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