Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations

TL;DR

This paper explores the stability of affine optimal control problems constrained by semilinear elliptic PDEs by analyzing metric subregularity and differentiability properties, with applications to error estimates in Tikhonov regularization.

Contribution

It introduces new stability analysis methods for control problems with semilinear elliptic PDE constraints, considering more general nonlinear perturbations under weaker assumptions.

Findings

Establishment of differentiability of the switching function.

Extension of stability results to broader nonlinear perturbations.

Application to error estimates in Tikhonov regularization.

Abstract

This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions, than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.