Stability Analysis For Impulsive Integro-differential Systems

In this paper,we mainly study the Lipschitz stability and W-stability for a class of impulsive Integro-differential systems where Tx=∫~t_toP(t,s,x(s))ds,P:R2+×S(P)â†'Rn,Z＋is the positive integerIn recent years,impulsive Integro-differential systems are encountered in many ar-eas of science,for example,in biology,the control of pest invasion speedï¼›in medicine,The diseases spread through the source and so on.Therefore,this type of systems has im-portant application value,and the research for this systems has aroused the interest and concern of many experts,at the same time,they have made some achievements[3-18].Among them,[3]has established the comparison results of stability for the systems (I)ï¼›[4-11]studied the stability and boundedness for this systems,and the direct results have been givenï¼›in particular,[11]has established some criterion of asymptotic stability for the systems,and emphasized the pulse effect on stabilityï¼›[12-13]have investigated the stability of the two measures from the viewpoint of perturbation for the impulsive Integro-differential systems.However,the research of the systems(I)is still not perfect, For example,under the action of pulse disturbance or pulse control whether or not can find the sufficient condition of Lipschitz stability for the systems?Can we find the suf-flcient conditions of W-stability and W-uniform stability for the systems(I)?can we use the comparative method to research uniform Lipschitz stability for the systems(I)? therefore.there are many problems to be solved.In this paper,we focus on the research work on dynamics analysis of the impulsive Integro-differential and have made a positive answer to the above question with the di-rect method and comparative method using Lyapunov function method and Razumikhin technique, the paper consists of two parts.From the two sides of pulse disturbance and pulse control, the first chapter of this ar-ticle research the uniform Lipschitz stability, global uniform Lipschitz stability, global uni-form Lipschitz asymptotical stability with Lyapunov function method and Razumikhin technique. And two examples are given to verify the validity of the results. At the same time, inspired by [27], we use scalar impulsive differential systems uniform Lipschitz stability get the uniform Lipschitz stability of (â… ).From the two sides of pulse disturbance and pulse control, the second chapter of this article research the W-uniform stability with Lyapunov function method and Razu-mikhin technique. also, obtain the sufficient condition of W-stability for the (â… ). Finally, two examples are given to verify the validity of the results. In the section four, we get the Lyapunov stability with the help of W-stability of impulsive integro-differential systems.In the theorems1.3.1,2.3.1of this paper, the conditions of derivative of the Lya-punov function is weakened, the Lyapunov function is no longer limited to the monotone decreasing along the solution, allowing appropriate increase at impulse points, as long as it could guarantee offset the reduction of pulse point, In the theorems1.3.3,1.3.5and2.3.4allow the Lyapunov function in the pulse points increase, but the decrease of pulse point could offset it, these stress the influence of impulse for the systems. As a result, the application scope is more extensive.In this paper, the method adopted is different from [4-9], but it have taken the thought of [19-22], discussing the pulse intervals with segmenting and combine the math-ematical induction, leaving out the categorization of whether pulse points or not. So there have more effective when we determine the stability of the systems (â… ).However, to our knowledge, there has been very little existing work on Lipschitz stability and W-stability of trivial solutions of impulsive Integro-differential systems(â… ).