  This paper focuses on the stability of solutions for a velocity-tracking
problem associated with the two-dimensional Navier-Stokes equations. The
considered optimal control problem does not possess any regularizer in the
cost, and hence bang-bang solutions can be expected. We investigate
perturbations that account for uncertainty in the tracking data and the initial
condition of the state, and analyze the convergence rate of solutions when the
original problem is regularized by the Tikhonov term. The stability analysis
relies on the H\"older subregularity of the optimality mapping, which stems
from the necessary conditions of the problem.
