Downer and Perron branches in interconnection topologies for coordination and control of multi-agent networks

In this paper, we devote to the study of inter-connection topologies for the coordinated control of multi-agent networks. It turns out that downer and Perron branches contribute to the understanding of coordinated behavior of multiple agents from the view point of interconnection topology structures. In particular, we show that the uncontrollability of multi-agent systems is equivalent to the existence of a downer branch when the interconnection graph is a tree. For general interconnection graph, it is shown that the existence of a Perron branch leads to the uncontrollability of the system in most cases. In the latter case, two equivalent conditions are also given. When there are edge failures occurred in the graph, a result is presented to cope with the robustness of the controllability. In all the results, the selection of leaders is outlined.