Existence of solution for an optimal control problem in a heterogeneous porous medium

The paper explores an optimal control problem concerning transport processes in a porous medium. The transport phenomena is governed by diffusion-reaction equations, which is basically a semi-linear parabolic system. An L²-cost (energy) functional is introduced, and control (distributed) is applied in the porous part of the medium. The primary objective is to characterize a given control to be an optimal control and analyse the relationship between optimal control and the adjoint state. The analysis commences with the microscopic description of the controlled system followed by upscaling the system via periodic homogenization.