The Local Structure of Time-Optimal Trajectories in Dimension Three under Generic Conditions

We consider the problem of time-optimal control for systems of the form x = f(x) + g(x)u, where f and g are smooth vector fields and admissible controls are measurable scalar functions u with values in - 1 u 1. Under the assumption that f, g and $[f,g]$ are independent, and that also one of the triples $(g[f,g],[f + g,[f,g]])$ or $(g[f,g],[f - g,[f,g]])$ consists of independent vectors, we show that generically every point has a neighborhood U such that time-optimal trajectories that lie in U are concatenations of at most six bang and singular arcs. This implies that globally time-optimal trajectories are finite concatenations of bang and singular arcs with a bound on the number of switchings; in particular, time-optimal controls are piecewise smooth. Results of this type are relevant for the existence of a regular synthesis.