# Demystifying Linear Operator Learning for Control Systems

arXiv: https://arxiv.org/abs/2609.19428 · HTML: https://mlsanity.com/paper/2609.19428/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 31 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Finite-data error bounds for Koopman-based prediction and control](https://mlsanity.com/paper/2108.07102/) (2021)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023)
- [An overview of Koopman-based control: From error bounds to closed-loop guarantees](https://mlsanity.com/paper/2509.02839/) (2025)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2406.09028/) (2024)
- [Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity](https://mlsanity.com/paper/2307.15368/) (2023)

## Abstract (arXiv)

This paper proposes a structured approach to learning linear operators for control systems from data. We address both structural and learning-theoretic aspects of the problem. To derive structural assumptions, we propose using the well-established framework of (semi)groups for evolution equations, as operators in control systems are of the same type. Further, we propose analyzing learning algorithms through the lens of the inverse problems framework. This reveals how a learned model depends on the data via error decompositions, convergence guarantees, and optimal regularization -- enabling us to compare existing methods and derive provably advantageous algorithms. In order to obtain these results, we restrict our scope to bounded operators on Hilbert spaces. Although this may appear restrictive, existing approaches often make this assumption implicitly to obtain matrix-like representations. We demonstrate the power of using these frameworks by deriving a convergent estimator for time-varying systems.