Multi-Agent Deployment in 3-D via PDE Control


This paper introduces a technique for modelling, analysis, and management style of a giant-scale system of agents deployed in 3-d house. The agents' communication graph may be a mesh-grid disk two-D topology in polar coordinates. Treating the agents as a continuum, we model the agents' collective dynamics by complex-valued reaction-diffusion two-D partial differential equations (PDEs) in polar coordinates, whose states represent the position coordinates of the agents. Due to the reaction term within the PDEs, the agents can achieve a made family of two-D deployment manifolds in three-D house that correspond to the PDEs' equilibrium as determined by the boundary conditions. Unfortunately, many of these deployment surfaces are open-loop unstable. To stabilize them, a heretofore open and difficult problem of PDE stabilization by boundary management on a disk has been solved during this paper, using a new category of specific backstepping kernels that involve the Poisson kernel. A dual observer, which is also express, allows to estimate the positions of all the agents, as needed in the leaders' feedback, by only measuring the position of their closest neighbors. Hence, an all-express management theme is found which is distributed in the way that every agent solely needs local information. Closed-loop exponential stability within the $L^2$, $H^1$, and $H^2 $ areas is proved for both full state and output feedback designs. Numerical simulations illustrate the proposed approach for three-D deployment of discrete agents.

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