Stability Analysis Of Impulsive Functional Diferential System At Fixed Time

Impulsive functional diferential system is an important branch of modern appliedmathematics, because the system fully considers the impact of time-delay and impulse,therefore it is widely used in neural networks, control theory, economics and epidemiology,etc [7-11]. The dynamic system has become a hot research topic for many mathemati-cians who have made some good researches [12-27]. But the researches are not enough,for example, if there are some general global exponential stability results on impulsivefunctional diferential system with infinite delay? If we can find some sufcient conditionsto ensure the W-uniform stability of impulsive functional diferential system under im-pulsive perturbations or impulsive control? If we can use partial components Lyapunovmethod to study the stability on the qualitative theory of impulsive functional diferentialsystem with infinite delays? and so on. Therefore, there is a lot of work to be done in thisfields. This paper is focused on the dynamics analysis of impulsive functional diferentialsystem and gives afrmative answer to some of the issues. This paper is divided intothree chapters.In chapter one, we study the uniform stability and global exponential stability ofimpulsive functional diferential system with infinite delayBy using Lyapunov functions and Razumikhin technique, the impulsive conditions andRazumikhin conditions are more general, some new results on uniform stability of im-pulsive functional diferential system with infinite delays and global exponential stabilityare established. The result of uniform stability can be applied to solve the stability ofa more general nonlinear and the condition (?) is very important to overcame the difculty from finite delay to infinite delay of the global exponentialstability result.In chapter two, we deal with the W-uniform stability of impulsive functional difer-ential system with finite delay.The W-uniform stability of the system are obtained from the two angles of pulse distur-bance and pulse control. It is worth mentioning that the results on W-uniform stabilityis really rare in this paper.In chapter three, partial components Lyapunov method is proposed to investigatethe stability of impulsive functional diferential system with infinite delaysActually, if people only interest in part of the variable element or they can not predict andcontrol another part of the variable element due to technical difculties, then the partialcomponents Lyapunov method becomes very important. Several direct and comparativeresults via partial components Lyapunov method are established in this chapter. Also,the results of infinite delay by using partial components Lyapunov method is really rare.