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

We consider a wave equation with damping coefficient where T ⩾ 2, the complex-valued functions p1, p2 ∊ C1[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.