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Time domain decomposition in final value optimal control of the Maxwell system. (English) Zbl 1063.78029
Summary: 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.

MSC:
78M50 Optimization problems in optics and electromagnetic theory
49M25 Discrete approximations in optimal control
35Q60 PDEs in connection with optics and electromagnetic theory
49K20 Optimality conditions for problems involving partial differential equations
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
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References:
[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 1609607 | Zbl 1043.78554 · Zbl 1043.78554 · doi:10.1090/S0025-5718-99-01013-3
[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 1750615 | Zbl 1121.93307 · Zbl 1121.93307 · doi:10.1051/cocv:2000108 · eudml:90568
[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 0879.35090 · Zbl 0879.35090 · doi:10.1016/S0764-4442(97)87887-1
[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 1702845 | Zbl 0946.49025 · Zbl 0946.49025 · doi:10.1023/A:1021882126367
[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 1468624 | Zbl 0884.65118 · Zbl 0884.65118 · doi:10.1006/jcph.1997.5742
[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 1085.65077 · Zbl 1085.65077 · doi:10.1137/S003614290139559X
[7] M. Heinkenschloss , Time domain decomposition iterative methods for the solution of distributed linear quadratic optimal control problems (submitted). Zbl 1075.65091 · Zbl 1075.65091 · doi:10.1016/j.cam.2004.03.005
[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 1817181 | Zbl 0979.93058 · Zbl 0979.93058
[9] J.E. Lagnese , Exact boundary controllability of Maxwell’s equation in a general region . SIAM J. Control Optim. 27 ( 1989 ) 374 - 388 . Zbl 0678.49032 · Zbl 0678.49032 · doi:10.1137/0327019
[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 1741151 | Zbl 0952.93010 · Zbl 0952.93010 · doi:10.1137/S0363012998333530
[11] J.E. Lagnese , A singular perturbation problem in exact controllability of the Maxwell system . ESAIM: COCV 6 ( 2001 ) 275 - 290 . Numdam | MR 1824104 | Zbl 1030.93025 · Zbl 1030.93025 · doi:10.1051/cocv:2001111 · numdam:COCV_2001__6__275_0 · eudml:90595
[12] J.-L. Lions , Virtual and effective control for distributed parameter systems and decomposition of everything . J. Anal. Math. 80 ( 2000 ) 257 - 297 . MR 1771528 | Zbl 0964.93043 · Zbl 0964.93043 · doi:10.1007/BF02791538
[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 1827521 | Zbl 1082.93581 · Zbl 1082.93581
[14] J.-L. Lions and O. Pironneau , Domain decomposition methods for C .A.D. C. R. Acad. Sci. Paris 328 ( 1999 ) 73 - 80 . MR 1674382 | Zbl 0937.68140 · Zbl 0937.68140 · doi:10.1016/S0764-4442(99)80015-9
[15] Kim Dang Phung, Contrôle et stabilisation d’ondes électromagnétiques. ESAIM: COCV 5 (2000) 87-137. Numdam | Zbl 0942.93002 · Zbl 0942.93002 · doi:10.1051/cocv:2000103 · www.edpsciences.org · eudml:90586
[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 0918.65083 · Zbl 0918.65083 · doi:10.1002/(SICI)1098-2426(199807)14:4<407::AID-NUM1>3.0.CO;2-O
[17] H. Schaefer , Über die Methode sukzessiver Approximationen . Jber Deutsch. Math.-Verein 59 ( 1957 ) 131 - 140 . Article | MR 84116 | Zbl 0077.11002 · Zbl 0077.11002 · eudml:146424
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