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

Abstract

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 $\mathcal{M}:=\{\mathbf{u}\in \mathbb{H}:\Vert \mathbf{u}{\Vert }_{\mathbb{H}}=1\}$. This geometric projection yields a constrained evolution system of finite codimension, ensuring that the state remains on $\mathcal{M}$ for all times. Using a Faedo-Galerkin approximation technique, we establish the global existence and uniqueness of strong solutions to the constrained Navier-Stokes equations with external controls $\mathbf{U}\in {\mathrm{L}}^2(0,T;\mathbb{H})$, together with uniform energy estimates. The associated optimal control problem, minimizing a quadratic cost functional subject to the manifold constraint, is shown to admit at least one optimal pair $({\mathbf{u}}^{\ast },{\mathbf{U}}^{\ast })$. We derive the first-order necessary optimality conditions by formulating an optimal control system whose state-control pair $({\mathbf{u}}^{\ast },{\mathbf{U}}^{\ast })$ evolves on the tangent bundle $T\mathcal{M}$ of the constraint manifold $\mathcal{M}$. In this setting, the control satisfies ${\mathbf{U}}^{\ast }(t)\in T_{{\mathbf{u}}^{\ast }(t)}\mathcal{M}$ for all $t\in [0,T]$, ensuring that the dynamics remain confined to the tangent bundle $T\mathcal{M}$. Furthermore, we establish the Fréchet differentiability of the control-to-state mapping $\mathbf{U}\mapsto {\mathbf{u}}_{\mathbf{U}}$ in $\mathrm{C}([0,T];\mathbb{H})\cap {\mathrm{L}}^2(0,T;\mathbb{V})$, with the corresponding linearized state characterized as the unique weak solution of a linearized constrained Navier-Stokes system preserving the tangency condition. These results provide a rigorous analytical foundation for the optimal control of fluid flows constrained to nonlinear manifolds.