A control barrier function certifies safety — it provides a mathematical guarantee that a system’s state will remain within a safe set, regardless of disturbances, as long as the control law satisfies certain conditions. The function defines a boundary, and the control keeps the system inside. The question: can such a function always be found for a given safe set?

No. The researchers extend Brockett’s 1983 theorem — which identified topological obstructions to continuous stabilizing control — to control barrier functions. The result: some safe sets, by their topological structure, cannot admit control barrier functions. The geometry of the set itself prevents the existence of the safety certificate, no matter how clever the control design.

The obstruction is topological, meaning it depends on the shape of the safe set (its holes, connectivity, fundamental group) rather than on the specific dynamics. A safe set with the wrong topology cannot be certified safe by a CBF even if the system can actually stay safe inside it. The barrier function might exist as a concept but not as a smooth function on that topology.

The application to nonholonomic systems is concrete. A car that cannot move sideways (nonholonomic constraint) has a state space with topological features that classical smooth barrier functions cannot accommodate. The constraint creates holes in the control authority that no smooth CBF can paper over.

The structural insight: safety certification and actual safety are different problems. A system can be safe (the controller keeps it in the set) without being certifiably safe (no CBF exists). The gap between “is safe” and “can prove it’s safe” is topological — it depends on the shape of what you’re trying to protect, not on how hard you try to protect it.

(arXiv:2603.18422)