Barrier certificate

A barrier certificate or barrier function is used to prove that a given region is forward invariant for a given ordinary differential equation or hybrid dynamical system. That is, a barrier function can be used to show that if a solution starts in a given set, then it cannot leave that set.

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 map the system state (for example, its position) to a scalar value for which certain ranges must correspond to either the safe or unsafe set, the specifics of this vary by the type of barrier function used. Assuming a function ${\displaystyle h}$ of zeroing type, the question of safety becomes that of determining whether ${\displaystyle {\dot {h}}}$ is non-negative for any state where ${\displaystyle h=0}$. As the unsafe set corresponds to all negative values of ${\displaystyle h}$, its inability to reach a value below 0 implies the system is unable to cross the safe set boundary outwards.

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.