# zbMATH — the first resource for mathematics

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

##### MSC:
 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
Full Text: