  This paper builds the theoretical foundations for dynamic mode decomposition
(DMD) of control-affine dynamical systems by leveraging the theory of
vector-valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control
Liouville operators and control occupation kernels are introduced to separate
the drift dynamics from the input dynamics. A given feedback controller is
represented through a multiplication operator and a composition of the control
Liouville operator and the multiplication operator is used to express the
nonlinear closed-loop system as a linear total derivative operator on RKHSs. A
spectral decomposition of a finite-rank representation of the total derivative
operator yields a DMD of the closed-loop system. The DMD generates a model that
can be used to predict the trajectories of the closed-loop system. For a large
class of systems, the total derivative operator is shown to be compact provided
the domain and the range RKHSs are selected appropriately. The sequence of
models, resulting from increasing-rank finite-rank representations of the
compact total derivative operator, are shown to converge to the true system
dynamics, provided sufficiently rich data are available. Numerical experiments
are included to demonstrate the efficacy of the developed technique.
