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Fractal Fract
Special Issues
Stability Analysis and Control of Fractional-Order Markovian Jump Systems
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Stability Analysis and Control of Fractional-Order Markovian Jump Systems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (20 June 2023) | Viewed by 6200

Special Issue Editor

Dr. M. Syed Ali

E-Mail Website
Guest Editor
Department of Mathematics, Thiruvalluvar University, Vellore 632115, India
Interests: fractional calculus; stability analysis; dynamical systems; neural networks

Special Issue Information

Dear Colleagues,

The study of fractional differential equations has recently emerged as a new subject of research in applied mathematics. In modeling the heredity and memory properties of many materials and processes, fractional-order models have been shown to be superior to the traditional integer-order models. Due to the unchanging persistence of researchers, some real-world applications of fractional-order systems have been discovered, including network approximation, state estimation and system identification, robotic manipulators, formation control, disease treatment, and so on. As a result, the stability analysis of fractional-order systems have been reported.

In the past few decades, Markov jump systems (MJSs) have been an active area of research. They switch from one mode to another in a random way. The switching between modes is governed by a Markovian process with discrete and finite state space. These models serve as convenient tools for analyzing plants that are subjected to random abrupt changes, which may result from random component failures, abrupt environment changes, disturbance, and changes in the interconnections of subsystems. As a dominant factor, the transition rates (TRs) in the jumping process determine the system behavior to a large extent, and so far, many analysis and synthesis results have been reported, assuming complete knowledge of the transition rates. In practice, it is difficult to precisely estimate the TRs. Therefore, developing analysis and synthesis methods for fractional-order systems with Markovian jumping parameters has received the attention of researchers.

This Special Issue is focused on the stability analysis and control of fractional-order dynamical systems with Markovian jumping parameters. Submissions focused on the robust stability of fractional-order complex systems, fractional-order stochastic systems, and the stochastic stabilization of fractional-order nonlinear dynamical systems with Markovian parameters. Potential topics include, but are not limited to, the following:

  • Stochastic fractional-order dynamic model development with Markovian parameters;
  • Robust stability analysis of fractional-order chaotic systems with Markovian parameters;
  • Novel stochastic stabilization of fractional-order nonlinear dynamical systems with Markovian parameters;
  • Stability analysis for stochastic fractional order systems with Markovian parameters;
  • Discrete-time fractional-order systems with Markovian parameters;
  • Fractional-order switching systems with Markovian parameters;
  • Stability and boundedness control on open fractional-order economy systems or ecology systems with Markovian parameters.

Dr. M. Syed Ali
Guest Editor

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. Fractal and Fractional is an international peer-reviewed open access monthly 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 2700 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-order systems
  • Markovian jumping parameters
  • stability
  • synchronization

Published Papers (4 papers)

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20 pages, 549 KiB  
Article
Synchronization of Discrete-Time Fractional-Order Complex-Valued Neural Networks with Distributed Delays
by R. Perumal, M. Hymavathi, M. Syed Ali, Batul A. A. Mahmoud, Waleed M. Osman and Tarek F. Ibrahim
Fractal Fract. 2023, 7(6), 452; https://doi.org/10.3390/fractalfract7060452 - 01 Jun 2023
Cited by 1 | Viewed by 959
Abstract
This research investigates the synchronization of distributed delayed discrete-time fractional-order complex-valued neural networks. The necessary conditions have been established for the stability of the proposed networks using the theory of discrete fractional calculus, the discrete Laplace transform, and the theory of fractional-order discrete [...] Read more.
This research investigates the synchronization of distributed delayed discrete-time fractional-order complex-valued neural networks. The necessary conditions have been established for the stability of the proposed networks using the theory of discrete fractional calculus, the discrete Laplace transform, and the theory of fractional-order discrete Mittag–Leffler functions. In order to guarantee the global asymptotic stability, adequate criteria are determined using Lyapunov’s direct technique, the Lyapunov approach, and some novel analysis techniques of fractional calculation. Thus, some sufficient conditions are obtained to guarantee the global stability. The validity of the theoretical results are finally shown using numerical examples. Full article
(This article belongs to the Special Issue Stability Analysis and Control of Fractional-Order Markovian Jump Systems)
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21 pages, 2946 KiB  
Article
From Stochastic to Rough Volatility: A New Deep Learning Perspective on Hedging
by Qinwen Zhu and Xundi Diao
Fractal Fract. 2023, 7(3), 225; https://doi.org/10.3390/fractalfract7030225 - 02 Mar 2023
Cited by 3 | Viewed by 1579
Abstract
The Black–Scholes model assumes that volatility is constant, and the Heston model assumes that volatility is stochastic, while the rough Bergomi (rBergomi) model, which allows rough volatility, can perform better with high-frequency data. However, classical calibration and hedging techniques are difficult to apply [...] Read more.
The Black–Scholes model assumes that volatility is constant, and the Heston model assumes that volatility is stochastic, while the rough Bergomi (rBergomi) model, which allows rough volatility, can perform better with high-frequency data. However, classical calibration and hedging techniques are difficult to apply under the rBergomi model due to the high cost caused by its non-Markovianity. This paper proposes a gated recurrent unit neural network (GRU-NN) architecture for hedging with different-regularity volatility. One advantage is that the gating network signals embedded in our architecture can control how the present input and previous memory update the current activation. These gates are updated adaptively in the learning process and thus outperform conventional deep learning techniques in a non-Markovian environment. Our numerical results also prove that the rBergomi model outperforms the other two models in hedging. Full article
(This article belongs to the Special Issue Stability Analysis and Control of Fractional-Order Markovian Jump Systems)
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16 pages, 1551 KiB  
Article
On the Variable Order Fractional Calculus Characterization for the Hidden Variable Fractal Interpolation Function
by Valarmathi Raja and Arulprakash Gowrisankar
Fractal Fract. 2023, 7(1), 34; https://doi.org/10.3390/fractalfract7010034 - 28 Dec 2022
Cited by 4 | Viewed by 1193
Abstract
In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus are investigated for hidden [...] Read more.
In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus are investigated for hidden variable fractal interpolation function. Moreover, the conditions for the variable fractional order μ on a specified range are also derived. It is observed that, under certain conditions, the Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus of the hidden variable fractal interpolation function are again the hidden variable fractal interpolation functions interpolating the new data set. Full article
(This article belongs to the Special Issue Stability Analysis and Control of Fractional-Order Markovian Jump Systems)
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10 pages, 291 KiB  
Article
Fractals via Controlled Fisher Iterated Function System
by C. Thangaraj and D. Easwaramoorthy
Fractal Fract. 2022, 6(12), 746; https://doi.org/10.3390/fractalfract6120746 - 19 Dec 2022
Cited by 2 | Viewed by 1469
Abstract
This paper explores the generalization of the fixed-point theorem for Fisher contraction on controlled metric space. The controlled metric space and Fisher contractions are playing a very crucial role in this research. The Fisher contraction on the controlled metric space is used in [...] Read more.
This paper explores the generalization of the fixed-point theorem for Fisher contraction on controlled metric space. The controlled metric space and Fisher contractions are playing a very crucial role in this research. The Fisher contraction on the controlled metric space is used in this paper to generate a new type of fractal set called controlled Fisher fractals (CF-Fractals) by constructing a system named the controlled Fisher iterated function system (CF-IFS). Furthermore, the interesting results and consequences of the controlled Fisher iterated function system and controlled Fisher fractals are demonstrated. In addition, the collage theorem on controlled Fisher fractals is established as well. The newly developing IFS and fractal set in the controlled metric space can provide the novel directions in the fractal theory. Full article
(This article belongs to the Special Issue Stability Analysis and Control of Fractional-Order Markovian Jump Systems)
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