# Learning Certified Control using Contraction Metric

arXiv: https://arxiv.org/abs/2011.12569 · HTML: https://mlsanity.com/paper/2011.12569/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2020.

## Built on by (most cited first, 30 of 49)

- [Adaptive-Control-Oriented Meta-Learning for Nonlinear Systems](https://mlsanity.com/paper/2103.04490/) (2021)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Learning Control Policies for Stochastic Systems with Reach-avoid Guarantees](https://mlsanity.com/paper/2210.05308/) (2022)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Safe Output Feedback Motion Planning from Images via Learned Perception Modules and Contraction Theory](https://mlsanity.com/paper/2206.06553/) (2022)
- [Backward Reachability Analysis of Neural Feedback Loops: Techniques for Linear and Nonlinear Systems](https://mlsanity.com/paper/2209.14076/) (2022)
- [SABLAS: Learning Safe Control for Black-box Dynamical Systems](https://mlsanity.com/paper/2201.01918/) (2022)
- [Control-oriented meta-learning](https://mlsanity.com/paper/2204.06716/) (2022)
- [Neural Termination Analysis](https://mlsanity.com/paper/2102.03824/) (2021)
- [How Generalizable Is My Behavior Cloning Policy? A Statistical Approach to Trustworthy Performance Evaluation](https://mlsanity.com/paper/2405.05439/) (2024)
- [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)
- [Learning-based Robust Motion Planning with Guaranteed Stability: A Contraction Theory Approach](https://mlsanity.com/paper/2102.12668/) (2021)
- [Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems](https://mlsanity.com/paper/2104.08695/) (2021)
- [Tensor-Compressed Back-Propagation-Free Training for (Physics-Informed) Neural Networks](https://mlsanity.com/paper/2308.09858/) (2023)
- [Conformal Methods for Quantifying Uncertainty in Spatiotemporal Data: A Survey](https://mlsanity.com/paper/2209.03580/) (2022)
- [Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems](https://mlsanity.com/paper/2212.06874/) (2022)
- [Neural Abstractions](https://mlsanity.com/paper/2301.11683/) (2023)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [State-Wise Safe Reinforcement Learning With Pixel Observations](https://mlsanity.com/paper/2311.02227/) (2023)
- [Predictability Enables Parallelization of Nonlinear State Space Models](https://mlsanity.com/paper/2508.16817/) (2025)
- [A Theoretical Overview of Neural Contraction Metrics for Learning-based Control with Guaranteed Stability](https://mlsanity.com/paper/2110.00693/) (2021)
- [Model-free Neural Lyapunov Control for Safe Robot Navigation](https://mlsanity.com/paper/2203.01190/) (2022)
- [Data-Driven Control with Inherent Lyapunov Stability](https://mlsanity.com/paper/2303.03157/) (2023)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Scalable Synthesis of Verified Controllers in Deep Reinforcement Learning](https://mlsanity.com/paper/2104.10219/) (2021)
- [Learning Control-Oriented Dynamical Structure from Data](https://mlsanity.com/paper/2302.02529/) (2023)
- [Efficient Motion Planning for Manipulators with Control Barrier Function-Induced Neural Controller](https://mlsanity.com/paper/2404.01184/) (2024)
- [Neural Continuous-Time Supermartingale Certificates](https://mlsanity.com/paper/2412.17432/) (2024)
- [Neural Contraction Metrics with Formal Guarantees for Discrete-Time Nonlinear Dynamical Systems](https://mlsanity.com/paper/2504.17102/) (2025)

## Abstract (arXiv)

In this paper, we solve the problem of finding a certified control policy that drives a robot from any given initial state and under any bounded disturbance to the desired reference trajectory, with guarantees on the convergence or bounds on the tracking error. Such a controller is crucial in safe motion planning. We leverage the advanced theory in Control Contraction Metric and design a learning framework based on neural networks to co-synthesize the contraction metric and the controller for control-affine systems. We further provide methods to validate the convergence and bounded error guarantees. We demonstrate the performance of our method using a suite of challenging robotic models, including models with learned dynamics as neural networks. We compare our approach with leading methods using sum-of-squares programming, reinforcement learning, and model predictive control. Results show that our methods indeed can handle a broader class of systems with less tracking error and faster execution speed. Code is available at https://github.com/sundw2014/C3M.