Lipschitz Stability Estimate for an Initial Wave Reconstruction Problem of Telegraph Type With Gaussian Noise

This work is devoted to the study of an initial wave reconstruction problem of telegraph type. Our specific goal is to determine the initial spikes of an idealized signal in a telegraph wire from two consecutive observations contaminated by Gaussian noise. Our main result shows that the initial reconstruction is Lipschitz stable in the expectation operator. The proof relies upon the integral representation of solution for Cauchy data, together with the use of the truncated Fourier approximator. Additionally, we prove that the reconstructed initial data are exactly observable. Some numerical tests are given to validate our theoretical findings.