Analysis of coordination in multi-agent systems through Partial difference Equations. Part I: The Laplacian control

In this first part of a two-parts paper we introduce the framework of Partial difference Equations (PdEs) over graphs for analyzing the behavior of multi-agent systems equipped with decentralized control schemes. We generalize the Vicsek's model by introducing errors in the agent dynamics and analyze agent alignment in leaderless and leader-follower models through the joint use of PdEs and automatic control tools. Moreover, we show that the resulting PdEs enjoy properties that are similar to those of well-known Partial Differential Equations (PDEs) like the heat equation, thus allowing to exploit physical-based reasoning for conjecturing properties of the collective dynamics