Control Barrier Functions: Theory and Applications

Safety is not a property you bolt onto a learned controller after the fact — it's a constraint you enforce, provably, at every step. Control barrier functions are the cleanest tool we know for that: define the safe set, and the CBF gives you a condition that keeps the system inside it, turning “please stay safe” into a constraint a controller must satisfy. For anyone making guarantees about a system that acts in the world, this is foundational.

What we value is the provable part. A CBF doesn't ask the model to learn safety and hope it generalizes; it wraps the controller in a filter that mathematically cannot let it cross the boundary, whatever the policy underneath is doing. That separation — a learned policy for capability, a proven filter for safety — is exactly the architecture we think safety-critical AI should have. The intelligence can be as empirical as you like, as long as the shell around it is not.

This survey is the map of that territory, and we treat it as a starting point rather than a finish line. The hard part in practice is the honest one the theory makes explicit: a CBF guarantees safety with respect to a model and a specified safe set, so the guarantee is only ever as good as the assumptions you're willing to write down and defend. That's a feature, not a bug — it forces you to state what “safe” means before you claim it.