An innovative method for stability assessment in fractional-order systems via higher-order Lyapunov derivatives

The Lyapunov method plays an important role in determining the stability properties of fractional-order systems (FOSs). However, the key challenge of this method is to compute a positive-definite function (PDF) that decreases over time. This article proposes a novel approach that simplifies this task by introducing new sufficient conditions, allowing the Lyapunov function candidate to increase in certain regions. Specifically, we consider inequalities involving higher-order Lyapunov derivatives, particularly when conventional Lyapunov-based stability analysis is infeasible. This innovative framework circumvents the key challenge of finding a strictly decreasing positive-definite Lyapunov function, which is a significant difficulty in traditional methods. By introducing new sufficient conditions that allow the Lyapunov function candidate $ V $ to increase in certain regions, our approach provides a revolutionary methodology that expands the application scope of stability analysis for FOSs. It is proven that if $ {}^{ \; \mathcal{C}}_{t_0}D^{p}_t V $, $ {}^{ \; \mathcal{C}}_{t_0}D^{2p}_t V $, and $ {}^{ \; \mathcal{C}}_{t_0}D^{3p}_t V $ satisfy a specific inequality, stability, asymptotic stability, and global asymptotic stability of the equilibrium point can still be inferred. Then, we apply this criterion to analyze stability properties and design controllers for FOSs, including an application to investigate the stability properties of a fractional-order mass-spring-damper system and to stabilize positive fractional-order electrical circuits. Finally, numerical simulations are provided to validate the proposed sufficient conditions.