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arXiv · 2009.11544

Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems

Yoshihiko Susuki·Alexandre Mauroy·Igor Mezic
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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.

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Yoshihiko Susuki, Alexandre Mauroy, Igor Mezic. 2020-09-24. Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems. https://doi.org/10.1137/20m1335935

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