We consider the problem of deploying a group of autonomous vehicles (agents) in a formation that has higher density near the source of a measurable signal and lower density away from the source. The spatial distribution of the signal and the location of the source are unknown, but the signal is known to decay with the distance from the source. The vehicles do not have the capability of sensing their own positions, but they are capable of sensing the distance between them and their neighbors. We design a control algorithm based on a combination of two components. One component of the control law is inspired by the heat partial differential equation (PDE) and results in the agents deploying between two anchor agents. The other component of the control law is based on extremum seeking and achieves higher vehicle density around the source. By using averaging theory for PDEs, we prove that the vehicle density will be highest around the source. We also quantify the density function of the agents' deployment position. By discretizing the model with respect to the continuous agent index, we obtain decentralized control laws for discrete agents and illustrate the theoretical results with simulations.
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