On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Order α∈(1,2)

Abstract

We study the Mittag-Leffler and class-K function stability of fractional differential equations with order ∈ (1,2). We also investigate the comparison between two systems with Caputo and Riemann-Liouville derivatives. Two examples related to fractional-order Hopfield neural networks with constant external inputs and a marine protected area model are introduced to illustrate the applicability of stability results.

The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

“…Proof. Using the fact that system (22) is strongly stabilizable by control (24), and inequality (25) yields z u (t) −→ 0 as t −→ +∞ (26) and…”

“…In [24], stability theorems for fractional differential systems, which include linear systems, time-delayed systems, and perturbed systems, are established, while in [25], Ge, Chen and Kou provide results on the Mittag-Leffler stability and propose a Lyapunov direct method, which covers the power law stability and the exponential stability. See also [26], where the Mittag-Leffler and the class-K function stability of fractional differential equations of order α ∈ (1, 2) are investigated. In 2018, the notion of regional stability was introduced for fractional systems in [27], where the authors study the Mittag-Leffler stability and the stabilization of systems with Caputo derivatives, but only on a sub-region of its geometrical domain.…”

Approximate Controllability of Fractional Nonlinear Hybrid Differential Systems via Resolvent Operators

“…Fractional differential systems have described many practical dynamical phenomena more efficiently than the corresponding integer-order systems; hence they have attracted the attention of many researchers in such fields (see [1][2][3][4][5][6][7][8][9][10] and references cited therein).…”

On stability analysis of semi‐linear fractional differential systems

“…Whereas, the author in studied the stability of n ‐dimensional fractional differential systems where is the Riemann‐Liouville fractional derivative. The other kind of stability analysis also has been studied by some authors, for example, Mittag‐Leffler stability in Li et al, Lyapunov direct method as in Li et al, Ulam stability in Ahmad et al, and K‐class function as in Matar and Abu Skhail . Recently, Matar and Abu Skhail have studied the Mittag‐Leffler and K‐function stability of nonautonomous perturbed semilinear fractional differential systems with Caputo, and Riemann‐Liouville fractional derivatives.…”

“…Some of recent contributions in these topics can be found in the articles [8][9][10][11][12] and references therein. Stability of the system is one of most important topic among qualitative properties of the solution of fractional differential system; hence, many interested researchers focused on this topic and then many of articles have been appeared with various types of stability (see other works [13][14][15][16][17][18][19][20][21] ). The stability of the solution for the fractional system depends on the existence of this solution; hence, it is basically to study the existence and uniqueness of the system.…”