  In this article, we derive \textit{a posteriori} error estimates for the
Dirichlet boundary control problem governed by Stokes equation. An energy-based
method has been deployed to solve the Dirichlet boundary control problem. We
employ an inf-sup stable finite element discretization scheme by using
$\mathbf{P}_1$ elements(in the fine mesh) for the velocity and control variable
and $P_0$ elements(in the coarse mesh) for the pressure variable. We derive an
\textit{a posteriori} error estimator for the state, adjoint state, and control
error. The control error estimator generalizes the standard residual type
estimator of the unconstrained Dirichlet boundary control problems, by
additional terms at the contact boundary addressing the non-linearity. We prove
the reliability and efficiency of the estimator. Theoretical results are
illustrated by some numerical experiments.
