Lipschitz stability for an inverse source problem of the wave equation with kinetic boundary conditions

TL;DR

The paper develops a global Lipschitz stability result for an inverse source problem in the wave equation with mixed kinetic boundary conditions by deriving a sharp Carleman estimate using a novel weight function that avoids a previously relied-upon boundary identity. This Carleman framework enables a Lipschitz bound for simultaneously recovering the interior and boundary forcing terms from a single boundary flux measurement, given a time horizon $T$ exceeding a geometry-dependent threshold $T_*$ and under the condition $\delta>d$. The authors further show that the improved Carleman estimate yields a sharp boundary controllability result via standard duality arguments. Compared with prior work, this approach removes the need for cut-offs and strengthens stability and controllability results for dynamic boundary conditions, while also outlining key open problems in extending Carleman estimates to broader dynamic-boundary settings.

Abstract

In this paper, we present a refined approach to establish a global Lipschitz stability for an inverse source problem concerning the determination of forcing terms in the wave equation with mixed boundary conditions. It consists of boundary conditions incorporating a dynamic boundary condition and Dirichlet boundary condition on disjoint subsets of the boundary. The primary contribution of this article is the rigorous derivation of a sharp Carleman estimate for the wave system with a dynamic boundary condition. In particular, our findings complete and drastically improve the earlier results established by Gal and Tebou [SIAM J. Control Optim., 55 (2017), 324-364]. This is achieved by using a different weight function to overcome some relevant difficulties. As for the stability proof, we extend to dynamic boundary conditions a recent argument avoiding cut-off functions. Finally, we also show that our developed Carleman estimate yields a sharp boundary controllability result.

Theorems & Definitions (14)

• Remark 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Theorem 3.1
• Remark 3.2
• Theorem 4.1
• Remark 4.2
• proof : Proof of Theorem \ref{['thm:stab']}
• Theorem 5.1