Complete Stability For Uniformly Distributed Delay Systems

Distributed delay systems are often used to model the time lag phenomenon in thermo dynamics,in ecology epidemiology like predator prey systems as well as traffic flow.Contrary to a point-wise delay system,the use of a distributed kernel is more likely to capture reality in some applications.Nevertheless,the stability problem of the distributed delay systems is much more difficult than the point-wise delay case.For different kernel functions there are different distributed time-delay systems and characteristic is different.This thesis is dedicated to a kind of common distributed time delay systems,uniformly distributed delay systems.This thesis analyzes the complete stability problem of the uniformly distributed delay systems by the characteristic function.First of all,the characteristic function of the uniformly distributed delay systems must be obtained.A time-delay system has infinitely many characteristic roots.For this reason,uniformly distributed delay systems represent a class of infinite-dimensional systems and the complete stability problem is complicated.The frequency-sweeping approach will be used to detect the critical imaginary roots and critical delays.The second issue,uniformly distributed delay systems retains infinitely dimensional at the minimum value of time delay,we will adopt an argument principle-based method to compute the number of the unstable roots.The third issue,in order to solve the complete stability problem,we have to further analyze the asymptotic behavior of a critical imaginary root.Since a critical imaginary root of a uniformly distributed delay system generally has infinitely many critical delays,it is important to study the invariance property concerning the asymptotic behavior of the critical imaginary root with respected to the infinitely many positive critical delays.The invariance property of uniformly distributed delay systems will be proved by the relationship between the Puiseux series and dual Puiseux series,the asymptotic behavior of a critical imaginary root must correspond to some Puiseux series and the asymptotic beha-vior of the frequency-sweeping curves must correspond to some dual Puiseux series.Then we can solve the complete stability problem and obtain the explicit expression of the unstable roots for the uniformly distributed delay systems.Based on the complete stability problem of the uniformly distributed delay systems,this paper analytic the stability of traffic flow behavior with distributed delays model.The model by introducing a distribution of delays is more realistic.We calculate exact stability regions in the parameter space of some realistic delay...