  Conventionally, data driven identification and control problems for higher
order dynamical systems are solved by augmenting the system state by the
derivatives of the output to formulate first order dynamical systems in higher
dimensions. However, solution of the augmented problem typically requires
knowledge of the full augmented state, which requires numerical differentiation
of the original output, frequently resulting in noisy signals. This manuscript
develops the theory necessary for a direct analysis of higher order dynamical
systems using higher order Liouville operators. Fundamental to this theoretical
development is the introduction of signal valued RKHSs. Ultimately, it is
observed that despite the added abstractions, the necessary computations are
remarkably similar to that of first order DMD methods using occupation kernels.
