A singular perturbation problem in exact controllability of the Maxwell system
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001) , pp. 275-289.

This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact controllability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection, for small values of the perturbation parameter, between observability estimates for the two systems, and between the optimality systems corresponding to the problem of norm minimum exact control of the solutions of the two systems from the rest state to a specified terminal state.

Classification : 93B05,  35Q60,  49N10,  93C20
Mots clés : Maxwell system, exact controllability, singular perturbation
@article{COCV_2001__6__275_0,
author = {Lagnese, John E.},
title = {A singular perturbation problem in exact controllability of the Maxwell system},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {275--289},
publisher = {EDP-Sciences},
volume = {6},
year = {2001},
zbl = {1030.93025},
mrnumber = {1824104},
language = {en},
url = {www.numdam.org/item/COCV_2001__6__275_0/}
}
Lagnese, John E. A singular perturbation problem in exact controllability of the Maxwell system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001) , pp. 275-289. http://www.numdam.org/item/COCV_2001__6__275_0/

[1] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV (to appear). | Numdam | MR 1836048 | Zbl 0992.93039

[2] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976). | MR 521262 | Zbl 0331.35002

[3] M. Eller (private communication).

[4] M. Eller, Exact boundary controllability of electromagnetic fields in a general region. Appl. Math. Optim. (to appear). | MR 1861468 | Zbl 0997.35099

[5] E. Hendrickson and I. Lasiecka, Numerical approximations and regularization of Riccati equations arising in hyperbolic dynamics with unbounded control operators. Comput. Optim. and Appl. 2 (1993) 343-390. | MR 1259415 | Zbl 0791.49007

[6] E. Hendrickson and I. Lasiecka, Finite dimensional approximations of boundary control problems arising in partially observed hyperbolic systems. Dynam. Cont. Discrete Impuls. Systems 1 (1995) 101-142. | MR 1361235 | Zbl 0876.93046

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

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

[9] J. Lagnese, The Hilbert Uniqueness Method: A retrospective, edited by K.-H. Hoffmann and W. Krabs. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 149 (1991). | MR 1178298 | Zbl 0850.93104

[10] I. Lasiecka and R. Triggiani, A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations. Proc. Amer. Math. Soc. 104 (1988) 745-755. | MR 964851 | Zbl 0699.47034

[11] R. Leis, Initial Boundary Value Problems in Mathematical Physics. B. G. Teubner, Stuttgart (1986). | MR 841971 | Zbl 0599.35001

[12] O. Nalin, Contrôlabilité exacte sur une partie du bord des équations de Maxwell. C. R. Acad. Sci. Paris 309 (1989) 811-815. | MR 1055200 | Zbl 0688.49041

[13] K.D. Phung, Contrôle et stabilisation d'ondes électromagnétiques. ESIAM: COCV 5 (2000) 87-137. | EuDML eudml.org/doc/90586 | Numdam | Zbl 0942.93002

[14] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189-211. | MR 341256 | Zbl 0274.35041

[15] M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. Preprint. | Zbl 1063.93026