  We consider optimization problems for interacting particle systems. We show
that critical points solve a Vlasov equation, and that in general no minimizers
exist despite continuity of the action functional. We prove an explicit
representation of the relaxation of the action functional. We show convergence
of N-particle minimizers to minimizers of the relaxed action, and finally
characterize minimizers of dynamic interacting particle optimal transport
problems as solutions to Hamilton-Jacobi-Bellman equations.
