耦合无限维系统的稳定、同步与控制

A coupled infinite dimensional system consists of partial differential equations, ordinary differential equations and integral equations coupled through certain conditions. Coupled systems have attracted increasing research activities in the field of the control theory since they are a convenient modeling paradigm for a variety of control applications and considerable progress has been made in the development of technologies for the design and manufacturing of the coupled systems. The properties of coupled systems depend on sub-systems, the match among the sub-systems, the coupling law, geometry of the domain, controllers, etc. Therefore, the analysis about the coupled infinite dimensional systems is very sophisticated. As one of the classical problems in control theory, stabilization of a system requires the design of feedback controllers and stability of the closed-loop system. Synchronization appears within a widespread field and means that all or part of states have same dynamical behavior. In this project, we study coupled systems which are governed by linear partial differential equations with physical and engineering background, such as flexible structures with distributed control, transmission systems, laminated beam and plate models, etc. First, based on the properties of sub-system and the coupling term, we design proper controllers to achieve the exponential stability, polynomial stability, logarithmic stability and strong stability of the closed-loop system. We also study the synchronization of some coupled infinite dimensional systems. A general method will be presented for this kind of problems by analyzing the controllability, observability and stabilization of the corresponding error system.

耦合无限维系统是由若干偏微分方程、常微分方程、积分方程等通过特定关系耦合而成的系统，更加贴近复杂多变的实际模型。随着科技的发展，新的耦合系统层出不穷，针对其展开的研究一直是控制领域的热点之一。由于耦合系统的性质取决于各子系统的特性、子系统间的匹配度、耦合的形式、区域几何特征、反馈控制等诸多因素，与其相关的研究异常复杂。系统的能稳性问题旨在寻找控制率，使得闭环系统稳定，是控制理论的基本问题之一。同步是自然界广泛存在的现象，它描述耦合系统中各子系统状态的趋同行为。本项目将研究一些具物理背景的耦合无限维系统，如具局部结构扰动的振动系统、联结系统、复合材料系统等。首先针对耦合系统特性，设计反馈控制，全面分析（闭环）系统的指数稳定、多项式稳定、对数稳定等性质。其次引入合适的耦合无限维系统同步的概念，并充分应用无限维系统能观、能控、能稳等相关结论与思想，搭建研究耦合无限维系统同步性质的框架和一般方法。

耦合无限维系统是由若干偏微分方程、常微分方程、积分方程等通过特定关系耦合而成的系统，更加贴近复杂多变的实际模型。随着科技的发展，新的耦合系统层出不穷，针对其展开的研究一直是控制领域的热点之一。由于耦合系统的性质取决于各子系统的特性、子系统间的匹配度、耦合的形式、区域几何特征、反馈控制等诸多因素，与其相关的研究异常复杂。系统的能稳性问题旨在寻找控制率，使得闭环系统稳定，是控制理论的基本问题之一。同步是自然界广泛存在的现象，它描述耦合系统中各子系统状态的趋同行为。本项目研究了一些具物理背景的耦合偏微分系统。针对具有局部结构扰动的弹性系统，项目建立了局部粘弹性扰动系数与系统的正则性、衰减率之间的关系。项目还详尽分析了一些耦合偏微分系统的镇定、同步行为与控制器位置、子系统性质之间的联系，得到粘性阻尼下复合材料梁模型的稳定性、衰减率和同步性质，以及热弹性联结系统、多孔弹性系统的控制与多项式稳定性质。项目研究了一类具有非负定反馈控制算子的抽象系统，建立易于验证的多项式稳定判据，从而突破了控制算子非负时，耗散的抽象系统衰减率分析的瓶颈，填补了具有局部耗散的抽象系统稳定性分析的空白；分析基于Cattaneo率的热弹性抽象系统，得到当波速相等时系统指数稳定、多项式稳定的参数图谱。