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This book provides a comprehensive study of wave equation stabilization under various damping mechanisms, including internal damping, Wentzell-type boundary damping, mixed configurations, and dynamic feedback controls. Its central focus is to examine how the presence and localization of dissipation affect long-term energy decay and the asymptotic stability of solutions. The work combines rigorous mathematical analysis with numerical simulations, applying tools such as energy methods, spectral and frequency-domain techniques, multiplier approaches, and Ingham-type inequalities to study decay rates ranging from polynomial to exponential stabilization. The theoretical results are reinforced by computational experiments that confirm the predicted behaviors. By analyzing the interplay between geometry, boundary conditions, and damping mechanisms, the book offers valuable insights into the control and stabilization of hyperbolic systems, making it relevant for graduate students, researchers, and professionals in applied mathematics, PDE analysis, control theory, and engineering fields involving vibration control and dissipative systems.