Converse Theorems for Certificates of Safety and Stability

TL;DR

This paper establishes theoretical conditions for the existence of control barrier functions and their extensions, which are crucial for ensuring safety and stability in nonlinear control systems, and explores their relationships and constructions.

Contribution

It provides the first set of converse theorems for control barrier functions, extending their definitions and linking them with control Lyapunov functions for safety and stability guarantees.

Findings

Existence conditions for control barrier functions given any safe set.

Extended notions of CBF that always exist if the set is safe.

Conditions for constructing compatible CLF-CBF pairs from non-compatible ones.

Abstract

Motivated by the key role of control barrier functions (CBFs) in assessing safety and enabling the synthesis of safe controllers in nonlinear control systems, this paper presents a suite of converse results on CBFs. Given any safe set, we first identify a set of general sufficient conditions which guarantee the existence of a CBF. Our technical analysis also enables us to define an extended notion of CBF which is always guaranteed to exist if the set is safe. We next turn our attention to the problem of joint safety and stability, and give conditions under which the notions of control Lyapunov-barrier function (CLBF) and compatible control Lyapunov function (CLF) and CBF pair are guaranteed to exist. Finally, we identify conditions under which a CLBF and a compatible CLF-CBF pair can be constructed from a non-compatible CLF-CBF pair. Throughout the paper, we intersperse different examples and counterexamples to motivate our results and position them within the state of the art.