Stability Study For Impulsive Control Systems With State-dependent Impulses

In this paper, we consider the stability property as well as the stability andboundedness property in terms of two measures about the following impulsivedifferential control systems with state-dependent impulsesIt is known that the theory of impulsive differential systems provides a nat-ural framework for mathematical model of many real-world phenomena, and wecall this mathematical model impulsive differential systems. Impulsive controlproblems which arise in a vide variety of applications, such as orbital transfer ofsatellite, optimal control of nerve network, as well as control of money supply ina financial market and so on have attracted the interest of many researchers inrecent years. Most of these mathematical model are called impulsive differentialcontrol systems.At present, the results about impulsive differential control systems mainlyconcentrate in the stability and boundedness property and most of the methodsare comparison of Lyapunov functions [16-18]. It should be noted that most of theresults are about the impulsive differential control systems with fixed impulses.Impulsive differential control systems with state-dependent impulses as anextent of systems with fixed impulses have more application. However, in theinvestigation of this kind systems, there arise a number of diffculties related tothe phenomena of beating and bifurcation etc. Up to now, there are some re-sults about impulsive differential systems with state-dependent impulses [8-13] andabout impulsive differential control systems with fixed impulses [16-18], but thereare few results about impulsive differential control systems with state-dependentimpulses.In view of the extensive application and theoretical background, we investi-gate the stability property as well as the stability and boundedness property in terms of two measures about the following impulsive differential control systemswith state-dependent impulses.This paper is divided into two chapters. In chapter one of this paper, firstin section 3, we construct a map which converts the impulsive differential controlsystems with state-dependent impulses to systems with fixed impulses equally,and then by constructing reduced systems we make sure the converted systemshave trivial solution. So we investigate the stability property of impulsive differ-ential control systems with state-dependent impulses by investigating the stabilityproperty of systems with fixed impulses ; Second in section 4, through construct-ing new control sets, we give some results about strict uniform stability propertyof impulsive differential control systems with state-dependent impulses .In chapter two of this paper, first in section 3 , we establish a comparison prin-ciple by vector Lyapunov functions and comparing with an ordinary differentialsystem , then apply it to investigate stability of systems. It should be noted thatthe phenomena beating can be allowed in the comparison principle of this section, finally an example is given to show the e?ectiveness of the theorems; Second insection 4,we apply comparison principle in section 3 to investigate boundednessof systems.