Time-Optimal Control Problem in the Space of Probability Measures

We are going to define a time optimal control problem in the space of probability measures. Our aim is to model situations in which the initial position of a particle is not exactly known, even if the evolution is assumed to be deterministic. We will study some natural generalization of objects commonly used in control theory, proving some interesting properties. In particular we will focus on a comparison result between the classical minimum time function and its natural generalization to the probability measures setting.