Optimal Control of 2D Navier--Stokes Equations with Constraints of Finite Codimension

We study an optimal control problem for the two-dimensional Navier-Stokes equations projected onto the tangent space of the $\mathbb{H}$-unit sphere manifold $\mathcal{M} := \{\boldsymbol{u} \in \mathbb{H} : \|\boldsymbol{u}\|_{\mathbb{H}}=1\}$. This geometric projection yields a constrained evoluti