IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i6p500-d236555.html
   My bibliography  Save this article

Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

Author

Listed:
  • Guotao Wang

    (School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, China
    College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China)

  • Jianfang Qin

    (School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, China)

  • Huanhe Dong

    (College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China)

  • Tingting Guan

    (School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, China)

Abstract

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

Suggested Citation

  • Guotao Wang & Jianfang Qin & Huanhe Dong & Tingting Guan, 2019. "Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System," Mathematics, MDPI, vol. 7(6), pages 1-10, June.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:500-:d:236555
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/6/500/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/6/500/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Wang, Zhen & Xie, Yingkang & Lu, Junwei & Li, Yuxia, 2019. "Stability and bifurcation of a delayed generalized fractional-order prey–predator model with interspecific competition," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 360-369.
    2. Wang, JinRong & Zhang, Yuruo, 2015. "Nonlocal initial value problems for differential equations with Hilfer fractional derivative," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 850-859.
    3. Yang, Xujun & Li, Chuandong & Huang, Tingwen & Song, Qiankun, 2017. "Mittag–Leffler stability analysis of nonlinear fractional-order systems with impulses," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 416-422.
    4. Gu, Haibo & Trujillo, Juan J., 2015. "Existence of mild solution for evolution equation with Hilfer fractional derivative," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 344-354.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nawapol Phuangthong & Sotiris K. Ntouyas & Jessada Tariboon & Kamsing Nonlaopon, 2021. "Nonlocal Sequential Boundary Value Problems for Hilfer Type Fractional Integro-Differential Equations and Inclusions," Mathematics, MDPI, vol. 9(6), pages 1-19, March.
    2. Ahmed, Hamdy M. & El-Borai, Mahmoud M., 2018. "Hilfer fractional stochastic integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 182-189.
    3. Surang Sitho & Sotiris K. Ntouyas & Ayub Samadi & Jessada Tariboon, 2021. "Boundary Value Problems for ψ -Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions," Mathematics, MDPI, vol. 9(9), pages 1-18, April.
    4. Athasit Wongcharoen & Sotiris K. Ntouyas & Jessada Tariboon, 2020. "Boundary Value Problems for Hilfer Fractional Differential Inclusions with Nonlocal Integral Boundary Conditions," Mathematics, MDPI, vol. 8(11), pages 1-11, October.
    5. Debbouche, Amar & Antonov, Valery, 2017. "Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 140-148.
    6. Nisar, Kottakkaran Sooppy & Jothimani, K. & Kaliraj, K. & Ravichandran, C., 2021. "An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    7. Gou, Haide & Li, Baolin, 2018. "Study on the mild solution of Sobolev type Hilfer fractional evolution equations with boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 168-179.
    8. Kavitha, K. & Vijayakumar, V. & Udhayakumar, R., 2020. "Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    9. Li, Xing-Yu & Wu, Kai-Ning & Liu, Xiao-Zhen, 2023. "Mittag–Leffler stabilization for short memory fractional reaction-diffusion systems via intermittent boundary control," Applied Mathematics and Computation, Elsevier, vol. 449(C).
    10. Jianguang Zhu & Kai Li & Binbin Hao, 2019. "Image Restoration by Second-Order Total Generalized Variation and Wavelet Frame Regularization," Complexity, Hindawi, vol. 2019, pages 1-16, March.
    11. Wu, Tianyu & Huang, Xia & Chen, Xiangyong & Wang, Jing, 2020. "Sampled-data H∞ exponential synchronization for delayed semi-Markov jump CDNs: A looped-functional approach," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    12. Ahmed, Hamdy M. & Zhu, Quanxin, 2023. "Exploration nonlocal controllability for Hilfer fractional differential inclusions with Clarke subdifferential and nonlinear noise," Statistics & Probability Letters, Elsevier, vol. 195(C).
    13. Wang, Xinhe & Lu, Junwei & Wang, Zhen & Li, Yuxia, 2020. "Dynamics of discrete epidemic models on heterogeneous networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    14. Ma, Tingting & Meng, Xinzhu & Hayat, Tasawar & Hobiny, Aatef, 2021. "Stability analysis and optimal harvesting control of a cross-diffusion prey-predator system," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    15. Ghanbari, Behzad, 2021. "On detecting chaos in a prey-predator model with prey’s counter-attack on juvenile predators," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    16. B. Radhakrishnan & T. Sathya, 2022. "Controllability of Hilfer Fractional Langevin Dynamical System with Impulse in an Abstract Weighted Space," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 265-281, October.
    17. Syed Ali, M. & Narayanan, Govindasamy & Shekher, Vineet & Alsulami, Hamed & Saeed, Tareq, 2020. "Dynamic stability analysis of stochastic fractional-order memristor fuzzy BAM neural networks with delay and leakage terms," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    18. Harshavarthini, S. & Sakthivel, R. & Ma, Yong-Ki & Muslim, M., 2020. "Finite-time resilient fault-tolerant investment policy scheme for chaotic nonlinear finance system," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    19. Huang, Chengdai & Liu, Heng & Chen, Xiaoping & Zhang, Minsong & Ding, Ling & Cao, Jinde & Alsaedi, Ahmed, 2020. "Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 554(C).
    20. Dhayal, Rajesh & Zhu, Quanxin, 2023. "Stability and controllability results of ψ-Hilfer fractional integro-differential systems under the influence of impulses," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:500-:d:236555. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.