可由有限维系统诱导的无穷维系统的混沌性质及控制

Chaos is the eternal topic in nonlinear science. Compared with compact dynamical systems, the obtained theoretical results for chaos of infinite-dimensional dynamical systems are relatively little for the well known reasons. There still exist a lot of infinite-dimensional systems governed by partial differential equations in which chaotic phenomena are routinely observed or numerically simulated, but rigorous proofs for these phenomena are still few and even there are no effective ways to describe such complex behaviors. In this project, we try to propose a new way to study the chaos and their control for a class of partial differential equation systems. Such kind of systems can be viewed as a particular infinite-dimensional systems which are induced by finite-dimensional systems.To do this end, we first study in general sense the relationship between the complex behaviors of a finite-dimensional dynamical system and that of its inducing infinite-dimensional dynamical system. Then we study chaotic behaiors for a class of partial differential equation systems arising in practice, especially for the wave equations with boundary feedback controls, and their control problems, such as the boundary feedback stabilization and observer design, etc., in terms of dynamical theoretical approach. In this project, we will provide a new possible way to investigate the chaos and some control problems for a class of partial differential equation systems. It is significant in the sense of both theory and applications.

混沌是非线性科学永恒的课题。相对于紧致系统而言，对无穷维系统混沌的研究起步较晚，所得理论结果相对较少。还存在大量由偏微分方程描述的无穷维系统，通过模拟（包括数值模拟或仿真）可观测到混沌现象，但理论上还没有严格的证明，甚至还没法给出合适的数学描述.本项目研究一类偏微分方程系统的混沌性质及其控制问题，该类系统都可以看成由有限维系统诱导的无穷维系统。所以我们首先从一般意义上研究有限维动力系统的复杂性与由它诱导的无穷维系统的复杂性之间的关系,得到一般理论性结果。然后研究一类具实际背景的偏微分方程系统，特别是带边界反馈控制的偏微分系统，的混沌特性，及其边界反馈能稳性和边界观测器设计等问题。为研究偏微分方程系统的复杂性和利用动力系统理论方法研究其控制问题提供一种新的途径，具有重要的理论和应用意义。

在实际中，存在一类由偏微分方程描述的无穷为系统，其解可有限维系统所决定。按研究计划，我们研究一类可由有限维系统诱导的无穷维系统（亦称为原系统的包咯系统）的动力学性质。理论上研究有限维动力系统的复杂性与由它诱导的无穷维系统的复杂性之间的关系。应用上研究一类具实际背景的偏微分方程系统，特别是带边界反馈控制的偏微分系统，的混沌特性，以及边界反馈混沌化等问题，主要研究内容包括如下几个方面：（1），研究一维系统（线段系统）与其诱导的无穷维系统复杂性之间的关系，得到了某些动力学性质（如若混合，混合，混沌等）的向上遗传性，并把所得结果应用到具体的偏微分方程系统中去；（2），讨论一般动力系统与由其诱导的包咯系统之间的关系，推广了已有的相关结果；（3），讨论了一类由偏微分方程组描述的无穷维系统的复杂性。此外，我们还讨论了线性切换系统稳定性和混沌特性，以及一类双线性系统的能控性等。所得结果具有摘要的理论意义。