Exponential stabilization of perturbed wave equations with localized internal damping: Theory and numerical validation

Abstract

This work investigates the exponential stability of a perturbed non-linear (or linear) wave equation in a sub-critical regime with respect to both the damping and perturbation terms. Using an energy-based analysis, we first establish the decay of the total energy over a finite time interval T > 0. When the damping region satisfies the geometric control condition (GCC), and the perturbations contribute to the dissipation mechanism, we prove that the energy of the system decays exponentially by combining the multiplier method with a unique continuation principle. In addition to the theoretical analysis, we present numerical simulations for one- and two-dimensional damped wave equations with localized internal damping. Finite difference schemes are employed to approximate the dynamics and quantify the dissipation of energy, and the results confirm the theoretical predictions: Introducing a localized damping region induces strong stabilization, and the total discrete energy decays exponentially over time.