发表机构
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究二维二级流体输运标量的最优边界混合，通过Navier-滑移边界控制切向牵引力，证明状态适定性与最优控制存在性，并在特定条件下获得唯一性。
AI 中文摘要
我们研究了由二维不可压缩二级流体输运的无扩散标量的最优边界混合。控制变量是Navier-滑移边界条件中的切向牵引力，主要目标是终端$(H^1(\Omega))'$混合范数，并辅以二次控制成本和可选的涡度奖励。二级本构定律引入了空间滤波加速度和广义涡度；在提升非齐次边界数据后，$\mathbf{g}$和$\partial_t\mathbf{g}$都进入状态方程，其中$\mathbf{g}$是边界控制变量。我们证明了被动标量的全局状态适定性和主动标量在共同的、与控制无关的局部区间上的适定性。然后，我们建立了最优控制的存在性、控制到状态映射在终端目标所需拓扑中的方向可微性，以及弱后向伴随公式。一个对偶恒等式导出了一阶变分不等式。在缺少摩擦系数$\beta$和涡度奖励权重$\zeta$（即$\beta=\zeta=0$）的情况下，额外的伴随正则性意味着对于所有足够大的控制惩罚$\gamma$，最优控制的唯一性。
英文摘要
We study optimal boundary mixing of a nondiffusive scalar transported by a two-dimensional incompressible second-grade fluid. The control is the tangential traction in a Navier-slip boundary condition, and the principal objective is the terminal $(H^1(Ω))'$ mix-norm, supplemented by a quadratic control cost and an optional enstrophy reward. The second-grade constitutive law introduces a spatially filtered acceleration and a generalized vorticity; after lifting the nonhomogeneous boundary data, both $\mathbf{g}$ and $\partial_t\mathbf{g}$ enter the state equation, where $\mathbf{g}$ is the boundary control variable. We prove global state well-posedness for a passive scalar and well-posedness on a common, control-independent local interval for an active scalar. We then establish existence of an optimal control, directional differentiability of the control-to-state map in the topology required by the terminal objective, and a weak backward adjoint formulation. A duality identity yields the first-order variational inequality. In the absence of the friction coefficient $β$ and enstrophy reward weight $ζ$ (that is, $β=ζ=0$), the additional adjoint regularity implies uniqueness of the optimal control for all sufficiently large control penalties $γ$.
