  In this paper, we consider the exact boundary controllability and the exact
boundary synchronization (by groups) for a coupled system of wave equations
with coupled Robin boundary controls. Owing to the difficulty coming from the
lack of regularity of the solution, we confront a bigger challenge than that in
the case with Dirichlet or Neumann boundary controls. In order to overcome this
difficulty, we use the regularity results of solutions to the mixed problem
with Neumann boundary conditions by Lasiecka and Triggiani to get the
regularity of solutions to the mixed problem with coupled Robin boundary
conditions. Thus we show the exact boundary controllability of the system, and
by a method of compact perturbation, we obtain the non-exact boundary
controllability of the system with fewer boundary controls on some special
domains. Based on this, we further study the exact boundary synchronization (by
groups) for the same system, the determination of the exactly synchronizable
state (by groups), as well as the necessity of the compatibility conditions of
the coupling matrices.
