Viability theorem for deterministic mean field type control systems

The mean field type control system is a dynamical system acting on the space of probabilities (Wasserstein space). In the paper, we introduce a tangent distribution to a subset of the Wasserstein space. This permits to formulate the viability theorem for the mean field type control system in the classical way: the set of probabilities is viable with respect to the mean field type control system if and only if the corresponding set of tangent distributions intersects with the set of distributions feasible by virtue of dynamics.