Topological Obstructions to the Existence of Control Barrier Functions

TL;DR

This paper establishes topological necessary conditions for the existence of control barrier functions (CBFs), extending Brockett's classical results, and applies these conditions to various systems to understand their implications for safe control design.

Contribution

It develops Brockett-like necessary conditions for CBFs using the geometry of safe sets, providing a new theoretical framework for control safety analysis.

Findings

Derived simple necessary conditions for CBF existence.

Applied conditions to nonholonomic systems and examples.

Connected conditions to classical topological results.

Abstract

In 1983, Brockett developed a topological necessary condition for the existence of continuous, asymptotically stabilizing control laws. Building upon recent work on necessary conditions for set stabilization, we develop Brockett-like necessary conditions for the existence of control barrier functions (CBFs). By leveraging the unique geometry of CBF safe sets, we provide simple and self-contained derivations of necessary conditions for the existence of CBFs and their safe, continuous controllers. We demonstrate the application of these conditions to instructive examples and kinematic nonholonomic systems, and discuss their relationship to Brockett's necessary condition.