Variance Optimization and Control Regularity for Mean-Field Dynamics

We study a family of optimal control problems in which one aims at minimizing cost that mixes quadratic penalization and the variance system, both for finitely many agents mean-field dynamics as their number goes to infinity. While solutions discrete problem always exist unique explicit form, behavior macroscopic counterparts is very sensitive magnitude time horizon parameter. When minimizes final variance, there exists Lipschitz-in-space controls infinite dimensional problem, can be obtained suitable extension finite-dimensional problems. The same holds true maximizations whenever sufficiently small. On contrary, large times (or equivalently small penalizations cost), it proven does not Lipschitz-regular problem.