Regularity properties of time-optimal trajectories of an analytic single- input control-linear system in dimension three (Q1094663)

We consider the problem of time-optimal control for systems of the form ẋ\(=f(x)+ug(x)\), \(x\in M\), \(| u| \leq 1\), where the state space M is a three-dimensional real-analytic manifold, f and g are real-analytic vector fields on M, and admissible controls are scalar measurable functions u(\(\cdot)\) with values in -1\(\leq u\leq 1\) a.e. We prove, for arbitrary f and g, that there exists an analytic subset A of M with positive codimension such that every point not in A has a neighborhood U such that time-optimal trajectories that lie in U are, in nondegenerate cases, bang-bang with at most two switchings or concatenations of at most a bang arc, followed by a singular arc and another bang arc; in degenerate cases, whenever \(q_ 1\in U\) can be steered to \(q_ 2\in U\) in time T in U, when there also exists a bang- bang trajectory with at most two switchings that steers \(q_ 1\) to \(q_ 2\) in time T.