New assumptions for stability analysis in elliptic optimal control problems

TL;DR

This paper establishes Lipschitz stability of optimal controls and states in semilinear elliptic control problems under new assumptions, despite challenges posed by non-monotone, non-coercive operators and implicit control appearances.

Contribution

It introduces novel assumptions enabling stability analysis in complex elliptic control problems with non-monotone operators and implicit control terms.

Findings

Proved Lipschitz stability of optimal solutions under perturbations.

Extended stability results to Tikhonov regularization parameters.

Addressed challenges from convection terms in elliptic equations.

Abstract

This paper is dedicated to the stability analysis of the optimal solutions of a control problem associated with a semilinear elliptic equation. The linear differential operator of the equation is neither monotone nor coercive due to the presence of a convection term. The control appears only linearly, or even it can not appear in an explicit form in the objective functional. Under new assumptions, we prove Lipschitz stability of the optimal controls and associated states with respect to perturbations in the equation and the objective functional as well as with respect to the Tikhonov regularization parameter.