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 finite-dimensional 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 . We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function...) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int. Game Theor. Rev. 10, 1-16, 2008). 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 Savar, in Ambrosio et al. [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system.