Accepted Paper

Inserted: 3 mar 2020
Last Updated: 3 mar 2020

Journal: Caclulus of Variations and PDE
Year: 2020

Abstract:

This paper concerns a class of optimal control problems, where a central planner aims to control a multi-agent system in $\mathbb R^d$ in order to minimize a certain cost of Bolza type. At every time and for each agent, the set of admissible velocities, describing hisher underlying microscopic dynamics, depends both on hisher position, and on the configuration of all the other agents at the same time. So the problem is naturally stated in the space of probability measures on $\mathbb R^d$ equipped with the Wasserstein distance. The main result of the paper gives a new characterization of the value function as the unique viscosity solution of a first order partial differential equation. We introduce and discuss several equivalent formulations of the concept of viscosity solutions in the Wasserstein spaces suitable for obtaining a comparison principle of the Hamilton Jacobi Bellman equation associated with the above control problem.

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