Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition

TL;DR

This paper introduces a new Hilbert space framework called occupation kernel Hilbert space (OKHS) for analyzing fractional order dynamical systems using spectral methods and a novel fractional order DMD routine.

Contribution

It develops the theoretical foundation of OKHS, enabling the application of spectral decomposition and DMD to fractional order operators within a unified Hilbert space setting.

Findings

Defined a new OKHS framework for fractional operators

Presented a fractional order DMD algorithm with finite rank representations

Demonstrated computational similarity to existing occupation kernel DMD methods

Abstract

This manuscript gives a theoretical framework for a new Hilbert space of functions, the so called occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex numbers. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD routine is presented, and the details of the finite rank representations are given. Significantly, despite the added theoretical content through the OKHS formulation, the resultant computations only differ slightly from that of occupation kernel DMD methods for integer order systems posed over RKHSs.