  This paper focuses on the control of collective dynamics in large-scale
multi-agent systems (MAS) operating in a 3-D space, with a specific emphasis on
compensating for the influence of an unknown delay affecting the actuated
leaders. The communication graph of the agents is defined on a mesh-grid 2-D
cylindrical surface. We model the agents' collective dynamics by a complex- and
a real-valued reaction-advection-diffusion 2-D partial differential equations
(PDEs) whose states represent the 3-D position coordinates of the agents. The
leader agents on the boundary suffer unknown actuator delay due to the
cumulative computation and information transmission time. We design a
delay-adaptive controller for the 2-D PDE by using PDE backstepping combined
with a Lyapunov functional method, where the latter is employed to design an
update law that generates real-time estimates of the unknown delay.
Capitalizing on our recent result on the control of 1-D parabolic PDEs with
unknown input delay, we use Fourier series expansion to bridge the control of
1-D PDEs to that of 2-D PDEs. To design the update law for the 2-D system, a
new target system is defined to establish the closed-loop local boundedness of
the system trajectories in $H^2$ norm and the regulation of the states to zero
assuming a measurement of the spatially distributed plant's state. We
illustrate the performance of delay-adaptive controller by numerical
simulations.
