Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs

¹ Department of Differential Equations, Dnipropetrovsk National University, Naukova str., 13, 49050 Dnipropetrovsk, Ukraine. p.kogut@i.ua
² Institüt für Angewandte Mathematik Lehrstuhl II, Universität Erlangen-Nürnberg Martensstr.3, 91058 Erlangen, Germany. Guenter.Leugering@am.uni-erlangen.de

Received: 7 January 2005
Revised: 20 January 2006
Revised: 8 December 2006

Abstract

We are concerned with the asymptotic analysis of optimal control problems for 1-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system, and its solution can be used as suboptimal controls for the original problem.