No-gap second-order conditions for optimization problems involving transport distances

We consider optimization problems in the space of measures. As a regularization term, the problem includes the transport distance to a given prior measure. For the derivation of second-order optimality conditions of no-gap type, the theory of weak-$\star$ second subderivatives is used which will lead to an equivalence with quadratic growth under additional assumptions on the smooth part of the objective and on the Kantorovich potential, i.e., the solution of the dual transport problem. Further, the weak-$\star$ second subderivative is calculated and weak-$\star$ epidifferentiability is proven. Finally, the results are applied to optimal control problems in measure space.