Optimal Control of the 2D Boussinesq Equations with Finite Codimension Constraints

Abstract

We study an optimal control problem (OCP) governed by the two-dimensional (2D) Boussinesq equations, where the right-hand sides of the velocity and temperature equations are projected onto the corresponding tangent spaces of the manifolds $\mathcal{M}_1:=\{y\in\mathbb{H}:\|y\|_{\mathbb{H}}=1\}$ and $\mathcal{M}_2:=\{θ\in H:\|θ\|_H=1\}$, respectively. The resulting evolution is subject to finite-codimensional constraints, under which the state $(y,θ)$ remains on $\mathcal{M}:=\mathcal{M}_1\times\mathcal{M}_2$ throughout the time interval. A Faedo--Galerkin construction provides a unique, global, strong solution to the constrained Boussinesq system for external controls $\mathfrak{H}\in L^2(0,T;\mathbb{H})$, along with uniform energy bounds. In addition, we establish the existence of an optimal triplet $(y^*,θ^*,\mathfrak{H}^*)$ that minimizes the quadratic cost subject to the constrained Boussinesq system. Necessary conditions of the first order are obtained through a coupled state--adjoint--control system. The optimal state $(y^*,θ^*)$ remains on $\mathcal{M}$, while the optimal control lies in the tangent space: $\mathfrak{H}^*(t)\in T_{y^*(t)}\mathcal{M}_1$, for all $t\in[0,T]$. We also show that the map $\mathfrak{H}\mapsto(y_{\mathfrak{H}},θ_{\mathfrak{H}})$ in $(C([0,T];\mathbb{H})\cap L^2(0,T;\mathbb{V}))\times(C([0,T];H)\cap L^2(0,T;V))$ is Fréchet differentiable. The linearized state is the unique weak solution to the constrained linearized Boussinesq system and satisfies the necessary tangency condition. Hence, the well-posedness and first-order optimality conditions are established for the Boussinesq system under nonlinear manifold constraints.