Solving Reach- and Stabilize-Avoid Problems Using Discounted Reachability

TL;DR

This paper develops a Hamilton-Jacobi reachability framework for infinite-horizon reach-avoid and stabilize-avoid problems in nonlinear systems, providing new value functions and verification methods.

Contribution

It introduces a Lipschitz continuous reach-avoid value function with a contractive Bellman operator and integrates it with control Lyapunov functions for stabilization.

Findings

Validated on a 3D Dubins car system demonstrating effectiveness.

Established the uniqueness of the viscosity solution for the Hamilton-Jacobi variational inequality.

Abstract

In this article, we consider the infinite-horizon reach-avoid (RA) and stabilize-avoid (SA) zero-sum game problems for general nonlinear continuous-time systems, where the goal is to find the set of states that can be controlled to reach or stabilize to a target set, without violating constraints even under the worst-case disturbance. Based on the Hamilton-Jacobi reachability method, we address the RA problem by designing a new Lipschitz continuous RA value function, whose zero sublevel set exactly characterizes the RA set. We establish that the associated Bellman backup operator is contractive and that the RA value function is the unique viscosity solution of a Hamilton-Jacobi variational inequality. Finally, we develop a two-step framework for the SA problem by integrating our RA strategies with a recently proposed Robust Control Lyapunov-Value Function, thereby ensuring both target reachability and long-term stability. We numerically verify our RA and SA frameworks on a 3D Dubins car system to demonstrate the efficacy of the proposed approach.