Asymptotic properties of non-coercive Hamiltonians with drift

We consider Hamiltonians associated with optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong bracket generation condition on the vector fields. We first prove the existence of a critical value of the Hamiltonian by means of the ergodic approximation. Next, we prove the existence of a possibly discontinuous viscosity solution to the critical equation. We show that the long-time behavior of solutions to the evolutive Hamilton-Jacobi equation is described in terms of the critical constant and a critical solution. As in the classical weak KAM theory, we find a fixed point of the Hamilton-Jacobi-Lax-Oleinik semigroup, although possibly discontinuous. Finally, we apply the existence and properties of the critical value to the periodic homogenization of stationary and evolutive H-J equations.