PDE-based deployment with communicating leaders for a large-scale multi-agent system

We study the deployment of a first-order multi-agent system (MAS) onto a curve in $\mathbb{R}^{n}$. The MAS has a chain topology and two types of agents: leaders and followers. The leaders know their positions relative to the target curve. Neighboring leaders can communicate with one another. Each follower is aware of the intended and existing differences between its state and the states of its two nearest neighbors. To solve the formation control problem, we derive a semi-linear parabolic PDE describing the system when the number of agents is sufficiently large. We derive the stability condition in terms of linear matrix inequalities (LMIs). Using numerical simulations, we demonstrate that increased connectivity between the leaders improves the deployment speed of the MAS.