Homogenization of the optimal control problem for the Dirichlet cost functional and the Poisson state problem with rapidly alternating boundary conditions in critical case

The paper studies the homogenization of the optimal control problem for the Dirichlet cost functional with the Poisson state equation specified in a bounded domain Omega. On part of the boundary partial derivative Omega, denoted by (partial derivative Omega)(0), rapidly alternating boundary conditions are considered. It is assumed that on subsets of (partial derivative Omega)(0), distributed with the period epsilon, of diameter of order O(epsilon(n-1/n-2)) a Robin type boundary condition is specified involving the large coefficient epsilon- (n-1/n-2). On the rest part of (partial derivative Omega)(0), we set the Neumann boundary condition. We suppose that parameters take the so-called critical values. The critical case is characterized by the fact that the effective problem contains the strange term.

Automated summary

This paper studies the homogenization of the optimal control problem for the Dirichlet cost functional with the Poisson state equation specified in a bounded domain Omega. It considers rapidly alternating boundary conditions on a part of the boundary and sets Robin type boundary condition with a large coefficient on subsets of this part. Neumann boundary condition is set on the remaining part of the boundary. The critical case, characterized by the presence of a strange term, is examined.