On asymptotic properties of solutions to fractional differential equations

TL;DR

This paper investigates the long-term behavior of solutions to Caputo fractional differential equations, introducing Mittag-Leffler stability and analyzing how solutions decay and relate to stability criteria.

Contribution

It introduces Mittag-Leffler stability for fractional systems and analyzes asymptotic properties using Lyapunov methods, advancing understanding of fractional differential equations.

Findings

Non-trivial solutions decay no faster than t^{- extalpha}

Mittag-Leffler stability is suitable for fractional systems

Relations between Lipschitz condition, stability, and decay speed

Abstract

We present some distinct asymptotic properties of solutions to Caputo fractional differential equations (FDEs). First, we show that the non-trivial solutions to a FDE can not converge to the fixed points faster than $t^{-\alpha}$, where $\alpha$ is the order of the FDE. Then, we introduce the notion of Mittag-Leffler stability which is suitable for systems of fractional-order. Next, we use this notion to describe the asymptotical behavior of solutions to FDEs by two approaches: Lyapunov's first method and Lyapunov's second method. Finally, we give a discussion on the relation between Lipschitz condition, stability and speed of decay, separation of trajectories to scalar FDEs.