Nonsmooth analysis for control problem in the space of probabilities

The note concerns the viability theorem for the systems driven by mean field type dynamics in the space of probabilities. We introduce a tangent cone to the given set of probabilities. Elements of this cone are distributions on the tangent bundle of the phase space. The viability theorem for the mean field type control system states that the given set of probabilities on the phase space is viable if and only if the tangent cone intersects with the set of distributions feasible by virtue of dynamics.