Topological Obstructions to the Existence of Control Barrier Functions
This provides theoretical insights for control theorists working on safety-critical systems, but it is incremental as it builds on prior work on set stabilization.
The paper tackles the problem of determining when control barrier functions (CBFs) exist for safe control by developing topological necessary conditions, similar to Brockett's condition, and applies these to examples like kinematic nonholonomic systems.
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.