LYAPUNOV FUNCTIONS AND ASYMPTOTIC EVENTUAL STABILITY FOR IMPULSIVE SYSTEMS WITH NONLINEAR TERMS

Abstract

This paper investigates the asymptotic eventual stability of a class of nonlinear impulsive differential equations with 
impulses occurring at fixed moments. The analysis is conducted 
within a Lyapunov framework by extending the classical concept of 
vector Lyapunov functions to a generalized class of piecewise 
continuous Lyapunov functions suitable for impulsive systems. This 
approach effectively captures both the continuous system evolution 
and the discrete effects introduced by impulses. By employing 
appropriate comparison principles, the behavior of the impulsive 
system is related to that of corresponding comparison systems, 
enabling the derivation of tractable stability criteria. Based on this 
methodology, sufficient conditions guaranteeing asymptotic eventual 
stability are established in terms of inequalities involving the 
proposed Lyapunov functions and system parameters. The results 
obtained extend and improve several existing stability criteria in the 
literature. In particular, the proposed conditions are less restrictive 
and applicable to a wider class of nonlinear impulsive differential 
systems, thereby enhancing the scope and effectiveness of Lyapunov
based methods for the qualitative analysis of impulsive dynamics.