Characterization of Safety and Conditional Invariance for Nonlinear Systems

This paper investigates sufficient and necessary conditions for safety (equivalently, conditional invariance) in terms of barrier functions. Relaxed sufficient conditions concerning the sign and the regularity of the barrier function are proposed. Furthermore, via a counterexample, the lack of existence of an autonomous and continuous barrier function certifying safety in a class of autonomous systems is shown. As a consequence, guided by converse Lyapunov theorems for only stability, time-varying barrier functions are proposed and infinitesimal conditions are shown to be both necessary as well as sufficient, provided that mild regularity conditions on the system's dynamics holds. Examples illustrate the results.