Linear-Quadratic Mean Field Control with Non-Convex Data

Abstract: In this manuscript, we study a class of linear-quadratic (LQ) mean field control problems with a common noise and their corresponding $N$-particle systems. The mean field control problems considered are not standard LQ mean field control problems in the sense that their dependence on the mean field terms can be non-linear and non-convex. Therefore, all the existing methods to deal with LQ mean field control problems fail. The key idea to solve our LQ mean field control problem is to utilize the common noise. We first prove the global well-posedness of the corresponding Hamilton-Jacobi equations via the non-degeneracy of the common noise. In contrast to the LQ mean field games master equations, the Hamilton-Jacobi equations for the LQ mean field control problems can not be reduced to finite-dimensional PDEs. We then globally solve the Hamilton-Jacobi equations for $N$-particle systems. As byproducts, we derive the optimal quantitative convergence results from the $N$-particle systems to the mean field control problem and the propagation of chaos property for the related optimal trajectories. This paper extends the results in [{\sc M. Li, C. Mou, Z. Wu and C. Zhou}, \emph{Trans. Amer. Math. Soc.}, 376(06) (2023), pp.~4105--4143] to the LQ mean field control problems.