  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.
