# Time domain decomposition in final value optimal control of the Maxwell system

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Lagnese.dvi ESAIM: Control, Optimisation and Calculus of Variations June 2002, Vol. 8, 775{799 URL: http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002042 TIME DOMAIN DECOMPOSITION IN FINAL VALUE OPTIMAL CONTROL OF THE MAXWELL SYSTEM ∗, ∗∗ John E. Lagnese1 and G. Leugering2 Abstract. 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. Mathematics Subject Classification. 65N55, 49M27, 35Q60. Received December 17, 2001. Revised February 16, 2002. 1. Introduction Problems of optimal control of electromagnetic waves arise in a variety of applications, e.g. in stealth technology, design and control of antennas, di�raction optics, magnetotellurics and related �elds. In many important cases the objective to be met relates to the �nal values of both the electric and the magnetic �elds involved. Hence, the problem of exact controllability has attracted considerable interest. See Lagnese [9] as an early paper and Phung [15] or Belishev and Glasman [2] for more recent ones on this topic. The list of references is far from being exhaustive. As a more realistic requirement, one typically asks for controls that bring the �nal states close to a given target con�guration, that is, one wants to achieve optimal controls. While the papers mentioned focus on the case of constant permeabilities and permittivities, modern applications require dealing with heterogeneous materials. In addition, in most cases real-time requirements are to be met. Therefore, in order to obtain adjoint-based gradients and sensitivities in real-time, the large scale (or

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