Optimal Control of the Laplace-Beltrami operator on compact surfaces -   concept and numerical treatment
  We consider optimal control problems of elliptic PDEs on hypersurfaces in 2- or 3-dimensional Euclidean space. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of the surface. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semismooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings. 
Michael Hinze and Morten Vierling
math.OC