In this paper, we study finite element approximations to some elliptic optimal control problems with controls acting on a lower dimensional manifold which can be a point, a curve, or a surface. We use piecewise linear finite elements to approximate state variables, while utilizing the variational discretization to approximate control variables. We derive several a priori error estimates for optimal controls from different cases depending on the dimensions of the computational domain and the manifold where controls act. We ends up with extensive numerical experiments which confirm our theoretical findings.