Solution stability of parabolic optimal control problems with fixed state-distribution of the controls

TL;DR

This paper investigates the stability of solutions in parabolic optimal control problems with fixed spatial control distribution, providing new conditions for optimality and stability analysis.

Contribution

It introduces strong metric subregularity conditions for the optimality mapping in control problems with fixed control distribution, extending existing optimality conditions.

Findings

Derived sufficient optimality conditions based on subregularity.

Extended known optimality conditions to fixed spatial control distributions.

Provided stability analysis results for the control solutions.

Abstract

The paper presents results about strong metric subregularity of the optimality mapping associated with the system of first-order necessary optimality conditions for a problem of optimal control of a semilinear parabolic equation. The control has a predefined spatial distribution and only the magnitude at any time is a subject of choice. The obtained conditions for subregularity imply, in particular, sufficient optimality conditions that extend the known ones. The paper is complementary to a companion one by the same authors, in which a distributed control is considered.