Multiscale asymptotic method of optimal control on the boundary for heat equations of composite materials

In this paper, we study the optimal control on the boundary for parabolic equations with rapidly oscillating coefficients arising from the heat transfer problems and the optimal control on the boundary of composite materials or porous media. The multiscale asymptotic expansion of the solution for the problem in the case without any constraints is presented. We derive the proofs of all convergence results.