Document ESAIM: COCV
DOI: 10.1051/cocv:2008037
Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs
Peter I. Kogut1 and Günter Leugering21 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 January 7, 2005. Revised January 20, 2006 and December 8, 2006. Published online June 24, 2008.
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.
Mathematics Subject Classification. 35B27, 35J25, 49J20, 93C20
Key words: Optimal control, homogenization, elliptic equation, periodic graph
© EDP Sciences, SMAI 2008