On the local structure of time-optimal controls in R3

The paper deals with the time-optimal control of a system ??: x = f(x) + ug(x), |u| ?? 1, x ?? IR3, u ?? IR where f and g are smooth vector fields. We consider the system near a point p where both triples (f(p), g(p), [f,g](p)) and (g(p),[f,g](p),[f,[f,g]](p)) are independent. We show that, for a generic system ??, p has a neighborhood U such that any time-optimal trajectory of ?? lying in U is a concatenation of at most 6 bang or singular arcs.