Mayer control problem with probabilistic uncertainty on initial positions

In this paper we introduce and study an optimal control problem in the Mayer's form in the space of probability measures on R-n endowed with the Wasserstein distance. Our aim is to study optimality conditions when the knowledge of the initial state and velocity is subject to some uncertainty, which are modeled by a probability measure on R-d and by a vector-valued measure on R-d, respectively. We provide a characterization of the value function of such a problem as unique solution of an Hamilton-Jacobi-Bellman equation in the space of measures in a suitable viscosity sense. Some applications to a pursuit-evasion game with uncertainty in the state space is also discussed, proving the existence of a value for the game. (C) 2017 Elsevier Inc. All rights reserved.