A Priori Error Estimates for Optimal Control Problems with Constraints on the Gradient of the State on Nonsmooth Polygonal Domains

In this article we are concerned with the finite element discretization of optimal control problems subject to a second order elliptic PDE and additional pointwise constraints on the gradient of the state.

We will derive error estimates for the convergence of the cost functional under mesh refinement. Subsequently error estimates for the control and state variable are obtained.

As an intermediate tool we will also analyze a Moreau-Yosida regularized version of the optimal control problem. In particular we will derive convergence rates for the cost functional and the primal variables. To this end we will employ new techniques in estimating the L ^∞-norm of the feasibility error which could also be used to improve existing estimates in the state constrained case.