  This paper is devoted to the numerical analysis of a control constrained
distributed optimal control problem subject to a time fractional diffusion
equation with non-smooth initial data. The solutions of state and co-state are
decomposed into singular and regular parts, and some growth estimates are
obtained for the singular parts. Following the variational discretization
concept, a full discretization is applied to the corresponding state and
co-state equations by using linear conforming finite element method in space
and piecewise constant discontinuous Galerkin method in time. By the growth
estimates, error estimates are derived with non-smooth initial data. In
particular, graded temporal grids are used to obtain the first-order temporal
accuracy. Finally, numerical experiments are performed to verify the
theoretical results.
