# Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods

arXiv: https://arxiv.org/abs/2202.11762 · HTML: https://mlsanity.com/paper/2202.11762/

## Key facts

- Cited by 151 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Feb 2022.

## Built on by (most cited first, 30 of 151)

- [Toward General-Purpose Robots via Foundation Models: A Survey and Meta-Analysis](https://mlsanity.com/paper/2312.08782/) (2023)
- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Iterative Reachability Estimation for Safe Reinforcement Learning](https://mlsanity.com/paper/2309.13528/) (2023)
- [Multi-Agent Reinforcement Learning for Autonomous Driving: A Survey](https://mlsanity.com/paper/2408.09675/) (2024)
- [A Survey of Robotic Language Grounding: Tradeoffs between Symbols and Embeddings](https://mlsanity.com/paper/2405.13245/) (2024)
- [Safety Certification for Stochastic Systems via Neural Barrier Functions](https://mlsanity.com/paper/2206.01463/) (2022)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Backward Reachability Analysis of Neural Feedback Loops: Techniques for Linear and Nonlinear Systems](https://mlsanity.com/paper/2209.14076/) (2022)
- [Closed-loop Analysis of Vision-based Autonomous Systems: A Case Study](https://mlsanity.com/paper/2302.04634/) (2023)
- [Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2303.10506/) (2023)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [AutoCost: Evolving Intrinsic Cost for Zero-violation Reinforcement Learning](https://mlsanity.com/paper/2301.10339/) (2023)
- [Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates](https://mlsanity.com/paper/2405.14058/) (2024)
- [Hamilton-Jacobi Reachability in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2407.09645/) (2024)
- [Fossil 2.0: Formal Certificate Synthesis for the Verification and Control of Dynamical Models](https://mlsanity.com/paper/2311.09793/) (2023)
- [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)
- [Compositional Policy Learning in Stochastic Control Systems with Formal Guarantees](https://mlsanity.com/paper/2312.01456/) (2023)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Stable and Safe Reinforcement Learning via a Barrier-Lyapunov Actor-Critic Approach](https://mlsanity.com/paper/2304.04066/) (2023)
- [Trajectory Generation, Control, and Safety with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.15512/) (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)
- [Online Control Barrier Functions for Decentralized Multi-Agent Navigation](https://mlsanity.com/paper/2303.04313/) (2023)
- [Temporal Logic Imitation: Learning Plan-Satisficing Motion Policies from Demonstrations](https://mlsanity.com/paper/2206.04632/) (2022)
- [LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attraction](https://mlsanity.com/paper/2403.10013/) (2024)
- [Real-Time Out-of-Distribution Failure Prevention via Multi-Modal Reasoning](https://mlsanity.com/paper/2505.10547/) (2025)
- [State-Wise Safe Reinforcement Learning With Pixel Observations](https://mlsanity.com/paper/2311.02227/) (2023)
- [CBF-LLM: Safe Control for LLM Alignment](https://mlsanity.com/paper/2408.15625/) (2024)

## Abstract (arXiv)

Learning-enabled control systems have demonstrated impressive empirical performance on challenging control problems in robotics, but this performance comes at the cost of reduced transparency and lack of guarantees on the safety or stability of the learned controllers. In recent years, new techniques have emerged to provide these guarantees by learning certificates alongside control policies -- these certificates provide concise, data-driven proofs that guarantee the safety and stability of the learned control system. These methods not only allow the user to verify the safety of a learned controller but also provide supervision during training, allowing safety and stability requirements to influence the training process itself. In this paper, we provide a comprehensive survey of this rapidly developing field of certificate learning. We hope that this paper will serve as an accessible introduction to the theory and practice of certificate learning, both to those who wish to apply these tools to practical robotics problems and to those who wish to dive more deeply into the theory of learning for control.