控制Liouville算子的动态模式分解
Dynamic Mode Decomposition with Control Liouville Operators
AI总结：
本文利用向量值再生核希尔伯特空间理论，通过控制Liouville算子与乘法算子复合，将控制仿射非线性闭环系统表示为线性全导数算子，进而实现动态模式分解，并证明在适当RKHS选择下算子紧性及模型收敛性，数值实验验证了其有效性。
摘要：
本文利用向量值再生核希尔伯特空间（RKHSs）的理论，为控制仿射动力系统的动态模式分解（DMD）奠定了理论基础。具体而言，引入了控制Liouville算子和控制占据核，以将漂移动态与输入动态分离。给定的反馈控制器通过乘法算子表示，并将控制Liouville算子与乘法算子的复合用于将非线性闭环系统表示为RKHSs上的线性全导数算子。对全导数算子的有限秩表示的谱分解产生了闭环系统的DMD。DMD生成的模型可用于预测闭环系统的轨迹。对于一大类系统，只要适当选择定义域和值域的RKHSs，全导数算子被证明是紧的。由紧全导数算子的增秩有限秩表示所产生的一系列模型，在数据足够丰富的情况下，被证明收敛于真实系统动态。文中包含数值实验以证明所开发技术的有效性。
英文摘要：
This paper builds the theoretical foundations for dynamic mode decomposition (DMD) of control-affine dynamical systems by leveraging the theory of vector-valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernels are introduced to separate the drift dynamics from the input dynamics. A given feedback controller is represented through a multiplication operator and a composition of the control Liouville operator and the multiplication operator is used to express the nonlinear closed-loop system as a linear total derivative operator on RKHSs. A spectral decomposition of a finite-rank representation of the total derivative operator yields a DMD of the closed-loop system. The DMD generates a model that can be used to predict the trajectories of the closed-loop system. For a large class of systems, the total derivative operator is shown to be compact provided the domain and the range RKHSs are selected appropriately. The sequence of models, resulting from increasing-rank finite-rank representations of the compact total derivative operator, are shown to converge to the true system dynamics, provided sufficiently rich data are available. Numerical experiments are included to demonstrate the efficacy of the developed technique.
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