  This work is devoted to the control of the Fokker-Planck equation, posed on a
smooth bounded domain of R^d, with a localized drift force. We prove that this
equation is locally controllable to regular nonzero trajectories. Moreover,
under some conditions, we explain how to reduce the number of controls around
the reference control. The results are obtained thanks to a standard
linearization method and the fictitious control method. The main novelties are
twofold. First, the algebraic solvability is performed and used directly on the
adjoint problem. We then prove a new Carleman inequality for the heat equation
with a space-time varying first-order term: the right-hand side is the gradient
of the solution localized on an open subset. We finally give an example of
regular trajectory around which the Fokker-Planck equation is not controllable
with a reduced number of controls, to highlight that our conditions are
relevant.
