  Control barrier functions have been widely used for synthesizing
safety-critical controls, often via solving quadratic programs. However, the
existence of Gaussian-type noise may lead to unsafe actions and result in
severe consequences. In this paper, we study systems modeled by stochastic
differential equations (SDEs) driven by Brownian motions. We propose a notion
of stochastic control barrier functions (SCBFs)and show that SCBFs can
significantly reduce the control efforts, especially in the presence of noise,
compared to stochastic reciprocal control barrier functions (SRCBFs), and offer
a less conservative estimation of safety probability, compared to stochastic
zeroing control barrier functions (SZCBFs). Based on this less conservative
probabilistic estimation for the proposed notion of SCBFs, we further extend
the results to handle high relative degree safety constraints using high-order
SCBFs. We demonstrate that the proposed SCBFs achieve good trade-offs of
performance and control efforts, both through theoretical analysis and
numerical simulations.
