# From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma

arXiv: https://arxiv.org/abs/2508.07494 · HTML: https://mlsanity.com/paper/2508.07494/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 61 references that are in the map.
- First posted on arXiv 10 Aug 2025.

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

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics](https://mlsanity.com/paper/2010.09961/) (2020)
- [A trajectory-based framework for data-driven system analysis and control](https://mlsanity.com/paper/1903.10723/) (2019)
- [Koopman Form of Nonlinear Systems with Inputs](https://mlsanity.com/paper/2207.12132/) (2022)
- [Robust and Kernelized Data-Enabled Predictive Control for Nonlinear Systems](https://mlsanity.com/paper/2206.01866/) (2022)
- [Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories](https://mlsanity.com/paper/2209.09977/) (2022)
- [Learning Parametric Koopman Decompositions for Prediction and Control](https://mlsanity.com/paper/2310.01124/) (2023)
- [Nonlinear Data-Enabled Prediction and Control](https://mlsanity.com/paper/2101.03187/) (2021)
- [Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity](https://mlsanity.com/paper/2307.15368/) (2023)
- [Basis functions nonlinear data-enabled predictive control: Consistent and computationally efficient formulations](https://mlsanity.com/paper/2311.05360/) (2023)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Data-based Control of Feedback Linearizable Systems](https://mlsanity.com/paper/2204.01148/) (2022)
- [Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method](https://mlsanity.com/paper/2208.00175/) (2022)
- [Data-Driven Optimal Control of Bilinear Systems](https://mlsanity.com/paper/2112.15510/) (2021)
- [Exploring the Links between the Fundamental Lemma and Kernel Regression](https://mlsanity.com/paper/2403.05368/) (2024)
- [Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems](https://mlsanity.com/paper/2103.16171/) (2021)
- [Control-Coherent Koopman Modeling: A Physical Modeling Approach](https://mlsanity.com/paper/2403.16306/) (2024)
- [Neural Data-Enabled Predictive Control](https://mlsanity.com/paper/2406.08003/) (2024)
- [On the approximability of Koopman-based operator Lyapunov equations](https://mlsanity.com/paper/2302.05187/) (2023)
- [A universal reproducing kernel Hilbert space for learning nonlinear systems operators](https://mlsanity.com/paper/2412.18360/) (2024)

## Built on by (most cited first, 2 of 2)

- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [On the Generalisation of Koopman Representations for Chaotic System Control](https://mlsanity.com/paper/2508.18954/) (2025)

## Abstract (arXiv)

The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. In this paper we derive a novel solution to these problems based on basis functions expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We identify relaxed invariance conditions that guarantee existence of a bounded linear operator, i.e., the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. Compared to classical Koopman invariance conditions, measure preservation is not required. Moreover, we derive a nonlinear fundamental lemma by exploiting the constructed exact infinite-dimensional bilinear Koopman representation and Hankel operators. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator and in predictive control of a soft-robotic manipulator model.