Forward and Control Invariance Analysis of Backup Control Barrier Function Induced Safe Sets for Online Safety of Nonlinear Systems

This work analyzes the forward invariance and control invariance properties of the safe set induced by the online backup control barrier function (CBF). This is achieved by proving that this approach to CBF enforcement induces a safe set that verifies the extension of Nagumo’s theorem for non-autonomous systems. The objective of this work is to be self-contained while addressing both theoretical questions on safety guarantees and practical questions regarding online computation and feasibility guarantees. Numerical examples are provided to reinforce the intuition generated throughout the analysis.