# Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes

arXiv: https://arxiv.org/abs/2010.05818 · HTML: https://mlsanity.com/paper/2010.05818/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Oct 2020.

## Built on by (most cited first, 20 of 20)

- [In-Distribution Barrier Functions: Self-Supervised Policy Filters that Avoid Out-of-Distribution States](https://mlsanity.com/paper/2301.12012/) (2023)
- [Reinforcement Learning for Safe Robot Control using Control Lyapunov Barrier Functions](https://mlsanity.com/paper/2305.09793/) (2023)
- [Data-Driven Distributionally Robust Safety Verification Using Barrier Certificates and Conditional Mean Embeddings](https://mlsanity.com/paper/2403.10497/) (2024)
- [Can Learning Deteriorate Control? Analyzing Computational Delays in Gaussian Process-Based Event-Triggered Online Learning](https://mlsanity.com/paper/2305.08169/) (2023)
- [Learning Control Barrier Functions and their application in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2404.16879/) (2024)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Formal Verification of Unknown Dynamical Systems via Gaussian Process Regression](https://mlsanity.com/paper/2201.00655/) (2021)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions](https://mlsanity.com/paper/2208.10733/) (2022)
- [Safety Certification in the Latent space using Control Barrier Functions and World Models](https://mlsanity.com/paper/2507.13871/) (2025)
- [Real-Time Adaptive Safety-Critical Control with Gaussian Processes in High-Order Uncertain Models](https://mlsanity.com/paper/2402.18946/) (2024)
- [Barrier Certificates for Unknown Systems with Latent States and Polynomial Dynamics using Bayesian Inference](https://mlsanity.com/paper/2504.01807/) (2025)
- [Safe Learning of Uncertain Environments](https://mlsanity.com/paper/2103.01413/) (2021)
- [Learning Safety Filters for Unknown Discrete-Time Linear Systems](https://mlsanity.com/paper/2111.00631/) (2021)
- [Safe Online Dynamics Learning with Initially Unknown Models and Infeasible Safety Certificates](https://mlsanity.com/paper/2311.02133/) (2023)
- [Learning Passive Continuous-Time Dynamics with Multistep Port-Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2510.00384/) (2025)
- [BarrierBench: Evaluating Large Language Models for Safety Verification in Dynamical Systems](https://mlsanity.com/paper/2511.09363/) (2025)
- [Kernel-Based Learning of Safety Barriers](https://mlsanity.com/paper/2601.12002/) (2026)
- [Data-Driven Synthesis of Probabilistic Controlled Invariant Sets for Linear MDPs](https://mlsanity.com/paper/2604.02727/) (2026)
- [Learning from Demonstration via Spatiotemporal Tubes for Unknown Euler-Lagrange Systems](https://mlsanity.com/paper/2607.00534/) (2026)

## Abstract (arXiv)

This paper focuses on the controller synthesis for unknown, nonlinear systems while ensuring safety constraints. Our approach consists of two steps, a learning step that uses Gaussian processes and a controller synthesis step that is based on control barrier functions. In the learning step, we use a data-driven approach utilizing Gaussian processes to learn the unknown control affine nonlinear dynamics together with a statistical bound on the accuracy of the learned model. In the second controller synthesis steps, we develop a systematic approach to compute control barrier functions that explicitly take into consideration the uncertainty of the learned model. The control barrier function not only results in a safe controller by construction but also provides a rigorous lower bound on the probability of satisfaction of the safety specification. Finally, we illustrate the effectiveness of the proposed results by synthesizing a safety controller for a jet engine example.