Multi-agent deployment in 3-D via reaction-diffusion system with radially-varying reaction

This paper considers the problem of the deployment of a set of agents distributed on a disk -shaped grid onto three-dimensional (3-D) profiles, by using a continuum approximation (valid in the limit for a large number of agents) and then a control methodology for partial differential equations (PDEs). The agents' collective behavior is modeled by a pair of radially -varying diffusion-reaction PDEs in polar coordinates, whose state determines the agents' position. Having a radially -varying reaction coefficient not only increases the challenge of kernel equations becoming singular in radius, but also brings more potential deployment manifolds. To stabilize and increase the convergence of the deployment, a boundary controller and a boundary observer are designed by combining an infinite -dimensional backstepping approach with a Fourier series decomposition technique, thus driving all agents to the desired profile. A key feature of the presented result is that the desired profile only needs to be known by the leaders, with the followers only needing to follow a simple control strategy which requires only the measurement of its current position and communication with its neighbors as defined by the multi -agent system topology. The method provides closed -loop exponential stability with any prescribed convergence rate in the L-2 norm. Simulation tests are shown to prove the effectiveness of the proposed algorithm. (c) 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY -NC -ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).