Stable recovery of a time dependent matrix potential for wave equation from arbitrary measurements

Abstract The identification of time-dependent parameters in partial differential equations is a fundamental challenge in inverse problems, with wide-ranging implications in fields such as quantum mechanics, wave propagation, and material science. In this paper, we investigate the inverse problem of stably recovering a time-dependent matrix-valued potential in the context of the wave equation. We consider a cylindrical domain with measurements taken on an arbitrary subset of the lateral surface. The matrix potential plays a crucial role in the regulation of wave dynamics and energy propagation, making its accurate reconstruction essential to understand complex physical phenomena. We establish a logarithmic stability estimate for determining this matrix potential from the partial Dirichlet-to-Neumann map. Our results contribute to the theoretical understanding of inverse problems and provide valuable insight into applications such as inverse scattering and time-dependent quantum field theory.