On the Local Structure of Time-Optimal Bang-Bang Trajectories in R³

We consider the problem of time-optimal control for systems of the form x = f(x) + ug(x) where f and g are smooth vector fields and admissible controls are measurable functions u with values in - 1 u 1. Under the assumption that f, g and $[f,g]$ are independent, we prove that generically every point has a neighborhood U such that bang-bang trajectories that lie in U and have more than 7 switchings are not time-optimal.