Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem

Abstract

We consider an elliptic optimal control problem with pointwise state constraints. The cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation. The corresponding minima are shown to converge in $L^2$ to the exact control as the discretization parameter tends to zero. Furthermore, error bounds for the control and the state are obtained in both two and three space dimensions. Finally, we present numerical examples which confirm our analytical findings.