Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T09:21:24.700Z Has data issue: false hasContentIssue false
  • English
  • Français

Contrôle et stabilisation d'ondesélectromagnétiques

Published online by Cambridge University Press:  15 August 2002

Kim Dang Phung
Kim Dang Phung*
Affiliation:
Centre de Mathématiques et leurs Applications, CMLA, École Normale Supérieure de Cachan, ENS, Cachan, France; phung@cmla.ens-cachan.fr.
Get access

Abstract

We consider the exact controllability and stabilization of Maxwell equation by using results on the propagation of singularities of the electromagnetic field. We will assume geometrical control condition and use techniques of the work of Bardos et al. on the wave equation. The problem of internal stabilization will be treated with more attention because the condition divE=0 is not preserved by the system of Maxwell with Ohm's law.

Keywords

Contrôlabilitéstabilisationéquation de Maxwell.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 87 - 137
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H. Barucq, Étude asymptotique du système de Maxwell avec conditions aux limites absorbantes. Thèse de l'université de Bordeaux I (1993). g
Burq, N., Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki. Asterisque 245 (1997) 167-195. g
Barucq, H. et Hanouzet, B., Étude asymptotique du système de Maxwell avec la condition aux limites absorbante de Silver-Müller II. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 1019-1024. g
Bardos, C., Halpern, L., Lebeau, G., Rauch, J. et Zuazua, E., Stabilisation de l'équation des ondes au moyen d'un feedback portant sur la condition aux limites de Dirichlet. Asymptot. Anal. 4 (1991) 285-291. g
Bardos, C., Lebeau, G. et Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. g CrossRef
M. Cessenat, Mathematical method in electromagnetism - linear theorie and applications. Ser. Adv. Math. Appl. Sci. 41 (1996). g
R. Dautray et J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Masson (1988). g
Komornik, V., Boundary stabilization, observation and control of Maxwell's equations. Panamer. Math. J. 4 (1994) 47-61. g
Lagnese, J., Exact boundary controllability of Maxwell's equations in a general region. SIAM J. Control Optim. 27 (1989) 374-388. g CrossRef
G. Lebeau, Contrôle et stabilisation hyperboliques. Séminaire E.D.P. École Polytechnique (1990). g
J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèmes distribués. RMA, Masson, Paris (1988). g
J.-L. Lions et E. Magenes, Problèmes aux limites non homogènes. Dunod (1968). g
Lebeau, G. et Robbiano, L., Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. g CrossRef
Melrose, R. et Sjöstrand, J., Singularities of boundary value problems I. Comm. Pure Appl. Math. 31 (1978) 593-617. g CrossRef
Melrose, R. et Sjöstrand, J., Singularities of boundary value problems II. Comm. Pure Appl. Math. 35 (1982) 129-168. g CrossRef
Nalin, O., Contrôlabilité exacte sur une partie du bord des équations de Maxwell. C. R. Acad. Sci. Paris Sér. I Math. 309 (1989) 811-815. g
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983). g
Phung, K.-D., Stabilisation frontière du système de Maxwell avec la condition aux limites absorbante de Silver-Müller. C. R. Acad. Sci. Paris Sér. I Math. 3233 (1995) 187-192. g
Phung, K.-D., Controlabilité exacte et stabilisation interne des équations de Maxwell. C. R. Acad. Sci. Paris Sér. I Math. 3233 (1996) 169-174. g
Ralston, J.V., Solutions of Wave equation with localized energy. Comm. Pure. Appl. Math. 22 (1969) 807-823. g CrossRef
Rauch, J. et Taylor, M., Penetration into shadow region and unique continuation properties in hyperbolic mixed problems. Indiana Univ. Mathematics J. 22 (1972) 277-284. g CrossRef
M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, N.J. (1981). g
Taylor, M., Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28 (1975) 457-478. g CrossRef
Taylor, M., Grazing rays and reflection of singularities of solutions to wave equations II. Comm. Pure Appl. Math. 29 (1976) 463-481. g CrossRef
N. Weck, Exact boundary controllability for a Maxwell problem, submitted to SIAM J. Control. Optim. g
Weck, N. et Witsch, K.J., Low frequency asymptotics for dissipative Maxwell's equations in bounded domains. Math. Methods Appl. Sci. 13 (1990) 81-93. g CrossRef
Yamamoto, K., Singularities of solutions to the boundary value problem for elastic and Maxwell's equations. Japan J. Math. (N.S.) 14 (1988) 119-163. CrossRef