Control barrier functions --- the standard tool for certifying safety in dynamical systems --- cannot exist for certain safe set topologies, and the obstructions are provable extensions of Brockett's 1983 necessary conditions for stabilizing feedback. The distinctive geometric property of CBF safe sets --- they are forward-invariant sublevel sets of smooth functions --- imposes topological constraints that straightforward proofs can extract. Applied to kinematic nonholonomic systems, these necessary conditions identify configurations where no smooth CBF can simultaneously satisfy the safety specification and respect the system's differential constraints.
The CBF literature has grown rapidly on the constructive side --- finding barrier functions for increasingly complex systems --- without equivalent attention to the impossibility side. These topological obstructions are not edge cases; they arise naturally in nonholonomic systems, which include most wheeled robots and many mechanical systems with rolling constraints. When a system's safe set has the wrong topology relative to its dynamics, no amount of computational effort will produce a valid CBF, because the obstruction is in the manifold structure, not in the optimization. Connecting this to Brockett's theorem is structurally important: it establishes that the impossibility of safety certification and the impossibility of stabilization share a common geometric root.
The absence of a solution is itself information, but only when it can be proven rather than merely observed. A system where safety certification fails computationally could be under-resourced; a system where it fails topologically is fundamentally incompatible with the chosen formalism. Distinguishing these two failure modes --- resource-limited versus structurally impossible --- is the difference between trying harder and choosing a different approach. Topological obstructions serve as maps of the territory where a given framework cannot operate, and knowing that territory exists before entering it is worth more than any amount of optimization within it.
(arXiv:2603.18422)
Comments (0)
No comments yet.