CIRM - Videos & books Library - Mean field type control with congestion

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Authors : Laurière, Mathieu (Author of the conference)
CIRM (Publisher )

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Abstract : 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.

MSC Codes :
35K40 - Second-order parabolic systems
35K55 - Nonlinear parabolic equations
35K65 - Parabolic equations of degenerate type
49K20 - Optimal control problems with PDE (optimality conditions)
65K10 - Optimization and variational techniques
65M06 - Finite difference methods (IVP of PDE)
35D30 - Weak solutions of PDE
49N70 - Differential games in calculus of variations

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2017-08-09_Lauriere.mp4

Information on the Event

Event Title : CEMRACS: Numerical methods for stochastic models: control, uncertainty quantification, mean-field / CEMRACS : Méthodes numériques pour équations stochastiques : contrôle, incertitude, champ moyen
Event Organizers : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme
Dates : 17/07/17 - 25/08/17
Event Year : 2017
Event URL : http://conferences.cirm-math.fr/1556.html

Citation Data

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.19205403
URI : http://dx.doi.org/10.24350/CIRM.V.19205403

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Bibliography

Achdou, Y., & Laurière, M. (2016). Mean field type control with congestion. Applied Mathematics and Optimization, 73(3), 393-418 - http://dx.doi.org/10.1007/s00245-016-9342-8

Achdou, Y., & Laurière, M. (2016). Mean field type control with congestion. II: An augmented Lagrangian method. Applied Mathematics and Optimization, 74(3), 535-578 - http://dx.doi.org/10.1007/s00245-016-9391-z

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