We deal with a singularly perturbed optimal control problem with slow and\udfast variable depending on a parameter ". We study the asymptotics, as " goes to 0, of the\udcorresponding value functions, and show convergence, in the sense of weak semilimits, to\udsub and supersolution of a suitable limit equation containing the eective Hamiltonian.\udThe novelty of our contribution is that no compactness condition are assumed on the\udfast variable. This generalization requires, in order to perform the asymptotic procedure,\udan accurate qualitative analysis of some auxiliary equations posed on the space of\udfast variable. The task is accomplished using some tools of Weak KAM theory, and in\udparticular the notion of Aubry set.
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