Koopman operator theory: fundamentals, control, and applications
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Abstract (arXiv)
The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.
Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 92 of its 198 references that are in the map. First posted on arXiv 2 Jul 2026.
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92 of 198 references.
References · 92
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- 2016Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive controlcited by 140
- 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 127
- 2020Kernel methods through the roof: handling billions of points efficientlycited by 89
- 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
- 2017On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operatorcited by 73
- 2022Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spacescited by 67
- 2019Uniform Error Bounds for Gaussian Process Regression with Application to Safe Controlcited by 64
- 2016Generalizing Koopman Theory to allow for inputs and controlcited by 53
- 2021Practical and Rigorous Uncertainty Bounds for Gaussian Process Regressioncited by 49
- 2021Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systemscited by 45
- 2021Koopman Operator Dynamical Models: Learning, Analysis and Controlcited by 42
- 2023Sharp Spectral Rates for Koopman Operator Learningcited by 36
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- 2023Koopman Kernel Regressioncited by 22
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- 2023Data-driven MPC with stability guarantees using extended dynamic mode decompositioncited by 7
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- 2025Koopman-Equivariant Gaussian Processescited by 6
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