A systematic study of optimal control problems for anisotropic electromagnetic media will be conducted. This project will focus on boundary controllability and internal controllability as well as stabilizability. The evolution of the electromagnetic field in a medium is governed by Maxwell's equations. These equations relate the two electric field quantities, the displacement and the electric field intensity, and the two magnetic field quantities, the magnetic induction and the magnetic field intensity. A medium is electrically anisotropic if the two electric field quantities point in different directions. The mathematical analysis of the anisotropic Maxwell equations poses a number of challenging problems. Contrary to the isotropic case, when the electric quantities and the magnetic quantities point into the same direction, Maxwell's equations cannot be reduced to the second order linear wave equation. Maxwell's equations form a first order system of partial differential equations. Closely connected with the study of optimal control properties is an ill-posed Cauchy problem. Another objective of this project is the investigation of ill-posed Cauchy problems for elliptic first order systems of partial differential equations.
Problems arising optimal control are both mathematically challenging and of practical relevance. For example, in the boundary control problem, the goal is to construct a current flux in the boundary of the medium that will drive the electromagnetic field to a desired target state. The significance of this project lies in the importance of anisotropic electromagnetic materials. Practically all metals and composite materials used in the production of microchips are anisotropic materials. Further examples include synthetic sapphire and some ceramics such as piezoelectric materials, which give off a measurable electrical discharge when acted upon by a mechanical force and stretch or bend when subjected to an electrical current. Most of the research conducted will also be relevant to other systems of mathematical physics such as the system of anisotropic elasticity, a problem of great importance in geophysics.
