AI 中文总结
该研究针对测度空间中含运输距离正则项的优化问题，利用弱-⋆二阶次导数理论推导无间隙二阶最优性条件，证明相关可微性并将结果应用于测度空间最优控制问题。
AI 中文摘要
我们考虑测度空间中的优化问题，该问题包含到给定先验测度的运输距离作为正则项。为推导无间隙型二阶最优性条件，我们使用弱-⋆二阶次导数理论，在目标函数光滑部分及 Kantorovich 势（即对偶运输问题的解）的附加假设下，这将导出与二次增长的等价性。此外，我们计算了弱-⋆二阶次导数，并证明了弱-⋆上微分可微性。最后，将所得结果应用于测度空间中的最优控制问题。
英文摘要
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.
