Optimal control of multiagent systems in the Wasserstein space

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 his/her underlying microscopic dynamics, depends both on his/her 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.