Topological Obstructions to the Existence of Control Barrier Functions
Systems and Control
2026-03-20 v1 Systems and Control
Optimization and Control
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.
Keywords
Cite
@article{arxiv.2603.18422,
title = {Topological Obstructions to the Existence of Control Barrier Functions},
author = {Massimiliano de Sa and Aaron D. Ames},
journal= {arXiv preprint arXiv:2603.18422},
year = {2026}
}
Comments
6 pages, 3 figures