Stability Analysis Of Impulsive Control Diferential Systems

The development of many things exist an abrupt change of state at certain momentsof time in the nature,this kind of phenomenon in mathematics called pulse.Pulse phe-nomenon widely exists in real life, such as, population growth model,control of moneysupply in a financial market, neural network model and so on. The research of thisthings.developments and changes promotes establishment and development of impul-sive diferential equations,and receives fruitful results[3][12].With the deepening of research on impulse, many scholars have turned to use im-pulse to control the stability of the system[13][17],[21][27],[29]. Because the impulse controlhas targeted, fast response, strong robustness and anti-interference ability etc. impulsivecontrol system makes a collection of advantages of impulse control and continuous con-trol,and has a wide application in analysis of the dynamic behavior.Up to now,the research of stability of impulsive control system has attracted theinterest of many researchers[8][14],[33].The control vector of impulsive control systemwhich are studied is defined the control set={u∈Rm: U(t, u)≤r(t), t≥t0},butthere are few results[1,2]on the control setE={u∈Rm: U(t, u)≤λ0(t), t≥t0}, whereλ0(t)is a given function.In addition to the impulse phenomenon, delay is inevitable in real life.The currentstate of the system may be efected by the past state of the system.Such as populationgrowth model.Many classical control theory can only apply to Linear time-invariant sys-tems,and can not be used to analyze nonlinear and time-varying systems. Xinzhi Liustudies The exponential stability of the impulsive diferential system with time delay in[14][17],[23],[24].Xiaodi Li further studied the global exponential stability of the system in[29].In the first chapter of this article,we will give a new comparison principle between and the following impulsive diferential systemsin the control set E,We also study the comparison principle whenu1∈, u2∈E. Afterthat we apply it to research asymptotic stability,strong practical stability and otherproperties in terms of two measures of impulsive control system(1.1)in the new controlset. At the end of this chapter we illustrate an application about the theorem.In the second chapter of this article,we study the exponential stability of impulsivediferential systemfrom two aspects. One is that we don’t know whether the original system is stable ornot,we make it stable through impulse control,and then we study The stability of impul-sive control system with distributed delay and time-varying impulsive control system.theother is that impulse exists as a perturbation,the original system can maintain its stabil-ity. At the end of this chapter we give examples to illustrate the useful of our theorem.