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 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.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper focuses on the study of multi-agent deterministic optimal control problems. The Lagrangian, Eulerian, and Kantorovich formulations of the problems, as well as their relaxations, are thoroughly analyzed. Equivalence results among the different representations are presented, and the respective value functions are compared. Techniques and ideas from optimal transportation, control theory, Young measures, and evolution equations in Banach spaces are combined to achieve these goals. Additionally, consistency results are obtained as the number of particles/agents tends to infinity by exploiting the connections between Lagrangian and Eulerian descriptions.