SIAM Journal on Optimization, Volume 30, Issue 2, Page 1724-1755, January 2020.
This paper studies generalized differentiability properties of the marginal function of parametric optimal control problems governed by semilinear elliptic partial differential equations. We establish some upper estimates for the regular and the limiting subgradients of the marginal function for Hilbert parametric spaces. In addition, we provide sufficient conditions for these upper estimates to be equalities. For the circumstance of parametric bang-bang optimal control problems, under some additional assumptions we show that the solution map of the perturbed optimal control problems has local upper Hölderian selections for both cases of Asplund parametric spaces and non-Asplund parametric spaces. This leads to explicit exact formulas for computing the regular and the limiting subdifferentials of the marginal function for the Asplund parametric spaces as well as lower estimates for the regular and the limiting subdifferentials of the marginal function with respect to the non-Asplund parametric spaces.

中文翻译：

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偏微分方程参数控制问题中边际函数的次梯度

SIAM优化杂志，第30卷，第2期，第1724-1755页，2020年1月。
本文研究了由半线性椭圆型偏微分方程控制的参数最优控制问题的边际函数的广义微分性质。我们为希尔伯特参数空间的边际函数的正则和极限子梯度建立了一些较高的估计。此外，我们为这些较高的估计值相等提供了充分的条件。对于参数bang-bang最优控制问题的情况，在一些附加假设下，我们表明，对于Asplund参数空间和非Asplund参数空间两种情况，扰动的最优控制问题的解图都有局部上Hölderian选择。