Recover all coefficients in second-order hyperbolic equations from finite sets of boundary measurements *

Abstract

Abstract We consider the inverse hyperbolic problem of recovering all spatial dependent coefficients, which are the wave speed, the damping coefficient, potential coefficient and gradient coefficient, in a second-order hyperbolic equation defined on an open bounded domain with smooth enough boundary. We show that by appropriately selecting finite pairs of initial conditions we can uniquely and Lipschitz stably recover all those coefficients from the corresponding boundary measurements of their solutions. The proofs are based on Carleman estimate, continuous observability inequality and regularity theory for general second-order hyperbolic equations.