Optimal boundary temperature control in the model of radiative heat transfer

An optimal control problem for a system of semilinear elliptic partial differential equations describing the radiative-conductive heat transfer is considered. The specific of this problem is that the control, the boundary temperature field, appears nonlinearly in boundary conditions. The solvability of this control problem is proven if the set of admissible controls is compact in a certain functional space. An example of nonexistence is given in the case where the set of admissible control is not compact. Necessary optimality conditions are obtained without any a priory smallness and regularity assumptions.