Speaker: Aldo Gutiérrez (Universidad de Chile)
Abstract: The framework of Mean-field optimal control, formally introduced in 2014 by M. Fornasier and F. Solombrino has been lately used to model different real-life situations, such as biological, social and economic phenomena. The focus is on the evolution of populations of similar individuals, where each agent senses the interaction with the others through an average term. In this case, when the number of individuals is very large, an aggregation effect occurs and the (discrete) collection of agents is generally replaced by their spatial density. In a recent work by G. Cavagnari et al, three formulations are introduced, the Lagrangian formulation treats the system as $N$ particles and follows the individual trajectories, the Eulerian formulation describes the system via a curve of probability measures and the Kantorovich formulation acts as a bridge between the previous two formulations through the use of the superposition principle. As far as we know, there are no constraints considered in the agents dynamics.
The aim of this talk is to study $N-$agents following a sweeping process dynamic, where each agent $x_i \in \{1,\dots\ N\}$ is governed by a controlled sweeping process, depending on $\mu_t^N,$ is the empirical distribution of the agents at time $t$, the moving set $C(\cdot)$ represents the constraints of the dynamics and $F^N$ that models the interaction between agents, each controlled by some $u_i$ depending only on the time. The differential inclusion falls into the Lagrangian formulation, we show that its limit falls into an Eulerian formulation with the form of a contrainsed continuity equation, where there is a measure associated to the (now feedback) control. This limit system, known as a measure sweeping process, was introduced, without control, in 2016 by S. Di Marino, B. Maury and F. Santambrogio and further developed to model crowd motion dynamics.
Fecha y hora: miércoles 12 de noviembre, 12:00hrs.
Lugar: Sala Seminario Felipe Álvarez (5to piso)
Link de zoom: https://uchile.zoom.us/j/93417604828?pwd=eSHafQqVcAegttXqKH3swXZsMseRrn.1