  In this paper, optimal control problems governed by diffusion equations with
Dirichlet and Neumann boundary conditions are investigated in the framework of
the gradient discretisation method. Gradient schemes are defined for the
optimality system of the control problem. Error estimates for state, adjoint
and control variables are derived. Superconvergence results for gradient
schemes under realistic regularity assumptions on the exact solution is
discussed. These super-convergence results are shown to apply to non-conforming
$\mathbb{P}_1$ finite elements, and to the mixed/hybrid mimetic finite
differences. Results of numerical experiments are demonstrated for the
conforming, nonconforming and mixed/hybrid mimetic finite difference schemes.
