Generalized Control Systems in the Space of Probability Measures

In this paper we formulate a time-optimal control problem in the space of probability measures. The main motivation is to face situations in nitedimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on R. We propose for this problem a generalized version of some concepts from classical control theory in nite dimensional systems (namely, target set, dynamic, minimum time function...) and formulate an Hamilton-JacobiBellman equation in the space of probability measure solved by the generalized minimum time function. We prove also some representation results linking the classical concept to the corresponding generalized ones. The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savaré, which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system.