  In recent years, the analysis of a control barrier function has received
considerable attention because it is helpful for the safety-critical control
required in many control application problems. While the extension of the
analysis to a stochastic system studied by many researchers, it remains a
challenging issue. In this paper, we consider sufficient conditions for
reciprocal and zeroing control barrier functions ensuring safety with
probability one and design a control law using the functions. Then, we propose
another version of a stochastic zeroing control barrier function to evaluate a
probability of a sample path staying in a safe set and confirm the convergence
of a specific expectation related to the attractiveness of a safe set. We also
show a way of deisgning a safety-critical control law based on our stochastic
zeroing control barrier function. Finally, we confirm the validity of the
proposed control design and the analysis using the control barrier functions
via simple examples with their numerical simulation.
