# Verification and Synthesis of Control Barrier Functions

arXiv: https://arxiv.org/abs/2104.14001 · HTML: https://mlsanity.com/paper/2104.14001/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Apr 2021.

## Built on by (most cited first, 13 of 13)

- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Exact Verification of ReLU Neural Control Barrier Functions](https://mlsanity.com/paper/2310.09360/) (2023)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [Synthesizing Stable Reduced-Order Visuomotor Policies for Nonlinear Systems via Sums-of-Squares Optimization](https://mlsanity.com/paper/2304.12405/) (2023)
- [Learning Conservative Neural Control Barrier Functions from Offline Data](https://mlsanity.com/paper/2505.00908/) (2025)
- [Scalable Verification of Neural Control Barrier Functions Using Linear Bound Propagation](https://mlsanity.com/paper/2511.06341/) (2025)
- [Safe Control using Learned Safety Filters and Adaptive Conformal Inference](https://mlsanity.com/paper/2604.18482/) (2026)
- [Pre-Trained Vision Models as Perception Backbones for Safety Filters in Autonomous Driving](https://mlsanity.com/paper/2410.22585/) (2024)
- [Safe Beyond the Horizon: Efficient Sampling-based MPC with Neural Control Barrier Functions](https://mlsanity.com/paper/2502.15006/) (2025)
- [Learning Neural Control Barrier Functions from Expert Demonstrations using Inverse Constraint Learning](https://mlsanity.com/paper/2510.21560/) (2025)
- [Fault Tolerant Neural Control Barrier Functions for Robotic Systems under Sensor Faults and Attacks](https://mlsanity.com/paper/2402.18677/) (2024)
- [Sample Efficient Certification of Discrete-Time Control Barrier Functions](https://mlsanity.com/paper/2509.03899/) (2025)

## Abstract (arXiv)

Control systems often must satisfy strict safety requirements over an extended operating lifetime. Control Barrier Functions (CBFs) are a promising recent approach to constructing simple and safe control policies. This paper proposes a framework for verifying that a CBF guarantees safety for all time and synthesizing CBFs with verifiable safety in polynomial control systems. Our approach is to show that safety of CBFs is equivalent to the non-existence of solutions to a family of polynomial equations, and then prove that this nonexistence is equivalent to a pair of sum-of-squares constraints via the Positivstellensatz of algebraic geometry. We develop this Positivstellensatz to verify CBFs, as well as generalization to high-degree systems and multiple CBF constraints. We then propose a set of heuristics for CBF synthesis, including a general alternating-descent heuristic, a specialized approach for compact safe regions, and an approach for convex unsafe regions. Our approach is illustrated on two numerical examples.