Multi-agent deployment in the plane using stochastic extremum seeking

We consider the problem of deployment of a group of N autonomous fully actuated vehicles (agents) in a non-cooperative manner in a planar signal field using the recently introduced method of stochastic extremum seeking. The spatial distribution of the signal is unknown to the vehicles but known to be convex. The vehicles are not able to sense their own positions but are capable of sensing the distance between their neighbors and themselves. Each vehicle employs a stochastic extremum seeking control law whose goal is to minimize the value of the measured signal, namely to be as close as possible to the bottom of the signal field, as well as to simultaneously minimizing a function of the distances between neighboring agents. Such a seemingly conflicting and mutually competitive nature of the agents' control laws produces a Nash equilibrium that depends on the agents' control parameters and the unknown signal distribution. We prove local exponential convergence, both almost surely and in probability, to a small neighborhood near the Nash equilibrium. The theoretical results are illustrated with simulations.