# Control Barrier Functions for Stochastic Systems

arXiv: https://arxiv.org/abs/2003.03498 · HTML: https://mlsanity.com/paper/2003.03498/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2020.

## Built on by (most cited first, 14 of 14)

- [Safe Perception-Based Control under Stochastic Sensor Uncertainty using Conformal Prediction](https://mlsanity.com/paper/2304.00194/) (2023)
- [Correct-by-construction reach-avoid control of partially observable linear stochastic systems](https://mlsanity.com/paper/2103.02398/) (2021)
- [Policy Verification in Stochastic Dynamical Systems Using Logarithmic Neural Certificates](https://mlsanity.com/paper/2406.00826/) (2024)
- [Decentralized Safe Multi-agent Stochastic Optimal Control using Deep FBSDEs and ADMM](https://mlsanity.com/paper/2202.10658/) (2022)
- [Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety](https://mlsanity.com/paper/2203.15474/) (2022)
- [Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions](https://mlsanity.com/paper/2208.10733/) (2022)
- [Provably Safe Generative Sampling with Constricting Barrier Functions](https://mlsanity.com/paper/2602.21429/) (2026)
- [Myopically Verifiable Probabilistic Certificates for Safe Control and Learning](https://mlsanity.com/paper/2404.16883/) (2024)
- [Transfer of Safety Controllers Through Learning Deep Inverse Dynamics Model](https://mlsanity.com/paper/2405.13735/) (2024)
- [CaRT: Certified Safety and Robust Tracking in Learning-based Motion Planning for Multi-Agent Systems](https://mlsanity.com/paper/2307.08602/) (2023)
- [Set Invariance with Probability One for Controlled Diffusion: Score-based Approach](https://mlsanity.com/paper/2507.22385/) (2025)
- [Robust Taylor-Lagrange Control for Safety-Critical Systems](https://mlsanity.com/paper/2602.20076/) (2026)
- [Response-Aware Risk-Constrained Control Barrier Function With Application to Vehicles](https://mlsanity.com/paper/2603.24598/) (2026)
- [SafeMind: A Risk-Aware Differentiable Control Framework for Adaptive and Safe Quadruped Locomotion](https://mlsanity.com/paper/2604.09474/) (2026)

## Abstract (arXiv)

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.