Reach-avoid Verification using Lyapunov Densities

TL;DR

This paper introduces new Lyapunov density-based conditions for verifying reach-avoid properties in systems modeled by differential equations, improving verification success rates for safety-critical applications.

Contribution

The paper proposes two novel, weaker Lyapunov density conditions for weak reach-avoid verification and extends them to all initial states, enhancing verification capabilities.

Findings

Two new sufficient conditions outperform existing ones

Conditions successfully verify reach-avoid properties in examples

Enhanced verification success for safety-critical systems

Abstract

Reach-avoid analysis combines the construction of safety and specific progress guarantees, and is able to formalize many important engineering problems. In this paper we study the reach-avoid verification problem of systems modelled by ordinary differential equations using Lyapunov densities. Firstly, the weak reach-avoid verification is considered. Given an initial set, a safe set and a target set, the weak reach-avoid verification is to verify whether the reach-avoid property (i.e., the system will enter the target set eventually while staying inside the safe set before the first target hitting time) holds for almost all states in the initial set. We propose two novel sufficient conditions using Lyapunov densities for the weak reach-avoid verification. These two sufficient conditions are shown to be weaker than existing ones, providing more possibilities of verifying weak reach-avoid properties successfully. Then, we generalize these conditions to the verification of reach-avoid properties for all states in the initial set. Finally, an example demonstrates theoretical developments of proposed conditions.