Abstract

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.