Control Problems for Strongly Coupled Non-Linear Partial Differential Equations

本研究旨在探讨具有界面的强耦合偏微分方程非齐次系统对相互作用结构的控制、优化、稳定性和长期行为的数学建模。单个偏微分方程描述了整体结构的组成部分以及它们之间的耦合，可以是线性的或非线性的、色散的和振荡的。brbr将对四种典型的交互结构激励类进行初步调查。它们旨在作为物理上重要的相互作用和非平凡耦合偏微分方程的一般主题的基准案例。它们是：(1)通过使用“智能材料/结构”技术，结构声室中的降噪问题；(2)浸没在亚音速或超音速三维气流中的二维机翼气动弹性颤振控制；(3)磁流体动力学方程控制中的等离子体加热和等离子体约束；(4)粘性不可压缩流体动力学中湍流的渐近抑制。

The present research project aims at investigating the properties of control, optimization, stability and long-time behavior of interactive structures, which are mathematically modeled by inhomogeneous systems of strongly coupled partial differential equations with an interface. Both the single partial differential equations describing each a constitutive component of the overall structure as well as the coupling between them, may be linear or non-linear, dispersive, and oscillatory. brbr Investigation will be carried out initially on four canonical motivating classes of interactive structures. They are intended to serve as benchmark cases for the general topic of physically significant interactive and non-trivially coupled Partial Differential Equations. They are: (1) the noise reduction problem in a structural acoustic chamber, by use of 'smart material/structure' technology; (2) flutter control of a 2-dimensional wing immersed in a subsonic or supersonic 3-dimensional gas flow in aeroelasticity; (3) plasma heating and plasma confinement in the control of magnetohydrodynamics equations; (4) asymptotic suppression of turbulence in viscous incompressible fluid dynamics.