发表机构
University of California, San Diego（加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对具有空间变化反扩散的 Kuramoto--Sivashinsky 方程，提出双输入 Fredholm 反步设计，利用预反馈和曲率输入实现快速镇定，并通过神经算子近似增益，数值验证了约 0.1% 的误差和稳定效果。
AI 中文摘要
我们提出了首个针对具有空间变化反扩散系数的 Kuramoto--Sivashinsky 方程快速镇定的反馈设计。对于常数系数，Coron 和 Lü (2015) 的单输入 Fredholm 设计排除了一个离散的值集合，在这些值处，重复的不稳定特征值导致可控性丧失。我们通过引入第二个边界输入并为两个输入分配不同的角色来克服这一障碍。关键思想受 Heymann 引理启发，即完全利用边界值 $u(0,t)$ 进行预反馈，使修改后的被控对象通过曲率输入 $u_{xx}(0,t)$ 变得可控。后者随后通过 Fredholm 反步变换镇定被控对象。我们证明两个输入足以实现可控性，并且当被控对象具有不稳定二重特征值时是必要的。然而，Fredholm 核仍必须近似以实现实现。因此，为了能够进行核和增益近似，我们证明了在紧致可接受设计类上系数到增益设计映射的连续性。与使用逐次逼近的基于 Volterra 的连续性证明不同，我们的证明使用模态表示来控制谱数据、逆系数系统以及核和增益级数的尾部。这产生了在整个类中达到任意规定 $L^2$ 精度的增益的单一神经算子近似。最后，我们在精确增益和足够精确的近似下建立了非线性闭环系统的快速局部镇定。我们以数值结果结束，这些结果说明了规定的衰减率和近似的计算成本。特别是，我们训练了一个傅里叶神经算子，它实现了大约 $0.1\%$ 的典型相对增益误差，并稳定了所有测试的保留案例，包括具有不稳定二重特征值的被控对象。
英文摘要
We develop the first feedback design for rapid stabilization of the Kuramoto--Sivashinsky equation with a spatially varying anti-diffusion coefficient. For constant coefficients, the single-input Fredholm design of Coron and Lü (2015) excludes a discrete set of values at which repeated unstable eigenvalues cause a loss of controllability. We overcome this obstruction by introducing a second boundary input and assigning the two inputs distinct roles. The key idea, inspired by Heymann's Lemma, is to use the boundary value $u(0,t)$ entirely for a pre-feedback that renders the modified plant controllable through the curvature input $u_{xx}(0,t)$. The latter input then stabilizes the plant through a Fredholm backstepping transformation. We show that two inputs suffice for controllability and are necessary when the plant has an unstable double eigenvalue. However, the Fredholm kernel still must be approximated for implementation. Hence, to enable kernel and gain approximation, we prove continuity of the coefficient-to-gain design map on compact admissible design classes. Unlike Volterra-based continuity proofs using successive approximations, our proof uses the modal representation to control the spectral data, the inverse coefficient system, and the tails of the kernel and gain series. This yields a single neural operator approximation of the gain to any prescribed $L^2$ accuracy across the class. Finally, we establish rapid local stabilization of the nonlinear closed-loop system under both the exact gains and sufficiently accurate approximations. We conclude with numerical results that illustrate prescribed decay rates and the computational cost of the approximations. In particular, we train a Fourier neural operator that achieves typical relative gain errors of approximately $0.1\%$ and stabilizes all held-out cases tested, including a plant with an unstable double eigenvalue.
Comments46 pages
