Georgia Institute of Technology College of Sciences

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 control problem set on the $d$-dimensional torus.  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. 

 

First, 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. Subsequently, our main result asserts that, whenever the cost functionals are smooth and convex, the optimal control, trajectory, and value function from the approximating problem converge to their counterparts in the original mean field control problem. Noticeably, we show that our method yields a polynomial convergence rate directly proportional to the data's regularity. This convergence rate is faster than the one achieved by the usual particles methods available in the literature, offering a more efficient alternative. Furthermore, our technique also provides an explicit method for constructing an approximate optimal control along with its corresponding trajectory. This talk is based on joint work with François Delarue.