Swarm Aggregation with Fading Attractions

The use of gradient descent for organizing multi-agent systems is widely appreciated in mathematics and in its real-world applications. Besides a demonstration of the existence of a local minimum point, in the context of formation control, gradient descent can be interpreted as providing decentralized control laws for pairs of neighboring agents. Often, the control laws between neighboring agents are designed so that two agents repel/attract each other at a short/long distance. Conventional techniques for proving convergence of a dynamical system over a Euclidean space are, for example, constructing a radially unbounded Lyapunov function and then appealing to the LaSalle's principle. When the attractions between neighboring agents are nonfading; then, it is well known that the potential function associated with the multi-agent system is radially unbounded, and hence using LaSalle's principle is enough for establishing the system convergence. However, if the attractions are fading; then, using only the LaSalle's arguments may not be sufficient. In this paper, we assume that interactions between neighboring agents have fading attractions. We develop, among other things, a new approach for proving the convergence of the resulting gradient flow; in particular, we introduce a class of partitions, termed dilute partitions, of frameworks. This is a rich question relating to classic algorithms such as $k$-mean clustering and its variants, and is useful for studying other multi-agent problems concerning about large size formations.