Stability Analysis of Short Memory Fractional Differential Equations

TL;DR

This paper introduces a new short memory fractional derivative, explores its properties, compares it with Caputo derivatives, and provides criteria for stability of related systems, supported by examples.

Contribution

It defines a novel short memory fractional derivative and establishes stability criteria for systems using this derivative, extending existing fractional calculus methods.

Findings

Established properties of the short memory fractional derivative

Derived stability criteria for short memory fractional systems

Validated results with three illustrative examples

Abstract

In this paper, a fractional derivative with short-term memory properties is defined, which can be viewed as an extension of Caputo fractional derivative. Then, some properties of the short memory fractional derivative are discussed. Also, a comparison theorem for a class of short memory fractional systems is shown, via which some relationship between short memory fractional systems and Caputo fractional systems can be established. By applying the comparison theorem and Lyapunov direct method, some sufficient criteria are obtained, which can ensure the asymptotic stability of some short memory fractional equations. Moreover, a special result is presented, by which the stability of some special systems can be judged directly. Finally, three examples are provided to demonstrate the effectiveness of the main results.