A singular perturbation problem in exact controllability of the Maxwell system

John E. Lagnese

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 275-289
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Lagnese, John E.. "A singular perturbation problem in exact controllability of the Maxwell system." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 275-289. <http://eudml.org/doc/90595>.

@article{Lagnese2001,
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.},
author = {Lagnese, John E.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Maxwell system; exact controllability; singular perturbation; robustness; minimum norm control},
language = {eng},
pages = {275-289},
publisher = {EDP-Sciences},
title = {A singular perturbation problem in exact controllability of the Maxwell system},
url = {http://eudml.org/doc/90595},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Lagnese, John E.
TI - A singular perturbation problem in exact controllability of the Maxwell system
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 275
EP - 289
AB - 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.
LA - eng
KW - Maxwell system; exact controllability; singular perturbation; robustness; minimum norm control
UR - http://eudml.org/doc/90595
ER -

References

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  1. [1] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV (to appear). Zbl0992.93039MR1836048
  2. [2] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976). Zbl0331.35002MR521262
  3. [3] M. Eller (private communication). 
  4. [4] M. Eller, Exact boundary controllability of electromagnetic fields in a general region. Appl. Math. Optim. (to appear). Zbl0997.35099MR1861468
  5. [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. Zbl0791.49007MR1259415
  6. [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. Zbl0876.93046MR1361235
  7. [7] V. Komornik, Boundary stabilization, observation and control of Maxwell’s equations. Panamer. Math. J. 4 (1994) 47-61. Zbl0849.35136
  8. [8] J. Lagnese, Exact boundary controllability of Maxwell’s equations in a general region. SIAM J. Control Optim. 27 (1989) 374-388. Zbl0678.49032
  9. [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). Zbl0850.93104MR1178298
  10. [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. Zbl0699.47034MR964851
  11. [11] R. Leis, Initial Boundary Value Problems in Mathematical Physics. B. G. Teubner, Stuttgart (1986). Zbl0599.35001MR841971
  12. [12] O. Nalin, Contrôlabilité exacte sur une partie du bord des équations de Maxwell. C. R. Acad. Sci. Paris 309 (1989) 811-815. Zbl0688.49041MR1055200
  13. [13] K.D. Phung, Contrôle et stabilisation d’ondes électromagnétiques. ESIAM: COCV 5 (2000) 87-137. Zbl0942.93002
  14. [14] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189-211. Zbl0274.35041MR341256
  15. [15] M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. Preprint. Zbl1125.93383

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