Reachability for multiagent control systems via Lyapunov functions

This paper concerns the problem of reachability of a given state for a multiagent control system in ${\mathbb{R}}^d$. In such a system, at every time each agent can choose his/her velocity which depends both on his/her position and on the position of the whole crowd of agents (modeled by a probability measure on ${\mathbb{R}}^d$). The main contribution of the paper is to study the above reachability problem with a given rate of attainability through a Lyapunov method adapted to the Wasserstein space of probability measures. As a byproduct we obtain a new comparison result for viscosity solutions of Hamilton Jacobi equations in the Wasserstein space.