Abstract
In this article, we survey the Lyapunov direct method for distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives. We provide various ways to determine the stability or asymptotic stability for these types of fractional differential systems. Some examples are applied to determine the stability of certain distributed-order systems.
Highlights
In recent years, distributed-order fractional calculus has played a significant role in many areas of science, engineering, and mathematics [1,2,3,4,5]
For the first time in 1969, the distributed-order fractional calculus with the Caputo fractional derivatives was surveyed by Caputo [6]
Duong et al [3] studied the deterministic analysis of distributed-order systems using operational matrix
Summary
In recent years, distributed-order fractional calculus has played a significant role in many areas of science, engineering, and mathematics [1,2,3,4,5]. We intend to survey the stability or asymptotic stability analysis of a distributed-order fractional differential/integral operator containing the Prabhakar fractional derivatives This type of fractional derivative was introduced by Garra et al [20] in that it is considered in terms of the generalized Mittag-Leffler function and can be considered as a generalization of the most popular definitions of fractional derivatives. In the field of stability and asymptotic stability, Abstract and Applied Analysis several papers have been published as follows: in [21], the Hyers-Ulam stability of the linear and nonlinear differential equations of fractional order with Prabhakar derivative by using the Laplace transform method is studied and the authors show that the fractional equation introduced is Hyers-Ulam stable, and in [22], the authors obtained the stability regions of differential systems of fractional order with the Prabhakar fractional derivatives 0, ð7Þ where m − 1 < RðμÞ < m and ρ, μ, ω, γ ∈ C
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