Control problems in Wasserstein space

In this talk we present recent results about the existence and uniqueness of the viscosity solution for a certain classes on Hamilton-Jacobi Equations in the Wasserstein space of probability measure, arising in problem of mean field control of multi-agent systems. We consider a multi-agent system subject to a centralized controller
aiming to minimize a cost function. The microscopic dynamics of each agent is given by a differential inclusion. We model the distribution of agents by a probability measure, and formulate the minimization problem
as a Mayer problem for a dynamics in the Wasserstein space represented by a controlled continuity equation describing the macroscopical evolution of the system. We prove that the value function V of the
problem solves a Hamilton-Jacobi equation in the Wasserstein space in a suitable viscosity sense, and prove a comparison principle for such an equation, thus characterizing V as the unique viscosity solution of the
Hamilton-Jacobi equation associated to the problem.