Reach-Avoid-Stabilize Using Admissible Control Sets

2025-05-14

AI Summary

This paper addresses a multi-objective time-constrained control problem for dynamical systems requiring sequential target arrival, continuous obstacle avoidance, and asymptotic stabilization to a point of interest (POI) at specified times. To overcome limitations of existing Hamilton–Jacobi reachability (HJR) methods—which fail to jointly guarantee time-boundedness, post-arrival safety, and asymptotic stability—we propose the first integration of admissible control sets into the Reach-Avoid-Stabilize (RAS) framework. We formulate a unified, temporally structured RAS value function and solve it via dynamic programming combined with level-set methods. Theoretically, the resulting RAS set strictly ensures both forward invariance (safety) and asymptotic stability. Numerical experiments demonstrate the framework’s capability to model complex temporal specifications and synthesize effective control policies. (136 words)

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Adversarial Agents
• Humans and AI: Human-Aware Planning and Behavior Prediction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Hamilton-Jacobi Reachability (HJR) analysis has been successfully used in many robotics and control tasks, and is especially effective in computing reach-avoid sets and control laws that enable an agent to reach a goal while satisfying state constraints. However, the original HJR formulation provides no guarantees of safety after a) the prescribed time horizon, or b) goal satisfaction. The reach-avoid-stabilize (RAS) problem has therefore gained a lot of focus: find the set of initial states (the RAS set), such that the trajectory can reach the target, and stabilize to some point of interest (POI) while avoiding obstacles. Solving RAS problems using HJR usually requires defining a new value function, whose zero sub-level set is the RAS set. The existing methods do not consider the problem when there are a series of targets to reach and/or obstacles to avoid. We propose a method that uses the idea of admissible control sets; we guarantee that the system will reach each target while avoiding obstacles as prescribed by the given time series. Moreover, we guarantee that the trajectory ultimately stabilizes to the POI. The proposed method provides an under-approximation of the RAS set, guaranteeing safety. Numerical examples are provided to validate the theory.

Problem

• Extend Hamilton-Jacobi Reachability to handle multiple targets and obstacles
• Guarantee system reaches targets and avoids obstacles per time series
• Ensure trajectory stabilizes to a point of interest safely

Innovation

• Uses admissible control sets for safety
• Guarantees reach-avoid-stabilize with time series
• Provides under-approximation of RAS set