Algebraic Criteria For Asymptotic Behavior Of Delayed Functional Differential Equations

Time-delay is a common phenomenon that is almost present in any dynamic systems.Time-delay is an important reason for the instability of the systems.Time-delay systems and its stability have been extensively studied in many disciplines.In the previous studies,from a new analytic curve perspective,a frequency-sweeping framework for stability analysis of time-delay systems can be uesd to detect all the critical imaginary roots(CIRs)and can analyze the asymptotic behavior of the critical imaginary roots by studying the algebraic properties of frequency-sweeping curves.However,from a theoretical and practical point of view,frequency-sweeping approach need to draw the frequency-sweeping curves by artificial analysis.It is very complicated and the efficiency is quite low in the case of large-scale systems or heavier tasks.Now we want to use the computer for the parameter scanning task.So we put forward a different approach-the algebraic study of the asymptotic behavior of the time-delay systems.Firstly,discussing a class of Retarded type time-delay systems which is completely given.We first derive a new algebraic criterion to address the complete stability.By this criterion,the critical imaginary roots(CIRs)are associated with ?NU which stands for the number change of unstable roots.So we can completely analyze the asymptotic behavior of the CIRs.Secondly,if a system changes in a space,it is necessary to consider the system of the classification in the parameter space from a more macro point of view.This is not enough based on the new algebraic criterion,so we introduce an additional tool-polynomial discrimination system,which can solve this problem with the proposed new algebra criterion.We can obtain the complete real root classification by the polynomial discrimination system.But according to the study,we need to get the complete positive real root classification.So we present an alternative method to obtain the positive root classification directly.However,this method is very complicated in the case where the system contains parameters,so we propose a new method for the complete positive real root classification in parameter space.We can determine the positive real root classification of the entire interval by just detecting one point of the interval that greatly reduces the amount of computation.Finally,the parameter scanning task can be realized by the computer algorithms,and we can obtain the complete stability analysis and expression in parameter space.