The theory of mean field type control developed mainly by Bensoussan, Frehse and Yam aims at describing the behaviour of a large number of interacting agents using a common feedback. A phenomenon that have raised a lot of interest recently concerns congestion effects, where the agents try to avoid crowded regions. One way to take into account such effects is to let the cost of displacement increase in the regions where the density is large. We will present a system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation describe respectively the evolution of the density of agents and the value function. We are able to prove the existence and uniqueness of suitably defined weak solutions, which are characterized as the optima of two optimal control problems in duality. This optimal control point of view also leads to a numerical method to solve the PDE system. The algorithm we will present is based on an augmented Lagrangian and an Alternate Direction Method of Multipliers. Numerical results will be presented. This is joint work with Yves Achdou.
Biography
Mathieu Laurière is a postdoctoral fellow at NYU Shanghai. His research interests are: Partial Differential Equations, Numerical Methods, Communication Complexity, Quantum Computing.