Finite-time stability for fractional-order systems with respect to another function

This paper introduces a novel framework for finite-time stability (FTS) analysis of fractional-order systems using the Caputo fractional derivative (CFD) with respect to another function-a derivative operator that has not been comprehensively explored in control theory, particularly in the context of FTS. By extending the well-established Caputo-Hadamard fractional derivative, we address its application to nonlinear systems and establish rigorous theoretical conditions for FTS. Our approach leverages Lyapunov-based techniques and Mittag-Leffler function properties to derive sufficient stability criteria, expressed via linear matrix inequalities (LMIs), ensuring system trajectories remain bounded within specified thresholds over finite intervals. The effectiveness of the proposed framework is showcased through three numerical examples, providing valuable insights for progress in fractional-order control systems.