We study an optimal control problem governed by diffusion-reaction equations in a periodic porous medium (bounded domain). Our control problem is equivalent to a convex minimization problem. We take a $L^2$-cost functional and pose controls on the mobile species present in the pore part of the domain. One of the main aims here is to characterize a given control to be an optimal control for the microscopic problem. We obtain the existence of solution of the control problem and analyse a relation between optimal control and its adjoint state. Then, we do the homogenization of the optimal control problem (diffusion-reaction model with cost functional) by a formal asymptotic analysis and then via rigorous two-scale convergence and periodic unfolding method.

In this article we show well-posedness for a relatively general semilinear parabolic system under nonhomogeneous Neumann boundary conditions and semi-linearities of (some) equilibrium reaction type. The result in here weakens previously made by Krautle and Mahato (Kra08, Kra11 and MB131, MB132, resp.) on the coefficients of the elliptic operator as well as on the boundary conditions considerably.