This study examines the stability properties of a dissipative wave equation through spectral theory, establishing a rigorous framework to analyze the system’s dynamic behavior and long-term response. The core theoretical contributions are twofold: first, we prove that the system operator generates a \(C_0-\) semigroup of contractions on the appropriate energy space; second, under a suitable geometric condition on the damping region, we establish exponential stability. This stability analysis is validated numerically through the finite differences method (FDM), with simulations quantifying the energy decay. We focus on the evolution of energy and its exponential decay over time, which are key indicators of the system’s dissipative nature. The numerical experiments and theoretical analysis conclusively demonstrate that our control strategy achieves exponential stabilization of the wave equation solution, while revealing fundamental characteristics of the system’s energy dissipation behavior.
