# Neural Koopman Lyapunov Control

arXiv: https://arxiv.org/abs/2201.05098 · HTML: https://mlsanity.com/paper/2201.05098/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 55 references that are in the map.
- First posted on arXiv 13 Jan 2022.

## Builds on (most cited first, 20 of 20)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [Actor-Critic Reinforcement Learning for Control with Stability Guarantee](https://mlsanity.com/paper/2004.14288/) (2020)
- [Learning Stability Certificates from Data](https://mlsanity.com/paper/2008.05952/) (2020)
- [Neural Certificates for Safe Control Policies](https://mlsanity.com/paper/2006.08465/) (2020)
- [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019)
- [Funnel Libraries for Real-Time Robust Feedback Motion Planning](https://mlsanity.com/paper/1601.04037/) (2016)
- [Learning Control Lyapunov Functions from Counterexamples and Demonstrations](https://mlsanity.com/paper/1804.05285/) (2018)
- [Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics](https://mlsanity.com/paper/2010.09961/) (2020)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems](https://mlsanity.com/paper/2010.05778/) (2020)
- [Power grid transient stabilization using Koopman model predictive control](https://mlsanity.com/paper/1803.10744/) (2018)
- [Feedback Stabilization Using Koopman Operator](https://mlsanity.com/paper/1810.00089/) (2018)
- [Parallel Learning of Koopman Eigenfunctions and Invariant Subspaces For Accurate Long-Term Prediction](https://mlsanity.com/paper/2005.06138/) (2020)
- [A convex data-driven approach for nonlinear control synthesis](https://mlsanity.com/paper/2006.15477/) (2020)
- [Koopman Operator Based Modeling for Quadrotor Control on SE(3)](https://mlsanity.com/paper/2103.03363/) (2021)
- [Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions](https://mlsanity.com/paper/2110.04967/) (2021)

## Built on by (most cited first, 11 of 11)

- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Leveraging KANs For Enhanced Deep Koopman Operator Discovery](https://mlsanity.com/paper/2406.02875/) (2024)
- [Computationally Efficient Data-Driven Discovery and Linear Representation of Nonlinear Systems For Control](https://mlsanity.com/paper/2309.04074/) (2023)
- [MPC-Inspired Reinforcement Learning for Verifiable Model-Free Control](https://mlsanity.com/paper/2312.05332/) (2023)
- [Deep Learning Based Dynamics Identification and Linearization of Orbital Problems using Koopman Theory](https://mlsanity.com/paper/2403.08965/) (2024)
- [Learning Deep Input-Output Stable Dynamics](https://mlsanity.com/paper/2206.13093/) (2022)
- [Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition](https://mlsanity.com/paper/2207.07719/) (2022)
- [FLEX: an Adaptive Exploration Algorithm for Nonlinear Systems](https://mlsanity.com/paper/2304.13426/) (2023)
- [DeepKoopFormer: A Koopman Enhanced Transformer Based Architecture for Time Series Forecasting](https://mlsanity.com/paper/2508.02616/) (2025)
- [KoopCast: Trajectory Forecasting via Koopman Operators](https://mlsanity.com/paper/2509.15513/) (2025)
- [Lyapunov Stability Learning with Nonlinear Control via Inductive Biases](https://mlsanity.com/paper/2511.01283/) (2025)

## Abstract (arXiv)

Learning and synthesizing stabilizing controllers for unknown nonlinear control systems is a challenging problem for real-world and industrial applications. Koopman operator theory allows one to analyze nonlinear systems through the lens of linear systems and nonlinear control systems through the lens of bilinear control systems. The key idea of these methods lies in the transformation of the coordinates of the nonlinear system into the Koopman observables, which are coordinates that allow the representation of the original system (control system) as a higher dimensional linear (bilinear control) system. However, for nonlinear control systems, the bilinear control model obtained by applying Koopman operator based learning methods is not necessarily stabilizable. Simultaneous identification of stabilizable lifted bilinear control systems as well as the associated Koopman observables is still an open problem. In this paper, we propose a framework to construct these stabilizable bilinear models and identify its associated observables from data by simultaneously learning a bilinear Koopman embedding for the underlying unknown control affine nonlinear system as well as a Control Lyapunov Function (CLF) for the Koopman based bilinear model using a learner and falsifier. Our proposed approach thereby provides provable guarantees of asymptotic stability for the Koopman based representation of the unknown control affine nonlinear control system as a bilinear system. Numerical simulations are provided to validate the efficacy of our proposed class of stabilizing feedback controllers for unknown control-affine nonlinear systems.