Lipschitz Stability for an Inverse Source Problem of the Wave Equation with Kinetic Boundary Conditions

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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 a 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), pp. 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.

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