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Based on recent works on transverse exponential stability, we establish somenecessary and sufficient conditions for the existence of a (locally)exponential synchronizing control law. We show that the existence of astructured synchronizer is equivalent to the existence of a stabilizer for theindividual linearized systems (on the synchronization manifold) by a linearstate feedback. This, in turn, is also equivalent to the existence of asymmetric covariant tensor field, which satisfies a Control Matrix Functioninequality. Based on this result, we provide the construction of suchsynchronizer via backstepping approaches. In some particular cases, we show howglobal exponential synchronization may be obtained.

Some results on exponential synchronization of nonlinear systems (long version)