发表机构
Indian Institute of Technology Roorkee（印度理工学院鲁尔基分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对投影到单位球面切空间的二维Navier-Stokes方程，提出有限余维约束最优控制问题，证明强解存在唯一性及最优对存在性，并推导一阶最优性条件与Fréchet可微性。
AI 中文摘要
我们研究了投影到$\mathbb{H}$-单位球面流形$\mathcal{M}:= \{\boldsymbol{u} \in \mathbb{H}: \\|\boldsymbol{u}\\|_{\mathbb{H}}=1\}$的切空间上的二维Navier-Stokes方程的最优控制问题。这种几何投影产生了一个有限余维的约束演化系统，确保状态在所有时刻保持在$\mathcal{M}$上。利用Faedo-Galerkin逼近技术，我们建立了具有外部控制$\boldsymbol{U} \in \mathrm{L}^2(0,T;\mathbb{H})$的约束Navier-Stokes方程的强解的全局存在性和唯一性，以及一致能量估计。相应的最优控制问题，即在流形约束下最小化二次代价泛函，被证明至少存在一个最优对$(\boldsymbol{u}^*,\boldsymbol{U}^*)$。我们通过制定一个最优控制系统来推导一阶必要最优性条件，该系统的状态-控制对$(\boldsymbol{u}^*,\boldsymbol{U}^*)$在约束流形$\mathcal{M}$的切丛$T\mathcal{M}$上演化。在此设置中，控制满足$\boldsymbol{U}^*(t)\in T_{\boldsymbol{u}^*(t)}\mathcal{M}$对所有$t\in[0,T]$成立，确保动力学保持在切丛$T\mathcal{M}$内。此外，我们建立了控制到状态映射$\boldsymbol{U}\mapsto \boldsymbol{u}_{\boldsymbol{U}}$在$\mathrm{C}([0,T];\mathbb{H})\cap\mathrm{L}^2(0,T;\mathbb{V})$中的Fréchet可微性，相应的线性化状态被表征为保持切触条件的线性化约束Navier-Stokes系统的唯一弱解。这些结果为非线性流形上约束流体流动的最优控制提供了严格的分析基础。
英文摘要
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 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 $\boldsymbol{U} \in \mathrm{L}^2(0,T;\mathbb{H})$, 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 $(\boldsymbol{u}^*,\boldsymbol{U}^*)$. We derive the first-order necessary optimality conditions by formulating an optimal control system whose state-control pair $(\boldsymbol{u}^*,\boldsymbol{U}^*)$ evolves on the tangent bundle $T\mathcal{M}$ of the constraint manifold $\mathcal{M}$. In this setting, the control satisfies $\boldsymbol{U}^*(t)\in T_{\boldsymbol{u}^*(t)}\mathcal{M}$ 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 $\boldsymbol{U}\mapsto \boldsymbol{u}_{\boldsymbol{U}}$ 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.
