In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discrete-time model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its \neighbors." In their paper, Vicsek et. al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.

In a recent paper [10], we provided a formal analysis for a distributed coordination strategy proposed in [17] for coordination of a set of agents moving in the plane with the same speed but variable heading direction. Each agents heading is updated as the average of its heading and a set of its nearest neighbors. As the agents move, the graph induced by the nearest neighbor relationship changes, resulting in switching. We recently demonstrated that by modelling the system as a discrete linear inclusion (in a discrete time setting) and a switched linear system (in continuous time setting), conditions for convergence of all headings to the same value can be provided. In this paper, we extend these results and demonstrate that in order for convergence to happen switching has to stop in a finite time. Moreover, we will show that a necessary and su#cient condition for convergence is that the switching stops on a connected graph. We also provide connections between this problem and Left convergent product (LCP) sets of matrices.