  We develop a new method which extends Dynamic Mode Decomposition (DMD) to
incorporate the effect of control to extract low-order models from
high-dimensional, complex systems. DMD finds spatial-temporal coherent modes,
connects local-linear analysis to nonlinear operator theory, and provides an
equation-free architecture which is compatible with compressive sensing. In
actuated systems, DMD is incapable of producing an input-output model;
moreover, the dynamics and the modes will be corrupted by external forcing. Our
new method, Dynamic Mode Decomposition with control (DMDc), capitalizes on all
of the advantages of DMD and provides the additional innovation of being able
to disambiguate between the underlying dynamics and the effects of actuation,
resulting in accurate input-output models. The method is data-driven in that it
does not require knowledge of the underlying governing equations, only
snapshots of state and actuation data from historical, experimental, or
black-box simulations. We demonstrate the method on high-dimensional dynamical
systems, including a model with relevance to the analysis of infectious disease
data with mass vaccination (actuation).
