Stability Analysis Of Impulsive Functional Differential Systems At Fixed Time

In nature,many evolution process may experience the change of status at fixed time,the study on the impulsive dynamic systems cause more and more attention.In the natural science and engineering technology,many mathematical models 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,interest rate controlling and work 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 wide research on these systems and obtain a lot of results. However,there are a lot of work need to be done in this field,so the research on impulsive functional differential systems is not enough.In recent years,the studies on the stability of zero solution for functional differen-tial systems have yielded relatively good results,the main method is to use Lyapunov functions and Razumikhin techniques.Variational Lyapunov function is derived from the parameters formula and Lyapunov function,which is very useful in the study of the nature of the solution for nonlinear systems.Variational Lyapunov function can research systems with the non-perturbation case of linear or smooth nonlinear.At the same time,the use of perturbation can better improve the nature of non-perturbed systems from the pertur-bation point of view. It can be said that variational Lyapunov function method is more general than the Lyapunov function method.In view of this,this paper will further investigate the stability property in terms of two measures for the functional differential systems by variational Lyapunov function method.In this paper,we hope to improve the existed variational comparison theorem, relaxing restrictions on the variational Lyapunov function appropriately.On this basis,we systematically research the stability in terms of two measures for the retarded functional differential system by combining the Razumikhin techniques.This paper is divided into two parts.In chapter one,we study the stability in terms of two measures of impulsive retarded functional differential system with finite delay as follows:In this chapter, We have created a variational comparison principle and got a few comparison results to ensure the stability in terms of two measures for system (1)by using the variational Lyapunov function and Razumikhin techniques.We have reduced the restrictions on the derivative conditions for the Lyapunov function,without requiring the Dini derivative of the solution negative or negative definite and allowing the derivative positive.The impact of impulse on the nature of the solution is highlighted.A number of Razumikhin type direct results are acquired by improving Razumikhin conditions, several known results are improved and generalized.An example is given to verify the reasonable of the theorem.In chapter two,we study the stability of functional differential systems with impulsive function depending on the delay:The difference from system(1)is the impulsive function depending on the delay r,which has essentially changed the nature of the solution.This chapter focuses on how to solve the impact of the delay in the impulsive function. We have set up the relation between system(2)and ordinary differential system, several comparison results of sys-tem(2)are got by using Lyapunov function and Razumikhin techniques.Then,we have re-searched the uniform stability and uniform practical stability property,some Razumikhin type direct results of stability in terms of two measures for system(2)are acquired. Our results generalize and improve several known ones.An example is given to verify the validity of the results.