Asymptotic Stability Of A Class Of Impulsive Control Systems With Nonlinear Impulsive Control

Impulsive control is a control paradigm based on a impulsive differential equation. Pulse control and its mathematical foundation were called impulsive differential equations, or the differential equation with effect. Its development has a long history and can be traced back to the beginning of the modern control theory. From the beginning of last century, many scholars study the impulsive control system, and achieve a lot of conclusions. Because the impulsive control was more effective than continuous control, even some system only can be stabe under impulsive control, so 他 the conclusions can be used in various fields, such as ecological system, economic system, a chaotic system, confidentiality communications system.Many scholars have studied the stability of linear impulsive control systems, but the study of the stability of nonlinear systems with nonlinear impulsive control is less. The asymptotic stability of nonlinear systems under nonlinear impulsive control is studied in this paper. By constructing a Lyapunov function and building its comparison system, if the comparison system is asymptotically stable, impulsive control syetem is asymptotically stable. Then it gives sufficient conditions for the asymptotic stability of four special forms of impulsive control system. Finally, the asymptotic stability of the Lorenz system under nonlinear impulsive control is introduced, at the same time, Lorenz systems also verify the conclusions on the effectiveness.