Partial Differential Equations
Uniqueness results for Stokes cascade systems and application to insensitizing controls
Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 831-835.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.09.008
Gueye, Mamadou 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
@article{CRMATH_2012__350_17-18_831_0,
     author = {Gueye, Mamadou},
     title = {Uniqueness results for {Stokes} cascade systems and application to insensitizing controls},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {831--835},
     publisher = {Elsevier},
     volume = {350},
     number = {17-18},
     year = {2012},
     doi = {10.1016/j.crma.2012.09.008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.09.008/}
}
TY  - JOUR
AU  - Gueye, Mamadou
TI  - Uniqueness results for Stokes cascade systems and application to insensitizing controls
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 831
EP  - 835
VL  - 350
IS  - 17-18
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.09.008/
DO  - 10.1016/j.crma.2012.09.008
LA  - en
ID  - CRMATH_2012__350_17-18_831_0
ER  - 
%0 Journal Article
%A Gueye, Mamadou
%T Uniqueness results for Stokes cascade systems and application to insensitizing controls
%J Comptes Rendus. Mathématique
%D 2012
%P 831-835
%V 350
%N 17-18
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.09.008/
%R 10.1016/j.crma.2012.09.008
%G en
%F CRMATH_2012__350_17-18_831_0
Gueye, Mamadou. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.09.008/

[1] Bodart, O.; Fabre, C. 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] Coron, J.-M.; Guerrero, S. 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] De Teresa, L.; Kavian, O. Unique continuation principle for systems of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010), pp. 247-274

[5] De Teresa, L.; Zuazua, E. 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] Díaz, J.; Fursikov, A. 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] Guerrero, S. 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] Lions, J.-L. 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] Lions, J.-L.; Zuazua, E. 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] Saut, J.C.; Scheurer, B. Unique continuation for some evolution equations, J. Differential Equations, Volume 66 (1987), pp. 118-139

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

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

Cited by Sources: