The Koopman operator provides a way to
transform a (potentially) nonlinear finite-dimensional dynamical
system into an infinite-dimensional linear system
by lifting the nonlinear state dynamics into a functional
space of observables, where the dynamics are linear. Previous
literature has claimed that it is not possible to represent
nonlinear dynamics with multiple isolated critical points
if the set of observables is finite and contains the state;
more precisely, such a set cannot be invariant under the
Koopman operator. In this paper, we investigate this claim
in more detail and provide an analytical counterexample to
disprove it. We also consider the convergence of discretetime
Koopman approximation error to the continuous-time
error: we show both how this convergence occurs in general
and how it can fail for systems with multiple isolated
critical points. In particular, discontinuities in Koopman
observables at the boundaries of basins of attraction may
cause the continuous-time error to diverge; the discretetime
error also suffers from this as the sampling time step
goes to zero.
Revised: May 1, 2020 |
Published: December 11, 2019
Citation
Bakker C., K.E. Nowak, and W.S. Rosenthal. 2019.Learning Koopman Operators for Systems with Isolated Critical Points. In Proceedings of the IEEE 58th Conference on Decision and Control, December 11-13, 2019, Nice, France, 7733-7739. Piscataway, New Jersey:IEEE.PNNL-SA-141977.doi:10.1109/CDC40024.2019.9029818