Control of synchronization of a group of nodes in directed networks

In this contribution we propose a technique to control a set of arbitrary nodes in a directed network such that they follow a synchronous trajectory which is in general not shared by the other units of the network. Our problem is inspired by those natural or artificial systems where the proper operation of the network is associated to the presence of clusters of synchronous nodes. Although typical control problems aim at leading all the units of the network either towards a common solution or to trajectories differentiated according to clusters, here we control only a subset of network nodes independently from the remaining units. The method we propose is based on introducing a set of distributed controllers modifying the topology of the connections in order to generate outer symmetries in the nodes to be controlled. Existence and stability of the solution are proved in the paper and a numerical example is discussed.