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

arXiv: https://arxiv.org/abs/2011.07117 · HTML: https://mlsanity.com/paper/2011.07117/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Nov 2020.

## Built on by (most cited first, 1 of 1)

- [High-dimensional Mean-Field Games by Particle-based Flow Matching](https://mlsanity.com/paper/2512.01172/) (2025)

## Abstract (arXiv)

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