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Stability Analysis of Fractional Systems
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Stability Analysis of Fractional Systems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 13647

Special Issue Editors

Prof. Dr. Andrey Zahariev

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Interests: fractional differential equations; functional-differential equations; impulsive differential equations; differential equations in Banach spaces; integral equations; integral inequalities; stability analysis; real and functional analysis; applied mathematics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hristo Kiskinov

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Interests: fractional differential equations; impulsive differential equations; functional-differential equations; differential equations in Banach spaces
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nowadays, fractional dynamical systems (FDSs) based on different kinds of fractional derivatives (Riemann–Liouville, Caputo, Gruenwald–Letnikov, and other fractional derivatives with a good mathematical and physical background) have become powerful tools for modelling real-world phenomena.

This Special Issue, “Stability Analysis of Fractional Systems”, invites papers that focus on the recent and novel developments in the stability theory of fractional dynamical systems of different types (with or without delays, impulsive or not, and so on). We expect high-quality articles concerning various types of stabilities, namely: Lyapunov’s type, finite time stability, Mittag-Leffler stability, robust stability, Hyers–Ulam–Rassias stability, and so on.

Works exploring the possibilities for using FDSs as relevant models in the applied sciences, for example, neural networks modeled with FDS, are welcome too.

Prof. Dr. Andrey Zahariev
Prof. Dr. Hristo Kiskinov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional dynamical systems
  • fractional differential systems
  • stability
  • fractional derivatives and integrals

Published Papers (7 papers)

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Research

12 pages, 331 KiB  
Article
On the Stability of Linear Incommensurate Fractional-Order Difference Systems
by Noureddine Djenina, Adel Ouannas, Iqbal M. Batiha, Giuseppe Grassi and Viet-Thanh Pham
Mathematics 2020, 8(10), 1754; https://doi.org/10.3390/math8101754 - 13 Oct 2020
Cited by 17 | Viewed by 2088
Abstract
To follow up on the progress made on exploring the stability investigation of linear commensurate Fractional-order Difference Systems (FoDSs), such topic of its extended version that appears with incommensurate orders is discussed and examined in this work. Some simple applicable conditions for judging [...] Read more.
To follow up on the progress made on exploring the stability investigation of linear commensurate Fractional-order Difference Systems (FoDSs), such topic of its extended version that appears with incommensurate orders is discussed and examined in this work. Some simple applicable conditions for judging the stability of these systems are reported as novel results. These results are formulated by converting the linear incommensurate FoDS into another equivalent system consists of fractional-order difference equations of Volterra convolution-type as well as by using some properties of the Z-transform method. All results of this work are verified numerically by illustrating some examples that deal with the stability of solutions of such systems. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems)
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17 pages, 321 KiB  
Article
Existence and Stability Analysis for Fractional Impulsive Caputo Difference-Sum Equations with Periodic Boundary Condition
by Rujira Ouncharoen, Saowaluck Chasreechai and Thanin Sitthiwirattham
Mathematics 2020, 8(5), 843; https://doi.org/10.3390/math8050843 - 22 May 2020
Cited by 2 | Viewed by 1548
Abstract
In this paper, by using the Banach contraction principle and the Schauder’s fixed point theorem, we investigate existence results for a fractional impulsive sum-difference equations with periodic boundary conditions. Moreover, we also establish different kinds of Ulam stability for this problem. An example [...] Read more.
In this paper, by using the Banach contraction principle and the Schauder’s fixed point theorem, we investigate existence results for a fractional impulsive sum-difference equations with periodic boundary conditions. Moreover, we also establish different kinds of Ulam stability for this problem. An example is also constructed to demonstrate the importance of these results. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems)
18 pages, 357 KiB  
Article
Asymptotic Stability of the Solutions of Neutral Linear Fractional System with Nonlinear Perturbation
by Andrey Zahariev and Hristo Kiskinov
Mathematics 2020, 8(3), 390; https://doi.org/10.3390/math8030390 - 10 Mar 2020
Cited by 8 | Viewed by 1691
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems)
14 pages, 335 KiB  
Article
The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems
by Hanaa Zitane, Ali Boutoulout and Delfim F. M. Torres
Mathematics 2020, 8(3), 353; https://doi.org/10.3390/math8030353 - 05 Mar 2020
Cited by 4 | Viewed by 2467
Abstract
We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and [...] Read more.
We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics are symmetric and uniformly elliptical and by using the properties of the Mittag–Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems)
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11 pages, 321 KiB  
Article
Existence of Bounded Solutions to a Modified Version of the Bagley–Torvik Equation
by Daniel Cao Labora and José António Tenreiro Machado
Mathematics 2020, 8(2), 289; https://doi.org/10.3390/math8020289 - 20 Feb 2020
Viewed by 1657
Abstract
This manuscript reanalyses the Bagley–Torvik equation (BTE). The Riemann–Liouville fractional differential equation (FDE), formulated by R. L. Bagley and P. J. Torvik in 1984, models the vertical motion of a thin plate immersed in a Newtonian fluid, which is held by a spring. [...] Read more.
This manuscript reanalyses the Bagley–Torvik equation (BTE). The Riemann–Liouville fractional differential equation (FDE), formulated by R. L. Bagley and P. J. Torvik in 1984, models the vertical motion of a thin plate immersed in a Newtonian fluid, which is held by a spring. From this model, we can derive an FDE for the particular case of lacking the spring. Here, we find conditions for the source term ensuring that the solutions to the equation of the motion are bounded, which has a clear physical meaning. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems)
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15 pages, 269 KiB  
Article
Impulsive Delayed Lasota–Wazewska Fractional Models: Global Stability of Integral Manifolds
by Gani Stamov and Ivanka Stamova
Mathematics 2019, 7(11), 1025; https://doi.org/10.3390/math7111025 - 31 Oct 2019
Cited by 9 | Viewed by 1868
Abstract
In this paper we deal with the problems of existence, boundedness and global stability of integral manifolds for impulsive Lasota–Wazewska equations of fractional order with time-varying delays and variable impulsive perturbations. The main results are obtained by employing the fractional Lyapunov method and [...] Read more.
In this paper we deal with the problems of existence, boundedness and global stability of integral manifolds for impulsive Lasota–Wazewska equations of fractional order with time-varying delays and variable impulsive perturbations. The main results are obtained by employing the fractional Lyapunov method and comparison principle for impulsive fractional differential equations. With this research we generalize and improve some existing results on fractional-order models of cell production systems. These models and applied technique can be used in the investigation of integral manifolds in a wide range of biological and chemical processes. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems)
13 pages, 803 KiB  
Article
On Dynamic Systems in the Frame of Singular Function Dependent Kernel Fractional Derivatives
by Thabet Abdeljawad, Fadila Madjidi, Fahd Jarad and Ndolane Sene
Mathematics 2019, 7(10), 946; https://doi.org/10.3390/math7100946 - 11 Oct 2019
Cited by 19 | Viewed by 1802
Abstract
In this article, we discuss the existence and uniqueness theorem for differential equations in the frame of Caputo fractional derivatives with a singular function dependent kernel. We discuss the Mittag-Leffler bounds of these solutions. Using successive approximation, we find a formula for the [...] Read more.
In this article, we discuss the existence and uniqueness theorem for differential equations in the frame of Caputo fractional derivatives with a singular function dependent kernel. We discuss the Mittag-Leffler bounds of these solutions. Using successive approximation, we find a formula for the solution of a special case. Then, using a modified Laplace transform and the Lyapunov direct method, we prove the Mittag-Leffler stability of the considered system. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems)
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