Delay-Compensated Control for α-Leader in PDE-Based Multi-agent 2-D Surface Formation Containment

This paper investigates the formation-containment control problem for PDE-based multi-agent systems under input delays and spatial information decay. A novel three-layer hierarchical control framework is proposed, with particular emphasis on stabilizing the α-leader–a key agent subject to delayed control inputs. An integral-type delay compensation law is constructed using kernel functions derived from backstepping transformations and equivalence principles. The leader dynamics are governed by a reaction–diffusion–advection partial differential equation (PDE). Rigorous theoretical results establish the boundedness, invertibility, and exponential stability of the closed-loop error system. Numerical simulations validate the effectiveness of the proposed strategy, demonstrating accurate convergence to the desired encirclement formation under dual damping boundary conditions and time-delay effects. The findings offer valuable insights into the design of robust and scalable formation control strategies for spatially distributed multi-agent systems operating under realistic constraints.