A Minimum Discounted Reward Hamilton-Jacobi Formulation for Computing Reachable Sets

TL;DR

This paper introduces a new Hamilton-Jacobi based method for computing reachable sets using a minimum discounted reward optimal control formulation, enabling efficient solutions and linking to reinforcement learning.

Contribution

It presents a novel Hamilton-Jacobi formulation for reachable sets that facilitates efficient numerical approximation and connects to reinforcement learning for systems with unknown dynamics.

Findings

Accurately approximates reachable sets for benchmark systems

Demonstrates computational efficiency over traditional methods

Shows potential for reinforcement learning applications

Abstract

We propose a novel formulation for approximating reachable sets through a minimum discounted reward optimal control problem. The formulation yields a continuous solution that can be obtained by solving a Hamilton-Jacobi equation. Furthermore, the numerical approximation to this solution can be obtained as the unique fixed-point to a contraction mapping. This allows for more efficient solution methods that could not be applied under traditional formulations for solving reachable sets. In addition, this formulation provides a link between reinforcement learning and learning reachable sets for systems with unknown dynamics, allowing algorithms from the former to be applied to the latter. We use two benchmark examples, double integrator, and pursuit-evasion games, to show the correctness of the formulation as well as its strengths in comparison to previous work.