Control Problems for Strongly Coupled Non-Linear Partial Differential Equations

本研究项目旨在研究相互作用结构的控制、最优化、稳定性和长期行为的性质，这些结构由带有界面的非齐次强耦合偏微分方程组进行数学建模。描述整体结构的本构成分以及它们之间的耦合的单个偏微分方程式可以是线性的或非线性的、色散的和振荡的。Brbr将首先对互动结构的四个典型激励类进行调查。它们的目的是作为物理上有意义的交互和非平凡耦合的偏微分方程组的一般主题的基准案例。它们是：(1)利用“智能材料/结构”技术在结构声室中的降噪问题；(2)气动弹性中的亚、超音速三维气流中二维机翼的颤振控制；(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.