Fourier Galerkin approximation of mean field control problems

Over the past twenty years, mean field control theory has been developed to study cooperative games between weakly interacting agents (particles). The limiting formulation of a (stochastic) mean field control problem, arising as the number of agents approaches infinity, is a control problem for trajectories with values in the space of probability measures.
The goal of this talk is to introduce a finite dimensional approximation of the solution to a mean field optimal control problem set on the d-dimensional torus, without relying on particle-based methods. Our approximation is obtained by means of a Fourier-Galerkin method, the main principle of which is to truncate the Fourier expansion of probability measures. However, this operation has the main feature not to leave the space of probability measures invariant, which drawback is know as Gibbs’ phenomenon.
We prove that the Fourier-Galerkin method induces a new finite-dimensional control problem with trajectories in the space of probability measures with a finite number of Fourier coefficients. We also present convergence results for the optimal control, trajectory, and value function, showing that our method achieves faster convergence rates than the usual particle approach, offering a more efficient alternative. This talk is based on joint work with François Delarue.