A Unified Stability Analysis of Safety-Critical Control using Multiple Control Barrier Functions

Ensuring liveness and safety of autonomous and cyber-physical systems remains a fundamental challenge, particularly when multiple safety constraints are present. This letter advances the theoretical foundations of safety-filter Quadratic Programs (QP) and Control Lyapunov Function (CLF)-Control Barrier Function (CBF) controllers by establishing a unified analytical framework for studying their stability properties. We derive sufficient feasibility conditions for QPs with multiple CBFs and formally characterize the conditions leading to undesirable equilibrium points at possible intersecting safe set boundaries. Additionally, we introduce a stability criterion for equilibrium points, providing a systematic approach to identifying conditions under which they can be destabilized or eliminated. Our analysis extends prior theoretical results, deepening the understanding of the conditions of feasibility and stability of CBF-based safety filters and the CLF-CBF QP framework.