Approximate Solutions of Optimal Control Problems for Ill-Posed Parabolic Problems on Thin Periodic Structures

Abstract

In this chapter, we focus our attention on control objects which are described by singular parabolic equations with Robin boundary conditions at the boundary of the holes, and with two different types of boundary controls – Dirichlet and Neumann controls – on the external boundary of the thin periodic structure Ω _ε . We allow for a blowing-up phenomenon in the original problem and we provide its asymptotic analysis as the small parameter ε tends to 0. It is shown that the structure of the limit problem depends essentially on how h tends to 0 as ε→0 (the so-called “scaling effect”). We derive conditions under which in the limit we do not obtain an optimal control problem (OCP), but rather some initial-boundary value problems with or without controls. Furthermore, we construct asymptotically suboptimal controls for the original problem and show an approximation property of such controls for small enough ε near the optimal characteristics.