Fractional differential equations with a constant delay: Stability and asymptotics of solutions

In the chapter two the most important properties of fractional order dynamical systems, namely, controllability and stability are presented. At the beginning the basic notations and the fundamental definitions are recalled. The first part of the chapter is devoted to controllability and contains the formulation of the problem, main hypotheses and theorems about controllability of semilinear fractional order systems with distributed and point multiplicities constant or variable delays in the state variables and controls. Next, using fixed point theorems approximate controllability problem in infinite dimensional spaces, in particular Banach or Hilbert space, are discussed. Second part of the chapter is devoted to the stability problem of fractional order systems. The problem of stability and the problem of the existence of solutions for linear and nonlinear fractional order systems are also presented.

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“…A function ϕ(•, φ) ∈ C([−r, T ]; R d ) is called a solution of the initial condition problem (15) and (16)…”

Section: Solution Existencementioning

confidence: 99%

Controllability and Stability of Semilinear Fractional Order Systems

Klamka

Babiarz

Czornik

et al. 2020

Automatic Control, Robotics, and Information Processing

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“…Abd-Elhameed and Youssri [1,2] introduced the fifth-kind orthonormal Chebyshev polynomial and the generalized Lucas polynomial sequence methods to obtain the numerical solution of FDEs. The stability and asymptotics of solutions of fractional differential equations with constant delay were investigated by Cermák et al [16].…”

Section: Ams Mathematics Subject Classification: 34k37 65l60 68r10mentioning

confidence: 99%

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

Kürkçü¹,

Aslan²,

Sezer³

2019

Turk J Math

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In this study, we introduce multidelay fractional differential equations with variable coefficients in a unique formula. A novel graph-operational matrix method based on the fractional Caputo, Riemann-Liouville, Caputo-Fabrizio, and Jumarie derivative types is developed to efficiently solve them. We also make use of the collocation points and matrix relations of the matching polynomial of the complete graph in the method. We determine which of the fractional derivative types is more appropriate for the method. The solutions of model problems are improved via a new residual error analysis technique. We design a general computer program module. Thus, we can explicitly monitor the usefulness of the method. All results are scrutinized in tables and figures. Finally, an illustrative algorithm is presented.

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“…Fractional-order calculus has gained much attention in recent three decades because of its widespread application, such as engineering, diffusion equations, control science, biology, calorifics, and so on [1][2][3][4][5]. As an important branch of fractional-order calculus, stability has been studied by many scholars [6][7][8][9][10][11][12][13][14][15]. In [16,17], the Mittag-Leffler stability for fractional nonlinear equation has been discussed.…”

Section: Introductionmentioning

confidence: 99%

Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays

Zhang

Liu

et al. 2019

Mathematics

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In this paper, the global asymptotical stability of Riemann-Liouville fractional-order neural networks with time-varying delays is studied. By combining the Lyapunov functional function and LMI approach, some sufficient criteria that guarantee the global asymptotical stability of such fractional-order neural networks with both discrete time-varying delay and distributed time-varying delay are derived. The stability criteria is suitable for application and easy to be verified by software. Lastly, some numerical examples are presented to check the validity of the obtained results.

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Fractional differential equations with a constant delay: Stability and asymptotics of solutions

Cited by 31 publications

References 16 publications

Controllability and Stability of Semilinear Fractional Order Systems

Controllability and Stability of Semilinear Fractional Order Systems

A novel graph-operational matrix method for solving multidelay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types

Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays

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