Swarm Aggregation under Fading Attractions

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 technical note 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 in a compact subset along the evolution, and converges to the set of equilibria.