Application of the Boundary Control Method to the Dynamical Inverse Problem for the Goupillaud Model

We apply Belishev’s Boundary Control method to the dynamical inverse problem for the Goupillaud model—a one-dimensional discrete layered medium with equal two-way wave travel times—providing a rigorous operator setting for this problem. The inverse problem consists of determining the layer reflection coefficients from the system response to an impulsive excitation. Since the model is classically specified as an input–output system without an explicit state space, we construct its operator realization as a controlled system, which is necessary for the applicability of this method. To achieve a unified description, we introduce a boundary parameter ϵ∈[−1,1] that unifies the free (ϵ=1) and non-free (ϵ=0) surface cases, while accommodating intermediate impedance loads. Our central result is an explicit formula, valid for any ϵ, that expresses the connecting operator through the response operator. We show that this representation leads to the Krein approach to the inverse problem, whereas its factorized form yields the Marchenko approach. Following the Boundary Control method, the inverse problem equations are formulated as a special direct control problem, providing a unified derivation of the discrete Krein and Marchenko equations along with layer-by-layer reconstruction algorithms. This work links control theory and inverse problems for layered media, offering a novel operator interpretation of classical algorithms.