On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements

Abstract

This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation $c\left(x\right){\partial }_{t}^{2}u-\Delta u=0$ in a bounded smooth domain in ${ℝ}^{d}$ from partial (on part of the boundary) dynamic boundary measurements. In this paper, we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset $\Gamma $ of the boundary determines explicitly the coefficient $c$ provided that $c$ is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient $c$ from the knowledge of the difference between the local Dirichlet-to-Neumann maps.