Suboptimal Boundary Control of Elliptic Equations in Domains with Small Holes

Abstract

Our prime interest in this chapter concerns the construction of suboptimal solutions to a class of boundary optimal control problems (OCPs) in ε-periodically perforated domains with small holes. We suppose that the support of controls is contained in the set of boundaries of the holes. This set is divided into two parts: On one part, the controls are of Dirichlet type; on the other one, the controls are of Neumann type. Using the ideas of the Γ-convergence theory and the concept of the variational convergence of constrained minimization problems, we show that the limit problem, as ε tends to 0, can be recovered in an explicit analytical form. However, in contrast to the case of thin periodic structures (see Chap. 9), the control through the boundary of small holes leads us in the limit to an OCP with drastically different properties and structures. In particular, we show that in this case, a “strange term” appears both in the limit equation and the cost functional, and that this term depends on the geometry of the holes. Moreover, the characteristic feature of the obtained limiting control problem is the fact that it contains two independent distributed control functions which can be used as suboptimal controls to the original one.