Stability Analysis Of Nonlinear Impulsive Functional Differential Systems

In the natural sciences and engineering technology,many mathematical model of practical problems can be described by impulsive functional differential systems,such as the circuit signal system,optical control in physics,the study of neural net,genetics and epidemics in the medical field,controlling interest rate and industry management in the economical field and so on.Because of the profound practical significance and wide application background that impulsive functional differential systems have, the study of these systems attracts many experts and scholars in and abroad and gradually become a hot research field. Many scientists do deep and widely research on these systems and obtained a lot of results ([6-16,23-24,34-40]). However,the research in those fields is not enough.For example, the research on W-stability is still rare.In[23],the author studied the W-stability of impulsive functional differential systems with finite delays. But there are little results about infinite delays.Hence there are a lot of work we need to be done in this field.In view of this,this paper will further investigate the stability property of the non-linear impulsive functional differential systems.This paper is divided into two parts.In chapter one,we study the W-stability of impulsive functional differential sys-tems with infinite delays as follows: Because there is greatly different between impulsive functional differential system with infinite delay and the system with finite delays,the study on the system (1) is more complicated. In the third section,some sufficient conditions for W-stability and W uniform stability of system (1) are gained by Lyapunov functional method or Razumikhin method.Unlike the past,due to the effect of impulsive,in theorem we weakens the restrict condition of the derivative of the Lyapunove function.The Dini derivative of Lyapunove along trajectories of the system (1) doesn't need to be required to be negative definite or negative and can be allowed to be positive,highlighting the importance that the impulsive played to the nature of solution of the system, In the end of this section,an example is given to illustrate the validity of the obtained result. In the fourth section,as a application of W-stability and W-uniform stability of the system (1), we gained some results of the uniform stability and the uniform asymptomatic stability for a class impulsive functional differential systems with infinite delays.In chapter two,we study the stability of impulsive functional differential systems with finite delays as follows: First,by using Lyapunov functions and Razumikhin technique,we obtain some new re-sult on uniform stability and uniform asymptotic stability. And then we generalize the extended generalized Halanay's delay differential inequality to impulsive delay differen-tial inequality in Dini derivative.Then we investigate the weak exponential stability and global exponential stability of system (2) by this inequality,and then an enample is given to illustrate the effectiveness of the result.In the last section of this chapter,by using Lyapunov functions and these obtained results about exponential stability, we study the global exponential stability of the equilibrium point of a class of impulsive neural net-works with variable coefficients and delays.At last,we give an example to illsutrate our result.