On the solution stability of parabolic optimal control problems

TL;DR

This paper analyzes the stability of solutions to semilinear parabolic optimal control problems, establishing conditions for solution dependence on perturbations and providing error estimates and regularization insights.

Contribution

It extends stability analysis to problems with affine structure in control and nonlinear perturbations, using new assumptions on the growth of the objective functional's variations.

Findings

Hölder and Lipschitz stability results for optimal solutions

Metric subregularity of the optimality conditions mapping

Lipschitz dependence of control on Tikhonov regularization parameter

Abstract

The paper investigates stability properties of solutions of optimal control problems for semilinear parabolic partial differential equations. H\"older or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may non-linearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper -- the proved metric subregularity of the mapping associated with the system of first-order necessary optimality conditions. This property also enables error estimates for approximation methods. Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product.