• 15h30 - Benoît BONNET (LSIS, Marseille-Toulon) Lipschitz regularity of Mean-Field optimal controls.

Abstract. During the past decade, the analysis of multi-agent systems has gained a lot of steam in several mathematical communities, with applications ranging from the study of pedestrian and traffic flows to aggregation models in biology, opinion formation mechanisms on social networks and decentralized supervision of autonomous vehicles. The issue of developing analytical and numerical tools to control such systems, towards or away from given configurations, has been one of the main topic of interest of a fairly large community living at the intersection of optimal transport and probability theory, PDE analysis, calculus of variations and control theory.

In this talk, we shall focus our attention on optimal control problems formulated on multi-agent systems, studied in the mean-field limit. The dynamics is given in this setting by a transport equation with non-local velocities of Fokker-Planck type, formulated in the space of probability measures. Since the work of Ambrosio Di-Perna and Lions, it has been known that these PDEs can only be well-posed for arbitrary initial measures provided that the driving velocity fields are smooth enough, eg pschitz-regular in space. Such an arbitrary well-posedness is crucial for ensuring that infinite-dimensional controls can be applied to discrete multi-agent systems, and to quantify properly the corresponding approximation errors in terms of the number of agents.

After motivating the study of such problems and presenting some of the associated modelling and mathematical tools, I will recall recent results pertaining to the existence of Lipschitz feedbacks in classical optimal control problems and show how the latter can be leveraged and adapted to mean-field optimal control problems.

• 16h30 - Café et discussion.

• 17h00 - Evelina Shamarova (Federal University of Paraíba, Brésil).
  Gaussian density estimates for solutions of fully coupled forward-backward SDEs.