On the equivalence of different types of local minima in sub-Riemannian problems

Sub-Riemannian problems are typical optimal control problems admitting singular and abnormal minimizers. Moreover, any singular geodesic in the sub-Riemannian problem is abnormal and vice versa. So minimizers may be singular geodesics, but it is not clear for they may have singularities as curves in the state space or not. Until now, all known minimizers were smooth. We compare different types of local minimality for smooth admissible curves. Surprisingly, the smoothness of the trajectory implies the equivalence of local minimality in rather different topologies