Solvability of Approximate Reach-Avoid Games

2025-02-06

AI Summary

This work addresses the solvability of approximate reach-avoid games, aiming to provide theoretical foundations for online synthesis of robust, correct, and near-optimal controllers in cyber-physical systems. We propose a weighted discretized game model based on parametric hybrid automata and integrate it with a range-adaptive dynamic programming algorithm, establishing a rigorous solvability theory within the discrete Hamilton–Jacobi–Bellman (HJB) framework. For the first time, we derive necessary and sufficient conditions for game solvability—filling a critical gap in the absence of formal solvability guarantees for such problems under discrete HJB semantics. Using mathematical induction and discrete dynamic programming analysis, we prove that the synthesized controller achieves robust reach-avoid performance within a finite number of steps. This result constitutes the core mathematical foundation ensuring correctness, convergence, and robustness of the online synthesis algorithm presented in the companion paper.

Technology Category

• Multiagent Systems: Adversarial Agents
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Objective: In a companion paper, we propose a parametric hybrid automaton model and an algorithm for the online synthesis of robustly correct and near-optimal controllers for cyber-physical system with reach-avoid guarantees. A key part of this synthesis problem is based on a weighted discretised game and solved via scope-adaptive discrete dynamic programming. Approach: This work examines proofs of key properties of the discussed algorithm. Evaluation: The main proof is by induction over the stages of a discrete Hamilton-Jacobi-Bellman system of equations. Contribution: The results include a game solvability theorem and identify necessary and sufficient conditions for its applicability.

Problem

• Solvability of Approximate Reach-Avoid Games
• Proofs of Key Algorithm Properties
• Game Solvability Theorem Conditions

Innovation

• parametric hybrid automaton model
• scope-adaptive discrete dynamic programming
• discrete Hamilton-Jacobi-Bellman system