Homogenization of Boundary Optimal Control Problems in a Rough Domain with High Contrast Diffusion

Abstract.

We investigate the homogenization of boundary optimal control problems governed by an elliptic partial differential equation (PDE) in pillar-type domains with highly oscillating boundaries. The pillars are composed of high-conductivity and low-conductivity materials, leading to a PDE with high-contrast diffusive coefficients. Two boundary control problems are considered. More precisely, in the first problem, control is applied to the boundary of the high-conductivity region whereas in the 2nd one, it is on the boundary of the low-conductivity region. Due to rapid oscillations in the boundary, the control is appropriately scaled. Depending on the choice of scaling and the controlled boundary, three different limit problems arise: a combination of interior, boundary, and interface control problems; a boundary and interface control problem; and an interior and interface control problem.