Smooth Converse Lyapunov-Barrier Theorems for Asymptotic Stability with Safety Constraints and Reach-Avoid-Stay Specifications

TL;DR

This paper develops a unified theoretical framework linking Lyapunov and barrier functions to stability and safety guarantees in dynamical systems, including those with control inputs, providing foundational insights for safety-critical control design.

Contribution

It introduces converse theorems that establish the existence of smooth Lyapunov and barrier functions for stability and safety properties, unifying these concepts under a single theoretical framework.

Findings

Existence of smooth Lyapunov functions for systems with safety guarantees.

Extension of results to systems with control inputs and reach-avoid specifications.

Theoretical foundation for computational approaches to safety-critical control.

Abstract

Stability and safety are two important aspects in safety-critical control of dynamical systems. It has been a well established fact in control theory that stability properties can be characterized by Lyapunov functions. Reachability properties can also be naturally captured by Lyapunov functions for finite-time stability. Motivated by safety-critical control applications, such as in autonomous systems and robotics, there has been a recent surge of interests in characterizing safety properties using barrier functions. Lyapunov and barrier functions conditions, however, are sometimes viewed as competing objectives. In this paper, we provide a unified theoretical treatment of Lyapunov and barrier functions in terms of converse theorems for stability properties with safety guarantees and reach-avoid-stay type specifications. We show that if a system (modeled as a dynamical system with measurable perturbations) possesses a stability with safety property, then there exists a smooth Lyapunov function to certify such a property. This Lyapunov function is shown to be defined on the entire set of initial conditions from which solutions satisfy this property. A similar but slightly weaker statement is made for reach-avoid-stay specifications. We show by a simple example that the latter statement cannot be strengthened without additional assumptions. We further extend the results for systems with control inputs and prove existence of converse Lyapunov-barrier functions for reach-and-avoid specifications. While the converse Lyapunov-barrier theorems are not constructive, as with classical converse Lyapunov theorems, we believe that the unified necessary and sufficient conditions with a single Lyapunov-barrier function are of theoretical interest and can hopefully shed some light on computational approaches.