Optimal control of elliptic surface PDEs with pointwise bounds on the state

We consider a linear-quadratic optimization problem with pointwise bounds on the state for which the constraint is given by the Laplace-Beltrami equation (to have uniqueness we add an lower order term) on a two-dimensional surface . By using finite elements we approximate the optimization problem by a family of discrete problems and prove convergence rates for the discrete controls and the discrete states. Furthermore, assuming (roughly spoken) a higher regularity for the control the order of convergence improves. This extends a result known in an Euclidean setting to the surface case.