[Contrôlabilité indirecte de systèmes localement couplés sous des conditions géométriques]
On sʼintéresse à des systèmes constitués de deux équations dʼondes, de la chaleur ou de Schrödinger, couplées par un terme dʼordre zéro, et dont seulement lʼune est controlée. En supposant que les zones de couplage et de contrôle satisfont toutes deux la Condition Géométrique de Contrôle, on montre un résultat de contrôle interne et frontière en dimension quelconque dʼespace. Ceci fournit de nombreux exemples pour lesquels ces deux régions ne sʼintersectent pas.
We consider systems of two wave/heat/Schrödinger-type equations coupled by a zero order term, only one of them being controlled. We prove an internal and a boundary null-controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition. This includes several examples in which these two regions have an empty intersection.
Accepté le :
Publié le :
@article{CRMATH_2011__349_7-8_395_0,
author = {Alabau-Boussouira, Fatiha and L\'eautaud, Matthieu},
title = {Indirect controllability of locally coupled systems under geometric conditions},
journal = {Comptes Rendus. Math\'ematique},
pages = {395--400},
publisher = {Elsevier},
volume = {349},
number = {7-8},
year = {2011},
doi = {10.1016/j.crma.2011.02.004},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2011.02.004/}
}
TY - JOUR AU - Alabau-Boussouira, Fatiha AU - Léautaud, Matthieu TI - Indirect controllability of locally coupled systems under geometric conditions JO - Comptes Rendus. Mathématique PY - 2011 SP - 395 EP - 400 VL - 349 IS - 7-8 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2011.02.004/ DO - 10.1016/j.crma.2011.02.004 LA - en ID - CRMATH_2011__349_7-8_395_0 ER -
%0 Journal Article %A Alabau-Boussouira, Fatiha %A Léautaud, Matthieu %T Indirect controllability of locally coupled systems under geometric conditions %J Comptes Rendus. Mathématique %D 2011 %P 395-400 %V 349 %N 7-8 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2011.02.004/ %R 10.1016/j.crma.2011.02.004 %G en %F CRMATH_2011__349_7-8_395_0
Alabau-Boussouira, Fatiha; Léautaud, Matthieu. Indirect controllability of locally coupled systems under geometric conditions. Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 395-400. doi : 10.1016/j.crma.2011.02.004. http://archive.numdam.org/articles/10.1016/j.crma.2011.02.004/
[1] Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim., Volume 41 (2002), pp. 511-541
[2] A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., Volume 42 (2003), pp. 871-906
[3] F. Alabau-Boussouira, M. Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var. (2011), in press.
[4] Null controllability of some reaction–diffusion systems with one control force, J. Math. Anal. Appl., Volume 320 (2006), pp. 928-943
[5] Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992), pp. 1024-1065
[6] Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, Volume 25 (2000), pp. 39-72
[7] Boundary controllability of parabolic coupled equations, J. Funct. Anal., Volume 259 (2010) no. 7, pp. 1720-1758
[8] Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., Volume 46 (2006), pp. 123-162
[9] Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var., Volume 16 (2010) no. 2, pp. 247-274
[10] Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., Volume 258 (2010), pp. 2739-2778
[11] Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées, vol. 8, Masson, Paris, 1988
[12] Controllability cost of conservative systems: resolvent condition and transmutation, J. Funct. Anal., Volume 218 (2005) no. 2, pp. 425-444
[13] The control transmutation method and the cost of fast controls, SIAM J. Control Optim., Volume 45 (2006) no. 2, pp. 762-772
[14] Observability and control of Schrödinger equations, SIAM J. Control Optim., Volume 40 (2001) no. 1, pp. 211-230
[15] Exact controllability of a cascade system of conservative equations, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 5–6, pp. 291-296
[16] A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. Appl. Math., Volume 52 (1973), pp. 189-221
Cité par Sources :
