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

S. A. Avdonin, G. Leugering, V. S. Mikhaylov

TL;DR

The paper addresses the inverse problem of recovering both the topology and physical properties of a planar tree-structured network of elastic strings from boundary measurements at leaves, with the root fixed. It employs the boundary-control method for wave equations on graphs, linking the dynamical response operator $R^T$ to the spectral Titchmarsh-Weyl matrix $ extbf{M}(oldsymbol{\\lambda})$, to achieve reconstruction. The authors provide constructive procedures for the two-edge case, star graphs, and a general iterative scheme for arbitrary trees, proving that diagonal TW components or diagonal response data suffices for recovery, including the branching angles. This work enables noninvasive identification of both geometry and material properties in tree-like networks, with potential applications in structural health monitoring and materials science involving hierarchical networks.

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.

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

TL;DR

The paper addresses the inverse problem of recovering both the topology and physical properties of a planar tree-structured network of elastic strings from boundary measurements at leaves, with the root fixed. It employs the boundary-control method for wave equations on graphs, linking the dynamical response operator to the spectral Titchmarsh-Weyl matrix , to achieve reconstruction. The authors provide constructive procedures for the two-edge case, star graphs, and a general iterative scheme for arbitrary trees, proving that diagonal TW components or diagonal response data suffices for recovery, including the branching angles. This work enables noninvasive identification of both geometry and material properties in tree-like networks, with potential applications in structural health monitoring and materials science involving hierarchical networks.

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.

Paper Structure

This paper contains 10 sections, 3 theorems, 88 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Forward dynamical and spectral problems for the two-velocity system on the tree.
  3. Spectral settings
  4. Dynamical settings.
  5. Inverse dynamical and spectral problems. Connection of the inverse data.
  6. Solution of the inverse problem. The case of two intervals.
  7. Solution of the inverse problem. The case of a star graph.
  8. Solution of the inverse problem. The case of an arbitrary tree.
  9. Conclusion and further work and open problems
  10. Acknowledgments

Key Result

Theorem 1

Let $\Omega$ be the tree consisted of two edges. Then the tree and the parameters of the systems (Eq1ch), (Eq2ch) and (SpEq1ch) are determined by the $2 \times 2$ matrix $M_{11}(\lambda)$ the diagonal element of the TW matrix, associated with the first boundary point.

Figures (2)

  • Figure 1: The tree
  • Figure 2: Representation of planar displacement

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Remark 3