Cascade of Fractional Differential Equations and Generalized Mittag-Leffler Stability

Subject Areas : International Journal of Mathematical Modelling & Computations

Ndolane Sene ¹

1 - Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, BP 5683 Dakar Fann, Senegal

Received: 2020-01-11 Accepted : 2020-03-08 Published : 2020-03-01

Keywords: ‎Mittag-Leffler stability, Generalized fractional derivatives, Input stability‎,

Abstract :

This paper address a new vision for the generalized Mittag-Leffler stability of the fractional differential equations. We mainly focus on a new method, consisting of decomposing a given fractional differential equation into a cascade of many sub-fractional differential equations. And we propose a procedure for analyzing the generalized Mittag-Leffler stability for the given fractional differential equation using the generalized Mittag-Leffler input stability of the sub-fractional differential equations. In other words, we prove a cascade of fractional differential equations, which are generalized Mittag-Leffler input stables and governed by a fractional differential equation, which is generalized Mittag-Leffler stable, is generalized Mittag-Leffler stable. We give Illustrative examples to illustrate our main results. Note in our paper; we use the generalized fractional derivative in Caputo-Liouville sense.