Stability of a one-dimensional discrete-time asynchronous swarm

In this article 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, that we call the comfortable position (with comfortable intermember distance). Our stability analysis employs some results on contractive mappings from the parallel and distributed computation literature.