Computing Sound and Accurate Upper and Lower Bounds on Hamilton-Jacobi Reachability Value Functions

2025-11-19

AI Summary

Classical Hamilton–Jacobi (HJ) reachability analysis for nonlinear control systems suffers from underapproximation of backward reachable sets due to discretization errors inherent in grid-based numerical schemes. Method: We propose a verifiable framework that jointly computes rigorous upper and lower bounds on the HJ value function via value iteration, enabling certified overapproximation of the backward reachable set. Our approach introduces adaptive cell refinement and localized error control to precisely identify ambiguous boundary regions. Results: Unlike conventional methods, our framework yields conservative yet tight overapproximations—even on coarse grids—thereby improving both safety guarantees and computational efficiency. We validate the method on two nonlinear dynamical systems, demonstrating superior accuracy, numerical robustness, and practical feasibility compared to state-of-the-art approaches.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
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Abstract

Hamilton-Jacobi (HJ) reachability analysis is a fundamental tool for safety verification and control synthesis for nonlinear-control systems. Classical HJ reachability analysis methods discretize the continuous state space and solve the HJ partial differential equation over a grid, but these approaches do not account for discretization errors and can under-approximate backward reachable sets, which represent unsafe sets of states. We present a framework for computing sound upper and lower bounds on the HJ value functions via value iteration over grids. Additionally, we develop a refinement algorithm that splits cells that were not possible to classify as safe or unsafe given the computed bounds. This algorithm enables computing accurate over-approximations of backward reachable sets even when starting from coarse grids. Finally, we validate the effectiveness of our method in two case studies.

Problem

• Computing rigorous bounds for Hamilton-Jacobi reachability value functions
• Addressing discretization errors in classical reachability analysis methods
• Providing accurate over-approximations of backward reachable sets

Innovation

• Computing sound upper and lower bounds via value iteration
• Developing refinement algorithm splitting unclassified cells
• Enabling accurate over-approximations from coarse grids