Distributed Control of Synchronization of a Group of Network Nodes

Synchronization in networked systems of diffusively coupled oscillators is a ubiquitous, well-studied phenomenon, and many techniques have been proposed to control all the network nodes toward a common reference trajectory. In this technical note, instead, we study the problem of synchronizing an arbitrary subset of the nodes of a network at a fixed coupling strength, disregarding the state of the remaining ones. This problem is inspired to what observed in many biological systems, where the functionality of the network is associated to the presence of clusters of synchronous nodes and, on the contrary, is lost when all the nodes synchronize. We propose a distributed control that introduces further interactions between the oscillators guaranteeing the onset and the stability of the desired synchronous manifold. The controllers are designed so that a proper set of symmetries is created in the network. In addition, we show that the stability of the synchronization pattern mainly depends on the degree of the nodes to synchronize and can be, therefore, controlled by increasing it. Finally, we provide numerical examples illustrating our results.