Aubry set for sub-Riemannian control systems

TL;DR

This paper studies the Aubry set for sub-Riemannian control systems, providing a variational formula for the critical constant, analyzing the properties of the Aubry set, and showing classical differentiability of solutions on it.

Contribution

It introduces a variational representation for the critical constant in sub-Riemannian systems and explores the properties of the Aubry set and solutions' differentiability.

Findings

A variational formula for the critical constant using adapted closed measures.

Topological and dynamical analysis of the Aubry set.

Critical solutions are horizontally differentiable and satisfy the Hamilton-Jacobi equation classically on the Aubry set.

Abstract

In the paper [P. Cannarsa, C. Mendico, Asymptotic analysis for Hamilton-Jacobi- Bellman equations on Euclidean space, (2021) Arxiv], we proved the existence of the limit as the time horizon goes to infinity of the averaged value function of an optimal control problem. For the classical Tonelli case such a limit is called the critical constant of the problem. In the special case of sub-Riemannian control systems, we also proved the existence of a critical solution, that is, a continuous solution to the Hamilton-Jacobi equation associated with such a constant, which also coincides with its Lax-Oleinik evolution. Here, we focus our attention on the sub- Riemannian case providing a variational representation formula for the critical constant which uses an adapted notion of closed measures. Having such a formula at our disposal, we define and study the Aubry set. First, we investigate dynamical and topological properties of such a set w.r.t. a suitable class of minimizing trajectories of the Lagrangian action. Then, we show that critical solutions to the Hamilton-Jacobi equation are horizontally differentiable and satisfy the equation in classical sense on the Aubry set.