New assumptions for stability analysis in elliptic optimal control problems
Eduardo Casas, Alberto Domínguez Corella, Nicolai Jork
TL;DR
The paper develops a stability theory for elliptic optimal control problems with semilinear state equations where the linear operator is non-monotone and non-coercive due to convection and the control may enter only linearly or implicitly in the objective. By introducing new structural assumptions, it establishes Lipschitz stability of both the optimal states and the optimal controls with respect to perturbations in the state equation, the objective, and the Tikhonov regularization parameter, and it extends the analysis to Tikhonov-perturbed and multi-parameter perturbations. The approach leverages a refined second-order framework with extended cones of critical directions, plus detailed PDE regularity and differentiability results for the state map and its adjoint. The results provide rigorous guarantees for stable recovery of controls and states under data perturbations and parameter variations, which is important for reliable numerical approximations and sensitivity analysis in nonlinear elliptic control problems.
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.