  We deal with a singularly perturbed optimal control problem with slow and
fast variable depending on a parameter {\epsilon}. We study the asymptotic, as
{\epsilon} goes to 0, of the corresponding value functions, and show
convergence, in the sense of weak semilimits, to sub and supersolution of a
suitable limit equation containing the effective Hamiltonian. The novelty of
our contribution is that no compactness condition are assumed on the fast
variable. This generalization requires, in order to perform the asymptotic
proce- dure, an accurate qualitative analysis of some auxiliary equations posed
on the space of fast variable. The task is accomplished using some tools of
Weak KAM theory, and in particular the notion of Aubry set.
