, France
2 Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La-Mancha, 13071 Ciudad Real, Spain
Comptes Rendus. Mathématique, Tome 351 (2013) no. 13-14, pp. 545-550.

Cette note concerne lʼapproximation de contrôles exactes pour le système de Stokes. Lʼexistence de contrôles L2 a été obtenue dans Fursikov et Imanuvilov (1996) [5], en utilisant des inégalités de type Carleman. On introduit et analyse une formulation de type moindres carrés et on montre quʼelle permet la construction de suites convergentes de fonctions vers des contrôles à zéro du système de Stokes.

This work deals with the approximation of distributed null controls for the Stokes equation. The existence of L2 controls has been obtained by Fursikov and Imanuvilov (1996) [5] via Carleman-type estimates. We introduce and analyze a least-squares formulation of the controllability problem, and we show that it allows the construction of convergent sequences of functions toward null controls for the Stokes system.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.07.019
Münch, Arnaud 1 ; Pedregal, Pablo 2

1 Laboratoire de mathématiques, Université Blaise-Pascal (Clermont-Ferrand-2), UMR CNRS 6620, campus des Cézeaux, 63177 Aubière, France
2 Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La-Mancha, 13071 Ciudad Real, Spain 
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Münch, Arnaud; Pedregal, Pablo. A least-squares formulation for the approximation of null controls for the Stokes system. Comptes Rendus. Mathématique, Tome 351 (2013) no. 13-14, pp. 545-550. doi : 10.1016/j.crma.2013.07.019. http://www.numdam.org/articles/10.1016/j.crma.2013.07.019/

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[8] A. Münch, P. Pedregal, Numerical null controllability of the heat equation through a variational approach, preprint.

[9] A. Münch, P. Pedregal, Numerical approximations of null controls for the Stokes through a least-squares approach, in preparation.

[10] Münch, A.; Zuazua, E. Numerical approximation of null controls for the heat equation: ill-posedness and remedies, Inverse Probl., Volume 26 (2010) no. 8, p. 085018 (39 p)

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