# Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularization

arXiv: https://arxiv.org/abs/1907.13122 · HTML: https://mlsanity.com/paper/1907.13122/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2019.

## Built on by (most cited first, 30 of 32)

- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
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- [Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems](https://mlsanity.com/paper/2104.08695/) (2021)
- [Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems](https://mlsanity.com/paper/2212.06874/) (2022)
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- [Data-Driven Control with Inherent Lyapunov Stability](https://mlsanity.com/paper/2303.03157/) (2023)
- [KCRL: Krasovskii-Constrained Reinforcement Learning with Guaranteed Stability in Nonlinear Dynamical Systems](https://mlsanity.com/paper/2206.01704/) (2022)
- [Learning Control-Oriented Dynamical Structure from Data](https://mlsanity.com/paper/2302.02529/) (2023)
- [ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction](https://mlsanity.com/paper/2302.05783/) (2023)
- [Multiscale Sensor Fusion and Continuous Control with Neural CDEs](https://mlsanity.com/paper/2203.08715/) (2022)
- [Fusion Dynamical Systems with Machine Learning in Imitation Learning: A Comprehensive Overview](https://mlsanity.com/paper/2403.19916/) (2024)
- [Discrete-time Contraction-based Control of Nonlinear Systems with Parametric Uncertainties using Neural Networks](https://mlsanity.com/paper/2105.05432/) (2021)
- [Learning to Act Safely with Limited Exposure and Almost Sure Certainty](https://mlsanity.com/paper/2105.08748/) (2021)
- [Adversarially Robust Stability Certificates can be Sample-Efficient](https://mlsanity.com/paper/2112.10690/) (2021)
- [Learning of Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2312.09734/) (2023)
- [Learning Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces and Random Features](https://mlsanity.com/paper/2404.07703/) (2024)
- [Stable Modular Control via Contraction Theory for Reinforcement Learning](https://mlsanity.com/paper/2311.03669/) (2023)
- [Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions](https://mlsanity.com/paper/2411.11405/) (2024)
- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [Ortho-ODE: Enhancing Robustness and of Neural ODEs against Adversarial Attacks](https://mlsanity.com/paper/2305.09179/) (2023)
- [Learning dissipative Hamiltonian dynamics with reproducing kernel Hilbert spaces and random Fourier features](https://mlsanity.com/paper/2410.18656/) (2024)
- [Contraction-Aware Reinforcement Learning for Nonlinear Control with Statistical Robustness](https://mlsanity.com/paper/2506.15700/) (2025)

## Abstract (arXiv)

We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key contribution is a control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, a constraint which guarantees the existence of robust tracking controllers for arbitrary open-loop trajectories generated with the learned system. Leveraging tools from contraction theory and statistical learning in Reproducing Kernel Hilbert Spaces, we formulate stabilizable dynamics learning as a functional optimization with convex objective and bi-convex functional constraints. Under a mild structural assumption and relaxation of the functional constraints to sampling-based constraints, we derive the optimal solution with a modified Representer theorem. Finally, we utilize random matrix feature approximations to reduce the dimensionality of the search parameters and formulate an iterative convex optimization algorithm that jointly fits the dynamics functions and searches for a certificate of stabilizability. We validate the proposed algorithm in simulation for a planar quadrotor, and on a quadrotor hardware testbed emulating planar dynamics. We verify, both in simulation and on hardware, significantly improved trajectory generation and tracking performance with the control-theoretic regularized model over models learned using traditional regression techniques, especially when learning from small supervised datasets. The results support the conjecture that the use of stabilizability constraints as a form of regularization can help prune the hypothesis space in a manner that is tailored to the downstream task of trajectory generation and feedback control, resulting in models that are not only dramatically better conditioned, but also data efficient.