Periodic Near Optimal Control

We consider the infinite time optimal control problem of minimizing an average cost functional for nonlinear singularly perturbed systems. In particular, we are interested in the existence of near optimal trajectories which are periodic. For this purpose, we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion by an iterated averaging procedure. We furthermore show that global tools, invariant control sets, and local tools, Lie algebra ranks of vector fields, are preserved under singular perturbations. The existence of periodic near optimal trajectories is verified under the condition that the fast subsystem and an unperturbed slow subsystem are controllable and that a standard Lie algebra rank condition for local accessibility holds