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 ℳ1:={y∈ℍ:|y|ℍ=1} and ℳ2:={θ∈ H:|θ|H=1}, respectively. The resulting evolution is subject to finite-codimensional constraints, under which the state (y,θ) remains on ℳ:=ℳ1×ℳ2 throughout the time interval. A Faedo--Galerkin construction provides a unique, global, strong solution to the constrained Boussinesq system for external controls H∈ L2(0,T;ℍ), along with uniform energy bounds. In addition, we establish the existence of an optimal triplet (y*,θ*,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 ℳ, while the optimal control lies in the tangent space: H*(t)∈ Ty*(t)ℳ1, for all t∈[0,T]. We also show that the map H↦(yH,θH) in (C([0,T];ℍ)∩ L2(0,T;𝕍))×(C([0,T];H)∩ L2(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.
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BibTeX
@misc{manna2026optimal,
title = {{Optimal Control of the 2D Boussinesq Equations with Finite Codimension Constraints}},
author = {Utpal Manna and Manil T. Mohan and Sangram Satpathi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33828},
eprint = {2610.33828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33828},
}
RIS
TY - UNPB
TI - Optimal Control of the 2D Boussinesq Equations with Finite Codimension Constraints
AU - Utpal Manna
AU - Manil T. Mohan
AU - Sangram Satpathi
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33828
UR - https://arxiv.org/abs/2610.33828
AB - 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 ℳ_1:={y∈ℍ:|y|_ℍ=1} and ℳ_2:={θ∈ H:|θ|_H=1}, respectively. The resulting evolution is subject to finite-codimensional constraints, under which the state (y,θ) remains on ℳ:=ℳ_1×ℳ_2 throughout the time interval. A Faedo--Galerkin construction provides a unique, global, strong solution to the constrained Boussinesq system for external controls H∈ L^2(0,T;ℍ), along with uniform energy bounds. In addition, we establish the existence of an optimal triplet (y^*,θ^*,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 ℳ, while the optimal control lies in the tangent space: H^*(t)∈ T_y^*(t)ℳ_1, for all t∈[0,T]. We also show that the map H↦(y_H,θ_H) in (C([0,T];ℍ)∩ L^2(0,T;𝕍))×(C([0,T];H)∩ 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.
ER -