偏微分网络系统的结构重构

Inverse problem of partial differential networks is a theoretically challenging topic and has various application background. It has been intensively discussed in the past ten years. This project committed to the structure reconstruction of the partial differential networks whose dynamic behavior is described by the one-dimensional wave equation. In this project, we try to reconstruct the shape of the networks through the observing spectral or scattering data. We intend to solve the following questions:.(1) By analysis of the corresponding characteristic equation of a series connected string network, network on a circuit, star-shaped network, we intend to obtain the structural properties of networks and give the corresponding characteristic equation of the abstract complex network;.(2) We will construct the corresponding Green function matrix of wave equation and develop the relationship between the characteristic equation and the Green function matrix;.(3) We analyze the algebraic and geometric properties on abstract networks in order to give general procedure for the reconstruction of the shape of networks. Furthermore, to get the graphical visualization.. The inverse problem of partial differential networks appears in the fields of communication, biology, medicine and etc. Meanwhile, its theoretical study is a good complement of graph theory and mechanics.

由于偏微分网络反问题在理论上有着鲜明的新颖性和挑战性, 在工程上有着广泛的应用背景，近十年来一直是国际上研究的热门课题。本项目主要研究偏微分网络系统的结构重构，它的动态行为由一维波方程来描述，研究目的是想通过观测到的谱信息或散射数据重构出网络的形状，即所谓的结构辨识。本项目拟解决下面三个问题：.（1）通过分析系列连接弦网络、环型网络、星型网络对应的特征方程来得到系统网络的结构性质，并试图给出抽象复杂网络对应的特征方程；.（2）构造波网络对应的Green函数矩阵，研究特征方程与Green函数矩阵之间的关系；.（3）研究抽象网络的代数性质与几何性质，给出重构网络形状的一般性步骤并实现图形可视化。. 偏微分网络反问题应用于通信、生物、医学等领域，它的理论研究是对图论及力学内容的很好补充。

偏微分网络系统广泛应用在工程、生物和医学领域。对偏微分网络反问题的研究主要集中在参数辨识与结构辨识。国内外对这类问题的研究很重视。我们针对网络的结构辨识进行了深入的研究。首先，我们给出了系列连接弦网络、环形网络、星型网络对应的特征方程，获得了系统网络的某些结构性质；对抽象网络的代数性质与几何性质进行了研究，在给出各边边界点的情况下，得到了重构网络形状的一般性步骤并实现了图形的可视化。为了更好的研究反问题，作者给出了偏微分耦合网络正问题的一些成果：研究了星形热弹性网络系统的稳定性及Riesz性质；通过频域方法得到了系列连接II型热弹性网络的指数稳定性；给出了单根热弹性系统超稳定性的一个充分条件。