A barrier certificate or barrier function is used to prove that a given region is forward invariant for a given

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

hybrid dynamical system A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine, automaton, or a differen ...

. That is, a barrier function can be used to show that if a solution starts in a given

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

, 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 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.

History

The first result in the field of barrier certificates was the Nagumo theorem by Mitio Nagumo in 1942. . English translation in The term "barrier certificate" was introduced later based on similar concept in

convex optimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems ...

called barrier functions. Barrier certificates were generalized to

hybrid systems A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine, automaton, or a differen ...

in 2004 by Stephen Prajna and Ali Jadbabaie.

Variants

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

S

. A barrier function that goes to zero as the input approaches the boundary of

S

is called a ''zeroing barrier function.'' A barrier function that goes to infinity as the inputs approach the boundary of

S

are called ''reciprocal barrier functions''. Here, "reciprocal" refers to the fact that a reciprocal barrier functions can be defined as the

multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...

of a zeroing barrier function.

References

{{Reflist Differential equations