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Generalizing Koopman Theory to allow for inputs and control

arXiv 1602.07647 ↗First posted 24 Feb 2016math.OCcited by 53

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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).

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

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 3; 2022: 6; 2023: 7; 2024: 11; 2025: 12; 2026: 14.

53 papers.

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  1. 2023Resurrecting Recurrent Neural Networks for Long Sequencescited by 408
  2. 2023Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inferencecited by 11
  3. 2024Koopman-Assisted Reinforcement Learningcited by 11
  4. 2023PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operatorcited by 10
  5. 2024Nonparametric Control Koopman Operatorscited by 7
  6. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  7. 2023End-to-End Reinforcement Learning of Koopman Models for Economic Nonlinear Model Predictive Controlcited by 6
  8. 2025InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamicscited by 6
  9. 2024SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systemscited by 5
  10. 2022Stable Invariant Models via Koopman Spectracited by 4
  11. 2022Online Estimation of the Koopman Operator Using Fourier Featurescited by 4
  12. 2024Online Control-Informed Learningcited by 4
38 more
  1. 2024Analysis, forecasting and system identification of a floating offshore wind turbine using dynamic mode decompositioncited by 3
  2. 2021Self-Supervised Decomposition, Disentanglement and Prediction of Video Sequences while Interpreting Dynamics: A Koopman Perspectivecited by 2
  3. 2024Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om methodcited by 2
  4. 2024Koopman Data-Driven Predictive Control with Robust Stability and Recursive Feasibility Guaranteescited by 2
  5. 2025Model-free system identification of surface ships in waves via Hankel dynamic mode decomposition with controlcited by 2
  6. 2025Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Enginecited by 2
  7. 2025From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemmacited by 2
  8. 2026Learning effective models from network dynamics data with multiple initial conditions using weak form SINDycited by 2
  9. 2021A purely data-driven framework for prediction, optimization, and control of networked processes: application to networked SIS epidemic modelcited by 1
  10. 2021Self-Reflective Terrain-Aware Robot Adaptation for Consistent Off-Road Ground Navigationcited by 1
  11. 2022Towards Data-driven LQR with Koopmanizing Flowscited by 1
  12. 2022Heterogeneous mixtures of dictionary functions to approximate subspace invariance in Koopman operatorscited by 1
  13. 2023Optimal Control of Nonlinear Systems with Unknown Dynamicscited by 1
  14. 2023Learning Nonautonomous Systems via Dynamic Mode Decompositioncited by 1
  15. 2024Multi-Step Embed to Control: A Novel Deep Learning-based Approach for Surrogate Modelling in Reservoir Simulationcited by 1
  16. 2025Leveraging Reinforcement Learning and Koopman Theory for Enhanced Model Predictive Control Performancecited by 1
  17. 2025MAKO: Meta-Adaptive Koopman Operators for Learning-based Model Predictive Control of Parametrically Uncertain Nonlinear Systemscited by 1
  18. 2026LC-SAC: Lyapunov-Constrained Soft Actor-Critic via Koopman Operator Theory for Trajectory Tracking and Stabilizationcited by 1
  19. 2026Koopman operator theory: fundamentals, control, and applicationscited by 1
  20. 2022Learning Invariant Subspaces of Koopman Operators--Part 1: A Methodology for Demonstrating a Dictionary's Approximate Subspace Invariancecited by 0
  21. 2022Learning Invariant Subspaces of Koopman Operators--Part 2: Heterogeneous Dictionary Mixing to Approximate Subspace Invariancecited by 0
  22. 2023Koopman Learning with Episodic Memorycited by 0
  23. 2024Cell reprogramming design by transfer learning of functional transcriptional networkscited by 0
  24. 2024Semantic Communication and Control Co-Design for Multi-Objective Distinct Dynamicscited by 0
  25. 2025Mitigating Traffic Oscillations in Mixed Traffic Flow with Scalable Deep Koopman Predictive Controlcited by 0
  26. 2025Deep Neural Koopman Operator-based Economic Model Predictive Control of Shipboard Carbon Capture Systemcited by 0
  27. 2025Deep Bilinear Koopman Model for Real-Time Vehicle Control in Frenet Framecited by 0
  28. 2025System Identification of a Moored ASV with Recessed Moon Pool via Deterministic and Bayesian Hankel-DMDccited by 0
  29. 2025Spatially Aware Dictionary-Free Koopman Eigenfunction Identification for Modeling and Controlcited by 0
  30. 2025Physics-Informed Neural Koopman Machine for Interpretable Longitudinal Personalized Alzheimer's Disease Forecastingcited by 0
  31. 2026Generalization Analysis and Method for Domain Generalization for a Family of Recurrent Neural Networkscited by 0
  32. 2026Reduced-Order Surrogates for Forced Flexible Mesh Coastal-Ocean Modelscited by 0
  33. 2026Interpreting Reinforcement Learning Model Behavior via Koopman with Controlcited by 0
  34. 2026Koopman-Based Nonlinear Identification and Model Predictive Control of a Turbofan Enginecited by 0
  35. 2026Bilinear Input Modulation for Mamba: Koopman Bilinear Forms for Memory Retention and Multiplicative Computationcited by 0
  36. 2026Efficient Reinforcement Learning using Linear Koopman Dynamics for Nonlinear Robotic Systemscited by 0
  37. 2026Reinforcement learning in linear embedding space unlocks generalizable control across soft robot configurationscited by 0
  38. 2026Learning the Koopman Operator using Attention Free Transformerscited by 0

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