  This work examines the distributed optimal control of generalized Oseen
equations with non-constant viscosity. We propose and analyze a new conforming
augmented mixed finite element method and a Discontinuous Galerkin (DG) method
for the velocity-vorticity-pressure formulation. The continuous formulation,
which incorporates least-squares terms from both the constitutive equation and
the incompressibility condition, is well-posed under certain assumptions on the
viscosity parameter. The CG method is divergence-conforming and suits any
Stokes inf-sup stable velocity-pressure finite element pair, while a generic
discrete space approximates vorticity. The DG scheme employs a stabilization
technique, and a piecewise constant discretization estimates the control
variable. We establish optimal a priori and residual-based a posteriori error
estimates for the proposed schemes. Finally, we provide numerical experiments
to showcase the method's performance and effectiveness.
