Subgradients of Marginal Functions in Parametric Control Problems of Partial Differential Equations

The paper studies generalized differentiability properties of the marginal function of parametric optimal control problems of semilinear elliptic partial differential equations. We establish upper estimates for the regular and the limiting subgradients of the marginal function. With some additional assumptions, we show that the solution map of the perturbed optimal control problems has local upper H\"{o}lderian selections. This leads to a lower estimate for the regular subdifferential of the marginal function.