Exact boundary controllability of Maxwell’s equations in a general region.

*(English)*Zbl 0678.49032This 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.

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 |