Existence of an optimal feedback control law for time-optimal control of two-input analytic nilpotent systems in dimension three

In this paper we analyze regularity properties of time-optimal trajectories for a class of two-input three-dimensional real-analytic systems whose accessibility Lie algebra is nilpotent of step 2, and establish the existence of an almost analytic optimal feedback control law. We prove that every time-optimal trajectory is replaceable by a bang-bang trajectory with at most three switchings, and show that there exist a large open dense subset of the state space and an optimal feedback control law which is analytic on the open dense subset