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 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).
An illustration about one of the goals of Koopman operator theory with or without inputs. The first row shows that there might be an unknown system evolving according to some dynamical system. The second row shows that we can measure the system experimentally, as in the case of optical systems, or historically, as in the case of historical infectious disease data. The last row shows one of the goals of Koopman operator theory: to discover an operator that can propagate forward in time a set of measurements for prediction and control.