Hamilton-Jacobi Reachability in Reinforcement Learning: A Survey

TL;DR

This survey advances the field by reviewing how Hamilton-Jacobi (HJ) reachability can be learned in tandem with reinforcement learning to provide safety guarantees for high-dimensional, data-driven control. It outlines model-free Bellman-based formulations (including discounted and reach-avoid variants), reachability-based safety shields, and optimization frameworks that combine reachability with control Lyapunov functions and CMDP-like objectives. The paper highlights techniques for deterministic and stochastic dynamics, robustness to real-world disturbances, and practical deployments via shielding and forward reachability. It also discusses current limitations—such as sample efficiency, transfer to unseen environments, and exact adherence to undiscounted HJB solutions—and lays out future directions, including lifelong learning, Koopman-Hopf hybrids, and closer integration with other certificate methods. Overall, the work provides a foundation for leveraging learned HJ reachability to achieve reliable, safe RL in complex systems.

Abstract

Recent literature has proposed approaches that learn control policies with high performance while maintaining safety guarantees. Synthesizing Hamilton-Jacobi (HJ) reachable sets has become an effective tool for verifying safety and supervising the training of reinforcement learning-based control policies for complex, high-dimensional systems. Previously, HJ reachability was restricted to verifying low-dimensional dynamical systems primarily because the computational complexity of the dynamic programming approach it relied on grows exponentially with the number of system states. In recent years, a litany of proposed methods addresses this limitation by computing the reachability value function simultaneously with learning control policies to scale HJ reachability analysis while still maintaining a reliable estimate of the true reachable set. These HJ reachability approximations are used to improve the safety, and even reward performance, of learned control policies and can solve challenging tasks such as those with dynamic obstacles and/or with lidar-based or vision-based observations. In this survey paper, we review the recent developments in the field of HJ reachability estimation in reinforcement learning that would provide a foundational basis for further research into reliability in high-dimensional systems.