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Generalizing Koopman Theory to Allow for Inputs and Control.

Joshua L Proctory^1,2, Steven L Bruntonz¹, J Nathan Kutzx²

¹Department of Mechanical Engineering and Department of Applied Mathematics, University of Washington, Seattle, WA 98195 (sbrunton@uw.edu).

SIAM Journal on Applied Dynamical Systems

|February 15, 2021

Summary

We present a new Koopman operator theory for nonlinear dynamical systems with control inputs. This advances dynamic mode decomposition (DMD) for input-output modeling and analyzing complex systems like disease spread.

Keywords:

37M10 37M99 37N10 37N25 37N35 65P99 DMD DMDc Koopman input-output spatio-temporal

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Area of Science:

Dynamical Systems Theory
Control Theory
Data-Driven Science

Background:

Koopman operator theory and dynamic mode decomposition (DMD) are powerful tools for analyzing nonlinear dynamical systems.
Standard Koopman analysis and DMD struggle with actuated systems, failing to produce input-output models and being corrupted by external forcing.
There is a need for theoretical frameworks that can handle nonlinear input-output characteristics in complex systems.

Purpose of the Study:

To develop a generalized Koopman operator theory that explicitly incorporates inputs and control for nonlinear dynamical systems.
To establish a rigorous connection between the new theoretical framework and dynamic mode decomposition with control (DMDC).
To demonstrate the applicability of the generalized theory on nonlinear systems, including epidemiological models with mass vaccination.

Main Methods:

Generalization of Koopman operator theory to include nonlinear input-output characteristics.
Theoretical development connecting the generalized Koopman theory to dynamic mode decomposition with control (DMDC).
Application and demonstration on nonlinear dynamical systems, including a susceptible-infectious-recovered (SIR) model with vaccination.

Main Results:

A novel generalization of Koopman operator theory capable of handling actuated nonlinear systems.
Demonstration of the rigorous link between the extended Koopman theory and DMDC.
Successful application to a SIR model, showcasing its utility for analyzing infectious disease dynamics with control interventions like mass vaccination.

Conclusions:

The developed generalized Koopman theory provides a robust framework for analyzing nonlinear dynamical systems with control inputs.
This advancement extends the capabilities of Koopman spectral analysis and DMD for data-driven modeling of complex, actuated systems.
The theory has significant implications for understanding and controlling phenomena in fields such as epidemiology and engineering.