Comptes Rendus
Partial Differential Equations
Uniqueness results for Stokes cascade systems and application to insensitizing controls
[Resultats dʼunicité pour des systèmes de Stokes en cascade et application au contrôle insensibilisant]
Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 831-835.

Dans cette Note, on sʼintéresse au problème dʼinsensibilisation pour le système de Stokes par un contrôle distribué unidirectionnel. On donne des résultats dʼinsensibilisation approchée avec un contrôle scalaire pour différentes configurations géométriques. Dʼautre part on donne un résultat négatif dʼinsensibilisation, par un contrôle scalaire, pour le système de Stokes posé sur une certaine variété bidimensionelle sans bord.

This Note is devoted to some insensitizing control problems for the Stokes system with a reduced number of controls. We give some ε-insensitization results with external unidirectional forces in different geometric configurations. We also provide a negative result of insensitization for the Stokes system in some 2-D manifold without boundary with one scalar control.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.09.008

Mamadou Gueye 1

1 Université Pierre-et-Marie-Curie-Paris 6, UMR 7598, laboratoire Jacques-Louis-Lions, boîte courrier 187, 4, place Jussieu 75252, Paris cedex 05, France
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Mamadou Gueye. Uniqueness results for Stokes cascade systems and application to insensitizing controls. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 831-835. doi : 10.1016/j.crma.2012.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.008/

[1] O. Bodart; C. Fabre Controls insensitizing the norm of the solution of a semilinear heat equation, J. Math. Anal. Appl., Volume 195 (1995) no. 3, pp. 658-683

[2] N. Carreño, M. Gueye, Insensitizing controls with one vanishing component for the Navier–Stokes system, preprint, 2012.

[3] J.-M. Coron; S. Guerrero Local null controllability of the two-dimensional Navier–Stokes system in the torus with a control force having a vanishing component, Journal de Mathématiques Pures et Appliquées (9), Volume 92 (2009) no. 5, pp. 528-545

[4] L. De Teresa; O. Kavian Unique continuation principle for systems of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010), pp. 247-274

[5] L. De Teresa; E. Zuazua Identification of the class of initial data for the insensitizing control of the heat equation, Commun. Pure Appl. Anal., Volume 8 (2009) no. 1, pp. 457-471

[6] J. Díaz; A. Fursikov Approximate controllability of the Stokes system on cylinders by external unidirectional forces, J. Math. Pures Appl., Volume 9 (1997) no. 76, pp. 353-375

[7] S. Guerrero Controllability of systems of Stokes equations with one control force: existence of insensitizing controls, Annales de lʼInstitut Henri Poincaré Analyse Non Linéaire, Volume 24 (2007) no. 6, pp. 1029-1054

[8] M. Gueye, Insensitizing controls for Navier–Stokes equations, preprint, 2011.

[9] J.-L. Lions Quelques notions dans lʼanalyse et le contrôle de systèmes à données incomplètes, Proceedings of the XI Congress on Differential Equations and Applications/First Congress on Applied Mathematics, Univ. Málaga, Málaga, 1989, pp. 43-54

[10] J.-L. Lions; E. Zuazua A Generic Uniqueness Result for the Stokes System and its Control Theoretical Consequences, Lecture Notes in Pure and Appl. Math., vol. 177, Dekker, New York, 1996 (pp. 221–235)

[11] R. Perez-García, Algunos resultados de control para algunos problemas parabólicos acoplados no lineales: Controlabilidad y controles insensibilizantes, Ph.D. thesis, University of Sevilla, Spain, 2004.

[12] J.C. Saut; B. Scheurer Unique continuation for some evolution equations, J. Differential Equations, Volume 66 (1987), pp. 118-139

[13] R. Temam Navier–Stokes Equations, Theory and Numerical Analysis, Stud. Math. Appl., vol. 2, North-Holland, Amsterdam–New York–Oxford, 1977

[14] K. Uhlenbeck Generic property of eigenfunctions, American J. Math., Volume 98 (1976) no. 4, pp. 1059-1078

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