Some controllability results for the relativistic Vlasov-Maxwell system
[Quelques résultats de contrôlabilité pour le système de Vlasov-Maxwell relativiste]
Han-Kwan, Daniel1
1 DMA Ecole Normale SupŽrieure 45 rue d’Ulm 75005 Paris France
Journées équations aux dérivées partielles (2012), article no. 5, 12 p.

L’objectif de cette note est de présenter les résultats récents concernant la contrôlabilité du système de Vlasov-Maxwell, qui sont prouvés dans le papier [10] écrit en collaboration avec Olivier Glass.

The goal of this note is to present the recent results concerning the controllability of the Vlasov-Maxwell system, which are proved in the paper [10] by Olivier Glass and the author.

DOI : 10.5802/jedp.88
Classification : 35Q83, 93B05
Keywords: Vlasov-Maxwell equations, controllability, geometric control condition
Mot clés : Equations de Vlasov-Maxwell, contrôlabilité, condition de contrôle géométrique
Han-Kwan, Daniel 1

1 DMA Ecole Normale SupŽrieure 45 rue d’Ulm 75005 Paris France 
@incollection{JEDP_2012____A5_0,
     author = {Han-Kwan, Daniel},
     title = {Some controllability results for the relativistic {Vlasov-Maxwell} system},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {5},
     pages = {1--12},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2012},
     doi = {10.5802/jedp.88},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.88/}
}
TY  - JOUR
AU  - Han-Kwan, Daniel
TI  - Some controllability results for the relativistic Vlasov-Maxwell system
JO  - Journées équations aux dérivées partielles
PY  - 2012
SP  - 1
EP  - 12
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.88/
DO  - 10.5802/jedp.88
LA  - en
ID  - JEDP_2012____A5_0
ER  - 
%0 Journal Article
%A Han-Kwan, Daniel
%T Some controllability results for the relativistic Vlasov-Maxwell system
%J Journées équations aux dérivées partielles
%D 2012
%P 1-12
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.88/
%R 10.5802/jedp.88
%G en
%F JEDP_2012____A5_0
Han-Kwan, Daniel. Some controllability results for the relativistic Vlasov-Maxwell system. Journées équations aux dérivées partielles (2012), article  no. 5, 12 p. doi : 10.5802/jedp.88. http://www.numdam.org/articles/10.5802/jedp.88/

[1] Asano, K. On local solutions of the initial value problem for the Vlasov-Maxwell equation, Comm. Math. Phys., Volume 106 (1986) no. 4, pp. 551-568 http://projecteuclid.org/getRecord?id=euclid.cmp/1104115851 | MR | Zbl

[2] Asano, K.; Ukai, S. On the Vlasov-Poisson limit of the Vlasov-Maxwell equation, Patterns and waves (Stud. Math. Appl.), Volume 18, North-Holland, Amsterdam, 1986, pp. 369-383 | MR | Zbl

[3] Bardos, C.; Lebeau, G.; Rauch, J. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992) no. 5, pp. 1024-1065 | DOI | MR | Zbl

[4] Burq, N.; Gérard, P. Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math., Volume 325 (1997) no. 7, pp. 749-752 | DOI | MR | Zbl

[5] Coron, J.-M. Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007 | MR | Zbl

[6] Degond, P. Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity, Math. Methods Appl. Sci., Volume 8 (1986) no. 4, pp. 533-558 | DOI | MR | Zbl

[7] Ervedoza, S.; Zuazua, E. A systematic method for building smooth controls for smooth data, Discrete Contin. Dyn. Syst. Ser. B, Volume 14 (2010) no. 4, pp. 1375-1401 | DOI | MR | Zbl

[8] Glass, O. On the controllability of the Vlasov-Poisson system, J. Differential Equations, Volume 195 (2003) no. 2, pp. 332-379 | DOI | MR | Zbl

[9] Glass, O. LA MÉTHODE DU RETOUR EN CONTRoLABILITÉ ET SES APPLICATIONS EN MÉCANIQUE DES FLUIDES, Séminaire Bourbaki (2010)

[10] Glass, O.; Han-Kwan, D. On the controllability of the relativistic Vlasov-Maxwell system, Preprint (2012) | MR

[11] Glass, O.; Han-Kwan, D. On the controllability of the Vlasov-Poisson system in the presence of external force fields, J. Differential Equations, Volume 252 (2012) no. 10, pp. 5453-5491 | MR | Zbl

[12] Glassey, R.; Strauss, W. Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., Volume 92 (1986) no. 1, pp. 59-90 | DOI | MR | Zbl

[13] Phung, K. D. Contrôle et stabilisation d’ondes électromagnétiques, ESAIM Control Optim. Calc. Var., Volume 5 (2000), p. 87-137 (electronic) | DOI | Numdam | MR | Zbl

[14] Rauch, J.; Taylor, M. Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., Volume 24 (1974), pp. 79-86 | MR | Zbl

[15] Schaeffer, J. The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., Volume 104 (1986) no. 3, pp. 403-421 http://projecteuclid.org/getRecord?id=euclid.cmp/1104115084 | MR | Zbl

[16] Wollman, S. An existence and uniqueness theorem for the Vlasov-Maxwell system, Comm. Pure Appl. Math., Volume 37 (1984) no. 4, pp. 457-462 | DOI | MR | Zbl

[17] Wollman, S. Local existence and uniqueness theory of the Vlasov-Maxwell system, J. Math. Anal. Appl., Volume 127 (1987) no. 1, pp. 103-121 | DOI | MR | Zbl

Cité par Sources :