  Based on recent works on transverse exponential stability, we establish some
necessary and sufficient conditions for the existence of a (locally)
exponential synchronizing control law. We show that the existence of a
structured synchronizer is equivalent to the existence of a stabilizer for the
individual linearized systems (on the synchronization manifold) by a linear
state feedback. This, in turn, is also equivalent to the existence of a
symmetric covariant tensor field, which satisfies a Control Matrix Function
inequality. Based on this result, we provide the construction of such
synchronizer via backstepping approaches. In some particular cases, we show how
global exponential synchronization may be obtained.
