  We consider an approximating control design for optimal mixing of a
non-dissipative scalar field $\theta$ in unsteady Stokes flows. The objective
of our approach is to achieve optimal mixing at a given final time $T>0$, via
the active control of the flow velocity $v$ through boundary inputs. Due to the
zero diffusivity of the scalar field $\theta$, establishing the well-posedness
of its G\^{a}teaux derivative requires $\sup_{t\in[0,T]}\|\nabla
\theta\|_{L^2}<\infty$, which in turn demands the flow velocity field to
satisfy the condition $ \int^{T}_{0}\|\nabla v\|_{L^{\infty}(\Omega)}\,
dt<\infty$. This condition results in the need to penalize the time derivative
of the boundary control in the cost functional. As a result, the optimality
system becomes difficult to solve \cite{hu2017boundary}. Our current
approximating approach will provide a more transparent optimality system, with
the set of admissible controls being $L^{2}$ in both time and space. This is
achieved by first introducing a small diffusivity to the scalar equation and
then establishing a rigorous analysis of convergence of the approximating
control problem to the original one as the diffusivity approaches to zero.
Uniqueness of the optimal solution is obtained for the two dimensional case.
