  In the present work, we develop a novel particle method for a general class
of mean field control problems, with source and terminal constraints. Specific
examples of the problems we consider include the dynamic formulation of the
p-Wasserstein metric, optimal transport around an obstacle, and measure
transport subject to acceleration controls. Unlike existing numerical
approaches, our particle method is meshfree and does not require global
knowledge of an underlying cost function or of the terminal constraint. A key
feature of our approach is a novel way of enforcing the terminal constraint via
a soft, nonlocal approximation, inspired by recent work on blob methods for
diffusion equations. We prove convergence of our particle approximation to
solutions of the continuum mean-field control problem in the sense of
Gamma-convergence. A byproduct of our result is an extension of existing
discrete-to-continuum convergence results for mean field control problems to
more general state and measure costs, as arise when modeling transport around
obstacles, and more general constraint sets, including controllable linear time
invariant systems. Finally, we conclude by implementing our method numerically
and using it to compute solutions the example problems discussed above. We
conduct a detailed numerical investigation of the convergence properties of our
method, as well as its behavior in sampling applications and for approximation
of optimal transport maps.
