Analytic and numerical exponential asymptotic stability of nonlinear impulsive differential equations

This paper deals with exponential stability of both analytic and numerical solutions to nonlinear impulsive differential equations. Instead of Lyapunov functions a new technique is used in the analysis. A sufficient condition is given under which the analytic solution is exponential asymptotically stable. The numerical solutions are calculated by Runge–Kutta methods and the corresponding stability properties are studied. It is proved that algebraically stable Runge–Kutta methods satisfying |1−bTA−1e|<1|1−bTA−1e|<1 can preserve the stability of the equation. Finally some numerical experiments are given to illustrate the conclusion.