New unexpected limit operators for homogenizing optimal control parabolic problems with dynamic reaction flow on the boundary of critically scaled particles

Abstract

We study the asymptotic behavior, as ε → 0, of the optimal control and the optimal state of an initial boundary value problem in a domain that is ε-periodically perforated by balls (or, equivalently it is the complementary to a set of spherical particles). On the boundary of the perforations (or of the particles) we assume a dynamic condition with a large growth coefficient in the time derivative. The control region is a possible small subregion and the cost functional includes a balance between the prize of the controls and the error with respect to a given target profile. We consider the so-called “critical case” concerning a certain relation between the structure’s period, the diameter of the balls, and the growth coefficient of the boundary condition. We show that the homogenized problem contains in the limit state equation a nonlocal "strange term", given as a solution to a suitable ordinary differential equation. We prove the weak convergence of the state and the optimal control to the state and the optimal control associated with the limit cost functional which now contains an unexpected new “strange” term.