Hamilton-Jacobi-Bellman Equation for a Time-Optimal Control Problem in the Space of Probability Measures

In this paper we formulate a time-optimal control problem in the space of probability measures endowed with the Wasserstein metric as a natural generalization of the correspondent classical problem in R^d where the controlled dynamics is given by a differential inclusion. The main motivation is to model situations in which we have only a probabilistic knowledge of the initial state. In particular we prove first a Dynamic Programming Principle and then we give an Hamilton-Jacobi- Bellman equation in the space of probability measures which is solved by a generalization of the minimum time function in a suitable viscosity sense.