Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler-Bernoulli beam and heat equations

TL;DR

This paper establishes logarithmic stability estimates for simultaneously recovering potential and damping coefficients in dissipative wave equations from boundary measurements, extending the approach to beam and heat equations.

Contribution

It introduces a novel method combining observability and spectral decomposition for inverse coefficient problems, applicable to wave, beam, and heat equations.

Findings

Logarithmic stability estimates for inverse problems

Method applicable to wave, beam, and heat equations

Extension of stability results to different PDEs

Abstract

We are concerned with the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. We establish stability estimates of logarithmic type when the measurements are given by the operator who maps the initial condition to Neumann boundary trace of the solution of the corresponding initial-boundary value problem. We build a method combining an observability inequality together with a spectral decomposition. We also apply this method to a clamped Euler-Bernoulli beam equation. Finally, we indicate how the present approach can be adapted to a heat equation.