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Optimal partial mass transport, which is a variant of the optimal transport problem, consists in transporting effectively a prescribed amount of mass from a source to a target. The problem was first studied by Caffarelli and McCann (2010) [6] and Figalli (2010) [12] with a particular attention to the quadratic cost. Our aim here is to study the optimal partial mass transport problem with Finsler distance costs including the Monge cost given by the Euclidian distance. Our approach is different and our results do not follow from previous works. Among our results, we introduce a PDE of Monge–Kantorovich type with a double obstacle to characterize active submeasures, Kantorovich potential and optimal flow for the optimal partial transport problem. This new PDE enables us to study the uniqueness and monotonicity results for the active submeasures. Another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper [14] (Igbida and Nguyen, 2018). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]