Boundary stabilization of Maxwell's equations with space-time variable coefficients
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 563-578.

We consider the stabilization of Maxwell's equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver-Müller boundary condition. This is based on some stability estimates that are obtained using the “standard” identity with multiplier and appropriate properties of the feedback. We deduce an explicit decay rate of the energy, for instance exponential, polynomial or logarithmic decays are available for appropriate feedbacks.

DOI : https://doi.org/10.1051/cocv:2003027
Classification : 93D15,  93B05,  93C20
Mots clés : Maxwell's system, boundary stabilization
@article{COCV_2003__9__563_0,
     author = {Nicaise, Serge and Pignotti, Cristina},
     title = {Boundary stabilization of {Maxwell's} equations with space-time variable coefficients},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {563--578},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003027},
     zbl = {1063.93041},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2003027/}
}
TY  - JOUR
AU  - Nicaise, Serge
AU  - Pignotti, Cristina
TI  - Boundary stabilization of Maxwell's equations with space-time variable coefficients
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2003
DA  - 2003///
SP  - 563
EP  - 578
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2003027/
UR  - https://zbmath.org/?q=an%3A1063.93041
UR  - https://doi.org/10.1051/cocv:2003027
DO  - 10.1051/cocv:2003027
LA  - en
ID  - COCV_2003__9__563_0
ER  - 
Nicaise, Serge; Pignotti, Cristina. Boundary stabilization of Maxwell's equations with space-time variable coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 563-578. doi : 10.1051/cocv:2003027. http://www.numdam.org/articles/10.1051/cocv:2003027/

[1] H. Barucq and B. Hanouzet, 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. | Zbl 0776.35073

[2] C. Castro and E. Zuazua, Localization of waves in 1-d highly heterogeneous media. Arch. Rational Mech. Anal. 164 (2002) 39-72. | MR 1921162 | Zbl 1016.35003

[3] M.G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces. Israel J. Math. 11 (1972) 57-94. | MR 300166 | Zbl 0249.34049

[4] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag, Vol. 3 (1990), Vol. 5 (1992).

[5] M. Eller, J.E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping. Comp. Appl. Math. 21 (2002) 135-165. | MR 2009950 | Zbl 1119.93402

[6] L.C. Evans, Nonlinear evolution equations in an arbitrary Banach space. Israel J. Math. 26 (1977) 1-42. | MR 440431 | Zbl 0349.34043

[7] P. Grisvard, Elliptic problems in nonsmooth Domains. Pitman, Boston, Monogr. Stud. Math. 21 (1985). | MR 775683 | Zbl 0695.35060

[8] T. Kato, Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19 (1967) 508-520. | MR 226230 | Zbl 0163.38303

[9] T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, CIME, II Ciclo. Cortona (1976) 125-191. | Zbl 0456.35052

[10] T. Kato, Abstract differential equations and nonlinear mixed problems. Accademia Nazionale dei Lincei, Scuola Normale Superiore, Lezione Fermiane, Pisa (1985). | MR 930267 | Zbl 0648.35001

[11] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. Masson-John Wiley, Collection RMA Paris 36 (1994). | MR 1359765 | Zbl 0937.93003

[12] V. Komornik, Boundary stabilization, observation and control of Maxwell's equations. Panamer. Math. J. 4 (1994) 47-61. | Zbl 0849.35136

[13] J.E. Lagnese, Exact controllability of Maxwell's equations in a general region. SIAM J. Control Optim. 27 (1989) 374-388. | Zbl 0678.49032

[14] C.-Y. Lin, Time-dependent nonlinear evolution equations. Differential Integral Equations 15 (2002) 257-270. | MR 1870643 | Zbl 1041.34049

[15] S. Nicaise, M. Eller and J.E. Lagnese, Stabilization of heterogeneous Maxwell's equations by nonlinear boundary feedbacks. EJDE 2002 (2002) 1-26. | Zbl 1030.93026

[16] S. Nicaise, Exact boundary controllability of Maxwell's equations in heteregeneous media and an application to an inverse source problem. SIAM J. Control Optim. 38 (2000) 1145-1170. | Zbl 0963.93041

[17] L. Paquet, Problèmes mixtes pour le système de Maxwell. Ann. Fac. Sci. Toulouse Math. 4 (1982) 103-141. | Numdam | MR 687546 | Zbl 0529.58038

[18] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag,, Appl. Math. Sci. 44 (1983). | MR 710486 | Zbl 0516.47023

[19] K.D. Phung, Contrôle et stabilisation d'ondes électromagnétiques. ESAIM: COCV 5 (2000) 87-137. | Numdam | Zbl 0942.93002

[20] C. Pignotti, Observability and controllability of Maxwell's equations. Rend. Mat. Appl. 19 (1999) 523-546. | Zbl 0979.93057

Cité par Sources :