Generalizing Koopman Theory to allow for inputs and control

TL;DR

This paper extends Koopman operator theory to include inputs and control, enabling better analysis of nonlinear systems with external influences, and demonstrates its application on epidemiological models.

Contribution

The authors develop a rigorous generalization of Koopman theory that incorporates control inputs, extending DMDc to nonlinear systems with input-output characteristics.

Findings

Successfully applied to infectious disease models with vaccination effects.

Provides a new framework for input-output analysis in nonlinear dynamical systems.

Enhances the capability of Koopman-based methods to handle controlled systems.

Abstract

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).