A Note About Stability of Fractional Retarded Linear Systems With Distributed Delays

Milev, Mariyan; Zlatev, Stoyan

doi:10.12732/ijpam.v115i4.21

Cited by 10 publications

(13 citation statements)

References 8 publications

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“…The results in this section are a generalization of the results concerning the autonomous case obtained in [10,15,16,25]. Theorem 1.…”

Section: Resultsmentioning

confidence: 57%

“…In the mentioned articles, first a formula for integral representation of the solutions of Cauchy problem is proved, and then, using the obtained result, sufficient conditions for finite time stability of the considered fractional delayed system are established. Furthermore, applying the same approach, in [16], the asymptotic stability properties of nonlinear perturbed linear fractional delayed systems are studied .…”

Section: Discussionmentioning

confidence: 99%

“…Regarding works concerning fractional differential systems with constant delays, we point out [10][11][12][13]. Concerning the retarded differential systems with variable or distributed delays-fundamental theory and application (stability properties)-we refer to [11,[14][15][16][17][18]. Neutral fractional systems with distributed delays are essentially studied less (see [19][20][21]).…”

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confidence: 99%

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Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

et al. 2020

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The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the particular case of fractional systems with constant delays and will be a useful tool for studying different kinds of stability properties. The proposed results coincide with the corresponding ones for first order neutral linear differential systems with integer order derivatives. MSC: 34A08; 34A12 Introduction and NotationsFractional Calculus has a long history, but it has attracted considerable attention recently as an important tool for modeling of various real problems, such as viscoelastic systems, diffusion processes, signal and control processing, and seismic processes. Detailed information about the fractional calculus theory and its applications can be found in the monographs [1][2][3][4]. Some results for fractional linear systems with delays are in given in the book [5]. The monograph [6] is devoted to the impulsive differential and functional differential equations with fractional derivatives, as well as to some of their applications.It is well known that the study of linear fractional equations (integral representation, several types of stability, etc.) is an evergreen theme for research. Concerning these fields of fundamental and qualitative investigations for linear fractional ordinary differential equations and systems we refer to [2,4,7] and the references therein. Using the Laplace transform method, several interesting results in this direction are obtained in [8,9] as well. Regarding works concerning fractional differential systems with constant delays, we point out [10][11][12][13]. Concerning the retarded differential systems with variable or distributed delays-fundamental theory and application (stability properties)-we refer to [11,[14][15][16][17][18]. Neutral fractional systems with distributed delays are essentially studied less (see [19][20][21]). Stability properties of retarded fractional systems with derivatives of distributed order are studied in [22]. One of the existing best applications of fractional order equations with delays is modeling human manual control, in which perceptual and neuromuscular delays introduce a delay term. As interesting studies, we refer to [23,24].

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“…The results in this section are a generalization of the results concerning the autonomous case obtained in [10,15,16,25]. Theorem 1.…”

Section: Resultsmentioning

confidence: 57%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

et al. 2020

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“…When q = n ∈ N, then a D q t 0 is the usual integer-order derivative. In the development of fractional calculus theory, the most widely recognized definitions of fractional derivatives include the Caputo, Riemann-Liouville (RL) and Grünwald-Letnikov (GL) definitions [37,38]. Caputo defines that whether it is a fractional-order differential equation or an integer-order differential equation, its initial condition can be consistent, and it has a clear explanation of the initial condition of integer-order, and has the advantage of zero initial value when applied to constants [39].…”

Section: Fractional Calculus Theorymentioning

confidence: 99%

Dynamical Analysis and Misalignment Projection Synchronization of a Novel RLCM Fractional-Order Memristor Circuit System

Liu,

Tian,

Wang

et al. 2023

Axioms

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In this paper, a simple and novel fractional-order memristor circuit is established, which contains only resistance, inductance, capacitance and memristor. By using fractional calculus theory and the Adomian numerical algorithm, special bifurcations, chaotic degradation, C0 and Spectral Entropy (SE) complexity under one-dimensional and two-dimensional parameter variations with different orders, parameters and initial memristor values of the system were studied. Meanwhile, in order to better utilize the applications of fractional-order memristor systems in communication and security, a misalignment projection synchronization scheme for fractional-order systems is proposed, which overcomes the shortcomings of constructing Lyapunov functions for fractional-order systems to prove stability and designing controllers for the Laplace transform matrix.

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“…Different types fractional differential equations and systems with delays (retarded and neutral) or without delays are studied for several types of stability. As works related to this theme we refer to [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Stability of the Solutions of Neutral Linear Fractional System with Nonlinear Perturbation

Zahariev¹,

Kiskinov²

2020

Mathematics

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In this article existence and uniqueness of the solutions of the initial problem for neutral nonlinear differential system with incommensurate order fractional derivatives in Caputo sense and with piecewise continuous initial function is proved. A formula for integral presentation of the general solution of a linear autonomous neutral system with several delays is established and used for the study of the stability properties of a neutral autonomous nonlinear perturbed linear fractional differential system. Natural sufficient conditions are found to ensure that from global asymptotic stability of the zero solution of the linear part of a nonlinearly perturbed system it follows global asymptotic stability of the zero solution of the whole nonlinearly perturbed system.

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A Note About Stability of Fractional Retarded Linear Systems With Distributed Delays

Cited by 10 publications

References 8 publications

Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

Dynamical Analysis and Misalignment Projection Synchronization of a Novel RLCM Fractional-Order Memristor Circuit System

Asymptotic Stability of the Solutions of Neutral Linear Fractional System with Nonlinear Perturbation

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