Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T14:24:06.216Z Has data issue: false hasContentIssue false
  • English
  • Français

A Singular Perturbation Problem in Exact Controllability of the Maxwell System

Published online by Cambridge University Press:  15 August 2002

John E. Lagnese
John E. Lagnese*
Affiliation:
Department of Mathematics, Georgetown University, Washington, DC 20057, U.S.A.; lagnese@math.georgetown.edu.
Get access

Abstract

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.

Keywords

Maxwell systemexact controllabilitysingular perturbation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 275 - 289
Copyright
© EDP Sciences, SMAI, 2001

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

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV (to appear).
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976).
M. Eller (private communication).
M. Eller, Exact boundary controllability of electromagnetic fields in a general region. Appl. Math. Optim. (to appear).
Hendrickson, E. and Lasiecka, I., Numerical approximations and regularization of Riccati equations arising in hyperbolic dynamics with unbounded control operators. Comput. Optim. and Appl. 2 (1993) 343-390. CrossRef
Hendrickson, E. and Lasiecka, I., Finite dimensional approximations of boundary control problems arising in partially observed hyperbolic systems. Dynam. Cont. Discrete Impuls. Systems 1 (1995) 101-142.
Komornik, V., Boundary stabilization, observation and control of Maxwell's equations. Panamer. Math. J. 4 (1994) 47-61.
Lagnese, J., Exact boundary controllability of Maxwell's equations in a general region. SIAM J. Control Optim. 27 (1989) 374-388. CrossRef
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).
Lasiecka, I. and Triggiani, R., 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. CrossRef
R. Leis, Initial Boundary Value Problems in Mathematical Physics. B. G. Teubner, Stuttgart (1986).
Nalin, O., Contrôlabilité exacte sur une partie du bord des équations de Maxwell. C. R. Acad. Sci. Paris 309 (1989) 811-815.
Phung, K.D., Contrôle et stabilisation d'ondes électromagnétiques. ESAIM: COCV 5 (2000) 87-137. CrossRef
Russell, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189-211. CrossRef
M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. Preprint.