Email this article ABOUT AN ESTIMATE FROM ABOVE OF THE FRACTIONAL DERIVATIVE OF THE COMPOSITION OF TWO FUNCTIONS
- Authors: Gomoyunov M.I.1
-
Affiliations:
- N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences
- Issue: Vol 23, No 122 (2018)
- Pages: 261-267
- Section: Articles
- URL: https://bakhtiniada.ru/2686-9667/article/view/297230
- DOI: https://doi.org/10.20310/1810-0198-2018-23-122-261-267
- ID: 297230
Cite item
Full Text
Abstract
In the paper, an estimate from above of the fractional Riemann-Liouville derivative of an order α∈0;1 of the composition of two functions is proved for the case when the inner function is assumed only to be represented by the fractional Riemann-Liouville integral of a measurable essentially bounded function. The necessity of such an estimate arises in control problems of dynamical systems described by differential equations with fractional derivatives.
Full Text
Пусть [a, b] ⊂ R, n ∈ N и заданы функции V : Rn → R и x : [a, b] → Rn. Рассмотрим их композицию v(t) = V (x(t)), t ∈ [a, b].×
About the authors
Mikhail Igorevich Gomoyunov
N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences
Email: m.i.gomoyunov@gmail.com
Candidate of Physics and Mathematics, Senior Researcher of the Dynamical Systems Department 16 S. Kovalevskaya St., Yekaterinburg 620990, Russian Federation
References
- Малкин И.Г. Теория устойчивости движения. М.: Наука, 1966. 530 c.
- Красовский Н.Н. Управление динамической системой. М.: Наука, 1985. 516 c.
- Кряжимский А.В, Осипов Ю.С. О моделировании управления в динамической системе // Известия АН СССР. Техническая кибернетика. 1983. № 2. С. 51-60.
- Самко С.Г., Килбас А.А., Маричев О.И. Интегралы и производные дробного порядка и некоторые их приложения. Минск: Наука и техника, 1987. 688 с.
- Podlubny I. Fractional differential equations. San Diego: Academic Press, 1999. 366 p.
- Idczak D., Kamocki R. On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in Rn // Frac. Calc. Appl. Anal. 2011. Vol. 14. № 4. P. 538-553.
- Tarasov V.E. On chain rule for fractional derivatives // Commun. Nonlinear Sci. Numer. Simulat. 2016. Vol. 30. P. 1-4.
- Алиханов А.А. Априорные оценки решений краевых задач для уравнений дробного порядка // Дифференциальные уравнения. 2010. Т. 46. № 5. С. 658-664.
- Aguila-Camacho N., Duarte-Mermoud M.A., Gallegos J.A. Lyapunov functions for fractional order systems // Commun. Nonlinear Sci. Numer. Simulat. 2014. Vol. 19. Issue 9. P. 2951-2957.
- Chen W., Dai H., Song Y., Zhang Z. Convex Lyapunov functions for stability analysis of fractional order systems // IET Control Theory Appl. 2017. Vol. 11. № 7. P. 1070-1074.
- Ross B., Samko S.G., Love E.R. Functions that have no first order derivative might have fractional derivatives of all orders less than one // Real Anal. Exchange. 1994-1995. Vol. 20. № 1. P. 140-157.
- Канторович Л.В., Акилов Г.П. Функциональный анализ. М.: Наука, 1984. 752 с.
Supplementary files
