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A **barrier certificate**^{[1]} or **barrier function** is used to prove that a given region is forward invariant for a given ordinary differential equation or hybrid dynamical system.^{[2]} 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.

This article may be too technical for most readers to understand. (September 2018) |

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]}

The first result in the field of barrier certificates was the Nagumo theorem by Mitio Nagumo in 1942.^{[4]}^{[5]} The term "barrier certificate" was introduced later based on similar concept in convex optimization called barrier functions.^{[4]}

Barrier certificates were generalized to hybrid systems in 2004 by Stephen Prajna and Ali Jadbabaie.^{[6]}

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.

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