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Boundary control of the Maxwell dynamical system : lack of controllability by topological reasons

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cocv133.dvi ESAIM: Control, Optimisation and Calculus of Variations March 2000, Vol. 5, p. 207{217 URL: http://www.emath.fr/cocv/ BOUNDARY CONTROL OF THE MAXWELL DYNAMICAL SYSTEM: LACK OF CONTROLLABILITY BY TOPOLOGICAL REASONS Mikhail Belishev1 and Aleksandr Glasman2 Abstract. The paper deals with a boundary control problem for the Maxwell dynamical system in a bounbed domain Ω � R3. Let ΩT � Ω be the subdomain �lled by waves at the moment T , T� the moment at which the waves �ll the whole of Ω. The following e�ect occurs: for small enough T the system is approximately controllable in ΩT whereas for larger T < T� a lack of controllability is possible. The subspace of unreachable states is of �nite dimension determined by topological characteristics of ΩT . AMS Subject Classi�cation. 93B05, 35B37, 35Q60, 78A25, 93C20. Received June 11, 1999. Revised December 30, 1999. Introduction Let Ω � R3 be a bounded domain with a smooth boundary Γ. We consider the Maxwell system "et = roth; �ht = −rote in Ω� (0; T ); div "e = 0; div�h = 0 in Ω; ejt=0 = 0; hjt=0 = 0; � � ejΓ�[0;T ] = f; where "; � are smooth positive scalar functions (permeabilities) given in Ω, � is a normal on Γ, f is a boundary control; let fef(x; t); hf(x; t)g be a solution (wave). Permeabilities determine the velocity c = ("�)1=2 and the optical metric d�2 = jdxj2 c2 ; which turns Ω into a Riemannian manifold; we denote distc the corresponding distance. Let ΩT := fx 2 Ω j distc(x;Γ) < Tg; T > 0 Keywords and phrases: Maxwell’s dynamical system, boundary control, unreachable states, topology of a domain. 1 Saint-Petersburg Department of Steklov Mathematical Institute, Fontanka 27, Saint-Petersburg 191011, Russia; e-mail: [email protected] Supported by RFBR, grant 98-01-00314. 2 Saint-Petersburg State University, Saint-Petersburg, Russia. Supported by RFBR, grant 99-01-00107. c© EDP Sciences, SMAI 2000 208 M. BELISHEV AND A. GLASMAN be a near-boundary layer of optical thickness T ; the surface ΓT :=

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