Swarm Aggregation under Fading Attractions

TL;DR

This paper studies multi-agent system convergence with fading attraction interactions, introducing dilute partitions to prove boundedness and convergence to equilibria.

Contribution

It introduces dilute partitions to analyze formations with fading attractions, extending convergence results beyond nonfading interaction scenarios.

Findings

Trajectories remain bounded during evolution.

Formations converge to equilibrium states.

Dilute partitions effectively analyze fading attraction effects.

Abstract

Gradient descent methods have been widely used for organizing multi-agent systems, in which they can provide decentralized control laws with provable convergence. Often, the control laws are designed so that two neighboring agents repel/attract each other at a short/long distance of separation. When the interactions between neighboring agents are moreover nonfading, the potential function from which they are derived is radially unbounded. Hence, the LaSalle's principle is sufficient to establish the system convergence. This paper investigates, in contrast, a more realistic scenario where interactions between neighboring agents have fading attractions. In such setting, the LaSalle type arguments may not be sufficient. To tackle the problem, we introduce a class of partitions, termed dilute partitions, of formations which cluster agents according to the inter- and intra-cluster interaction strengths. We then apply dilute partitions to trajectories of formations generated by the multi-agent system, and show that each of the trajectories remains bounded along the evolution, and converges to the set of equilibria.