Source Reconstruction by Partial Measurements for a Class of Hyperbolic Systems in Cascade

Abstract

We consider a system of two inhomogeneous wave equations coupled in cascade. The source terms are of the form σ ₁(t)f(x), and σ ₂(t)g(x), where the σ _i ’s are known functions whereas the sources f and g are unknown and have to be reconstructed. We investigate the reconstruction of these two space-dependent sources from a single boundary measurement of the second component of the state-vector. We prove identification and stability estimates for all sufficiently large times T under a smallness condition on the norm of (σ ₁ −σ ₂)′ in L ²([0, T]) in the class of coupling coefficients that keep a constant sign in the spatial domain. We give sharper conditions if one of the two kernels σ _i ’s is positive definite. Furthermore, we give examples of coupling coefficients that change sign within the domain for which identification fails. Our approach is based on suitable observability estimates for the corresponding free coupled system established in Alabau-Boussouira (Math Control Signals Syst 26:1–46, 2014; Math Control Relat Fields 5:1–30, 2015) and the approach based on control theory developed in Puel and Yamamoto (Inverse Probl 12:995–1002, 1996).