Generalizing Koopman Theory to Allow for Inputs and Control

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 coherent, spatio-temporal modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture for the study of complex systems. For 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 to a recent development called dynamic mode decomposition with control. 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).