Singular Perturbation Analysis of a Coupled System Involving the Wave Equation

Abstract

This article considers a system coupling an ordinary differential equation with a wave equation through its boundary data. The existence of a small parameter in the wave equation (as a factor multiplying the time derivative) suggests the idea of applying a singular perturbation method to get the stability of the full system by analyzing the stability of some appropriate subsystems given by the method. However, for infinite-dimensional systems, it is known that in some cases, this method does not work. Indeed, one cannot be sure of the stability of the full system even if the given subsystems are stable. In this article, we prove that the singular perturbation method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which is the first of this kind for systems involving the wave equation. Simulations are performed to show the applicability of our results.