Rigorous homogenisation of an optimal control problem for multispecies
diffusion-reaction equations
Abstract
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.