On an inverse problem for tree-like networks of elastic strings

TL;DR

This paper addresses an inverse problem for elastic strings arranged in a tree-like network, aiming to recover physical properties and topology using boundary measurements and wave equation control methods.

Contribution

It introduces a method to simultaneously recover the lengths and topology of a tree network of elastic strings from boundary data using the boundary control method.

Findings

Successful recovery of string lengths and topology under generic conditions

Application of boundary control method to complex network structures

Recovery achieved with measurements at all leaves of the tree

Abstract

We consider the in-plane motion of elastic strings on tree-like network, observed from the 'leaves'. We investigate the inverse problem of recovering not only the physical properties i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. To this end use the boundary control method for wave equations established in~\cite{AK,B}. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed.