Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence

This paper is devoted to the study of multi-agent deterministic optimal control problems. We initially provide a thorough analysis Lagrangian, Eulerian and Kantorovich formulations problems, as well their relaxations. Then we exhibit some equivalence results among various representations compare respective value functions. To do it, combine techniques ideas from transportation, theory, Young measures evolution equations in Banach spaces. further exploit connections Lagrangian descriptions derive consistency number particles/agents tends infinity. that purpose prove an empirical version Superposition Principle obtain suitable Gamma-convergence for controlled systems.