Gradient flows for organizing multi-agent system

In this paper, we consider a class of gradient flows that model rules by which a multi-agent system might approach to an equilibrium. The rules are quite simple to state, in fact they depend on a single attraction/repulsion function, but in the generality assumed here the analysis of the resulting flow presents several challenges. In part, these challenges arise from the natural invariance with respect to the Euclidean group of an equilibrium state, implying that it is just the shape of the configuration and not the Euclidean coordinates of the individual agents that matters. We establish, among other things, a metric property of the gradient flow and give conditions under which the paths of the individual agents remain bounded as the flow evolves. We give a parametrized definition of clustering which induces a partial order that reflects the granularity of the clustering and establish important properties of the lattice defined in this way. We also explain significant properties of the clusters related to the attraction/repulsion function. Finally, we note some generic properties of the class of attraction/repulsion functions considered here.