Optimal control problems for complex heat transfer equations with Fresnel matching conditions

A class of optimal control problems for a system of nonlinear elliptic equations simulating radiative heat transfer with Fresnel matching conditions on the surfaces of discontinuity of the refractive index is considered. Based on estimates for the solution of the boundary value problem, the solvability of the optimal control problems is proved. The existence and uniqueness of the solution of a linearized problem with the matching conditions is analyzed, and the nondegeneracy of the optimality conditions is proved. As an example, a control problem with boundary observation is considered and the relay-like character of the optimal control is shown.