A Novel Lyapunov Asymptotic Eventual Stability Approach for Nonlinear Impulsive Caputo Fractional Differential Equations

This paper investigates the asymptotic eventual stability (AE-S) for nonlinear impulsive Caputo fractional differential equations (ICFDEs) with fixed impulse moments, employing auxiliary Lyapunov functions (ALF) which are specifically constructed as analogues of vector Lyapunov functions (VLF). A novel derivative tailored for VLF is introduced, offering a more robust framework than existing approaches based on scalar Lyapunov functions (SLF). Adequate conditions for AE-S involving ICFDEs are provided. We also used the predictor corrector method to implement a numerical solution for a given impulsive Caputo fractional differential equation. These findings extend and improve upon existing results, providing significant advancements in the stability analysis of systems with memory effects and impulsive dynamics. The study holds practical relevance for modeling and analyzing real-world systems, including control processes, biological systems, and economic dynamics where fractional-order behavior and impulses play a crucial role.