Graphon-based control has recently been proposed and developed to solve
control problems for dynamical systems on networks which are very large or
growing without bound (see Gao and Caines, CDC 2017, CDC 2018). In this paper,
spectral representations, eigenfunctions and approximations of graphons, and
their applications to graphon-based control are studied. First, spectral
properties of graphons are presented and then approximations based on Fourier
approximated eigenfunctions are analyzed. Within this framework, two classes of
graphons with simple spectral representations are given. Applications to
graphon-based control analysis are next presented; in particular, the
controllability of systems distributed over very large networks is expressed in
terms of the properties of the corresponding graphon dynamical systems.
Moreover, spectral analysis based upon real-world network data is presented,
which demonstrates that low-dimensional spectral approximations of networks are
possible. Finally, an initial, exploratory investigation of the utility of the
spectral analysis methodology in graphon systems control to study the control
of epidemic spread is presented.

通过谱分解、特征函数和逼近方式，探讨了基于图子的控制的特性，阐释了两类具有简单谱表示的图子，并简单探讨了谱分析方法在防疫传播方面的应用。