Analysis of an optimization method for solving the problem of complex heat transfer with Cauchy boundary conditions

An optimization method is proposed for solving a boundary value problem with Cauchy conditions for the equations of radiative-conductive heat transfer in the $P_1$-approximation of the radiative transfer equation. Theoretical analysis of the corresponding problem of boundary optimal control is carried out. It is shown that a sequence of solutions of extremal problems converges to the solution of the boundary value problem with the Cauchy conditions for temperature. The results of theoretical analysis are illustrated with numerical examples.