On the necessity of barrier certificates

A methodology for safety verification of nonlinear systems using barrier certificates has been proposed recently. The condition was stated in a sufficiency form: if there exists a barrier certificate, then the system is safe, in the sense that there is no trajectory starting from a given set of initial states that reaches some given unsafe region. Using the concepts of convex duality and density, in this paper we show that the existence of a barrier certificate is also necessary for safety