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Learning-based control schemes have recently shown great efficacy performingcomplex tasks for a wide variety of applications. However, in order to deploythem in real systems, it is of vital importance to guarantee that the systemwill remain safe during online training and execution. Among the currently mostpopular methods to tackle this challenge, Control Barrier Functions (CBFs)serve as mathematical tools that provide a formal safety-preserving controlsynthesis procedure for systems with known dynamics. In this paper, we firstintroduce a model-uncertainty-aware reformulation of CBF-based safety-criticalcontrollers using Gaussian Process (GP) regression to bridge the gap between anapproximate mathematical model and the real system. Compared to previousapproaches, we study the feasibility of the resulting robust safety-criticalcontroller. This feasibility analysis results in a set of richness conditionsthat the available information about the system should satisfy to guaranteethat a safe control action can be found at all times. We then use theseconditions to devise an event-triggered online data collection strategy thatensures the recursive feasibility of the learned safety-critical controller.Our proposed methodology endows the system with the ability to reason at alltimes about whether the current information at its disposal is enough to ensuresafety or if new measurements are required. This, in turn, allows us to provideformal results of forward invariance of a safe set with high probability, evenin a priori unexplored regions. Finally, we validate the proposed framework innumerical simulations of an adaptive cruise control system and a kinematicvehicle.

Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions