# Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity

arXiv: https://arxiv.org/abs/2307.15368 · HTML: https://mlsanity.com/paper/2307.15368/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jul 2023.

## Built on by (most cited first, 5 of 5)

- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma](https://mlsanity.com/paper/2508.07494/) (2025)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors](https://mlsanity.com/paper/2604.01459/) (2026)
- [Demystifying Linear Operator Learning for Control Systems](https://mlsanity.com/paper/2609.19428/) (2026)

## Abstract (arXiv)

This paper introduces the Koopman Control Family (KCF), a mathematical framework for modeling general (not necessarily control-affine) discrete-time nonlinear control systems with the aim of providing a solid theoretical foundation for the use of Koopman-based methods in systems with inputs. We demonstrate that the concept of KCF captures the behavior of nonlinear control systems on a (potentially infinite-dimensional) function space. By employing a generalized notion of subspace invariance under the KCF, we establish a universal form for finite-dimensional models, which encompasses the commonly used linear, bilinear, and linear switched models as specific instances. In cases where the subspace is not invariant under the KCF, we propose a method for approximating models in general form and characterize the model's accuracy using the concept of invariance proximity. We end by discussing how the proposed framework naturally lends itself to data-driven modeling of control systems.