Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints and reach-avoid-stay specifications. (October 2022)

• Record Type: Journal Article
• Authors: Meng, Yiming; Li, Yinan; Fitzsimmons, Maxwell; Liu, Jun
• Is Part Of: Automatica. Volume 144(2022)
• Publication Date: 2022-10
• Subjects: Lyapunov functions -- Barrier functions -- Reachability -- Stability -- Safety -- Reach-avoid-stay specifications -- Stability with safety guarantees -- Converse theorems; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2022.110478
• Languages: English
• ISSNs: 0005-1098

Abstract

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. One clear limitation of the results of this paper is that the converse Lyapunov-barrier theorems are not constructive, as with classical converse Lyapunov theorems. We believe, however, 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.