Stability of a one-dimensional discrete-time asynchronous swarm

In this correspondence, we consider a discrete time one-dimensional asynchronous swarm. First, we describe the mathematical model for motions of the swarm members. Then, we analyze the stability properties of that model. The stability concept that we consider, which matches exactly with stability of equilibria in control theory, characterizes stability of a particular position (relative arrangement) of the swarm members. We call that position the comfortable position (with comfortable intermember distances). Our swarm model and stability analysis are different from other asynchronous swarm models considered in the literature. In particular, in our analysis we employ results on contractive mappings from the parallel and distributed computation literature. The application of these results to the swarm coordination problem is important by itself since they might prove useful also in n-dimensional swarms.