  Control Barrier Functions (CBFs) aim to ensure safety by constraining the
control input at each time step so that the system state remains within a
desired safe region. This paper presents a framework for CBFs in stochastic
systems in the presence of Gaussian process and measurement noise. We first
consider the case where the system state is known at each time step, and
present reciprocal and zero CBF constructions that guarantee safety with
probability 1. We extend our results to high relative degree systems with
linear dynamics and affine safety constraints. We then develop CBFs for
incomplete state information environments, in which the state must be estimated
using sensors that are corrupted by Gaussian noise. We prove that our proposed
CBF ensures safety with probability 1 when the state estimate is within a given
bound of the true state, which can be achieved using an Extended Kalman Filter
when the system is linear or the process and measurement noise are sufficiently
small. We propose control policies that combine these CBFs with Control
Lyapunov Functions in order to jointly ensure safety and stochastic stability.
Our results are validated via numerical study on a multi-agent collision
avoidance example.
