Cucker-smale flocking under rooted leadership with fixed and switching topologies

In recent years, a number of articles have focused on mathematical models for emergent phenomena, for instance, the flocking of birds or the schooling of fish. In 2007, Cucker and Smale proposed an ingenious model which captures many of the observed features of moving animals. Subsequently, Shen extended the result to hierarchically structured flocks. Motivated by these works, in this paper we study the discrete Cucker-Smale flocking under rooted leadership, which means that there exists an overall leader such that any other agent is led, directly or indirectly, by the leader. The feature of our proposal, departing from the existing models, is that both the assumption of symmetry and the partial ordering of a hierarchy are dropped. The rooted leadership topology is a necessary condition for the group to converge towards a single leader's fixed constant velocity. The rates of convergence are established for flocks with fixed and switching topologies. The results may reveal the applicability and advantage of cooperation, or exchange of information, inside the group.