Hilbert Spaces from Occupation Kernels and Learning in Nonlinear and Nonlocal Dynamical Systems
Abstract
Methods for data driven modeling of nonlinear fractional order and nonlocal dynamical systems are currently underdeveloped as compared to their integer order counterparts. This proposal aims to directly address learning and modeling problems related to these dynamical systems through the introduction of a new Hilbert space and associated nonlocal operators. The proposed work addresses three longstanding problems in the field; data representation for nonlinear fractional order dynamical systems, operator theoretic approaches for the data driven analysis of fractional order systems, and unbiased determination of the fractional order of a dynamical system. The particular approach adopted in this proposal is centered around Dynamic Mode Decomposition (DMD) analysis that has seen a tremendous amount of success in the integer order case. To adapt the operator theoretic methods of DMD to the fractional order case, the Koopman operator must be abandoned in light of generalizations of the Liouville operator. Moreover, special Hilbert spaces are constructed that are based on continuous functions on signals and are related to what are called occupation kernels, which previously enabled the convergence of DMD routines for continuous time integer order systems in the operator norm in combination with scaled Liouville operators. Successful implementation of the proposal will provide a fractional order model that may be expressed as a sum of dynamic modes and exponential or Mittag-Le?er functions. This follows heuristically by taking the view of casting a nonlinear dynamical system over a finite dimensional space as a linear dynamical system over an infinite dimensional space, which aligns with the perspective of DMD on integer order dynamical systems. The tools of this proposal will enable new perspectives on the optimal control of fractional order dynamical systems, and directly addresses new representations of Liouville operators for higher order integer dynamical systems.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Mar 07, 2023
- Source ID
- FA95502110134
Entities
People
- Joel A. Rosenfeld
Organizations
- Air Force Office of Scientific Research
- United States Air Force
- University of South Florida