  In this article, we study the internal stabilization and control of the
critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a
geometric assumption slightly stronger than the classical geometric control
condition, we prove exponential decay for some solutions bounded in the energy
space but small in a lower norm. The proof combines profile decomposition and
microlocal arguments. This profile decomposition, analogous to the one of
Bahouri-G\'erard on $\R^3$, is performed by taking care of possible geometric
effects. It uses some results of S. Ibrahim on the behavior of concentrating
waves on manifolds.
