Uniform asymptotic stability of \(q\)-deformed conformable fractional systems with delay and application

Volume 30, Issue 1, pp 38--47 https://doi.org/10.22436/jmcs.030.01.05

Publication Date: November 25, 2022 Submission Date: July 21, 2022 Revision Date: August 12, 2022 Accteptance Date: September 09, 2022

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Authors

N. Kamsrisuk - Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand. P. Srisilp - Rail System Institute of Rajamangala University of Technology Isan, Nakhon Ratchasima 30000, Thailand. T. Botmart - Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand. J. Tariboon - Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand. J. Piyawatthanachot - Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand. W. Chartbupapan - Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand. K. Mukdasai - Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand.

Abstract

In this article, we initiate the study of new concepts of conformable \(q\)-fractional calculus. The conformable fractional \(q\)-derivative and \(q\)-integral are defined and their fundamental theorems are also proved. The uniform asymptotic stability of the \(q\)-deformed conformable fractional system with constant delay is investigated by using the Lyapunov-Razumikhin method. For application, a new asymptotic stability necessary condition for the conformable \(q\)-fractional linear system with constant delay is obtained in term of linear matrix inequality (LMI). A numerical example is demonstrated for the results given to illustrate the effectiveness.

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Keywords

• \(q\)-deformed conformable fractional system
• Lyapunov-Razumikhin theorem
• uniform asymptotic stability
• linear matrix inequality

MSC

•  34K20
•  34K25
•  34K37

References

• [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] T. Abdeljawad, D. Baleanu, Caputo q-fractional initial value problems and a q-analogue mittag-leffler function, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4682–4688
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] T. Akram, M. Abbas, A. Ali, A. Iqbal, D. Baleanu, A Numerical Approach of a Time Fractional Reaction-Diffusion Model with a Non-Singular Kernel, Symmetry, 12 (2020), 16 pages
  • View Article
  • Google Scholar
  • MATH


• [4] T. Akram, M. Abbas, A. Iqbal, D. Baleanu, J. H. Asad, Novel Numerical Approach Based on Modified Extended Cubic B-Spline Functions for Solving Non-Linear Time-Fractional Telegraph Equation, Symmetry, 12 (2020), 11 pages
  • View Article
  • Google Scholar


• [5] M. Amin, M. Abbas, M. K. Iqbal, D. Baleanu, Numerical Treatment of Time-Fractional Klein-Gordon Equation Using Redefined Extended Cubic B-Spline Fuctions, Front. Phys., 8 (2020), 12 pages
  • View Article
  • Google Scholar


• [6] R. D. Carmichael, The general theory of linear q-difference equations, Amer. J. Math., 32 (1912), 147–168
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] W. S. Chung, On the q-deformed conformable fractional calculus and the q-deformed generalized conformable fractional calculus, preprint, 2016 (2016), 11 pages
  • View Article
  • Google Scholar


• [8] S. Dashkovskiy, L. Naujok, Lyapunov-Razumikhin and Lyapunov-Krasovskii theorems for interconnected ISS time-delay systems, In Proceedings of the 19th international symposium on mathematical theory of networks and systems (MTNS), 2010 (2010), 5–9
  • View Article
  • Google Scholar


• [9] E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Birkh¨auser/Springer, Cham (2014)
  • View Article
  • Google Scholar
  • MATH


• [10] X. Han, M. Hymavathi, S. Sanober, B. Dhupia, M. S. Ali, Robust Stability of Fractional Order Memristive BAM Neural Networks with Mixed and Additive Time Varying Delays, Fract. Fract., 6 (2022), 20 pages
  • View Article
  • Google Scholar


• [11] O. Herscovici, T. Mansour, q-Deformed conformable fractional natural tranform, arXiv, 2018 (2018), 16 pages
  • View Article
  • Google Scholar


• [12] C. H. Hou, J. X. Qian, On an estimate of the decay rate for applications of Razumikhin-type theorems, IEEE Trans. Automat. Control, 43 (1998), 958–960
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] F. H. Jackson, q-difference equations, Amer. J. Math., 32 (1910), 305–314
  • View Article
  • MathSciNet
  • Google Scholar


• [14] D. O. Jackson, T. Fukuda, O. Dunn, E. Majors, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203
  • View Article
  • Google Scholar
  • MATH


• [15] F. Jarad, T. Abdeljawad, D. Baleanu, Stability of q-fractional non-autonomous systems, Nonlinear Anal. Real World Appl., 14 (2013), 780–784
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York (2001)
  • View Article
  • MathSciNet
  • Google Scholar


• [17] R. Khalil, A. L. M. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70
  • View Article
  • MathSciNet
  • Google Scholar


• [18] I. Koca, E. Demirci, On local asymptotic stability of q-fractional nonlinear dynamical systems, Appl. Appl. Math., 11 (2016), 174–183
  • View Article
  • Google Scholar
  • MATH


• [19] C. P. Li, F. R. Zhang, A survey on the stability of fractional differential equations, Eur. Phys. J. Special Topics, 193 (2011), 27–47
  • View Article
  • Google Scholar


• [20] T. D. Liu, F. Wang, W. C. Lu, X. H. Wang, Global stabilization for a class of nonlinear fractional-order systems, Int. J. Model. Simul. Sci. Comput., 10 (2019), 11 pages
  • View Article
  • Google Scholar


• [21] L. Liu, S. Zhong, Finite-time stability analysis of fractional-order with multi-state time delay, Int. J. Math. Comput. Sci., 5 (2011), 9 pages
  • View Article
  • Google Scholar


• [22] A. Majeed, M. Kamran, M. Abbas, M. Y. Bin Misro, An Efficient Numerical Scheme for The Simulation of Time- Fractional Nonhomogeneous Benjamin-Bona-Mahony-Burger Model, Phys. Scr., 8 (2021), 10 pages
  • View Article
  • Google Scholar


• [23] A. Majeed, M. Kamran, M. Abbas, J. Singh, An Efficient Numerical Technique for Solving Time-Fractional Generalized Fisher’s Equation, Front. Phys., 8 (2020), 8 pages
  • View Article
  • Google Scholar


• [24] M. Musraini, E. Rustam, L. Endang, H. Ponco, Classical properties on conformable fractional calculus, Pure Appl. Math. J., 8 (2019), 83–87
  • View Article
  • Google Scholar


• [25] A. Souahi, A. Ben Makhlouf, M. A. Hammami, Stability analysis of conformable fractional-order nonlinear systems, Indag. Math. (N.S.), 28 (2017), 1265–1274
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] M. Syed Ali, M. Hymavathi, S. A. Kauser, G. Rajchakit, P. Hammachukiattikul, N. Boonsatit, Synchronization of Fractional Order Uncertain BAM Competitive Neural Networks, Fract. Fract., 6 (2022), 17 pages
  • View Article
  • Google Scholar


• [27] M. Syed Ali, G. Narayanan, V. Shekher, A. Alsaedi, B. Ahmad, Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays, Commun. Nonlinear Sci. Numer. Simul., 83 (2020), 22 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] M. Syed Ali, G. Narayanan, V. Shekher, H. Alsulami, T. Saeed, Dynamic stability analysis of stochastic fractional-order memristor fuzzy BAM neural networks with delay and leakage terms, Appl. Math. Comput., 369 (2020), 23 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] M. Syed Ali, G. Narayanan, V. Shekher, S. Arik, Global stability analysis of fractional-order fuzzy BAM neural networks with time delay and impulsive effects, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Difference Equ., 2013 (2013), 19 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] A. R. Teel, Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Trans. Automat. Control, 43 (1998), 960–964
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] M. Tenenbaum, H. Pollard, Ordinary differential equations, Dover Publications, New York (1986)
  • View Article


• [33] F. Usta, M. Z. Sarikaya, Explicit bounds on certain integral inequalities via conformable fractional calculus, Cogent Math., 4 (2017), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] F. Usta, M. Z. Sarikaya, Some improvements of conformable fractional integral inequalities, Int. J. Anal. Appl., 14 (2017), 162–166
  • View Article
  • Google Scholar
  • MATH


• [35] X. H. Wang, Mittag-Leffler stabilization of fractional-order nonlinear systems with unknown control coefficients, Adv. Differ. Equ., 16 (2018), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH