Inverse Problem for a Structural Acoustic Interaction

TL;DR

This paper addresses an inverse problem in structural acoustics, establishing uniqueness and stability for source identification in a coupled PDE system using boundary measurements, Carleman estimates, and operator theory.

Contribution

It provides the first stability and uniqueness results for source identification in a coupled acoustic-structure PDE model with boundary data.

Findings

Proved uniqueness of the source term from boundary measurements.

Derived stability estimates using Carleman inequalities.

Established regularity conditions for initial data.

Abstract

In this work, we consider an inverse problem of determining a source term for a structural acoustic partial differentia equation (PDE) model, comprised of a two or three-dimensional interior acoustic wave equation coupled to a Kirchoff plate equation, with the coupling being accomplished across a boundary interface. For this PDE system, we obtain the uniqueness and stability estimate for the source term from a single measurement of boundary values of the "structure". The proof of uniqueness is based on Carleman estimate. Then, by means of an observability inequality and a compactness/uniqueness argument, we can get the stability result. Finally, an operator theoretic approach gives us the regularity needed for the initial conditions in order to get the desired stability estimate.