PDE-based deployment of multi-agent systems with enhanced leader communication and Wentzell boundary control

We present a deployment strategy for a multi-agent system (MAS) with a chain topology, where leader agents communicate and direct the system using controllers designed from a Partial Differential Equation (PDE) model. Both the leader and follower agents implement a consensus protocol, resulting in a system that can be modeled by a semilinear parabolic PDE. Based on this model, we design global controllers that utilize the information available to the leader agents, who can assess their deviation from the target position. By enhancing communication between the leaders, we achieve more accurate state estimation, which improves the performance of the global controllers. Additionally, we introduce a Wentzell-type boundary control, enabling the boundary agents to balance swarm cohesion with the deployment objective. We establish sufficient conditions for successful deployment, expressed as Linear Matrix Inequalities (LMIs). Numerical simulations show that the proposed global controllers outperform those used by non-communicating leaders.