# Learning Performance-Oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation

arXiv: https://arxiv.org/abs/2401.05629 · HTML: https://mlsanity.com/paper/2401.05629/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 47 references that are in the map.
- First posted on arXiv 11 Jan 2024.

## Builds on (most cited first, 26 of 26)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [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)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Learning Safe Multi-Agent Control with Decentralized Neural Barrier Certificates](https://mlsanity.com/paper/2101.05436/) (2021)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [How to Train Your Neural Control Barrier Function: Learning Safety Filters for Complex Input-Constrained Systems](https://mlsanity.com/paper/2310.15478/) (2023)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Multi-Layered Safety for Legged Robots via Control Barrier Functions and Model Predictive Control](https://mlsanity.com/paper/2011.00032/) (2020)
- [Rule-based Optimal Control for Autonomous Driving](https://mlsanity.com/paper/2101.05709/) (2021)
- [SABLAS: Learning Safe Control for Black-box Dynamical Systems](https://mlsanity.com/paper/2201.01918/) (2022)
- [Refining Control Barrier Functions through Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2204.12507/) (2022)
- [Learning Differentiable Safety-Critical Control using Control Barrier Functions for Generalization to Novel Environments](https://mlsanity.com/paper/2201.01347/) (2022)
- [Composing Control Barrier Functions for Complex Safety Specifications](https://mlsanity.com/paper/2309.06647/) (2023)
- [Integral Control Barrier Functions for Dynamically Defined Control Laws](https://mlsanity.com/paper/2006.16936/) (2020)
- [Soft-Minimum and Soft-Maximum Barrier Functions for Safety with Actuation Constraints](https://mlsanity.com/paper/2305.10620/) (2023)
- [Compositions of Multiple Control Barrier Functions Under Input Constraints](https://mlsanity.com/paper/2210.01354/) (2022)
- [Data-Efficient Control Barrier Function Refinement](https://mlsanity.com/paper/2303.05973/) (2023)
- [Soft-Minimum Barrier Functions for Safety-Critical Control Subject to Actuation Constraints](https://mlsanity.com/paper/2304.00693/) (2023)
- [Time-Varying Soft-Maximum Control Barrier Functions for Safety in an A Priori Unknown Environment](https://mlsanity.com/paper/2310.05261/) (2023)
- [Learning Feasibility Constraints for Control Barrier Functions](https://mlsanity.com/paper/2303.09403/) (2023)
- [Learning a Better Control Barrier Function Under Uncertain Dynamics](https://mlsanity.com/paper/2310.04795/) (2023)

## Built on by (most cited first, 4 of 4)

- [Steering Dialogue Dynamics for Robustness against Multi-turn Jailbreaking Attacks](https://mlsanity.com/paper/2503.00187/) (2025)
- [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)
- [VertexCBF: Improving Neural Control Barrier Functions via Vertex-Restricted Control Search](https://mlsanity.com/paper/2609.12831/) (2026)

## Abstract (arXiv)

Control Barrier Functions (CBFs) provide an elegant framework for constraining nonlinear control system dynamics to remain within an invariant subset of a designated safe set. However, identifying a CBF that balances performance-by maximizing the control invariant set-and accommodates complex safety constraints, especially in systems with high relative degree and actuation limits, poses a significant challenge. In this work, we introduce a novel self-supervised learning framework to comprehensively address these challenges. Our method begins with a Boolean composition of multiple state constraints that define the safe set. We first construct a smooth function whose zero superlevel set forms an inner approximation of this safe set. This function is then combined with a smooth neural network to parameterize the CBF candidate. To train the CBF and maximize the volume of the resulting control invariant set, we design a physics-informed loss function based on a Hamilton-Jacobi Partial Differential Equation (PDE). We validate the efficacy of our approach on a 2D double integrator (DI) system and a 7D fixed-wing aircraft system (F16).