【作者】 狄成宽；

【导师】 王在华；

【作者基本信息】 南京航空航天大学 ， 应用数学， 2007， 硕士

【摘要】 本文研究了滞后型时滞动力系统的α-稳定性及其稳定性区域。对于α-稳定性分析,问题转化为具有与时滞相关的系数的时滞系统的渐近稳定性。我们采用的是稳定性切换思想,研究随着某个参数的变化,特征方程的根是否由左半复平面跨过虚轴到右半复平面或由右半复平面跨过虚轴到左半复平面;采用Beretta和Kuang提出的几何方法,引入S j=τ-τj确定临界参数值,进而判断特征根是否穿越虚轴。在稳定性区域分析中,为了确定保证时滞系统α-稳定性的一个稳定性区域,本文提出了“稳定性切换点法”的计算方法。在一阶时滞系统情形,本文分别采用该方法和应用Hayes定理、LambertW函数法进行分析、比较,得到的结果是相同的。为了说明该方法的有效性,我们将其进一步应用到向日葵方程,比较方便地得到了稳定性的一个区域。更多还原

【Abstract】 In this thesis, theα-Stability and its stable region in the parameter space are investigated for some delay differential equations of retarded type.The first part of the thesis is devoted to theα-stability analysis, which is justified if a certain characteristic quasi-polynomial with delay-dependent coefficients has roots with negative real parts only. On the basis of the graphical test of stability switches for time-delayed systems, proposed by Bretta and Kuang, an auxiliary function S j=τ-τj is introduced to find the critical values of the control parameter, for which the characteristic quasi-polynomial has a pair of conjugate roots on the imaginary axis, and to determine whether this pair of roots passes through the imaginary axis as the control parameter varies.In the second part of the thesis, a new method, based on the principal of stability switch, is proposed to determine the stable region for theα-stability of time-delay systems. Two examples are given to show the efficiency of the method. One is a first-order delay differential equation, and the other is the sunflower equation of second order. In the former example, a comparison is made between the proposed method and the available methods on the basis of Hayes Theorem and Lambert W function, the three routines yield the same stable region.更多还原