On an Inverse Problem for a One-Dimensional Two-Velocity Dynamical System

Abstract

The evolution of the dynamical system under consideration is governed by the wave equation ρu_tt − (γu_x)_x + Au_x + B_u = 0, x>0, t > 0, with the zero initial Cauchy data and Dirichlet boundary control at x = 0. Here, ρ, γ, A, B are smooth 2 × 2–matrix-valued functions of x; ρ = diag {ρ₁, ρ₂} and γ = diag {γ₁, γ₂} are matrices with positive entries; u = u(x, t) is a solution (an ℝ²-valued function). In applications, the system corresponds to one-dimensional models, in which there are two types of wave modes, which propagate with different velocities and interact with each other. The “input→state” correspondence is realized by the response operator R : u(0, t) _→ γ(0)u_x(0, t), t ≥ 0, which plays the role of inverse data. The representations for the coefficients A and B, which are used for their determination via the response operator, are derived. An example of two systems with the same response operator is given, where in the first system the wave modes do not interact, whereas in the second one the interaction does occur. Bibliography: 3 titles.