Stability Analysis Of Two Classes Of Nonlinear Impulsive Differential Systems

In the study of engineering technology and natural science, many mathematical models of practical problems always can be described by impulsive functional differen-tial systems, such as the study of epidemics,neural net in the medical field, the optical control,circuit signal system in physics.Because of this important practical significance and wide application background, now many scholars at home and abroad take up with this study and acquire a lot of results. But these results mostly focus on the systems in which impulsive conditions not depend on time delays or the time delays are finite, however, in practical application, particularly in the design of quick search network capacity and the neutral network optimization calculation, the impulsive condi-tions always depend in time delays or affected by them indirectly. In addition, impulsive functional differential systems with p-delay contain many impulsive functional differen-tial systems with finite delays as well as with unbounded delay which are more general. Hence, the research on stability theory of these two classes of nonlinear impulsive func-tional differential systems has a very important theoretical significance and application value.One important method of studying the stability of nonlinear impulsive functional differential systems is to use Lyapounv functions together with Razuminkin techniques. By combining the Lyapounv functions with the method of variational of parameters, the variational Lyapounv functions method has been derived in [11]. This method is always used when unperturbed terms are linear or when they are certain smoothness though possibly nonlinear. Now many results have been developed by applying the variational Lyapounv functions method to the stability theory.Based on the ideas above, we study the stability of these two classes of nonlinear impulsive functional differential systems by using the variational Lyapounv functions method. This paper is divided into two chapters.In chapter one, we consider the following functional differential systems in which impulsive functions depend on the time delays:In this chapter, we first introduce the related conceptions of the variational Lya-pounv functions and stability, give the idea of the variational Lyapounv functions method. Afterwards, we establish a variational comparison principle comparing with a scalar ordinary differential system by using variational Lyapounv functions and Razuminkin techniques,and then the comparison results of uniform stability in terms of two mea-sures for the systems are acquired, here the delays are included in perturbations and the unperturbed terms have no delays,we can derive the stability properties from the corresponding stability properties of relevant ordinary differential systems. In the fifth section, we gain some direct results on the uniform asymptotically stability,uniformly strict stability,uniformly strict practical stability by the method of variational Lyapounv functions and Razuminkin techniques, finally, an example is given.The restrictions on the Dini derivative of Lyapounv functions are weakened,can be positive,which is convenient in applications.In chapter two, we consider the following impulsive functional differential systems with p-delay:In this chapter, we first introduce the related conceptions of p-function. Then we create a variational comparison principle comparing with a scalar impulsive differential system by using variational Lyapounv functions and Razuminkin techniques,from which we get the comparison criteria on stability of system (4). These results improve and generalize some of the earlier comparison results of IFDE with finite time delay and infinite time delay, therefore these are more extensive in applications. Afterwards, some Razurninkin-type direct results of uniform stability in terms of two measures are got. An example is given at last.