Solving Reach- and Stabilize-Avoid Problems Using Discounted Reachability

2025-05-14

AI Summary

This paper addresses the reach-avoid (RA) and stabilize-avoid (SA) zero-sum differential games for nonlinear continuous-time systems under bounded disturbances over an infinite horizon. We propose a Lipschitz-continuous RA value function whose zero-sublevel set precisely characterizes the RA region. A Hamilton–Jacobi (HJ) equation incorporating a variational inequality is formulated, and the existence and uniqueness of its viscosity solution are rigorously established. We introduce a novel two-step SA framework that synergistically integrates the RA strategy with a robust control Lyapunov function. Theoretical analysis proves that the associated Bellman operator is a contraction, ensuring numerical convergence. The approach is validated on a 3D Dubins car system, demonstrating significant improvements in simultaneous satisfaction of state constraints, target reachability, and long-term stability under persistent disturbances.

Technology Category

• Multiagent Systems: Adversarial Agents
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Solve infinite-horizon reach-avoid and stabilize-avoid problems for nonlinear systems
• Design Lipschitz continuous value function to characterize reach-avoid sets
• Integrate reach-avoid strategies with control Lyapunov functions for stability

Innovation

• Designed Lipschitz continuous RA value function
• Established contractive Bellman backup operator
• Integrated RA strategies with Robust Control Lyapunov-Value Function