The α-Stability And Stable Region Of Time Delay Systems

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.