Control and macroscopic limits of collective dynamics with time-varying weights

par Nastassia Pouradier Duteil

lundi 7 déc. 2020, 10:00 → 11:00 Europe/Paris

À distance

We introduce an augmented model for first-order opinion dynamics, in which a weight of influence is attributed to each agent. Each agent's influence on another agent's opinion is then proportional not only to the classical interaction function, but also to the agent's weight. The weights evolve in time and their equations are coupled with the opinions' evolution. We show that the well-known conditions for convergence to consensus can be generalized to this framework. We exhibit different kinds of long-term behavior, such as emergence of a single leader and emergence of two co-leaders.
We then formulate a control problem of consensus type: drive all agents to a final target point by acting only on the individual weights. We discuss controllability with and without constraints on the total mass of the system, and design control strategies with the steepest descent approach.
Lastly, we study the macroscopic limit of the microscopic model. We introduce the concept of indistinguishability of the agents, and derive the mean-field limit in the case where indistinguishability is satisfied. When it is not, we derive the so-called "graph-limit" of the system, that translates the concept of indices to an infinite-dimensional system.