Showing that a set is forward invariant is an aspect of safety, which is the property where a system is guaranteed to avoid obstacles specified as an unsafe set.
Barrier certificates play the analogical role for safety to the role of Lyapunov functions for stability. For every ordinary differential equation that robustly fulfills a safety property of a certain type there is a corresponding barrier certificate.[3]
There are several different types of barrier functions. 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 is called a zeroing barrier function.[7] A barrier function that goes to infinity as the inputs approach the boundary of are called reciprocal barrier functions.[7] Here, "reciprocal" refers to the fact that a reciprocal barrier functions can be defined as the multiplicative inverse of a zeroing barrier function.
References
^Prajna, Stephen, and Ali Jadbabaie. "Safety verification of hybrid systems using barrier certificates." International Workshop on Hybrid Systems: Computation and Control. Springer, Berlin, Heidelberg, 2004.