  In this paper, we study a regularised relaxed optimal control problem and, in
particular, we are concerned with the case where the control variable is of
large dimension. We introduce a system of mean-field Langevin equations, the
invariant measure of which is shown to be the optimal control of the initial
problem under mild conditions. Therefore, this system of processes can be
viewed as a continuous-time numerical algorithm for computing the optimal
control. As an application, this result endorses the solvability of the
stochastic gradient descent algorithm for a wide class of deep neural networks.
