# On Safety and Liveness Filtering Using Hamilton-Jacobi Reachability Analysis

arXiv: https://arxiv.org/abs/2312.15347 · HTML: https://mlsanity.com/paper/2312.15347/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2023.

## Built on by (most cited first, 11 of 11)

- [SAFE-GIL: SAFEty Guided Imitation Learning for Robotic Systems](https://mlsanity.com/paper/2404.05249/) (2024)
- [A Physics-Informed Machine Learning Framework for Safe and Optimal Control of Autonomous Systems](https://mlsanity.com/paper/2502.11057/) (2025)
- [Preemptive Detection and Steering of LLM Misalignment via Latent Reachability](https://mlsanity.com/paper/2509.21528/) (2025)
- [One Filter to Deploy Them All: Robust Safety for Quadrupedal Navigation in Unknown Environments](https://mlsanity.com/paper/2412.09989/) (2024)
- [Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions](https://mlsanity.com/paper/2505.11755/) (2025)
- [Permissive Safety Through Trusted Inference: Verifiable Belief-Space Neural Safety Filters for Assured Interactive Robotics](https://mlsanity.com/paper/2606.02562/) (2026)
- [Feasibility-Aware Pessimistic Estimation: Toward Long-Horizon Safety in Offline RL](https://mlsanity.com/paper/2505.08179/) (2025)
- [MAD-PINN: A Decentralized Physics-Informed Machine Learning Framework for Safe and Optimal Multi-Agent Control](https://mlsanity.com/paper/2509.23960/) (2025)
- [Robust Verification of Controllers under State Uncertainty via Hamilton-Jacobi Reachability Analysis](https://mlsanity.com/paper/2511.14755/) (2025)
- [Distributed Control of Network Systems in the Space of Stabilizing Graph Neural Network Policies](https://mlsanity.com/paper/2512.18540/) (2025)
- [Large Language Models in the Loop: A Stability- and Network-Aware Survey in Networked Control, Cyber-Physical, and Multi-Agent Systems](https://mlsanity.com/paper/2609.16599/) (2026)

## Abstract (arXiv)

Hamilton-Jacobi (HJ) reachability-based filtering provides a powerful framework to co-optimize performance and safety (or liveness) for autonomous systems. Under this filtering scheme, a nominal controller is minimally modified to ensure system safety or liveness. However, the resulting controllers can exhibit abrupt switching and bang-bang behavior, which is not suitable for applications of autonomous systems in the real world. This work presents a novel, unifying framework to design safety and liveness filters through reachability analysis. We explicitly characterize the maximal set of control inputs that ensures safety (or liveness) at a given state. Different safety filters can then be constructed using different subsets of this maximal set along with a projection operator to modify the nominal controller. We use the proposed framework to design three safety filters, each balancing performance, computation time, and smoothness differently. We highlight their relative strengths and limitations by applying these filters to autonomous navigation and rocket landing scenarios and on a physical robot testbed. We also discuss practical aspects associated with implementing these filters on real-world autonomous systems. Our research advances the understanding and potential application of reachability-based controllers on real-world autonomous systems.