波类型PDE系统的混沌及相关控制问题

There are lots of complex phenomena in infinite dimensional systems, especially in the systems governed by partial differential equations(PDE), such as the turbulence in the famous Navier-Stokes equation. Theoretical description of chaos, as well as rigorous analysis of onset of chaos, are very difficult and challenging, and have very important theoretical and practical significance. This project mainly studies chaos in systems governed by higher dimensional wave-type PDE, and tries to prove the onset of chaos and show its visualization. Furthermore, this project considers the observer design problem of the system studied here. Since the nonlinear boundary term, the classical methods of observer design cannot be directly applied to this kind of infinite systems, this project tries to give a new method of observer design and rigorously prove the effectiveness of this method, which has many possible applications.

对于无穷维系统，特别在由偏微分方程描述的无穷维系统中，存在着大量的复杂现象，比如著名的Navier-Stokes 方程中的湍流现象。从理论上描述和分析这些复杂现象或混沌现象具有十分重要的理论和实际意义，同时也是十分具有挑战性的。本项目将PDE系统混沌化的研究拓展到高维状态空间，并尝试证明，分析和可视化几种高维波类型PDE系统的混沌。此外本项目还考虑一类混沌系统的观测器问题，这类混沌系统带有非线性的边界项，经典的观测器设计方法已经失效，本项目尝试给出一种新的观测器设计方法，并给出严格的分析，无论是理论意义还是实际背景，都有着广泛的应用前景。

本项目主要研究了如下问题：（a）二维弱双曲系统的混沌振动，给出了混沌的刻画，这对高维波方程系统混沌的研究具有重要的意义。（b）从能量函数的角度来研究一维波动方程中的混沌，我们确定了两种不同的方式都可以引起系统的混沌。这可以帮住我们弄清系统能量和系统混沌之间的关系。（c）利用算子扰动来研究一类带有位移反馈的波方程的混沌问题；这里取得的成果可以帮助我们了解这类混沌系统的结构稳定性。（d）一类混沌波方程系统的观测器设计；我们引入了时滞反馈的方法，此方法可以应用到其他混沌系统的观测器设计中。（e）一维波方程系统的反馈稳定性问题—最佳阻尼率的研究。