Recover all Coefficients in Second-Order Hyperbolic Equations from Finite Sets of Boundary Measurements

TL;DR

This paper demonstrates that all coefficients in a second-order hyperbolic equation can be uniquely and stably recovered from finite boundary measurements using Carleman estimates and observability inequalities.

Contribution

It introduces a method to recover multiple coefficients in hyperbolic equations from finite boundary data with proven uniqueness and stability.

Findings

Unique and Lipschitz stable recovery of all coefficients.

Use of Carleman estimates for inverse problems.

Applicability to general second-order hyperbolic equations.

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 sharp Carleman estimate, continuous observability inequality and regularity theory for general second-order hyperbolic equations.