# Regularity of the minimum time and of viscosity solutions of degenerate eikonal equations via generalized Lie brackets

## TL;DR

This paper extends the regularity theory for degenerate eikonal equations by employing set-valued Lie brackets, establishing conditions for the Hölder continuity of solutions and the minimum time function in optimal control.

## Contribution

It introduces a novel approach using set-valued Lie brackets to relax regularity assumptions and proves Hölder regularity results for solutions of degenerate eikonal equations.

## Key findings

- Hölder regularity of the minimum time function under new conditions
- Hölder continuity of solutions to the Dirichlet problem with low regularity coefficients
- Conditions for regularity are shown to be essentially necessary in certain cases

## Abstract

In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of { set-valued} iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hoelder regularity of the minimum time function in optimal control. We then apply such result to prove H\"older continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient conditions for the H\"older regularity are essentially necessary, at least for smooth vector fields and target.