Patching Approximately Safe Value Functions Leveraging Local Hamilton-Jacobi Reachability Analysis

Safe value functions, such as control barrier functions, characterize a safe set and synthesize a safety filter, overriding unsafe actions, for a dynamic system. While function approximators like neural networks can synthesize approximately safe value functions, they typically lack formal guarantees. In this paper, we propose a local dynamic programming-based approach to "patch" approximately safe value functions to obtain a safe value function. This algorithm, HJ-Patch, produces a novel value function that provides formal safety guarantees, yet retains the global structure of the initial value function. HJ-Patch modifies an approximately safe value function at states that are both (i) near the safety boundary and (ii) may violate safety. We iteratively update both this set of "active" states and the value function until convergence. This approach bridges the gap between value function approximation methods and formal safety through Hamilton-Jacobi (HJ) reachability, offering a framework for integrating various safety methods. We provide simulation results on analytic and learned examples, demonstrating HJ-Patch reduces the computational complexity by 2 orders of magnitude with respect to standard HJ reachability. Additionally, we demonstrate the perils of using approximately safe value functions directly and showcase improved safety using HJ-Patch.