On a hyperbolic coefficient inverse problem via partial dynamic boundary measurements

TL;DR

This paper demonstrates that the unknown coefficient in a hyperbolic PDE can be uniquely identified and explicitly reconstructed from partial boundary measurements, using geometrical control methods and difference of Dirichlet-to-Neumann maps.

Contribution

It provides a new uniqueness and reconstruction method for hyperbolic inverse problems with partial boundary data, extending previous results to more general settings.

Findings

Unique determination of the coefficient c from partial boundary data.

Explicit reconstruction formula for c using boundary measurements.

Application of geometrical control methods to inverse hyperbolic problems.

Abstract

This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation $c(x)\partial_{t}^{2}u - \Delta u = 0$ in a bounded smooth domain in $\R^{d}$ from partial (on part of the boundary) dynamic boundary measurements. In this paper we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset $\Gamma$ of the boundary determines explicitly the coefficient $c$ provided that $c$ is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient $c$ from the knowledge of the difference between the local Dirichlet to Neumann maps.