Hostname: page-component-cb9f654ff-h4f6x Total loading time: 0 Render date: 2025-08-27T23:50:02.881Z Has data issue: false hasContentIssue false
  • English
  • Français

Time Domain Decomposition in Final Value Optimal Controlof the Maxwell System

Published online by Cambridge University Press:  15 August 2002

John E. Lagnese  and
G. Leugering
John E. Lagnese
Affiliation:
Department of Mathematics, Georgetown University, Washington, DC 20057, USA; lagnese@math.georgetown.edu.
G. Leugering
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany; leugering@mathematik.tu-darmstadt.de.
Get access

Abstract

We consider a boundary optimal control problem for the Maxwell system with afinal value cost criterion. We introduce a time domain decomposition procedurefor the corresponding optimality system which leads to a sequence ofuncoupled optimality systems of local-in-time optimal control problems. Inthe limit full recovery of the coupling conditions is achieved, and, hence,the local solutions and controls converge to the global ones. The process isinherently parallel and is suitable for real-time control applications.

Keywords

Maxwell systemoptimal controldomain decomposition.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 775 - 799
Copyright
© EDP Sciences, SMAI, 2002

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.)

Article purchase

Temporarily unavailable

References

Alonso, A. and Valli, A., An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68 (1999) 607-631. CrossRef
Belishev, M. and Glasman, A., Boundary control of the Maxwell dynamical system: Lack of controllability by topological reason. ESAIM: COCV 5 (2000) 207-218. CrossRef
Benamou, J.-D., Décomposition de domaine pour le contrôle de systèmes gouvernés par des équations d'évolution. C. R. Acad. Sci Paris Sér. I Math. 324 (1997) 1065-1070. CrossRef
Benamou, J.-D., Domain decomposition, optimal control of systems governed by partial differential equations and synthesis of feedback laws. J. Opt. Theory Appl. 102 (1999) 15-36. CrossRef
Benamou, J.-D. and Desprès, B., A domain decomposition method for the Helmholtz equation and related optimal control problems. J. Comp. Phys. 136 (1997) 68-82. CrossRef
M. Gander, L. Halpern and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation. École Polytechnique, Palaiseau, Rep. 469 (2001).
M. Heinkenschloss, Time domain decomposition iterative methods for the solution of distributed linear quadratic optimal control problems (submitted).
J.E. Lagnese, A nonoverlapping domain decomposition for optimal boundary control of the dynamic Maxwell system, in Control of Nonlinear Distributed Parameter Systems, edited by G. Chen, I. Lasiecka and J. Zhou. Marcel Dekker (2001) 157-176.
Lagnese, J.E., Exact boundary controllability of Maxwell's equation in a general region. SIAM J. Control Optim. 27 (1989) 374-388. CrossRef
J.E. Lagnese and G. Leugering, Dynamic domain decomposition in approximate and exact boundary control problems of transmission for the wave equation. SIAM J. Control Optim. 38/2 (2000) 503-537.
Lagnese, J.E., A singular perturbation problem in exact controllability of the Maxwell system. ESAIM: COCV 6 (2001) 275-290. CrossRef
Lions, J.-L., Virtual and effective control for distributed parameter systems and decomposition of everything. J. Anal. Math. 80 (2000) 257-297. CrossRef
J.-L. Lions, Decomposition of energy space and virtual control for parabolic systems, in 12th Int. Conf. on Domain Decomposition Methods, edited by T. Chan, T. Kako, H. Kawarada and O. Pironneau (2001) 41-53.
Lions, J.-L. and Pironneau, O., Domain decomposition methods for C.A.D. C. R. Acad. Sci. Paris 328 (1999) 73-80. CrossRef
Kim, Dang Phung, Contrôle et stabilisation d'ondes électromagnétiques. ESAIM: COCV 5 (2000) 87-137.
J.E. Santos, Global and domain decomposed mixed methods for the solution of Maxwell's equations with applications to magneotellurics. Num. Meth. for PDEs 14/4 (2000) 407-438.
Schaefer, H., Über die Methode sukzessiver Approximationen. Jber Deutsch. Math.-Verein 59 (1957) 131-140.