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

Abstract

We consider the problem of time-optimal control for systems of the form $$\dot x = f(x) + ug(x),x \in M,|u| \leqslant 1$$ , where the state spaceM is a three-dimensional real-analytic manifold,f andg are real-analytic vector fields onM, and admissible controls are scalar measurable functionsu(·) with values in ź1≤u≤1 a.e. We prove, for arbitraryf andg, that there exists an analytic subsetA ofM with positive codimension such that every point not inA has a neighborhoodU such that time-optimal trajectories that lie inU 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, wheneverq1 źU can be steered toq2źU in timeT inU, then there also exists a bang-bang trajectory with at most two switchings that steersq1 toq2 in timeT.