Stability Study For Impulsive Differential Systems With State-Dependent Impulses

In this paper, we consider the stability property in terms of two measures about the following impulsive differential systems with state-dependent impulsesIt is known that the theory of impulsive differential systems provides a natural framework for mathematical modelling of many real-world phenomena, such as project, control, communication, biology, economy, and so on. There has been a significant development in the theory of impulsive differential equations in the past 10 yearst^{[1-10,13,15,19-33]}, especially in the area where impulses are fixed.Impulsive differential systems with state-dependent impulses as an extention of systems with fixed impulses have more application. However, in the investigation of this kind of systems, there arise a number of difficulties related to the phenomena of beating and bifurcation etc. Up to now, the results about impulsive differential systems with state-dependent impulses are really few, and most of them are about the existence of solutions^[1,10,22-24]. It should be pointed that, in [1], V.Lakshmikantham founded the basic theory of such systems and discussed the stability of them, but the conditions of those results are strong. On the other hand, the comparison results are as yet not well-developed, and most are comparable with an impulsive differential system with variable impulse and under the assumption that every solution meets each hypersurface exactly once^[5-9].In view of the extensive application and theoretical background, we investigate the stability property in terms of two measures about the impulsive differential systems with state-dependent impulses. The concepts in terms of two measures describe initial value and state of solution separately by means of two measures, through the different forms of two measures, it enable us to unify avariety of stability notions found in the literature, such as stability of the trivial solution, invariant set or periodic solution. This paper is divided into two chapters.In chapter one of this paper, first in section 3, we establish a comparison principle by vector Lyapunov functions and comparing with an ordinary differential system, then apply it to investigate stability of the systems. It should be noticed that the phenomena beating can be allowed in the comparison principle of this section. Second in section 4, through constructing some new sets, several interesting direct results which allow the Lyapunov function to be increase or decrease along trajectories between successive impulses and to switch back and forth between them at each impulse are established. But they are under the assumption of absence of beating. At the same time, some examples are worked out. Through them we can see that impulses do contribute to yield stability properties even the corresponding continuous system do not enjoy any stability behavior.In chapter two, using the variational measure method, we discuss the property of the perturbation systems with state-dependent impulses by considering the property of the unperturbed systems without impulses. It is very well known that the variational Lyapunov method is useful to study non-linear systems, it combines the method of variation of parameters and the Lyapunov's second method, constructs the relationship between perturbation systems and unperturbed systems by the generalized variation of parameters formula, then researches the behavior of perturbation systems through the behavior of unperturbed systems. In the study of many practical problems (such as the classical stability of zero), the measure function have the character of Lyapunov function, so we can replace the Lyapunov function with the measure function in the variational Lyapunov method, this is the basic thinking of variational measure method. In the case, we use the property of unperturbed system and the connection between the two measures to obtain some results. Finally, an example is given to show the effectiveness of the theorems.