Control of multiagent systems viewed as dynamical systems on the Wasserstein space

This talk is devoted to an overview of recent results on the optimal control of dynamical systems on probability measures modelizing the evolution of a large number of agents.

The system is composed by a number of agents so huge, that at each time only a statistical description of the state is available. A common way to model such kind of system is to consider a macroscopic point of view, where the state of the system is
described by a (time-evolving) probability measure on $R^d$ (which the underlying space where the agents move). So we are facing to a two-level system where the mascroscopic dynamic concerns probability measure while the microscopic dynamic - which describes the evolution of an individual agent - is a controlled differential equation on $ R^d$.

Associated to this dynamics on the Wasserstein space, one can associate a cost which allows to define a value function. We discuss the characterization of this value function through a Hamilton Jacobi Bellman equation stated on the Wasserstein space. We also discuss the problem of compatibility of state constraints with a multiagent control system. Since the Wasserstein space can be also viewed as the set of the laws of random variables in a suitable $L^2$ space, one can hope to reduce our problems to $L^2$ analysis. We discuss when this is possible.
This overview talk is based on several works in collaboration with I. Averboukh, P. Cardaliaguet, G. Cavagnari, C. Jimenez and A. Marigonda.