Boundary controllability of Maxwell's equations with nonzero conductivity and an application to an inverse source problem
Krigman, Steven Slava
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This thesis studies the question of control of Maxwell's equations in a medium with positive conductivity by means of boundary surface currents. Two types of domains and media are considered in connection with this question. First is a bounded simply connected star-shaped domain in R^3 which is made up of a heterogeneous medium with small conductivity, with controls being applied over the entire boundary. Using the Hilbert Uniqueness Method of Lions, the exact boundary controllability over a sufficiently long time period is established for this case, provided the conductivity is small enough to satisfy a certain technical inequality. It is also found that the requirement for the conductivity term to be very small remains in place even if the medium considered is homogenous. In order to remove this constraint, a special domain type is considered next - a cube - made up of a homogenous medium where the conductivity is allowed to take on any non-negative value. An additional restriction imposed here in order to make this problem more suitable for practical implementations is that the controls are applied over only one face of the cube. Employing the Method of Moments the spectral controllability is established for this case. It is also established that the exact controllability fails for this geometry regardless of the size of the conductivity term. This thesis will also consider the question of reconstructing the source of electromagnetic radiation, which is related to the controllability problem.
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