发表机构
University of California, San Diego（加利福尼亚大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次将Fredholm反推用于线性化2D Poiseuille通道流，在任意雷诺数下稳定流动并指定各波数衰减率，在雷诺数30000下验证，使流动减慢约二十倍。
AI 中文摘要
我们在任意雷诺数下稳定线性化的二维Poiseuille通道流动，并将每个波数的衰减率指派到用户指定的任意速率。执行器配置与Vazquez--Krstic（IEEE TAC, 2007，下文简称VK'07）相同：仅在一个壁面上的两个速度分量。这是首次将Fredholm反推设计应用于Navier--Stokes系统。在VK'07中，壁法向输入使系统空间因果化，随后切向输入闭合Volterra反推回路。此处，法向输入携带一个任意小增益的有限秩反馈，其唯一作用（依据Heymann引理（TAC, 1968））是赋予系统简单谱，并使每个模态均可由切向速度控制。切向输入随后施加一个稳定化反馈，该反馈通过可逆的Fredholm变换获得，变换到向左平移任意衰减的Stokes目标偏微分方程。不假设Orr--Sommerfeld算子特征值的简单性，并消除了伴随Fredholm反推的植物系数值排除条件。我们不加修饰地陈述与Volterra设计的比较：Fredholm架构强大，但其增益是谱形式的，而VK'07是闭式解，两者中更简单，且Fredholm方法从中获取其模型、执行器和目标。这里展示的是Fredholm方法在其迄今最具挑战性的系统上的可达范围。该反馈在雷诺数30000下测试，闭环谱的主导部分被指派为将流动减慢约二十倍。
英文摘要
We stabilize the linearized 2-D Poiseuille channel flow at arbitrary Reynolds numbers, and assign the decay of every wavenumber to any user-prescribed rate. The actuation is as in Vazquez--Krstic (IEEE TAC, 2007), VK'07 below: the two velocity components at only one wall. This is the first Fredholm backstepping design for a Navier--Stokes system. In VK'07 the wall-normal input renders the plant spatially causal, after which the tangential input closes a Volterra backstepping loop. Here the normal input carries a finite-rank feedback of arbitrarily small gain, whose sole role, in the sense of Heymann's lemma (TAC, 1968), is to give the plant simple spectrum and make every mode controllable from the tangential velocity. The tangential input then applies a stabilizing feedback, obtained through an invertible Fredholm transform to a Stokes target PDE, shifted left by an arbitrary decay. No simplicity of eigenvalues of the Orr--Sommerfeld operator is assumed and the exclusions of plant coefficient values that have accompanied Fredholm backstepping are removed. We state the comparison with the Volterra design without embellishment: the Fredholm architecture is powerful, but its gain is spectral where VK'07 is closed-form, the simpler of the two, and one from which the Fredholm approach takes its model, its actuation, and its target. What is demonstrated here is the reach of the Fredholm method on its most demanding plant to date. The feedback is tested at Reynolds number $30000$, with the dominant part of the closed-loop spectrum assigned so that it amounts to slowing the flow about twentyfold.
