In this talk we are interested in asymptotic behavior of singularly perturbedcontrol system in the non-periodic setting. More precisely, we consider the value function of finite horizon optimal control problem (Bolza form) associated with singularly perturbed control system, and aim at characterizing its weak semilimits as viscosity sub- and supersolutions of a limiting Hamilton-Jacobi-Bellman equation (also called effective HJB equation). This PDE approach is extensively studied in a series of papers by Alvarez and Bardi in the periodic setting ([AB03], [AB10]). Our contribution is to extend the results of Alvarez and Bardi to the nonperiodic case. The key idea is to replace the periodicity on the datum by coercivity on the running cost, and we only need the local version of boundedtime controllability used in [AB10]. The remarkable novelty of our work is to approximate the Bellman Hamiltonian (convex, but non-coercive in the momentum) by a suitable sequence of convex, coercive Hamiltonians and then use some basic tools of Aubry-Mather theory developed by Fathi and Siconolfi (see [FS05]) for these convex, coercive Hamiltonians. We finally obtain some similar results as those of Alvarez and Bardi. Joint work with Antonio Siconolfi.