Fractional coupled Halanay inequality and its applications

TL;DR

This paper develops a generalized fractional Halanay inequality to analyze the stability of time fractional differential equations, providing new criteria for stability and decay behavior with applications to coupled systems and neutral equations.

Contribution

It introduces a novel fractional Halanay inequality and applies it to establish stability criteria for fractional coupled systems and neutral differential equations.

Findings

Established asymptotic stability criteria for fractional coupled systems.

Proved contractility and dissipativity of fractional neutral equations.

Demonstrated polynomial decay behavior in time fractional systems.

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.