Analysis and Control of Large-Scale Multi-agent System using Partial Differential Equations

Multi-agent system (MAS) consist of multiple interacting agents that cooperate to accomplish tasks that cannot be efficiently achieved by a single agent. Such systems arise in numerous applications across engineering, biology, and the social sciences, motivating their systematic study. Traditional approaches study the dynamics of each agent individually, typically modelling them using ordinary differential equations (ODEs) with interactions represented through graph-theoretic structures. As the number of agents increases, this results in a large system of ODEs whose dimension grows with the population size, making analysis and control design increasingly challenging.
To address this limitation, several studies adopt a continuum perspective in which the collective behaviour of a large population of agents is approximated using partial differential equations (PDEs). Building upon this framework, this thesis investigates the problem of deploying a large-scale multi-agent system onto a predefined target. A leader–follower architecture is considered in which a subset of agents, referred to as leaders, can assess their deviation from the target, while follower agents rely only on local interactions with their neighbours. The resulting collective behaviour can be described by a semilinear parabolic PDE. Based on this model, global control strategies are designed that utilize information available to the leaders. In particular, communication between leaders is introduced to improve the estimation of the global error state, which enhances the effectiveness of the control strategy. Furthermore, a Wentzell-type boundary control is proposed to regulate the behaviour of boundary agents, allowing them to balance the deployment objective with the need to preserve swarm cohesion. Sufficient conditions for successful deployment are derived in the form of Linear Matrix Inequalities (LMIs). Numerical simulations confirm that the proposed control strategies achieve faster convergence and improved swarm cohesion compared with approaches that assume non-communicating leaders.