Singular perturbation of optimal control problems on multidomains

The goal of this paper is to study a singular perturbation problem in the framework of optimal control on multidomains. We consider an optimal control problem in which the controlled system contains a fast and a slow variable. This problem is reformulated as a Hamilton-Jacobi-Bellman equation. The main difficulty comes from the fact that the fast variable lives in a multidomain. The geometric singularity of the multidomains leads to the discontinuity of the Hamiltonian. Under a controllability assumption on the fast variable, the limit equation (as the velocity of the fast variable goes to infinity) is obtained via a PDE approach and by means of the tools of the control theory.