The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. They describe respectively the evolution of the distribution of the agents' states and the evolution of the value function. Since it comes from a control problem, this PDE system differs in general from the one arising in mean field games. Recently, this kind of model has been applied to crowd dynamics. More precisely, in this talk we will be interested in modeling congestion effects: the agents move but try to avoid very crowded regions. One way to take into account such effects is to let the cost of displacement increase in the regions where the density of agents is large. The cost may depend on the density in a non-local or in a local way. We will present one class of models for each case and study the associated PDE systems. The first one has classical solutions whereas the second one has weak solutions. Numerical results based on the Newton algorithm and the Augmented Lagrangian method will be presented. This is joint work with Yves Achdou.
Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 10/08/17 Date de captation : 09/08/17 Collection : Research talks Format : MP4 Durée : 00:47:28 Domaine : PDE ; Control Theory & Optimization ; Numerical Analysis & Scientific Computing Audience : Chercheurs ; Doctorants , Post - Doctorants Download : http://videos.cirm-math.fr/2017-08-09_Lauriere.mp4
Informations sur la rencontre
Nom du congrès : CEMRACS: Numerical methods for stochastic models: control, uncertainty quantification, mean-field / CEMRACS : Méthodes numériques pour équations stochastiques : contrôle, incertitude, champ moyen Organisteurs Congrès : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme Dates : 17/07/17 - 25/08/17 Année de la rencontre : 2017 URL Congrès : http://conferences.cirm-math.fr/1556.html
DOI : 10.24350/CIRM.V.19205403 Cite this video as: Laurière, Mathieu (2017). Mean field type control with congestion. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19205403URI : http://dx.doi.org/10.24350/CIRM.V.19205403