Time domain decomposition in final value optimal control of the Maxwell system
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 775-799.

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

DOI : 10.1051/cocv:2002042
Classification : 65N55, 49M27, 35Q60
Mots clés : Maxwell system, optimal control, domain decomposition 
@article{COCV_2002__8__775_0,
     author = {Lagnese, John E. and Leugering, G.},
     title = {Time domain decomposition in final value optimal control of the {Maxwell} system},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {775--799},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002042},
     mrnumber = {1932973},
     zbl = {1063.78029},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2002042/}
}
TY  - JOUR
AU  - Lagnese, John E.
AU  - Leugering, G.
TI  - Time domain decomposition in final value optimal control of the Maxwell system
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
SP  - 775
EP  - 799
VL  - 8
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2002042/
DO  - 10.1051/cocv:2002042
LA  - en
ID  - COCV_2002__8__775_0
ER  - 
%0 Journal Article
%A Lagnese, John E.
%A Leugering, G.
%T Time domain decomposition in final value optimal control of the Maxwell system
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 775-799
%V 8
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2002042/
%R 10.1051/cocv:2002042
%G en
%F COCV_2002__8__775_0
Lagnese, John E.; Leugering, G. Time domain decomposition in final value optimal control of the Maxwell system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 775-799. doi : 10.1051/cocv:2002042. http://archive.numdam.org/articles/10.1051/cocv:2002042/

[1] A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68 (1999) 607-631. | MR | Zbl

[2] M. Belishev and A. Glasman, Boundary control of the Maxwell dynamical system: Lack of controllability by topological reason. ESAIM: COCV 5 (2000) 207-218. | Numdam | MR | Zbl

[3] J.-D. Benamou, 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. | Zbl

[4] J.-D. Benamou, Domain decomposition, optimal control of systems governed by partial differential equations and synthesis of feedback laws. J. Opt. Theory Appl. 102 (1999) 15-36. | MR | Zbl

[5] J.-D. Benamou and B. Desprès, A domain decomposition method for the Helmholtz equation and related optimal control problems. J. Comp. Phys. 136 (1997) 68-82. | MR | Zbl

[6] M. Gander, L. Halpern and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation. École Polytechnique, Palaiseau, Rep. 469 (2001). | Zbl

[7] M. Heinkenschloss, Time domain decomposition iterative methods for the solution of distributed linear quadratic optimal control problems (submitted). | Zbl

[8] 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. | MR | Zbl

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

[10] 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. | MR | Zbl

[11] J.E. Lagnese, A singular perturbation problem in exact controllability of the Maxwell system. ESAIM: COCV 6 (2001) 275-290. | Numdam | MR | Zbl

[12] J.-L. Lions, Virtual and effective control for distributed parameter systems and decomposition of everything. J. Anal. Math. 80 (2000) 257-297. | MR | Zbl

[13] 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. | MR | Zbl

[14] J.-L. Lions and O. Pironneau, Domain decomposition methods for C.A.D. C. R. Acad. Sci. Paris 328 (1999) 73-80. | MR | Zbl

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

[16] 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. | Zbl

[17] H. Schaefer, Über die Methode sukzessiver Approximationen. Jber Deutsch. Math.-Verein 59 (1957) 131-140. | MR | Zbl

Cité par Sources :