Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities
Abstract
:1. Introduction
- denotes fractional integrals. The left superscript of denotes the type of integral, while the right superscript represents the order, and the right subscript indicates the lower limit of the integral:
- –
- is the Riemann–Liouville fractional integral of order .
- –
- is the Atangana–Baleanu fractional integral of Riemann–Liouville type of order .
- –
- is the fractional-order integral operator with general analytic kernel with orders and .
- denotes fractional derivatives. The left superscript of denotes the type of derivative, while the right superscript represents the order, and the right subscript indicates the lower limit of the set where the operator is being applied:
- –
- is the Riemann–Liouville fractional differential operator of order .
- –
- is the Caputo differential operator of order .
- –
- is the Caputo–Fabrizio differential operator of order .
- –
- is the Atangana–Baleanu differential operator of Caputo type of order .
- –
- is the Atangana–Baleanu differential operator of Riemann–Liouville type of order .
- In the case of the differential operators with general analytic kernel, both left super- and subscripts are considered:
- –
- is the Riemann–Liouville fractional differential operator with general analytic kernel with orders and .
- –
- is the Caputo fractional differential operator with general analytic kernel with orders and .
- For a candidate Lyapunov function , [40].
2. Preliminaries
- (a)
- If , , and , then
- (b)
- If , , and , then
- (c)
- If , , and , then
- (d)
- If , , and , then
3. Some Results with the General Analytic Kernel Operators
4. Laplace Transform and Generalized Lyapunov Direct Method
- (1)
- The series is uniformly convergent and satisfies for .
- (2)
- .
5. Useful Inequalities for Lyapunov Stability Analysis
- (a)
- If , , and , then for all
- (b)
- If , , and , then for all
- (c)
- If , , and , then for all
6. Convex Lyapunov Functions and Stability
- (a)
- If , , and , then, for all
- (b)
- If , , and , then for all
- (a)
- If , , , and , then, for all ,
- (b)
- If , , and , then, for all ,
- (c)
- If , , and , then, for all ,
7. Some Representative Examples
7.1. Scalar Systems
7.2. Second Order Systems
7.3. A Spacecraft Modeled by Generalized Dynamics
7.4. Financial Analysis
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Podlubny, I. Fractional Differential Equations; Academic Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach Sci. Publishers: Singapore, 1993. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific Singapore: Singapore, 2000. [Google Scholar]
- Oldham, K.; Spanier, J. The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order; Elsevier: Amsterdam, The Netherlands, 1974. [Google Scholar]
- Tavazoei, M.; Asemani, M.H. On Robust Stability of Incommensurate Fractional-Order Systems. Commun. Nonlinear Sci. Numer. Simul. 2020, 90, 105344. [Google Scholar] [CrossRef]
- Liu, S.; Yang, R.; Zhou, X.F.; Jiang, W.; Li, X.; Zhao, X.W. Stability analysis of fractional delayed equations and its applications on consensus of multi-agent systems. Commun. Nonlinear Sci. Numer. Simul. 2019, 73, 351–362. [Google Scholar] [CrossRef]
- Lenka, B.K.; Banerjee, S. Sufficient conditions for asymptotic stability and stabilization of autonomous fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 2018, 56, 365–379. [Google Scholar] [CrossRef]
- Lenka, B.K. Fractional comparison method and asymptotic stability results for multivariable fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 2019, 69, 398–415. [Google Scholar] [CrossRef]
- Martínez-Fuentes, O.; Martínez-Guerra, R. A high-gain observer with Mittag–Leffler rate of convergence for a class of nonlinear fractional-order systems. Commun. Nonlinear Sci. Numer. Simul. 2019, 79, 104909. [Google Scholar] [CrossRef]
- Baleanu, D.; Fernandez, A. On fractional operators and their classifications. Mathematics 2019, 7, 830. [Google Scholar] [CrossRef] [Green Version]
- De Oliveira, E.C.; Tenreiro Machado, J.A. A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014, 2014. [Google Scholar] [CrossRef] [Green Version]
- Fernández-Anaya, G.; Nava-Antonio, G.; Jamous-Galante, J.; Muñoz-Vega, R.; Hernández-Martínez, E. Asymptotic stability of distributed order nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 2017, 48, 541–549. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Fernandez, A.; Baleanu, D. Some new fractional-calculus connections between Mittag–Leffler functions. Mathematics 2019, 7, 485. [Google Scholar] [CrossRef] [Green Version]
- Hilfer, R.; Luchko, Y. Desiderata for fractional derivatives and integrals. Mathematics 2019, 7, 149. [Google Scholar] [CrossRef] [Green Version]
- Jarad, F.; Uğurlu, E.; Abdeljawad, T.; Baleanu, D. On a new class of fractional operators. Adv. Differ. Equ. 2017, 2017, 247. [Google Scholar] [CrossRef]
- Ren, J.; Zhai, C. Stability analysis for generalized fractional differential systems and applications. Chaos Solitons Fractals 2020, 139, 110009. [Google Scholar] [CrossRef]
- Akkurt, A.; Yildirim, M.; Yildirim, H. A new Generalized fractional derivative and integral. Konuralp J. Math. 2017, 5, 248–259. [Google Scholar]
- Zhao, D.; Luo, M. General conformable fractional derivative and its physical interpretation. Calcolo 2017, 54, 903–917. [Google Scholar] [CrossRef]
- Restrepo, J.E.; Ruzhansky, M.; Suragan, D. Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions. Appl. Math. Comput. 2021, 403, 126177. [Google Scholar]
- Luchko, Y. General fractional integrals and derivatives with the Sonine kernels. Mathematics 2021, 9, 594. [Google Scholar] [CrossRef]
- Fernandez, A.; Özarslan, M.A.; Baleanu, D. On fractional calculus with general analytic kernels. Appl. Math. Comput. 2019, 354, 248–265. [Google Scholar] [CrossRef] [Green Version]
- Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1, 73–85. [Google Scholar]
- Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci. 2016, 20, 763–769. [Google Scholar] [CrossRef] [Green Version]
- Owolabi, K.; Atangana, A. Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann–Liouville sense. Chaos Solitons Fractals 2017, 99, 171–179. [Google Scholar] [CrossRef]
- Al-Refai, M.; Jarrah, A. Fundamental results on weighted Caputo-Fabrizio fractional derivative. Chaos Solitons Fractals 2019, 126, 7–11. [Google Scholar] [CrossRef]
- Zheng, X.; Wang, H.; Fu, H. Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative. Chaos Solitons Fractals 2020, 138, 109966. [Google Scholar] [CrossRef]
- Kumar, A.; Pandey, D.N. Existence of mild solution of Atangana–Baleanu fractional differential equations with non-instantaneous impulses and with non-local conditions. Chaos Solitons Fractals 2020, 132, 109551. [Google Scholar] [CrossRef]
- Yadav, S.; Pandey, R.K. Numerical approximation of fractional burgers equation with Atangana–Baleanu derivative in Caputo sense. Chaos Solitons Fractals 2020, 133, 109630. [Google Scholar] [CrossRef]
- Sadeghi, S.; Jafari, H.; Nemati, S. Operational matrix for Atangana–Baleanu derivative based on Genocchi polynomials for solving FDEs. Chaos Solitons Fractals 2020, 135, 109736. [Google Scholar] [CrossRef]
- Shaikh, A.; Nisar, K. Transmission dynamics of fractional order Typhoid fever model using Caputo-Fabrizio operator. Chaos Solitons Fractals 2019, 128, 355–365. [Google Scholar] [CrossRef]
- Ali, F.; Ali, F.; Sheikh, N.A.; Khan, I.; Nisar, K.S. Caputo–Fabrizio fractional derivatives modeling of transient MHD Brinkman nanoliquid: Applications in food technology. Chaos Solitons Fractals 2020, 131, 109489. [Google Scholar] [CrossRef]
- Baleanu, D.; Jajarmi, A.; Mohammadi, H.; Rezapour, S. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Solitons Fractals 2020, 134, 109705. [Google Scholar] [CrossRef]
- Sadeghi, S.; Jafari, H.; Nemati, S. Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model. Chaos Solitons Fractals 2016, 89, 552–559. [Google Scholar]
- Taneco-Hernández, M.; Vargas-De-León, C. Stability and Lyapunov functions for systems with Atangana-Baleanu Caputo derivative: An HIV/AIDS epidemic model. Chaos Solitons Fractals 2020, 132, 109586. [Google Scholar] [CrossRef]
- Wei, Q.; Zhou, H.; Yang, S. Non-Darcy flow models in porous media via Atangana-Baleanu derivative. Chaos Solitons Fractals 2020, 141, 110335. [Google Scholar] [CrossRef]
- Ali, G.; Gómez-Aguilar, J.; Kamrana. Approximation of partial integro differential equations with a weakly singular kernel using local meshless method. Alex. Eng. J. 2020, 59, 2091–2100. [Google Scholar]
- Hoan, L.V.C.; Akinlar, M.A.; Inc, M.; Gómez-Aguilar, J.; Chu, Y.M.; Almohsen, B. A new fractional-order compartmental disease model. Alex. Eng. J. 2020, 59, 3187–3196. [Google Scholar] [CrossRef]
- Hidalgo-Reyes, J.; Gomez-Aguilar, J.; Alvarado-Martinez, V.; Lopez-Lopez, M.; Escobar-Jimenez, R. Battery state-of-charge estimation using fractional extended Kalman filter with Mittag–Leffler memory. Alex. Eng. J. 2020, 59, 1919–1929. [Google Scholar] [CrossRef]
- Glendinning, P. Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo type; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Gorenflo, R.; Kilbas, A.A.; Mainardi, F.; Rogosin, S.V. Mittag-Leffler Functions, Related Topics and Applications; Springer: Berlin, Germany, 2014. [Google Scholar]
- Li, Y.; Chen, Y.; Podlubny, I. Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 2009, 45, 1965–1969. [Google Scholar] [CrossRef]
- Liu, S.; Wu, X.; Zhou, X.; Jiang, W. Asymptotical stability of Riemann-Liouville fractional nonlinear systems. Nonlinear Dyn. 2016, 86, 65–71. [Google Scholar] [CrossRef]
- Aguila-Camacho, N.; Duarte-Mermoud, M.A.; Gallegos, J.A. Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 2014, 19, 2951–2957. [Google Scholar] [CrossRef]
- Martínez-Fuentes, O.; Delfín-Prieto, S. Stability of Fractional Nonlinear Systems with Mittag-Leffler kernel and Design of State Observers; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2020. [Google Scholar]
- Chen, W.; Dai, H.; Song, Y.; Zhang, Z. Convex Lyapunov functions for stability analysis of fractional order systems. IET Control Theory Appl. 2017, 11, 1070–1074. [Google Scholar] [CrossRef]
- Badri, V.; Tavazoei, M. Stability analysis of fractional order time-delay systems: Constructing new Lyapunov functions from those of integer order counterparts. IET Control Theory Appl. 2019, 13, 2476–2481. [Google Scholar] [CrossRef]
- Nesterov, Y. Introductory Lectures on Convex Optimization: A Basic Course; Springer Science & Business Media: New York, NY, USA, 2003; Volume 87. [Google Scholar]
- Salahshour, S.; Ahmadian, A.; Salimi, M.; Panserad, B.A.; Ferrarad, M. A new Lyapunov stability analysis of fractional-order systems with nonsingular kernel derivative. Alex. Eng. J. 2020, 59, 2985–2990. [Google Scholar] [CrossRef]
- Khalil, H.K. Nonlinear Control; Pearson Higher Ed.: Hoboken, NJ, USA, 2014. [Google Scholar]
- Perkins, W.R.; Cruz, J.B. Engineering of Dynamic Systems; John Wiley & Sons: New York, NY, USA, 1969. [Google Scholar]
- Dadras, S.; Momeni, H.R. Control of a fractional-order economical system via sliding mode. Phys. A Stat. Mech. Appl. 2010, 389, 2434–2442. [Google Scholar] [CrossRef]
- Qian, C.; Lin, W. Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization. Syst. Control Lett. 2001, 42, 185–200. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Martínez-Fuentes, O.; Meléndez-Vázquez, F.; Fernández-Anaya, G.; Gómez-Aguilar, J.F. Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities. Mathematics 2021, 9, 2084. https://doi.org/10.3390/math9172084
Martínez-Fuentes O, Meléndez-Vázquez F, Fernández-Anaya G, Gómez-Aguilar JF. Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities. Mathematics. 2021; 9(17):2084. https://doi.org/10.3390/math9172084
Chicago/Turabian StyleMartínez-Fuentes, Oscar, Fidel Meléndez-Vázquez, Guillermo Fernández-Anaya, and José Francisco Gómez-Aguilar. 2021. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities" Mathematics 9, no. 17: 2084. https://doi.org/10.3390/math9172084
APA StyleMartínez-Fuentes, O., Meléndez-Vázquez, F., Fernández-Anaya, G., & Gómez-Aguilar, J. F. (2021). Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities. Mathematics, 9(17), 2084. https://doi.org/10.3390/math9172084