An Update to the Level Set Theorems in Hamilton-Jacobi Reachability Analysis

Dylan Hirsch; William McEneaney; Jaime Fisac; Claire Tomlin; Sylvia Herbert

An Update to the Level Set Theorems in Hamilton-Jacobi Reachability Analysis

Systems and Control 2026-07-19 v1

Authors: Dylan Hirsch , William McEneaney , Jaime Fisac , Claire Tomlin , Sylvia Herbert

Abstract

Hamilton-Jacobi Reachability (HJR) is an important framework for controlling safety-critical systems despite uncertainty. Its theoretical underpinnings are rooted in Hamilton-Jacobi Partial Differential Equations, which provide the value function used for controller synthesis. The Level Set Theorems of HJR allow one to interpret the value function in terms of satisfaction of a qualitative goal (e.g. goal-reaching or obstacle-avoidance). We here provide a technical update regarding additional criteria needed for these theorems to hold.

Keywords

Hamilton-Jacobi reachability level set theorems safety-critical systems optimal control value function technical update

Cite

@article{arxiv.2607.17435,
  title  = {An Update to the Level Set Theorems in Hamilton-Jacobi Reachability Analysis},
  author = {Dylan Hirsch and William McEneaney and Jaime Fisac and Claire Tomlin and Sylvia Herbert},
  journal= {arXiv preprint arXiv:2607.17435},
  year   = {2026}
}