On the stability of solutions of fractional-like equations of perturbed motion

TitleOn the stability of solutions of fractional-like equations of perturbed motion
Publication TypeJournal Article
Year of Publication2018
AuthorsMartynyuk, AA
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2018.06.009
Issue6
SectionMathematics
Pagination9-16
Date Published6/2018
LanguageRussian
Abstract

The application of a fractional-like derivative of the Lyapunov function for the dynamic analysis of solutions of the equations of perturbed motion with a fractional-like derivative of the state vector is discussed. The main theorems of the direct Lyapunov method for a given class of equations of motion are presented.

Keywordsasymptotic stability, fractional-like system of equations, instability, Lyapunov direct method, stability
References: 
  1. Lyapunov, A. M. (1935). The general problem of motion stability. Leningrad, Moscow: ONTI (in Russian).
  2. Podlybny, I. (1999). Fractional differential equations. London, Acad. Press.
  3. Kilbas, A., Srivastava, M.H. & Trujillo, J. J. (2006). Theory and application on fractional differential equations. Amsterdam: North Holland.
  4. Lakshmikantham, V., Leela, S. & Devi, J. V. (2009). Theory of fractional dynamic systems. Cambridge: Cambridge Scientific Publ.
  5. Martynyuk, A. A. (2016). On the stability of a system of equations with fractional derivatives with respect to two measures. J. Math. Sci., 217, No. 4, pp. 468-475. doi: https://doi.org/10.1007/s10958-016-2986-8
  6. Khalil, R., Al Horani, M., Yousef, A. & Sababheh, M. (2014). A new definition of fractional derivative. J. Comput. Appl. Math., 264, pp. 65-70. doi: https://doi.org/10.1016/j.cam.2014.01.002
  7. Abdeljawad, T. (2015). On conformable fractional calculus. J. Comput. Appl. Math., 279, pp. 57-66. doi: https://doi.org/10.1016/j.cam.2014.10.016
  8. Ortigueira, M. D. & Machado, J. A. T. (2015). What is a fractional derivative? J. Comput. Phys., 293, pp. 4-13. doi: https://doi.org/10.1016/j.jcp.2014.07.019
  9. Pospíšil, M., Pospíšilová Škripková, L. (2016). Sturm's theorems for conformable fractional differential equation. Math. Commun., 21, pp. 273-281.
  10. Caputo, M. (1969). Elasticita e dissipazione. Bologna: Zanichelli.
  11. Aguila-Camacho, N., Duarte-Mermoud, M. A. & Gallegos, J. A. (2014). Liapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simulat., 19, pp. 2951-2957. doi: https://doi.org/10.1016/j.cnsns.2014.01.022