The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier–Stokes equations in the vicinity of an unstable equilibrium solution, by means of a ‘minimal’ and ‘least’ invasive feedback strategy which consists of a control pair \(\{ v,u \}\) (Lasiecka and Triggiani in Nonlinear Anal 121:424–446, 2015). Here v is a tangential boundary feedback control, acting on an arbitrary small part \({\widetilde{\varGamma }}\) of the boundary \(\varGamma \); u is a localized, interior feedback control, acting tangentially on an arbitrarily small subset \(\omega \) of the interior supported by \({{\widetilde{\varGamma }}}\). The ideal strategy of taking \(u = 0\) on \(\omega \) is not sufficient. A question left open in the literature is: can such feedback control v of the pair \(\{ v,u \}\) be asserted to be finite dimensional also in dimension \(d = 3\)? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control v, it is here then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with a “right" Besov space setting of lower regularity. These spaces are ‘close’ to \(L^3(\varOmega )\) for \(d = 3\). This functional setting is significant. It is in line with recent well-posedness results in the full space of the non-controlled N–S equations (Escauriaza et al. in Math Subj Classif 35K:76D, 1991; Rusin and Sverak in Minimal initial data for potential Navier–Stokes singularities. arXiv:0911.0500; Jia and Šverák in SIAM J Math Anal 45(3):1448–1459, 2013; Gallagher et al. in Math Ann 355(4):1527–1559, 2013). A double key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions while having a sufficiently high topological level to handle the 3d-nonlinearity in the analysis of well-posedness and uniform stabilization. The proof is constructive and is “optimal” also regarding the “minimal” number of tangential boundary feedback controllers needed. The new setting requires the solution of novel technical and conceptual issues. These include establishing maximal regularity up to \(T = \infty \) in the required suitably identified “right" Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. This result is also a new contribution to the area of maximal regularity as the operator to which it applies incorporates a boundary feedback control term rather than homogeneous boundary conditions. It escapes direct use of perturbation theory. Finally, the very ability to stabilize even the finite dimensional unstable projected system is linked to a Unique Continuation Property of a suitably over-determined (adjoint) Oseen eigenproblem, which requires the presence of the interior tangential-like control u acting on \(\omega \).

本论文通过“最小”和“最小”侵入性反馈，为在不稳定平衡解附近的 3 维纳维-斯托克斯方程的均匀稳定理论中公认的开放问题提供了肯定的解决方案由控制对\(\{ v,u \}\)组成的策略（Lasiecka 和 Triggiani in Nonlinear Anal 121:424–446, 2015）。这里v是切向边界反馈控制，作用于边界\(\varGamma \)的任意小部分\({\widetilde{\varGamma }} \)；u是一个局部的内部反馈控制，切向作用于由以下支持的内部的任意小子集\(\omega \)\({{\widetilde{\varGamma }}}\)。在\(\omega \)上取\(u = 0\)的理想策略是不够的。文献中一个悬而未决的问题是：对\(\{ v,u \}\) 的这种反馈控制v能否在维度\(d = 3\) 中也被断言为有限维？我们在这里对这个问题给出了肯定的回答，从而建立了一个最优结果。为了实现反馈切向边界控制v的所需有限维数，这里有必要放弃过去文献的 Hilbert-Sobolev 函数设置，并用较低正则的“正确”Besov 空间设置替换它。这些空间是“关闭的” ' 到\(L^3(\varOmega )\)对于\(d = 3\). 这个功能设置很重要。它与最近在非受控 N-S 方程的整个空间中的适定结果一致（Escauriaza 等人在 Math Subj Classif 35K:76D, 1991 中；Rusin 和 Sverak 在 Minimal initial data for potential Navier-Stokes arXiv:0911.0500；Jia 和 Šverák in SIAM J Math Anal 45(3):1448–1459, 2013；Gallagher et al. in Math Ann 355(4):1527–1559, 2013)。这种具有紧指数的 Besov 空间的双重关键特征是它们不能识别兼容性条件，同时具有足够高的拓扑级别来处理适定性和均匀稳定性分析中的 3d 非线性。该证明是建设性的，并且对于所需的切向边界反馈控制器的“最小”数量也是“最佳的”。新环境需要解决新的技术和概念问题。这些包括建立最大的规律性\(T = \infty \)在整个闭环线性化问题所需的适当识别的“正确” Besov 设置中，在边界上应用切向反馈控制。这个结果也是对最大规律性区域的新贡献，因为它所应用的算子结合了边界反馈控制项而不是齐次边界条件。它避免了直接使用微扰理论。最后，即使是有限维不稳定投影系统的稳定能力也与适当过-确定的（伴随）Oseen 特征问题，它需要存在作用于\(\omega \)的内部切线状控制u。