Fractional coupled Halanay inequality and its applications
La Van Thinh, Hoang The Tuan, Dongling Wang, Yin Yang
TL;DR
This work introduces a generalized fractional Halanay-type inequality for Caputo derivatives with coupled delays and leverages the sub-additive property of Mittag-Leffler functions to derive explicit algebraic decay rates. By combining this inequality with a positive representation framework, it establishes Mittag-Leffler stability for linear fractional delay systems and proves contractivity and dissipativity for fractional neutral FDEs. The results unify and strengthen prior fractional Halanay-type analyses, offering concise conditions for stability and long-time behavior with practical implications for numerical study. The approach highlights a clear path to assess asymptotic dynamics in a broad class of time-fractional systems exhibiting memory and delays.
Abstract
This paper introduces a generalized fractional Halanay-type coupled inequality, which serves as a robust tool for characterizing the asymptotic stability of diverse time fractional functional differential equations, particularly those exhibiting Mittag-Leffler type stability. Our main tool is a sub-additive property of Mittag-Leffler function and its optimal asymptotic decay rate estimation. Our results further optimize and improve some existing results in the literature. We illustrate two significant applications of this fractional Halanay-type inequality. Firstly, by combining our results in this work with the positive representation method positive representation of delay differential systems, we establish an asymptotic stability criterion for a category of linear fractional coupled systems with bounded delays. This criterion extends beyond the traditional boundaries of positive system theory, offering a new perspective on stability analysis in this domain. Secondly, through energy estimation, we establish the contractility and dissipativity of a class of time fractional neutral functional differential equations. Our analysis reveals the typical long-term polynomial decay behavior inherent in time fractional evolutionary equations, thereby providing a solid theoretical foundation for subsequent numerical investigations.