Study For Impulsive Stabilization Of Third-Order Delay Differential Systems And Stability Of Impulsive Functional Differential Systems

In this paper.we study impulsive stabilization of third-order delay differential system as followsand the stability of impulsive functional differential system with p-delay as followsImpulsive effects exist in many evolution processes in which states are changed abruptly at certain moments of time.which is applied widely in each field of modern technology .Thus impulsive differential systems appear as a natural description of observed evolution phenomena for several real word problems.We all know impulses can make unstable systems stable or. otherwise.stable systems can become unstable after impulse effects.So the problem of stabilizing the solutions by imposing proper impulsive control for differential systems (which is called impulsive stabilization)has attracted some scholars attentions.So far, there are many results of stability theory and control by impulses for delay differential systems.for example.see[6-9].But most of the results are about of the study for low-order linear delay differential systems or the non-linear delay differential system which is stable,for instance, see[10-11]. And that.the achievements of stability theory and control by impulses about researching the higher-order nonlinear delay differential systems haven't been seen widely.therefore also has many work must do.In chapter one,we prove that one kind of the third-order non-linear delay differential systems which is unstable can be stabilized by the imposition of proper impulsive control.Firstly,we define exponential stabilization by impulse and exponential stabilization by periodic impulse.Then by means of Lyapunov functions and Lyapunov functionals with analysis methods,we establish sufficient conditions for the stability of solutions about the system(1) by imposing proper impulse controls . And the impulsive controling functions with simple expressions are given.The results in this chapter improve and generalize some of the earlier foundings.therefore the applications are more extensive.Finally,we give one example to illustrate the application of our results.With the rapid development of science and technology,the study for the stability of impulsive functional differential systems gradually become a hotspot.In the last several years,the basic theory on impulsive functional differential systems with p-delay is just established [16].Impulsive functional differential systems with p-delay are very important impulsive functional differential systems which contain many impulsive functional differential systems with finite delay as well as with unbound delay.In past research,people usually make use of one order of derivatives in Lyapunov function to discuss various kinds of nature of the impulsive differential systems,and always to setting up the condition on continuous portion and dispersed portion of the syetems independdently. But the article [17] proposed one new method-the method of generalized second order derivatives of Lyapunov function.namely the derivative of Lyapunov function along the system rail line no longer confine to negative or definitely negative,and allow the continuous part of Lyapunov function along the system rail line increase progressively.or after jumping increasing in pulse.but must set up the condition to guarantee it can't increase too fast.They meet under the ground prerequisite of certain terms the generalized second order derivatives of Lyapunov function.through to setting up the condition of mixing on continuous portion and dispersed portion of the systems to estimate synthetically.So,when the symbols questions of one order of derivative for Lyapunov function are uncertain,and the generalized second order derivatives of system exist and the symbol is confirmed,use this method study impulsive functional differential systems very much effective.In recent years,the article of stability of impulsive differential system employing generalized second order derivative method have been many,but the article of applying this method research stability of impulsive functional differential systems is very rare[25].This article uses this method studies the stability of system(2).In the second part of chapter two,according to the results of article[19],we get the stability in terms of two measures of system(2) employing the generalized second order derivatives of Lyapunov function and unifies the Razumikin skill: In the third part of this chapter.we also gain several new results of uniform stability and uniformly asymptotic stabiliy in terms of two measures by the method of Lyapunov functions and Razumikin skill.which improve and generalize some of the earlier foundings.therefore the applications are more extensive. And finally we give an example to explain the theorem usability.