# Safe Neural Control for Non-Affine Control Systems with Differentiable Control Barrier Functions

arXiv: https://arxiv.org/abs/2309.04492 · HTML: https://mlsanity.com/paper/2309.04492/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 29 references that are in the map.
- First posted on arXiv 6 Sep 2023.

## Builds on (most cited first, 12 of 12)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Control Barrier Functions for Systems with High Relative Degree](https://mlsanity.com/paper/1903.04706/) (2019)
- [Learning Interactive Driving Policies via Data-driven Simulation](https://mlsanity.com/paper/2111.12137/) (2021)
- [On the Forward Invariance of Neural ODEs](https://mlsanity.com/paper/2210.04763/) (2022)
- [Integral Control Barrier Functions for Dynamically Defined Control Laws](https://mlsanity.com/paper/2006.16936/) (2020)
- [Sufficient Conditions for Feasibility of Optimal Control Problems Using Control Barrier Functions](https://mlsanity.com/paper/2011.08248/) (2020)
- [Inverse Optimal Safety Filters](https://mlsanity.com/paper/2112.08225/) (2021)
- [Control Barrier Functions for Systems with Multiple Control Inputs](https://mlsanity.com/paper/2203.07978/) (2022)

## Built on by (most cited first, 3 of 3)

- [End-to-End Training of High-Dimensional Optimal Control with Implicit Hamiltonians via Jacobian-Free Backpropagation](https://mlsanity.com/paper/2510.00359/) (2025)
- [Leveraging Analytic Gradients in Provably Safe Reinforcement Learning](https://mlsanity.com/paper/2506.01665/) (2025)
- [Differentiable Policy Transport over Multi-Layer Network Feasibility Geometry](https://mlsanity.com/paper/2609.26068/) (2026)

## Abstract (arXiv)

This paper addresses the problem of safety-critical control for non-affine control systems. It has been shown that optimizing quadratic costs subject to state and control constraints can be sub-optimally reduced to a sequence of quadratic programs (QPs) by using Control Barrier Functions (CBFs). Our recently proposed High Order CBFs (HOCBFs) can accommodate constraints of arbitrary relative degree. The main challenges in this approach are that it requires affine control dynamics and the solution of the CBF-based QP is sub-optimal since it is solved point-wise. To address these challenges, we incorporate higher-order CBFs into neural ordinary differential equation-based learning models as differentiable CBFs to guarantee safety for non-affine control systems. The differentiable CBFs are trainable in terms of their parameters, and thus, they can address the conservativeness of CBFs such that the system state will not stay unnecessarily far away from safe set boundaries. Moreover, the imitation learning model is capable of learning complex and optimal control policies that are usually intractable online. We illustrate the effectiveness of the proposed framework on LiDAR-based autonomous driving and compare it with existing methods.