In this work, we analyze three hybridized discontinuous Galerkin finite element methods for the control constrained Oseen equations with a non-constant viscosity. This formulation reduces globally coupled degrees of freedom and provides a divergence-conforming and pointwise divergence-free velocity field. We use a variational discretization approach for the approximation of optimal control by using a optimize-then-discretize strategy. An optimal convergence of \(O(h^{2})\) in the diffusion-dominated regime and a sub-optimal \(O(h^{3/2})\) in the convection-dominated regime is established for the velocity and control in the \(L^{2}\) -norm for all three schemes in a unified setting. Additionally, the optimal error estimates in discrete \(H^1\) -norm for velocity and in the \(L^{2}\) -norm for pressure variable are derived. We further derive a novel residual-based a posteriori error estimator in the energy-norm and a seminorm associated with the convective term that is both reliable and efficient for the proposed HDG scheme. Finally, several numerical experiments and benchmark problems in two and three dimensions are presented to demonstrate the accuracy, robustness, and overall performance of the proposed schemes.
