Partial Lipschitz regularity of the minimum time function for sub-Riemannian control systems

TL;DR

This paper investigates the regularity of the minimum time function in sub-Riemannian control systems with real-analytic vector fields, showing Lipschitz continuity in dimensions 2 and 3 outside measure-zero sets.

Contribution

It establishes partial Lipschitz regularity results for the minimum time function in low-dimensional sub-Riemannian systems, highlighting differences between dimensions 2 and 3.

Findings

Minimum time function is locally Lipschitz in dimension 2.

In dimension 3, it is Lipschitz outside a measure-zero set.

Examples show failure of Lipschitz regularity in some 3D cases.

Abstract

In Euclidean space of dimension 2 or 3, we study a minimum time problem associated with a system of real-analytic vector fields satisfying H\"ormander's bracket generating condition, where the target is a nonempty closed set. We show that, in dimension 2, the minimum time function is locally Lipschitz continuous while, in dimension 3, it is Lipschitz continuous in the complement of a set of measure zero. In particular, in both cases, the minimum time function is a.e. differentiable on the complement of the target. In dimension 3, in general, there is no hope to have the same regularity result as in dimension 2. Indeed, examples are known where the minimum time function fails to be locally Lipschitz continuous.