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Conventionally, data driven identification and control problems for higherorder dynamical systems are solved by augmenting the system state by thederivatives of the output to formulate first order dynamical systems in higherdimensions. However, solution of the augmented problem typically requiresknowledge of the full augmented state, which requires numerical differentiationof the original output, frequently resulting in noisy signals. This manuscriptdevelops the theory necessary for a direct analysis of higher order dynamicalsystems using higher order Liouville operators. Fundamental to this theoreticaldevelopment is the introduction of signal valued RKHSs and new operators posedover these spaces. Ultimately, it is observed that despite the addedabstractions, the necessary computations are remarkably similar to that offirst order DMD methods using occupation kernels.

Theoretical Foundations for the Dynamic Mode Decomposition of High Order Dynamical Systems