  This paper studies the benefits of pressure-robust discretizations in the
scope of optimal control of incompressible flows. Gradient forces that may
appear in the data can have a negative impact on the accuracy of state and
control and can only be correctly balanced if their $L^2$-orthogonality onto
discretely divergence-free test functions is restored. Perfectly orthogonal
divergence-free discretizations or divergence-free reconstructions of these
test functions do the trick and lead to much better analytic a priori estimates
that are also validated in numerical examples.
