Inverse problem on a tree-shaped network

TL;DR

This paper establishes uniqueness and stability results for inverse coefficient problems on a tree-shaped network for wave, heat, and Schrödinger equations, using Carleman estimates and observability techniques.

Contribution

It provides the first known proof of uniqueness and stability for inverse problems on tree-shaped networks for multiple PDE types, employing Carleman estimates and observability methods.

Findings

Uniqueness of potential determination on network edges.

Stability estimates for the inverse wave problem.

Application of Carleman estimates to network inverse problems.

Abstract

In this article, we prove a uniqueness result for a coefficient inverse problems regarding a wave, a heat or a Schr\"odinger equation set on a tree-shaped network, as well as the corresponding stability result of the inverse problem for the wave equation. The objective is the determination of the potential on each edge of the network from the additional measurement of the solution at all but one external end-points. Our idea for proving the uniqueness is to use a traditional approach in coefficient inverse problem by Carleman estimate. Afterwards, using an observability estimate on the whole network, we apply a compactness-uniqueness argument and prove the stability for the wave inverse problem.