Razumikhin-type theorems for impulsive differential equations with piecewise constant argument of generalized type

Abstract

Abstract In this paper, we focus on developing Razumikhin technique for stability analysis of impulsive differential equations with piecewise constant argument. Based on the Lyapunov–Razumikhin method and impulsive control theory, we obtain some Razumikhin-type theorems on uniform stability, uniform asymptotic stability, and global exponential stability, which are rarely reported in the literature. The significance and novelty of the results lie in that the stability criteria admit the existence of piecewise constant argument and impulses, which may be either slight at infinity or persistently large. Examples are given to illustrate the effectiveness and advantage of the theoretical results.

Keywords

• Impulsive differential equations
• Piecewise constant argument of generalized type
• Lyapunov–Razumikhin method
• Asymptotic stability
• Exponential stability