Ergodicity of flocking systems for infinite-dimensional multi-agent coordination

In this paper, we address the fundamental questions on the existence of solutions to flocking systems having an infinite number of agents and their ergodic properties. Ergodic theory is a modern dynamical system theory which primarily deals with averaging problems and general qualitative questions. Now it is also a powerful amalgam of methods used for the analysis of statistical properties of dynamical systems. Our method is based on various tools developed from ergodic theory and statistical mechanics. The basic idea is to view the infinite-dimensional flocking system as an ideal gas system consisting of an infinite number of agents. We show that the solution exists for a class of flocking systems in a subset of the phase space by constructing this subset and using the limiting dynamics in the proof. This is the first attempt to using “real” ergodic theory to address such an issue for multi-agent systems.