  Although the mathematical study on the synchronization of wave equations at
finite horizon has been well developed, there was few results on the
synchronization of wave equations for long-time horizon. The aim of the paper
is to investigate the uniform synchronization at the infinite horizon for one
abstract linear second order evolution system in a Hilbert space.
  First, using the classical compact perturbation theory on the uniform
stability of semigroups of contractions, we will establish a lower bound on the
number of damping, necessary for the uniform synchronization of the considered
system. Then, under the minimum number of damping, we clarify the algebraic
structure of the system as well as the necessity of the conditions of
compatibility on the coupling matrices. We then establish the uniform
synchronization by the compact perturbation method and then give the dynamics
of the asymptotic orbit. Various applications are given for the system of wave
equations with boundary feedback or (and) locally distributed feedback, and for
the system of Kirchhoff plate with distributed feedback. Some open questions
are raised at the end of the paper for future development.
  The study is based on the synchronization theory and the compact perturbation
of semigroups.
