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

TL;DR

This paper investigates the generalized differentiability of the marginal function in parametric control problems governed by semilinear elliptic PDEs, providing estimates for subgradients and conditions for local Hölderian selections.

Contribution

It establishes upper and lower estimates for subgradients of the marginal function and analyzes the solution map’s regularity under additional assumptions.

Findings

Upper estimates for regular and limiting subgradients

Conditions for local Hölderian selections of the solution map

Lower bounds for the regular subdifferential of the marginal function

Abstract

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.