偏微分网络系统的控制

项目名称： 偏微分网络系统的控制

项目编号： No.60874034

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 生物科学

项目作者： 许跟起

作者单位： 天津大学

项目金额： 30万元

中文摘要： 偏微分网络系统是区别于点对点网络系统的一种更精确描述系统力学行为的模式，它广泛出现在工程、生物及医学等领域。本项以常见的弦、Euler-Bernoulli梁、Timoshenko梁为基本动力学方程描述的偏微分网络系统为研究对象，研究网络系统的节点连接问题与变系数网络系统控制问题。通过研究网络在节点的一般连接方式，分别给出了三类网络的局部顶点条件与结构平衡条件。研究了系统在节点和边界点的反馈控制, 通过设计适当的反馈控制器，选择控制器位置使闭环系统达到稳定，特别地研究了控制器设计与稳定性分析的几何方法。同时考虑了控制器的时滞问题与异位控制问题，在一定条件得到系统的稳定性或指数稳定性。本项还对网络逆问题进行了初步探索。作为项目的辅助结果，我们还研究了系统的可靠性问题。本项研究主要采用频率分析方法，对反馈闭环系统所确定的算子给出较完整的谱分析，给出算子谱的渐近分布，本征值的重数估计，并研究系统解按照根向量系统的展开性质。本项研究内容具有广泛的应用背景，结果可应用到实际问题包括网络设计。其中解决节点连接问题，算子的谱分布和重数问题也是对力学和数学内容的补充

中文关键词： 偏微分网络；控制；节点条件；反馈镇定；谱

英文摘要： Partial differential networks which is distinguishing from the system of point to point are the style of exactly describing the dynamic behaviour of systems in mini world, that have extensive application in engineering, biology and medicine. This project studied the partial differential networks described by three kinds of elastic systems whose dynamical equations are governed by the 1-d wave equation, Euler-Bernoulli beam and Timoshenko beam, respectively. Firstly one discussed the join region of networks and gave a general characterization in local vertex condition and global condition of structure equilibrium, and then investigated the stabilization problem of these networks, in particular, the networks whose dynamical equations with variable coefficients. By a suitable choice of the feedback controllers and their locations, one stabilized the networks. As an easy applicable method one discussed the geometrical approach of design of controllers and stability analysis. At same time the time-delay in controller and non-collocated feedback problem are consider in this project, some results of exponential stabilization are obtained. Moreover, one explored the inverse problem of networks. As auxiliary results one discussed the reliability of the system. In this project, the stability analysis of the closed loop system is mainly carried out via the spectral analysis method. By a detailed asymptotical analysis of the spectrum, one gave spectral distribution of the operator determined by the closed loop system, under certain condition one got the basis property of the root vectors of the system operator. The results obtained in this project are applicable to the practice problems, they are complementary for theory of operator in mathematics and mechanics.

英文关键词： Partial differential networks; control; vertex condition; feedback stabilization; spectrum