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

JACKSON  ANTE; UBONG  AKPAN; JEREMIAH  ATSU; MARSHAL  SAMPSON; SAMUEL  ESSANG; MICHAEL  OGAR-ABANG

JACKSON ANTE DEPARTMENT OF MATHEMATICS, AKWA IBOM STATE UNIVERSITY, IKOT AKPADEN, MKPAT-ENIN.
UBONG AKPAN DEPARTMENT OF MATHEMATICS, AKWA IBOM STATE UNIVERSITY, IKOT AKPADEN, MKPAT-ENIN.
JEREMIAH ATSU DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CROSS RIVER STATE, CALABAR.
MARSHAL SAMPSON DEPARTMENT OF MATHEMATICS AKWA IBOM STATE UNIVERSITY, IKOT AKPADEN, MKPAT-ENIN.
SAMUEL ESSANG DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, ARTHUR JARVIS UNIVERSITY, AKPABUYO, CROSS RIVER STATE
MICHAEL OGAR-ABANG DEPARTMENT OF PHYSICS, ARTHUR JARVIS UNIVERSITY, AKPABUYO, CROSS RIVER STATE

Keywords: Asymptotic eventual stability, Impulse, differential equations, Vector Lyapunov functions

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.

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