Representation of nonlinear dynamical systems as infinite-dimensional linear operators over a Hilbert space of functions provides a means to analyze nonlinear systems via spectral methods for linear operators. In this paper, we provide a novel representation for discrete, control-affine nonlinear dynamical systems as operators acting on a Hilbert space of functions. We also demonstrate that this representation can be used to predict the behavior of a discrete-time nonlinear system under a known feedback law; thereby extending the predictive capabilities of dynamic mode decomposition to discrete nonlinear systems affine in control. The method requires only snapshots from the dynamical system. We validate the method in two numerical experiments by predicting the response of a controlled Duffing oscillator to a known feedback law, as well as demonstrating the advantage of our method to existing techniques in the literature.
