Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

Deng, Jingwei; Ma, Weiyuan; Deng, Kejian; Li, Yingxing

doi:10.1155/2020/7962542

Cited by 18 publications

(20 citation statements)

References 37 publications

(52 reference statements)

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“…Remark 2. Note that, in some works (for example, see [8][9][10]), the so-called tempered fractional integral and tempered fractional derivative are applied and defined by the following:…”

Section: Preliminary Resultsmentioning

confidence: 99%

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Almeida

Agarwal

Hristova

et al. 2021

Axioms

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A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

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“…Remark 2. Note that, in some works (for example, see [8][9][10]), the so-called tempered fractional integral and tempered fractional derivative are applied and defined by the following:…”

Section: Preliminary Resultsmentioning

confidence: 99%

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Almeida

Agarwal

Hristova

et al. 2021

Axioms

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“…If x ( t ) ∈ ℝ n is a continuously differentiable vector function, then

{{}_{0}^{C}D}_{t}^{α, λ} ((), x^{T} ((), t) italicQx ((), t)) \leq 2 x^{T} ((), t) Q {{}_{0}^{C}D}_{t}^{α, λ} ((), x ((), t)),

where t ≥ 0, 0 < α < 1, λ ≥ 0, and Q is a symmetric positive definite matrix.Lemma (Li et al 29 ). The Laplace transform of

{{}_{0}^{C}D}_{t}^{α, λ} x ((), t)

is given as

L \{{{}_{t_{0}}^{C}D}_{t}^{α, λ} x (t)\} = {((), s + λ)}^{α} X (s) - \sum_{k = 0}^{n - 1} {((), s + λ)}^{α - k - 1} [\frac{d^{k}}{{dt}^{k}} (e^{italicλt} x (t))] | t = 0,

where

X ((), s) = scriptL ({}, x ((), t))

denotes the Laplace transform of x ( t ).Lemma (Deng et al 30 ). Suppose that x ( t ) ∈ ℝ and y ( t ) ∈ ℝ are continuously differentiable functions and satisfy

{{}_{t_{0}}^{C}D}_{t}^{α, λ} x ((), t) \geq {{}_{t_{0}}^{C}D}_{t}^{α, λ} y ((), t),

where x (0) = y (0), 0 < α < 1, λ ≥ 0, then x ( t ) ≥ y ( t ).…”

Section: Preliminariesmentioning

confidence: 99%

“…Lemma 4. (Deng et al 30 ). Suppose that x (t) ∈ R and y (t) ∈ R are continuously differentiable functions and satisfy…”

Section: Preliminariesmentioning

confidence: 99%

Partial topology identification of tempered fractional‐order complex networks via synchronization method

2021

Math Methods in App Sciences

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This paper studies the problem of partial topology identification of tempered fractional complex networks. By tempered fractional calculus theory and pinning controlling techniques of network synchronization, we propose a strategy that can greatly reduce the expense of the partial topology identification. This method can also identify the whole topology of tempered fractional complex network. Sufficient conditions are given to guarantee partial topology identification of tempered fractional complex networks by designing suitable controllers and parameters update law. This method can also achieve synchronization between the drive network and the response network. Finally, some numerical experiments are presented to verify the validity of the method.

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“…e definitions of Mittag-Leffler stability and generalized Mittag-Leffler stability are proposed in [12,13]. Subsequently, some investigations focus on Mittag-Leffler stability [14][15][16][17][18][19][20][21]. Global Mittag-Leffler stability and synchronization analysis of discrete fractional-order complexvalued neural networks with time delay are given in [14].…”

Section: Introductionmentioning

confidence: 99%

Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

Yan

Qiao

Duan

et al. 2020

Mathematical Problems in Engineering

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In this paper, the global Mittag–Leffler stabilization of fractional-order BAM neural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. Third, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

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Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

Cited by 18 publications

References 37 publications

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Partial topology identification of tempered fractional‐order complex networks via synchronization method

Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

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