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

We consider a linear–quadratic optimal control problem for elliptic surface partial differential equations (PDEs) with additional state constraints. We approximate the optimization problem by a family of discrete problems and prove convergence rates for the discrete controls and the discrete states. With this we extend results known in the Euclidean setting to the surface case. We present numerical examples confirming our theoretical findings, with measures concentrated in points and measures concentrated on a line.