# The Aronsson equation, Lyapunov functions and local Lipschitz regularity of the minimum time function

## TL;DR

This paper investigates the Aronsson equation, showing that $C^1$-solutions serve as Lyapunov functions and establish local Lipschitz regularity of the minimum time function, with explicit examples for certain vector fields.

## Contribution

It introduces a new approach to analyze $C^1$-solutions of the Aronsson equation as Lyapunov functions, proving local Lipschitz regularity of the minimum time function away from singularities.

## Key findings

- $C^1$-solutions are absolutely minimizing functions.
- Such solutions serve as Lyapunov functions for control systems.
- Minimum time function is locally Lipschitz outside singular manifold.

## Abstract

We define and study $C^1-$solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that $C^1-$solutions are absolutely minimizing functions. We discuss how $C^1-$supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results show that it should only be H\"older continuous unless appropriate conditions hold. We provide two examples for H\"ormander and Grushin families of vector fields where we construct $C^1-$solutions (even classical) explicitly.