Feedback control for fluid mixing via advection
Feedback control for fluid mixing via advection
复制标题
DOI：
10.1016/j.jde.2023.07.009
复制
发表时间：
2023-11
影响因子：
2.4
通讯作者：
W. Hu;C. N. Rautenberg;Xiaoming Zheng
中科院分区：
文献类型：
--
作者：
W. Hu;C. N. Rautenberg;Xiaoming Zheng
本文研究流体平流混合问题的非线性反馈控制设计。整体动力学由一个开有界连通域Ω <$Rd（d= 2或d= 3）中的输运方程和斯托克斯方程控制。基于瞬时控制的思想，以及直接近似的最优性系统来自最优开环控制问题的反馈法律。结果表明，在适当的数值离散方案下，两种方法产生相同的次优反馈律。另一方面，不同的离散方案可能导致不同规律性的反馈律，这决定了不同的混合结果。利用H1（Ω）的对偶空间（H1（Ω））′的Sobolev范数作为混合范数，根据已知的弱收敛性质对混合进行量化.主要的挑战是遇到的闭环系统的渐近行为的分析，由于没有扩散的传输方程，连同它的非线性耦合与流动方程。为了解决这些问题，我们首先建立了速度的衰减特性，这反过来又有助于获得标量混合及其长时间行为的估计。最后，混合连续Galerkin（CG）和间断Galerkin（DG）方法离散的闭环系统。数值实验证明了我们的想法，并比较了不同的反馈律的有效性。
This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain Ω⊂ R d, with d= 2 or d= 3. The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space (H 1 (Ω))′ of H 1 (Ω) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.