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

Minghui Bi, Yixian Gao

TL;DR

This work addresses the problem of simultaneously recovering a spatially varying density $\rho(x)$ and the initial displacement $f$ in a damped biharmonic wave equation from boundary measurements of $\Delta u$ and $\partial_n(\Delta u)$. The authors establish well-posedness of the forward problem via a contraction semigroup generated by a suitably defined operator, derive a multiplier-based observability inequality under a geometric control condition, and use these tools to obtain explicit Lipschitz stability estimates for the inverse problem. The stability constants are shown to depend on the damping coefficient through a factor $ (1+\gamma)^{1/2} $, and the results provide the first explicit Lipschitz bounds for this coupled multi-parameter inverse problem in a higher-order damped wave setting. The findings have practical implications for non-destructive testing and dynamic inversion, where stable simultaneous identification of material parameters and initial states is essential.

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, \(Δu |_{\partial Ω}\) and \( \partial_{n}(Δu)|_{\partial Ω}.\) 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 \( (1 + γ)^{1/2} \). This work provides a rigorous theoretical basis for applications in non-destructive testing and dynamic inversion.

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

TL;DR

This work addresses the problem of simultaneously recovering a spatially varying density and the initial displacement in a damped biharmonic wave equation from boundary measurements of and . The authors establish well-posedness of the forward problem via a contraction semigroup generated by a suitably defined operator, derive a multiplier-based observability inequality under a geometric control condition, and use these tools to obtain explicit Lipschitz stability estimates for the inverse problem. The stability constants are shown to depend on the damping coefficient through a factor , and the results provide the first explicit Lipschitz bounds for this coupled multi-parameter inverse problem in a higher-order damped wave setting. The findings have practical implications for non-destructive testing and dynamic inversion, where stable simultaneous identification of material parameters and initial states is essential.

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, and \( \partial_{n}(Δu)|_{\partial Ω}.\) 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 \( (1 + γ)^{1/2} \). This work provides a rigorous theoretical basis for applications in non-destructive testing and dynamic inversion.
Paper Structure (6 sections, 9 theorems, 125 equations)

This paper contains 6 sections, 9 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Properties of the operator $\mathcal{A}$
  3. Well-posedness and Observability
  4. Proof of the Main Theorems
  5. Proof of Theorem \ref{['th1']}
  6. Proof of Theorem \ref{['th2']}

Key Result

Theorem 1.1

Let $u_{1}$ and $u_{2}$ be the solutions of system a corresponding to two admissible parameter sets $(\rho_{1},f_{1},g_{1})$ and $(\rho_{2},f_{2},g_{2})$, and denote by $u:=u_{1}-u_{2}$. Assume that the acceleration of the second solution satisfies for some constant $M>0$. Then there exists a constant $C=C(M,T,\rho_{\min},\rho_{\max},\Omega)>0$ such that

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Dissipativity
  • proof
  • Lemma 2.2: Maximality
  • proof
  • Lemma 2.3: Closedness
  • proof
  • Lemma 2.4: Denseness
  • proof
  • ...and 8 more