  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).
