  We consider finite element solutions to quadratic optimization problems,
where the state depends on the control via a well-posed linear partial
differential equation. Exploiting the structure of a suitably reduced
optimality system, we prove that the combined error in the state and adjoint
state of the variational discretization is bounded by the best approximation
error in the underlying discrete spaces. The constant in this bound depends on
the inverse square-root of the Tikhonov regularization parameter. Furthermore,
if the operators of control-action and observation are compact, this
quasi-best-approximation constant becomes independent of the Tikhonov parameter
as the meshsize tends to $0$ and we give quantitative relationships between
meshsize and Tikhonov parameter ensuring this independence. We also derive
generalizations of these results when the control variable is discretized or
when it is taken from a convex set.
