Regularity properties of time-optimal trajectories for two-input real-analytic systems without drift in dimension three

In this paper we prove that for two-input three-dimensional driftless analytic systems, there exists an open dense subset Ω of the state space, whose complement is an analytic subset of positive codimension, such that for every point p∈Ω there exist a positive integer N and a neighborhood U of p with the property that every time-optimal trajectory an U is either a concatenation of bang and singular arcs with at most N pieces or replaceable by a bang-bang trajectory with at most two switchings