Extendability of sub-Riemannian geodesics

Abstract

Sub-Riemannian length-minimizing curves that satisfy a geodesic equation can be extended while preserving local optimality. Understanding global optimality requires study of the cut locus, which is known only in very select cases. However there are also length-minimizers that do not satisfy a geodesic equation, and in such cases even the existence of locally optimal extensions is delicate. In the other extreme, the existence of an unbounded globally optimal extension implies rather restrictive asymptotic behaviour of the curve. In this talk I will give a brief glimpse into these perspectives on sub-Riemannian optimal control. The talk is based on joint works with Andrei Ardentov and with Enrico Le Donne.