Asymptotic Stability Of Impulsive Systems

In the fields of chemistry,biology and finance,the coexistence of differential and impulse is very common,which is called impulse differential system.Stability is the precondition for the normal operation of impulsive differential systems.Therefore,it is necessary to study the asymptotic stability of impulsive differential systems.Most scholars use the comparative principle to analyze the stability of impulsive systems.The form of comparative systems is linear homogeneous.In order to study the asymptotic stability of the nonlinear impulsive system,a linear non-homogeneous impulsive system is established for the nonlinear impulsive system based on the comparison principle by constructing a suitable Lyapunov function.The stability of the non-linear impulsive system is obtained by analyzing and comparing the stability of the system.The linear non-homogeneous comparison system adds a series of constants to the linear homogeneous comparison system.The column constants directly affect the stability of the impulsive system.By controlling the column constants,the stability of the impulsive system can be controlled.In this paper,we study the asymptotic stability of five different forms of impulsive systems.Three of them are the extraction of part or all of the linear parts from the non-linear functions to form a new form.This method can improve the flexibility of judging the stability of impulsive systems.In this paper,sufficient conditions for asymptotic stability of five types of impulsive systems are obtained,and the validity of the conclusions is verified by numerical examples.