Control and Inverse Problems for Differential Equations on Graphs

图上微分方程的控制与反问题

• 批准号: 1411564
• 负责人: Sergei Avdonin
• 金额: $ 14.65万
• 依托单位: University of Alaska Fairbanks Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411564&HistoricalAwards=false
• 关键词: Control; Inverse; Problems; Differential; Equations

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.

这个项目关注的是图上微分方程的控制和反问题。网络结构在许多科学和工程问题中起着基础性的作用。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。该模型描述了桥梁、空间结构、天线、输电线路杆、钢格栅钢筋和其他典型的土木工程对象。最近，小得多的规模上的应用已经成为焦点。特别是，分层材料，如陶瓷或金属泡沫，渗滤网络和碳和石墨烯纳米管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、树枝状树）、生物物理学、环境科学、死亡控制中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。量子图的控制与逆理论是近年来迅速发展的应用数学领域--图的分析的一个重要组成部分。它们对于所有上述应用都非常重要。最近PI和他的合作者开发了一种新的有效的剥叶方法，用于求解树（无圈图）上微分方程的反问题。该项目将把这种方法扩展到一般的图，包括树和带圈的图。 我们将专注于微分方程的反问题，这在科学和工程应用中很重要。沿着边上的方程的未知系数，将恢复图的拓扑和几何参数（边的长度，以及在相关情况下，相邻边之间的角度）。相应的动力系统的精确可控性的图与周期也将建立。剥叶法是基于数学物理反问题的边界控制法。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。这个属性使剥叶方法优于其他方法，并允许我们的方法扩展到图的周期。