Self-organization of autonomous swarms via Langevin equation

A new approach to self-organization of autonomous swarms is presented based on the Langevin equation. The motion of each mobile node is determined jointly by the gradient of a global potential function, evaluated locally, and by a decaying Wiener process (continuous annealing). The motivation of this work is to achieve global swarming goals without being trapped at local minima of the potential energy. However, dictated by the local computability and interaction requirements, the potential function is typically not differentiable. This results in a discontinuous drift in the stochastic differential equation, and presents challenges in the analysis of the asymptotic swarm distribution. We introduce the notion of a relaxed solution into the current context, and establish the existence of relaxed solutions which converge weakly to the Gibbs distribution under a fixed annealing temperature. This provides the first step towards understanding the Langevin equation with nonsmooth potential and proving the optimality of the asymptotic configuration under annealing. Design of local interaction rules and practical implementation issues are also considered in this paper.