Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems
Abstract
The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems -- the Koopman semigroup and its associated Koopman generator -- plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on non-stationary Koopman modes.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- arXiv:
- arXiv:2009.11544
- Bibcode:
- 2020arXiv200911544S
- Keywords:
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- Mathematics - Dynamical Systems;
- Electrical Engineering and Systems Science - Systems and Control;
- Mathematics - Optimization and Control
- E-Print:
- 24 pages, 2 figures