Asymptotical stability of numerical methods for semi-linear impulsive differential equations

Abstract

This paper is concerned with asymptotical stability of a class of semi-linear impulsive ordinary differential equations. First of all, sufficient conditions for asymptotical stability of the exact solutions of semi-linear impulsive differential equations are provided. Under the sufficient conditions, some explicit exponential Runge-Kutta methods can preserve asymptotically stability without additional restriction on stepsizes. Moreover, it is proved that some explicit Runge-Kutta methods can preserve asymptotical stability without additional restriction on stepsizes under stronger conditions.

Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

“…Remark 3. Most of the literature on the stability analysis of the numerical method for impulsive differential equations is based on the classic Lipschitz or one-sided Lipschitz conditions in the sense of the standard inner product norm (see [9,13,20]). If the value of the one-sided Lipschitz constant of the problem is very large (see problem (42)), these classic stability theories will fail.…”

Optimal Controls for a Class of Impulsive Katugampola Fractional Differential Equations with Nonlocal Conditions

“…Impulsive differential equation, which provides a natural description of observed evolution processes, is an important mathematical tool to solve some practical problems. The theory of impulsive differential equations of integer order has been widely used in practical mathematical modeling and has become an important area of research in recent years, which steadily receives attention of many authors (see [21][22][23][24][25][26][27][28][29][30][31][32][33][34]). Sun et al [35] considered a class of impulsive fractional differential equations with Riemann-Liouville fractional derivative, the existence of solution was proved by using Darbo-Sadovskii's fixed-point theorem, and the optimal control results were obtained.…”