A multi-agent system (MAS) refers to a group of interacting robots or, more generally, dynamical systems. Typical examples are robot swarms, traffic flows, smart grids, communication networks, and transportation systems. These systems tend to grow rapidly in size and complexity due to the continuous introduction of new agents. As a result, standard control methods fail, and we witness massive system breakdowns such as power blackouts, traffic jams, internet outages, and products delivery failures. Thus, there is a demand for new scalable solutions for the analysis and control of multi-agent systems.
Partial differential equations (PDEs) are a natural framework for studying large-scale multi-agent systems that provides a solution to the scalability problem. When cooperating agents are identical, and their communication topology is uniform, one can take the continuum limit of the ordinary differential equations (ODEs) describing the multi-agent system and obtain a PDE. The form of the PDE depends on the agents’ dynamics and network structure but does not depend on the number of agents. Consequently, such PDEs provide analysis methods whose complexity does not change when the number of agents grows.
This project aims to explore the possibility of using continuum mechanics models to derive stability conditions and design controllers for multi-agent systems in general and robot swarms in particular. This problem is inverse to the one solved in numerical methods. Namely, to approximate a solution of a PDE, one discretises it and solves the resulting set of ODEs. Here, we will approximate ODEs with a single PDE to simplify the theoretical analysis. This research topic is full of challenging questions: What is the best way to relate the state of a multi-agent system with the state of a PDE? How many agents justify the usage of PDEs? How to interpret the results derived for the PDE in terms of the original multi-agent system? What are the conditions on the local dynamics and communication topology that make it possible to associate a well-posed PDE with a given multi-agent system?