For every ordinary differential equation that robustly fulfills a safety property of a certain type there is a corresponding barrier certificate.
[4] Barrier certificates were generalized to hybrid systems in 2004 by Stephen Prajna and Ali Jadbabaie.
One distinguishing factor is the behavior of the barrier function at the boundary of the forward invariant set
A barrier function that goes to zero as the input approaches the boundary of
[7] A barrier function that goes to infinity as the inputs approach the boundary of