Lipschitz Stability for an Inverse Problem of Biharmonic Wave Equations with Damping

TL;DR

This paper proves Lipschitz stability for recovering a variable density and initial displacement in a damped biharmonic wave equation using boundary data, with implications for non-destructive testing.

Contribution

It establishes the first Lipschitz stability estimates for this inverse problem, highlighting the stabilizing effect of the biharmonic structure and damping.

Findings

Generated a contraction semigroup ensuring well-posedness.

Derived an observability inequality via multiplier techniques.

Obtained explicit stability estimates showing dependence on damping.

Abstract

This paper establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient and the initial displacement in a damped biharmonic wave equation. The data consist of the boundary Cauchy data for the Laplacian of the solution, \(\Delta u |_{\partial \Omega}\) and \( \partial_{n}(\Delta u)|_{\partial \Omega}.\) We first prove that the associated system operator generates a contraction semigroup, which ensures the well-posedness of the forward problem. A key observability inequality is then derived via multiplier techniques. Building on this foundation, explicit stability estimates for the inverse problem are obtained. These estimates demonstrate that the biharmonic structure inherently enhances the stability of parameter identification, with the stability constants exhibiting an explicit dependence on the damping coefficient via the factor \( (1 + \gamma)^{1/2} \). This work provides a rigorous theoretical basis for applications in non-destructive testing and dynamic inversion.