Exact boundary controllability of Maxwell’s equations in a general region. (English) Zbl 0678.49032

This paper deals with the exact controllability of solutions of Maxwell’s equations for an electric field E and a magnetic field H in a general region \(\Omega\) with boundary \(\Gamma\) by means of currents flowing tangentially to the boundary. The author uses the Hilbert uniqueness method (HUM) introduced by J. L. Lions [cf. C. R. Acad. Sci., Paris, Sér I 302, 471-475 (1986; Zbl 0589.49022); SIAM Rev. 30, No.1, 1-68 (1988; Zbl 0644.49028); one can also refer to the book in French: Controlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1: Controlabilité exacte. (1988; Zbl 0653.93002)].
The method needs some regularity results for the adjoint homogeneous system: these are obtained for initial data in a \(H^ 2(\Omega)\)-type Sobolev space. Furthermore a priori estimates on these initial data with respect to some boundary observations, are derived. By use of HUM, these lead to different exact controllability results: by \({\mathcal L}^ 2(\Gamma \times]0,T[)\) controls it is possible to steer to zero any initial condition E(0), H(0) in an \({\mathcal L}^ 2(\Omega)\)-type space for T larger than a certain \(T_ 0\), if the domain \(\Omega\) is star shaped. The space of controllable initial states may be enlarged if \((H^ 1(0,T;{\mathcal L}^ 2(\Gamma)))\) controls are allowed. This last result is generalized by weakening the geometric assumptions on \(\Gamma\), which does not seem possible for the first one.
Reviewer: J.Henry


93B03 Attainable sets, reachability
93C20 Control/observation systems governed by partial differential equations
35L50 Initial-boundary value problems for first-order hyperbolic systems
93B05 Controllability
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