Control and Inverse Problems for Differential Equations on Graphs

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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.