Variational interacting particle systems and Vlasov equations

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.