Superposition Principle for Differential Inclusions

We prove an extension of the Superposition Principle by Ambrosio-Gigli-Savaré in the context of a control problem. In partic- ular, we link the solutions of a finite-dimensional control system, with dynamics given by a differential inclusion, to a solution of a continuity equation in the space of probability measures with admissible vector field. We prove also a compactness and an approximation result for admissible trajectories in the space of probability measures.