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This paper focuses on the control of collective dynamics in large-scalemulti-agent systems (MAS) operating in a 3-D space, with a specific emphasis oncompensating for the influence of an unknown delay affecting the actuatedleaders. The communication graph of the agents is defined on a mesh-grid 2-Dcylindrical surface. We model the agents' collective dynamics by a complex- anda real-valued reaction-advection-diffusion 2-D partial differential equations(PDEs) whose states represent the 3-D position coordinates of the agents. Theleader agents on the boundary suffer unknown actuator delay due to thecumulative computation and information transmission time. We design adelay-adaptive controller for the 2-D PDE by using PDE backstepping combinedwith a Lyapunov functional method, where the latter is employed to design anupdate law that generates real-time estimates of the unknown delay.Capitalizing on our recent result on the control of 1-D parabolic PDEs withunknown input delay, we use Fourier series expansion to bridge the control of1-D PDEs to that of 2-D PDEs. To design the update law for the 2-D system, anew target system is defined to establish the closed-loop local boundedness ofthe system trajectories in $H^2$ norm and the regulation of the states to zeroassuming a measurement of the spatially distributed plant's state. Weillustrate the performance of delay-adaptive controller by numericalsimulations.

Delay-Adaptive Compensator for 3-D Space Formation of Multi-Agent Systems with Leaders Actuation