On the inverse problem of the two-velocity tree-like graph
S. A. Avdonin, A. Choque Rivero, G. Leugering, V. S. Mikhaylov
TL;DR
The paper studies inverse problems for planar tree-like networks of elastic strings governed by a two-velocity wave system. It shows that the graph topology (edge lengths, degrees, and inter-edge angles) and spatially varying densities can be uniquely recovered from boundary measurements, using the Titchmarsh-Weyl matrix $\mathbf{M}(\lambda)$ or the dynamical response operator $\mathbf{R}(t)$. The authors develop a leaf-peeling strategy coupled with boundary control methods to progressively identify edge properties, first on star graphs and then on arbitrary trees, via reduced data and a sequence of subgraph recoveries. The framework unifies spectral and dynamical data and yields a stable algorithm with explicit controllability times, offering a pathway for practical reconstruction in networks of planar elastic strings.
Abstract
In this article the authors continue the discussion in \cite{ALM} about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincar{é} operator for the dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree.