PDE-Based 2-D Surface Formation-Containment Control Multi-Agent Systems With α- Leader Delay-Compensated Input

Formation-containment control in multi-agent systems faces a critical challenge: how to stabilize all formation leaders when the
$\alpha -$
leader’s influence diminishes over topology distance, especially under input delays? This challenge is particularly acute in PDE-based models, where information decay is intrinsic, making it difficult for the
$\alpha -$
leader—a leader among leaders—to affect far-away agents, especially under speed constraints.Our work tackles this challenge by proposing a three-layer hierarchical control framework based on Partial Differential Equations (PDEs). Our key innovation is an integral-type delayed compensation input specifically designed for the
$\alpha -$
leader. This control law uses kernel functions derived from backstepping transformation and equivalence principles, tailored to compensate for input delays. By providing the
$L^{2}$
space expansion of these kernel functions, we show that formation control error for finite agents is acceptable, even with approximations.We organize formation leaders into a one-dimensional chain-like topology, categorizing them as leaders, anchors, or
$\alpha -$
leaders based on their roles. For other agents, we derive distributed control laws from the discrete form of PDEs governing formation deployment. Followers converge to the convex hull spanned by leaders, forming an internal two-dimensional surface. This PDE-based approach ensures invariance in translation, rotation, and expansion.Our work also contributes to the mathematical foundations of PDE-based multi-agent systems. We discuss the numerical solution of kernel functions, offer a distributional interpretation, and—critically—analyze the discretization error between PDE and ODE models. This analysis reveals the relationship between stability, time step, spatial step, and control laws, addressing an often-ignored issue in the field.Simulation examples validate our theoretical findings, showing effective formation-containment control under our framework.