Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

Abstract

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

Stability analysis of distributed order of Hilfer nonlinear systems

“…The next result provides sufficient conditions for DMLH stability of the (IVPDH) system. This generalizes the Lemma 4.3 in Rezazadeh et al, 13 the Theorem 4.1 in Taghavian and Tavazoei, 63 the Theorem 1 in Wang et al, 7 and the Theorem 3 in Fernández‐Anaya et al 61 …”

“…The following theorem generalizes the Theorem 4.2 in Taghavian and Tavazoei 63 and the Theorem 2 in Wang et al 7 …”

“…Recently, several articles on systems with the Hilfer derivative have been published, 1‐8 for example, in Hilfer and Luchko, 8 they have proposed six desired properties that a fractional derivative must satisfy and precisely the Hilfer derivative fulfill them. Several applications of this class of fractional derivatives have been proposed 9‐12 .…”

“…A first interpretation of the Hilfer fractional derivative was in simulations of dielectric relaxation in glass‐forming materials and generalized fractional diffusion. More recently, in Wang et al, 7 generalized Mittag‐Leffler stability of Hilfer fractional nonautonomous systems is presented. Stability analysis of Caputo distributed order linear differential systems with respect to a nonnegative density function is presented in literature 2,53 .…”

Generalized Mittag-Leffler stability of fractional impulsive differential system

“…The work of Stamova [16] on impulsive fractional order is a classical result. Even though stability analysis has been done for Hilfer fractional system by Rezazadeh et al [15], Wang et al [18], in order to step further, stability analysis of impulsive differential system with Hilfer fractional derivative is a much needed topic to understand the Hilfer derivative in a much deeper sense. Present work studies the Generalized Mittag-Leffler stability of a Hilfer fractional differential system involving both, instantaneous and non-instantaneous impulsive conditions using Lyapunov approach.…”