Lyapunov Direct Method for Nonlinear Hadamard-Type Fractional Order Systems

Abstract

In this paper, a rigorous Lyapunov direct method (LDM) is proposed to analyze the stability of fractional non-linear systems involving Hadamard or Caputo–Hadamard derivatives. Based on the characteristics of Hadamard-type calculus, several new inequalities are derived for different definitions. By means of the developed inequalities and modified Laplace transform, the sufficient conditions can be derived to guarantee the Hadamard–Mittag–Leffler (HML) stability of the systems. Lastly, two illustrative examples are given to show the effectiveness of our proposed results.

Mittag‐Leffler stability of neural networks with Caputo–Hadamard fractional derivative

“…As in integer‐order differential equations, in the fractional‐order differential equations, a suitable Lyapunov function is the main tool for the stability investigations. The following useful inequality is presented in [22].…”

“…The function $$$ {E}_{\alpha } $$$, defined by $$$$ {E}_{\alpha }(z)=\sum \limits_{k=0}^{\infty}\frac{z^k}{\Gamma \left(\alpha k+1\right)}, $$$$ whenever the series converges, is called Mittag‐Leffler function with one parameter $$$ \alpha $$$. The proof of the following lemma is stated in [22, 23].…”

“…In 2019, comparison results were given by Ma [21]. In recent works, Lyapunov's method has been used for the stability analysis of Caputo-Hadamard fractional (CHF) differential equations [22][23][24].…”

A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

“…Caputo fractional derivatives (CFD) are a modification of RLFDs that deals with initial conditions more conveniently for some applications. CFDs are defined as follows [12]:…”