  We consider an elliptic optimal control problem where the objective
functional contains an integral along a surface of codimension 1, also known as
a hypersurface. In particular, we use a fidelity term that encourages the state
to take certain values along a curve in 2D or a surface in 3D. In the
discretisation of this problem, which uses piecewise linear finite elements, we
allow the hypersurface to be approximated e.g. by a polyhedral hypersurface.
This can lead to simpler numerical methods, however it complicates the
numerical analysis. We prove a priori $L^2$ error estimates for the control and
present numerical results that agree with these. A comparison is also made to
point control problems.
