Finite Element Error Analysis and Solution Stability of Affine Optimal Control Problems

Abstract

We consider affine optimal control problems subject to semilinear elliptic PDEs. The results are two-fold; first, we continue the analysis of solution stability of control problems under perturbations appearing jointly in the objective functional and the PDE. In regard to this, we prove that a coercivity-type property, that appears in the context of optimal control problems where the optimal control is of bang-bang structure, is sufficient for solution stability estimates for the optimal controls. The second result is concerned with the obtainment of error estimates for the numerical approximation for a finite element and a variational discretization scheme. The error estimates for the optimal controls and states are obtained under several conditions of different strengths, that appeared recently in the context of solution stability. The approaches used for the proofs are motivated by the structure of the assumptions and enable an improvement of the error estimates for the finite element scheme for the optimal controls and states under a H¨older-type growth condition.