# Generalizing Koopman Theory to allow for inputs and control

arXiv: https://arxiv.org/abs/1602.07647 · HTML: https://mlsanity.com/paper/1602.07647/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2016.

## Built on by (most cited first, 30 of 53)

- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference](https://mlsanity.com/paper/2308.13819/) (2023)
- [Koopman-Assisted Reinforcement Learning](https://mlsanity.com/paper/2403.02290/) (2024)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [End-to-End Reinforcement Learning of Koopman Models for Economic Nonlinear Model Predictive Control](https://mlsanity.com/paper/2308.01674/) (2023)
- [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Stable Invariant Models via Koopman Spectra](https://mlsanity.com/paper/2207.07475/) (2022)
- [Online Estimation of the Koopman Operator Using Fourier Features](https://mlsanity.com/paper/2212.01503/) (2022)
- [Online Control-Informed Learning](https://mlsanity.com/paper/2410.03924/) (2024)
- [Analysis, forecasting and system identification of a floating offshore wind turbine using dynamic mode decomposition](https://mlsanity.com/paper/2411.07263/) (2024)
- [Self-Supervised Decomposition, Disentanglement and Prediction of Video Sequences while Interpreting Dynamics: A Koopman Perspective](https://mlsanity.com/paper/2110.00547/) (2021)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [Koopman Data-Driven Predictive Control with Robust Stability and Recursive Feasibility Guarantees](https://mlsanity.com/paper/2405.01292/) (2024)
- [Model-free system identification of surface ships in waves via Hankel dynamic mode decomposition with control](https://mlsanity.com/paper/2502.15782/) (2025)
- [Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine](https://mlsanity.com/paper/2505.10438/) (2025)
- [From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma](https://mlsanity.com/paper/2508.07494/) (2025)
- [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026)
- [A purely data-driven framework for prediction, optimization, and control of networked processes: application to networked SIS epidemic model](https://mlsanity.com/paper/2108.02005/) (2021)
- [Self-Reflective Terrain-Aware Robot Adaptation for Consistent Off-Road Ground Navigation](https://mlsanity.com/paper/2111.06742/) (2021)
- [Towards Data-driven LQR with Koopmanizing Flows](https://mlsanity.com/paper/2201.11640/) (2022)
- [Heterogeneous mixtures of dictionary functions to approximate subspace invariance in Koopman operators](https://mlsanity.com/paper/2206.13585/) (2022)
- [Optimal Control of Nonlinear Systems with Unknown Dynamics](https://mlsanity.com/paper/2305.15188/) (2023)
- [Learning Nonautonomous Systems via Dynamic Mode Decomposition](https://mlsanity.com/paper/2306.15618/) (2023)
- [Multi-Step Embed to Control: A Novel Deep Learning-based Approach for Surrogate Modelling in Reservoir Simulation](https://mlsanity.com/paper/2409.09920/) (2024)
- [Leveraging Reinforcement Learning and Koopman Theory for Enhanced Model Predictive Control Performance](https://mlsanity.com/paper/2505.08122/) (2025)
- [MAKO: Meta-Adaptive Koopman Operators for Learning-based Model Predictive Control of Parametrically Uncertain Nonlinear Systems](https://mlsanity.com/paper/2510.09042/) (2025)
- [LC-SAC: Lyapunov-Constrained Soft Actor-Critic via Koopman Operator Theory for Trajectory Tracking and Stabilization](https://mlsanity.com/paper/2602.04132/) (2026)

## Abstract (arXiv)

We develop a new generalization of Koopman operator theory that incorporates the effects of inputs and control. Koopman spectral analysis is a theoretical tool for the analysis of nonlinear dynamical systems. Moreover, Koopman is intimately connected to Dynamic Mode Decomposition (DMD), a method that discovers spatial-temporal coherent modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture allowing investigation of complex systems. In actuated systems, standard Koopman analysis and DMD are incapable of producing input-output models; moreover, the dynamics and the modes will be corrupted by external forcing. Our new theoretical developments extend Koopman operator theory to allow for systems with nonlinear input-output characteristics. We show how this generalization is rigorously connected and generalizes a recent development called Dynamic Mode Decomposition with control (DMDc). We demonstrate this new theory on nonlinear dynamical systems, including a standard Susceptible-Infectious-Recovered model with relevance to the analysis of infectious disease data with mass vaccination (actuation).