Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem

Abstract

We consider a second-order hyperbolic equation on an open bounded domain Ω in R n for n ≥ 2 , with C 2 -boundary Γ = ∂ Ω = Γ 0 ∪ Γ 1 ¯ , Γ 0 ∩ Γ 1 = 0̸ , subject to non-homogeneous Neumann boundary conditions on the entire boundary Γ . We then study the inverse problem of determining both the interior damping and potential coefficients of the equation in one shot by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit sub-portion Γ 1 of the boundary Γ , and over a computable time interval T > 0 . Under sharp conditions on the complementary part Γ 0 = Γ ∖ Γ 1 , T > 0 , and under weak regularity requirements on the data, we establish the two canonical results of the inverse problem: (i) global uniqueness and (ii) stability. The latter (ii) is the main result of the paper. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the H 1 × L 2 -level for second-order hyperbolic equations (Lasiecka et al. (2000) [3]); (b) a correspondingly implied continuous observability inequality at the same energy level [3]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Neumann boundary data (Lasiecka and Triggiani (1990, 1991, 1994) [20,21,29] and Tataru (1998) [24]). The proof of the linear uniqueness result (Section 4, step 5) also takes advantage of a convenient tactical route “post-Carleman estimates” suggested by Isakov (2006) in [12, Thm. 8.2.2, p. 231] .