2021/03/05 by Suddhasattwa Das, Dimitrios Giannakis, Joanna Slawinska · 3 citations
Engineering · Physics and Astronomy · #Control Systems and Identification #Model Reduction and Neural Networks #Stability and Controllability of Differential Equations
paper · pdf · doi:10.1016/j.acha.2021.02.004
openalex created_date 2018/08/22 · openalex publication_date 2021/03/05 · openalex updated_date 2026/08/01
A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). The operator Wτ is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, Wτ generates a unitary evolution group etWτt∈R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of Wτ as τ→0+, as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rössler system, are presented.