  Mean-field control problems have received continuous interest over the last
decade. Despite being more intricate than in classical optimal control, the
linear-quadratic setting can still be tackled through Riccati equations.
Remarkably, we demonstrate that another significant attribute extends to the
mean-field case: the existence of an intrinsic reproducing kernel Hilbert space
associated with the problem. Our findings reveal that this Hilbert space not
only encompasses deterministic controlled push-forward mappings but can also
represent of stochastic dynamics. Specifically, incorporating Brownian noise
affects the deterministic kernel through a conditional expectation, to make the
trajectories adapted. Introducing reproducing kernels allows us to rewrite the
mean-field control problem as optimizing over a Hilbert space of trajectories
rather than controls. This framework even accommodates nonlinear terminal
costs, without resorting to adjoint processes or Pontryagin's maximum
principle, further highlighting the versatility of the proposed methodology.
