  This paper focuses on the quasi-optimality of an adaptive nonconforming FEM
for a distributed optimal control problem governed by the Stokes equations. The
nonconforming lowest order Crouzeix-Raviart element and piecewise constant
spaces are used to discretize the velocity and pressure variables,
respectively. The control variable is discretized using a variational approach.
We present error equivalence results at both continuous and discrete levels,
leading to a priori and a posteriori error estimates. The quasi-optimal
convergence rates of the adaptive algorithm are established based on a general
axiomatic framework that includes stability, reduction, discrete reliability,
and quasi-orthogonality.
