Abstract
The aim of this paper is to discuss recent progress on the convergence problem in mean-field control theory and the study of associated nonlinear PDEs.
We are interested in optimal control problems involving a large number of interacting particles subject to independent Brownian noise. When the number of particles tends to infinity, the problem simplifies to a McKean-Vlasov-type optimal control problem for a typical particle.
I will present recent results concerning the quantitative analysis of this convergence. More specifically, I will discuss an approach based on the analysis of associated value functions. These functions are solutions of high-dimensional Hamilton-Jacobi equations, and the convergence problem translates into a stability problem for the limit equation, which is posed on the space of probability measures on Euclidean space.
I will also discuss the well-posedness of this limit equation, the study of which seems to escape the usual techniques for infinite-dimensional Hamilton-Jacobi equations.
