New cost terms through the homogenization of an optimal control problem under dynamic boundary conditions on the microscopic particles

TL;DR

This paper investigates the asymptotic behavior of an optimal control problem on a heterogeneous, periodically structured body with dynamic boundary conditions, revealing new non-local terms in the limit equations and cost functional.

Contribution

It introduces a novel homogenization analysis for optimal control problems with dynamic boundary conditions, uncovering non-local 'strange terms' in the limit equations and cost functional.

Findings

New non-local in time 'strange terms' appear in the limit equations and cost functional.

Microscopic localized controls induce unique terms not seen with full-body controls.

Critical case analysis reveals the influence of structure size, particle diameter, and boundary growth on the limit behavior.

Abstract

Given an optimal control problem on a heterogeneous body with a periodical structure of particles depending on a small parameter e, we study the asymptotic behavior, as e converges to zero, of the optimal control functional and the optimal state when the initial problem is of parabolic type, and when on the particles' boundary, we assume a dynamic condition and the actuation of some controls for some subset of the particles. We show, in 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 particles boundary condition), the appearance of some new non-local in time "strange terms", not only in the limit parabolic equation but also in the limit cost functional. Microscopic localized controls generate peculiar terms in both the limit equation and the cost functional that do not appear in the case of controls applied to the entire set of particles or when the boundary condition on the particles is of Robin type.