Strong contrasting diffusivity in general oscillating domains: Homogenization of optimal control problems

Abstract

The composites of materials with high contrasting properties is an interesting topic to study as it has applications. In this article, we wish to study problems in high oscillating domains, where the oscillatory part is made of two materials with high contrasting conductivities (or diffusivity). Thus the low contrast material acts as near insulation in-between the conducting materials. In the first part, we study the homogenization problem of an elliptic equation. The main discussion in this article is the study of optimal control problems based on the unfolding method. The interesting result is the difference in the limit behavior of the optimal control problem, which crucially depends on the action of the control, whether it is on the conductivity part or insulating part. In both cases, we derive the two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We do obtain the homogenized equation for the state, but the two-scale cost functional remains as it is.