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

TL;DR

This paper establishes necessary and sufficient conditions for exponential synchronization of nonlinear systems, linking structured synchronizers to stabilizers of linearized systems and providing constructive methods via backstepping.

Contribution

It introduces a novel framework connecting synchronization control laws to stabilizers and tensor field conditions, with constructive backstepping approaches.

Findings

Conditions for local exponential synchronization are derived.

Structured synchronizers are shown to be equivalent to stabilizers of linearized systems.

Global exponential synchronization can be achieved in specific cases.

Abstract

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.