Stability Theory For Impulsive Integro-Differential Systems

In this paper, we study stability and boundedness for impulsive integro-differential systems as followsQua a important embranchment of nonlinear impulsive di?erential systems[1, 27],impulsive differential systems have extensive applications in nature-science. Forexample, mathematic models of circuit simulation in physics and neuronal net-works in biology remain with impulsive di?erential systems to analyze and dis-cuss. So it is valuable to be studied and also attracts many expert's attention andintersting[2-6]. In the process of the study of this system, article [5] got the com-parison criteria of stability of the trivial solution of impulsive integro-di?erentialsystems, and article [2-4, 6] studied boundedness of solutions of this system andgot some direct results. Howerver, the study of stability of this system is inunderway phase, and there are many problems which are not solved. Therefore,we have a large number of work to do. In this paper we study the properties ofstability and boundedness of the systems, and we get some new results.In chapter one, firstly we mainly investigate the stability of the trivial solu-tion of the system(1) by the idea of using Lyapunov functions coupled with Razu-mikhin technique which are used in the study of impulsive functional di?erentialsystems[7?23, 28, 34] and get five theorems. In theorem 1.3.1-1.3.4, it weakens therestrict condition of V function at impulsive point. Moreover, in theorem 1.3.5,the derivative of Lyapunov function along trajectories of system(1) doesn't needto be required to be negative definite, so they are not only e?ective but suitablefor many applications. Finally we give an example to illustrate the practicabil-ity of the theorems in section 3 of this chapter. Secondly because stability andasymptotic stability of the trivial solution does not guarantee any informationabout the rate of decay of the solutions and various definitions of stability aretherefore one-sided estimates, we need introduce the conceptions of strict sta-bility [22?24]. In section 4 of this chapter, we firstly give the definitions of strictstability of system(1). Then we also make use of the idea of Lyapunov functionscoupled with Razumikhin technique to get two direct results of the strict stabilityof the trivial solution of system(1).In chapter two, we study the boundedness properties in terms of two mea-sures of system(1) in the same way via setting up proper Lyapunov functions andusing Razumikhin technique and get five theorems which are all direct rusults.In theorem 2.3.2 and theorem 2.4.2, it weakens the restrict condition of V func-tion at impulsive point. Moreover, in theorem 2.4.3, the derivative of Lyapunovfunction along trajectories of system(1) doesn't need to be required to be nega-tive definite, and we also can get direct results such as (h0,h)?uniform ultimateboundedness of the system(1). In provement, this paper di?ers from the article[3,4,6]. By the idea of discussing with impulsive section coupled with mathe-matic concluding method, the provement leaves out the case of judging that thefounded point is impulsive point or not so can be more compact and transparent.Similarly an example is given finally to show the e?ectiveness of the theorems inthis chapter.In chapter three, we firstly give the defination of the cone and order rela-tion on the cone. Then we introduce the conception of cone-valued Lyapunovfunctions and its derivative along the solution of system(1). There is the require-ment of quasimonotone nondecreasing property of the comparison system on R+nin Lyapunov functions method. However, it is not necessary. By using cone-valued Lyapunov functions, we make the method more useful[29?33, 35]. In thischapter, we firstly give two Lemmas from which we get the comparison criteria ofstability and boundedness in terms of two measures of system(1). What we mustpoint out is that in practise we always meet the situation that some state of thesystem may not stable in the viewpoint of mathematics. But because the systemmoves in the area which is su?cient small round the state it can be received.This phenomena makes us study practical stability of the system. Therefore,this chapter also gets the comparison rusult of practical stability trivial solutionof the system(1).