Boundary stabilization of Maxwell's equations with space-time variable coefficients

Nicaise, Serge; Pignotti, Cristina

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.

Classification : 93D15, 93B05, 93C20
Mots clés : Maxwell's system, boundary stabilization

[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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

[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

[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

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

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

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

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