# Safe Control Under Input Limits with Neural Control Barrier Functions

arXiv: https://arxiv.org/abs/2211.11056 · HTML: https://mlsanity.com/paper/2211.11056/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 50 references that are in the map.
- First posted on arXiv 20 Nov 2022.

## Builds on (most cited first, 27 of 27)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fast is better than free: Revisiting adversarial training](https://mlsanity.com/paper/2001.03994/) (2020)
- [Adversarial Training for Free!](https://mlsanity.com/paper/1904.12843/) (2019)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Safe Exploration in Continuous Action Spaces](https://mlsanity.com/paper/1801.08757/) (2018)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Learning Control Barrier Functions from Expert Demonstrations](https://mlsanity.com/paper/2004.03315/) (2020)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Understanding Catastrophic Overfitting in Single-step Adversarial Training](https://mlsanity.com/paper/2010.01799/) (2020)
- [Learning Safe Multi-Agent Control with Decentralized Neural Barrier Certificates](https://mlsanity.com/paper/2101.05436/) (2021)
- [Learning Stability Certificates from Data](https://mlsanity.com/paper/2008.05952/) (2020)
- [Control Barrier Functions for Systems with High Relative Degree](https://mlsanity.com/paper/1903.04706/) (2019)
- [Safe Control with Neural Network Dynamic Models](https://mlsanity.com/paper/2110.01110/) (2021)
- [Learning Control Lyapunov Functions from Counterexamples and Demonstrations](https://mlsanity.com/paper/1804.05285/) (2018)
- [Safe Control Synthesis via Input Constrained Control Barrier Functions](https://mlsanity.com/paper/2104.01704/) (2021)
- [Verification and Synthesis of Control Barrier Functions](https://mlsanity.com/paper/2104.14001/) (2021)
- [Backup Control Barrier Functions: Formulation and Comparative Study](https://mlsanity.com/paper/2104.11332/) (2021)
- [BarrierNet: A Safety-Guaranteed Layer for Neural Networks](https://mlsanity.com/paper/2111.11277/) (2021)
- [Probabilistic Safety-Assured Adaptive Merging Control for Autonomous Vehicles](https://mlsanity.com/paper/2104.14159/) (2021)
- [Comparative Analysis of Control Barrier Functions and Artificial Potential Fields for Obstacle Avoidance](https://mlsanity.com/paper/2010.09819/) (2020)
- [Reactive and Safe Road User Simulations using Neural Barrier Certificates](https://mlsanity.com/paper/2109.06689/) (2021)
- [Learning Control Barrier Functions with High Relative Degree for Safety-Critical Control](https://mlsanity.com/paper/2011.10721/) (2020)
- [A Robust, Multiple Control Barrier Function Framework for Input Constrained Systems](https://mlsanity.com/paper/2205.13726/) (2022)
- [Control-Lyapunov and Control-Barrier Functions based Quadratic Program for Spatio-temporal Specifications](https://mlsanity.com/paper/1903.06972/) (2019)
- [Safe-by-Design Control for Euler-Lagrange Systems](https://mlsanity.com/paper/2009.03767/) (2020)

## Built on by (most cited first, 29 of 29)

- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Safe Reinforcement Learning on the Constraint Manifold: Theory and Applications](https://mlsanity.com/paper/2404.09080/) (2024)
- [Exact Verification of ReLU Neural Control Barrier Functions](https://mlsanity.com/paper/2310.09360/) (2023)
- [Safety Filters for Black-Box Dynamical Systems by Learning Discriminating Hyperplanes](https://mlsanity.com/paper/2402.05279/) (2024)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [Verification-Aided Learning of Neural Network Barrier Functions with Termination Guarantees](https://mlsanity.com/paper/2403.07308/) (2024)
- [ModelVerification.jl: a Comprehensive Toolbox for Formally Verifying Deep Neural Networks](https://mlsanity.com/paper/2407.01639/) (2024)
- [Value Functions are Control Barrier Functions: Verification of Safe Policies using Control Theory](https://mlsanity.com/paper/2306.04026/) (2023)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Learning Performance-Oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation](https://mlsanity.com/paper/2401.05629/) (2024)
- [Learning Predictive Safety Filter via Decomposition of Robust Invariant Set](https://mlsanity.com/paper/2311.06769/) (2023)
- [Handling Long-Term Safety and Uncertainty in Safe Reinforcement Learning](https://mlsanity.com/paper/2409.12045/) (2024)
- [Certified Training with Branch-and-Bound for Lyapunov-stable Neural Control](https://mlsanity.com/paper/2411.18235/) (2024)
- [Designing Control Barrier Function via Probabilistic Enumeration for Safe Reinforcement Learning Navigation](https://mlsanity.com/paper/2504.21643/) (2025)
- [Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions](https://mlsanity.com/paper/2505.11755/) (2025)
- [ORN-CBF: Learning Observation-conditioned Residual Neural Control Barrier Functions via Hypernetworks](https://mlsanity.com/paper/2509.16614/) (2025)
- [Parallel Differentiable Reachability for Learning and Planning with Certified Neural Dynamics and Controllers](https://mlsanity.com/paper/2605.25346/) (2026)
- [The Feasibility Theory of Constrained Reinforcement Learning: A Tutorial Study](https://mlsanity.com/paper/2404.10064/) (2024)
- [Domain Adaptive Safety Filters via Deep Operator Learning](https://mlsanity.com/paper/2410.14528/) (2024)
- [Safe PDE Boundary Control with Neural Operators](https://mlsanity.com/paper/2411.15643/) (2024)
- [Neural Control Barrier Functions from Physics Informed Neural Networks](https://mlsanity.com/paper/2504.11045/) (2025)
- [CN-CBF: Composite Neural Control Barrier Function for Robot Navigation in Dynamic Environments](https://mlsanity.com/paper/2603.06921/) (2026)
- [Provably Safe, Yet Scalable Reinforcement Learning](https://mlsanity.com/paper/2606.14536/) (2026)
- [Fault Tolerant Neural Control Barrier Functions for Robotic Systems under Sensor Faults and Attacks](https://mlsanity.com/paper/2402.18677/) (2024)
- [Pareto Control Barrier Function for Inner Safe Set Maximization Under Input Constraints](https://mlsanity.com/paper/2410.04260/) (2024)
- [Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures](https://mlsanity.com/paper/2505.19521/) (2025)
- [Efficient Verification of Neural Control Barrier Functions with Smooth Nonlinear Activations](https://mlsanity.com/paper/2605.07757/) (2026)
- [VertexCBF: Improving Neural Control Barrier Functions via Vertex-Restricted Control Search](https://mlsanity.com/paper/2609.12831/) (2026)
- [Safe Reinforcement Learning with Receding-Horizon Safety Critic and Differentiable Action Shields](https://mlsanity.com/paper/2610.20255/) (2026)

## Abstract (arXiv)

We propose new methods to synthesize control barrier function (CBF)-based safe controllers that avoid input saturation, which can cause safety violations. In particular, our method is created for high-dimensional, general nonlinear systems, for which such tools are scarce. We leverage techniques from machine learning, like neural networks and deep learning, to simplify this challenging problem in nonlinear control design. The method consists of a learner-critic architecture, in which the critic gives counterexamples of input saturation and the learner optimizes a neural CBF to eliminate those counterexamples. We provide empirical results on a 10D state, 4D input quadcopter-pendulum system. Our learned CBF avoids input saturation and maintains safety over nearly 100% of trials.