  This manuscript provides a theoretical foundation for the Dynamic Mode
Decomposition (DMD) of control affine dynamical systems through vector valued
reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville
operators and control occupation kernels are introduced to separate the drift
dynamics from the control effectiveness components. Given a known feedback
controller that is represented through a multiplication operator, a DMD
analysis may be performed on the composition of these operators to make
predictions concerning the system controlled by the feedback controller.
