论文标题
koopman操作员的部分微分方程分析
Spectral Analysis of the Koopman Operator for Partial Differential Equations
论文作者
论文摘要
我们提供了Koopman操作员分析的概述，以描述稳定静止状态的场变量放松的一类偏微分方程。我们介绍了系统的Koopman本特征功能，并使用结合概念来发展Koopman运营商的光谱扩展。对于诸如扩散方程的线性系统，Koopman特征功能可以表示为字段变量的线性函数。惯性歧管的概念表明对应于Koopman eigenFunctionals的关节零级集，而等构物的概念定义为最慢的衰减Koopman eigenfunctunctions的水平集。线性扩散方程，非线性汉堡方程和非线性相扩散方程被分析为示例。
We provide an overview of the Koopman operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the notion of conjugacy to develop spectral expansion of the Koopman operator. For linear systems such as the diffusion equation, the Koopman eigenfunctionals can be expressed as linear functionals of the field variable. The notion of inertial manifolds is shown to correspond to joint zero level sets of Koopman eigenfunctionals, and the notion of isostables is defined as the level sets of the slowest decaying Koopman eigenfunctional. Linear diffusion equation, nonlinear Burgers equation, and nonlinear phase-diffusion equation are analyzed as examples.
