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.