Koopman Operator Theory for Nonautonomous and Stochastic Systems
Koopman Operator Theory for Nonautonomous and Stochastic Systems
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DOI：
10.1007/978-3-030-35713-9_6
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
S. Macesic;N. Črnjarić-Žic
中科院分区：
文献类型：
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作者：
S. Macesic;N. Črnjarić-Žic
在实践中，受时间相关或随机强迫影响的开放系统的动态性比自治系统的动态性更加存在。因此，将库普曼算子理论及其应用扩展到此类系统具有重要意义。同时，它为现有的非自治和随机动力系统理论带来了新的观点，特别是基于库普曼的数据驱动算法的应用。在本章中，我们首先基于两个标准的非自治动力系统定义（偏斜积和过程）回顾非自治库普曼算子族。然后，我们陈述算子的基本属性，并在两种定义的上下文中比较 DMD 和 Arnoldi 型算法的性能。对于随机动力系统（RDS），我们引入了相关的随机库普曼算子族。我们证明，当 RDS 是马尔可夫时，该族满足半群性质，并且我们给出了由随机微分方程生成的 RDS 的一些性质。最后，我们讨论随机框架中的数据驱动算法，并通过数值示例说明其性能。
In practice, the dynamics of open systems subject to time-dependent or random forcing is much more present than the dynamics of autonomous systems. Therefore, extension of the Koopman operator theory and applications to such systems is of great importance. At the same time, it brings a new viewpoint to the existing theory of nonautonomous as well as random dynamical systems, particularly, with application of Koopman-based data-driven algorithms. In this chapter, we first review the nonautonomous Koopman operator family based on the two standard nonautonomous dynamical system definitions: skew product and process. Then, we state basic properties of the operator and compare performance of the DMD and Arnoldi-type algorithms in the context of both definitions. In the case of the random dynamical systems (RDS), we introduce the associated stochastic Koopman operator family. We show that when RDS is Markovian, this family satisfies semigroup property and we present some properties for RDS generated by the stochastic differential equations. Finally, we discuss data-driven algorithms in the stochastic framework and illustrate their performance on numerical examples.