论文标题
Koopman Resolvent：非线性自主动力学系统的拉普拉斯域分析
Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems
论文作者
论文摘要
我们研究的动机是建立一种拉普拉斯域理论，该理论提供了分析和合成非线性动力学系统的原理和方法。针对非线性自主动力学系统定义的组成操作员的半群（Koopman Semigroup及其相关的Koopman Generator）在这项研究中起着核心作用。我们介绍了Koopman Generator的分解，我们称之为Koopman Resolvent，并为三种非线性动力学提供了光谱表征：吸引子上的Ergodic Evolution，收敛到稳定的平衡点，并收敛到（Quasi-）稳定的极限周期。这表明Koopman分解提供了这种非线性自主动力学的拉普拉斯域表示。还讨论了Laplace域表示的计算方面，重点是非平稳的Koopman模式。
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.
