Localization of effective pinning control in complex networks of dynamical systems

This paper concerns pinning control in complex networks of dynamical systems, where an external forcing signal is applied to the network in order to align the state of all the systems to the forcing signal. By considering the control signal as the state of a virtual dynamical system, this problem can be studied in a synchronization framework. Prior studies have determined that a single controller can pin an entire network under certain conditions. This paper aims to further this study by looking at sufficient and necessary conditions for the possible locations where pinning control can be applied. We also study how the ease of control is influenced by the topology and in particular the algebraic connectivity of the network. In particular, we show that for systems with locally connected coupling it is harder to achieve pinning control than for systems with random or fully connected coupling.We also show that when pinning control is applied to a single system and the underlying topology of the network is a vertex-balanced graph, then the amount of control needed to effect pinning control grows at least as fast as the number of vertices. Furthermore, in order to achieve pinning control in systems coupled via locally connected graphs, as the number of systems grows, both the pinning control and the coupling among all systems need to increase.