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 ℍ-unit sphere manifold ℳ := {u ∈ ℍ : |u|ℍ=1}. This geometric projection yields a constrained evolution system of finite codimension, ensuring that the state remains on 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 U ∈ L2(0,T;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 (u*,U*). We derive the first-order necessary optimality conditions by formulating an optimal control system whose state-control pair (u*,U*) evolves on the tangent bundle Tℳ of the constraint manifold ℳ. In this setting, the control satisfies U*(t)∈ Tu*(t)ℳ for all t∈[0,T], ensuring that the dynamics remain confined to the tangent bundle Tℳ. Furthermore, we establish the Fréchet differentiability of the control-to-state mapping U↦ uU in C([0,T];ℍ)∩L2(0,T;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.
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BibTeX
@misc{mohan2026optimal,
title = {{Optimal Control of 2D Navier--Stokes Equations with Constraints of Finite Codimension}},
author = {Manil T. Mohan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00249},
eprint = {2610.00249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00249},
}
RIS
TY - UNPB
TI - Optimal Control of 2D Navier--Stokes Equations with Constraints of Finite Codimension
AU - Manil T. Mohan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00249
UR - https://arxiv.org/abs/2610.00249
AB - We study an optimal control problem for the two-dimensional Navier-Stokes equations projected onto the tangent space of the ℍ-unit sphere manifold ℳ := {u ∈ ℍ : |u|_ℍ=1}. This geometric projection yields a constrained evolution system of finite codimension, ensuring that the state remains on 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 U ∈ L^2(0,T;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 (u^*,U^*). We derive the first-order necessary optimality conditions by formulating an optimal control system whose state-control pair (u^*,U^*) evolves on the tangent bundle Tℳ of the constraint manifold ℳ. In this setting, the control satisfies U^*(t)∈ T_u^*(t)ℳ for all t∈[0,T], ensuring that the dynamics remain confined to the tangent bundle Tℳ. Furthermore, we establish the Fréchet differentiability of the control-to-state mapping U↦ u_U in C([0,T];ℍ)∩L^2(0,T;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.
ER -