Control and Inverse Problems for Differential Equations on Graphs

本课题涉及图上微分方程的控制和逆问题。类网络结构在许多科学和工程问题中起着重要作用。这里的经典问题来自于由弦、梁、索和支柱组成的柔性结构的振动。这些模型描述了桥梁、空间结构、天线、输电线、钢网加固和其他土木工程的典型对象。最近，规模小得多的应用成为关注的焦点。特别是陶瓷或金属泡沫、渗透网络、碳和石墨烯纳米管等分层材料引起了人们的广泛关注。量子图作为化学（共轭分子的自由电子理论）、生物学（遗传网络、树突状树）、地球物理学、环境科学、疾病控制等各种现象的自然模型而出现，甚至与互联网（互联网或网络断层扫描）有关。量子图的控制和逆理论是应用数学中迅速发展的图分析领域的重要组成部分。它们对于上述所有应用程序都非常重要。提出的研究的主要目标是为这些理论开发新的方法和途径。最近，PI和他的合作者开发了一种新的有效的叶子剥离方法，用于解决树上微分方程的逆问题（无循环的图）。这个项目将把这个方法扩展到一般的图，包括树和带循环的图。我们将集中讨论微分方程的逆问题，这在科学和工程应用中是很重要的。随着边上方程的未知系数，图的拓扑结构和几何参数（边的长度，在相关情况下，相邻边之间的角度）将被恢复。相应的动力系统在带循环的图上的精确可控性也将被建立。剥叶法是在数学物理反问题边界控制方法的基础上提出的。这些方法的特点是它们的局部性：恢复子图的拓扑和其他参数只需要与该子图相关的数据。这个性质使得叶子剥离法比其他方法更有优势，并且允许将我们的方法扩展到带循环的图。

This project concerns control and inverse problems for differential equations on graphs. Network-like structures play a fundamental role in many problems of science and engineering. The classical problem here comes from oscillations of flexible structures made of strings, beams, cables, and struts. The models describe bridges, space-structures, antennas, transmission-line posts, steel-grid reinforcements and other typical objects of civil engineering. More recently, applications on a much smaller scale have come into focus. In particular, hierarchical materials like ceramic or metallic foams, percolation networks and carbon and grapheme nano-tubes have attracted much attention. Quantum graphs arise as natural models of various phenomena in chemistry (free-electron theory of conjugated molecules), biology (genetic networks, dendritic trees), geophysics, environmental science, decease control, and are even relevant in connection with the Internet (Internet or network tomography). Control and inverse theories for quantum graphs constitute an important part of the rapidly developing area of applied mathematics --- analysis on graphs. They are tremendously important for all aforementioned applications. The main goal of the proposed research is to develop new methods and approaches to these theories.Recently a new effective leaf-peeling method has been developed by the PI and his collaborators for solving inverse problems for differential equations on trees (graphs without cycles). The project will extend this method to general graphs which include trees and graphs with cycles. We will focus on inverse problems for differential equations that are important for applications in science and engineering. Along with unknown coefficients of the equations on the edges, the topology of the graph and geometrical parameters (the lengths of the edges and, in relevant cases, the angles between neighboring edges) will be recovered. Exact controllability of the corresponding dynamical systems on graphs with cycles will also be established. The leaf-peeling method is based on the Boundary Control method for inverse problems of mathematical physics. The characteristic feature of these methods is their locality: recovering the topology and other parameters of a subgraph requires only the data related to that subgraph. This property gives the leaf-peeling method an advantage over other methods and allows to extend our approach to graphs with cycles.