The Gel’fand-Levitan theory for one-dimensional hyperbolic systems with impulsive inputs

We consider a wave equation with damping coefficient ∂ 2 u ∂t 2 (x, t) = ∂ 2 u ∂x 2 (x, t) + p1(x) ∂u ∂t (x, t) + p2(x) ∂u ∂x (x, t), 0 < x < 1, −T < t < T , u(x, 0) = 0, ∂u ∂t (x, 0) = δ(x), 0 ≤ x ≤ 1, ∂u ∂x (0, t) = ∂u ∂x (1, t) = 0, −T ≤ t ≤ T where T ≥ 2, the complex-valued functions p1, p2 ∈ C 1 [0, 1] and δ(x) is the Dirac delta function. We discuss an inverse problem of determining simultaneously the coefficients p1(x) and p2(x), 0 ≤ x ≤ 1 from observation data u(0, t), −T ≤ t ≤ T. We prove a reconstruction formula for p1(x) and p2(x) from u(0, t) by establishing an intrinsic relation with the inverse spectral theory.