ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Asymptotic Stability of Fractional Langevin Systems

Journal of Applied Nonlinear Dynamics 11(3) (2022) 635--650 | DOI:10.5890/JAND.2022.09.008

Venkatesan Govindaraj$^1$, Sivaraj Priyadharsini$^2$, Pitchaikkannu Suresh Kumar$^3$, Krishnan Balachandran$^4$

$^1$ Department of Mathematics, National Institute of Technology Puducherry, Karaikkal - 609 609, India

$^2$ Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore - 641 008, India

$^3$ Department of Mathematics, National Institute of Technology, Calicut - 673 601, India

$^4$ Department of Mathematics, Bharathiar University, Coimbatore - 641 046, India

Abstract

In the paper, we present a method based on eigenvalue criterion to test the asymptotic stability of fractional linear Langevin systems represented by the fractional differential equation in the sense of Caputo fractional derivative. Also, this method is extended to nonlinear equations and finally some sufficient conditions ensuring asymptotical stability of fractional-order nonlinear Langevin systems are proposed. Some numerical examples are provided to illustrate the effectiveness of the proposed method.

References

1.  [1] Bagley, R.L. and Torvik, P.J. (1984), On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics, 51, 294-298.
2.  [2] Magin, R. (2006), Fractional Calculus in Bioengineering, Begell House Inc.: Redding.
3.  [3] Sousa, J.V.D.C., Tenreiro, M.J.A., and Oliveira, D.D.S. (2020), The $\psi$-Hilfer fractional calculus of variable order and its applications, Computational and Applied Mathematics, 39, 296, 35 pages.
4.  [4] Sousa, J.V.D.C. and Oliveira, D.D.S. (2019), Leibniz type rule: $\psi$-Hilfer fractional operator, Communication in Nonlinear Science and Numerical Simulation, 77, 305-311.
5.  [5] Sousa, J.V.D.C. and Oliveira, D.D.S. (2018), On the $\psi$ -Hilfer fractional derivative, Communication in Nonlinear Science and Numerical Simulation, 60, 72-91.
6.  [6] Li, Y., Chen, Y.Q., and Podlubny, I. (2010), Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Computational and Mathematical Applications, 59, 1810-1821.
7.  [7] Balachandran, K., Govindaraj, V., Rodriguez-Germa, L., and Trujillo, J.J. (2014), Stabilizability of fractional dynamical systems, Fractional Calculus $\&$ Applied Analysis, 17, 511-531.
8.  [8] Li, Y., Chen, Y., and Podlubny, I. (2009), Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 1965-1969.
9.  [9] Li, C.P. and Zhang F.R. (2011), A survey on the stability of fractional differential equations, The European Physical Journal, Special Topics, 193, 27-47.
10.  [10] Matignon, D. (1996), Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems Applications, 2, 963-968.
11.  [11] Petra$\check{\rm s}$, I. (2009), Stability of fractional order systems with rational orders: A survey, Fractional Calculus $\&$ Applied Analysis, 12, 269-298.
12.  [12] Priyadharsini, S. and Govindaraj, V. (2018), Asymptotic stability of caputo fractional singular differential systems with multiple delays, Discontinuity, Nonlinearity, and complexity, 7, 243-251.
13.  [13] Rivero, M., Rogosin, S.V., Tenreiro Machado, J.A., and Trujillo, J.J. (2013), Stability of fractional order systems, Mathematical Problems in Engineering, Article ID: 356215, 14 pages.
14.  [14] Sousa, J.V.D.C., Oliveira, D.D.S., and Capelas de Oliveira, E. (2018), On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation, Mathematical Methods and Applied Sciences, 1-13.
15.  [15] Liu, S., Li, X., Zhou, X.F., and Jiang, W. (2016), Lyapunov stability analysis of fractional nonlinear systems, Applied Mathematical Letters, 51, 13-19.
16.  [16] Sachin, B. and Varsha, G.D. (2011), A predictor-corrector Scheme for solving nonlinear delay differential equations of fractional order, Journal of Fractional Calculus Applications, 1, 1-9.
17.  [17] Kou, S.C. and Xie, X.S. (2004), Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule, Physical Revised Letters, 93, 180-191.
18.  [18] Lim, S.C., Li, M., and Teo, L.P. (2008), Langevin equation with two fractional orders, Physics Letters A, 372, 6309-6311.
19.  [19] Sau Fa, K. (2007), Fractional Langevin equation and Riemann-Liouville fractional derivative, European Physics Journal, 24, 139-143.
20.  [20] Vinalesm A.D. and Despositom M.A. (2006), Anomalous diffusion: Exact solution of the generalized Langevin equation for harmonically bounded particle, Physical Reports, 73, 12-16.
21.  [21] Figueiredo Camargo, R., Chiacchio, A.O., Charnet, R., and Capelas de Oliveira, E. (2009), Solution of the fractional Langevin equation and the Mittag-Leffler functions, Journal of Mathematical Physics, 50, 53-67.
22.  [22] Mathai, A.M., Saxena, R.K., and Haubold, H.J. (2006), A certain class of Laplace transform with applications to reaction and reaction-diffusion equations, Astrophysics Space Sciences, 305 283-286.
23.  [23] Sureshkumar, P., Govindaraj, V., Balachandran, K., and Annapoorani, N. (2019), Controllability of nonlinear fractional langevin systems, Discontinuity, Nonlinearity and Complexity, 8, 89-99.
24.  [24] Wang, J. and Li, X. (2015), Ulam-Hyers stability of fractional Langevin equations, Applied Mathematics and Computation, 258, 72-83.
25.  [25] Gao, Z. and Yu, X. (2017), Stability of nonlocal fractional Langevin differential equations involving fractional integrals, Journal of Applied Mathematics and Computing, 53, 599-611.
26.  [26] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equation, Elsevier: Amsterdam.
27.  [27] Podlubny, I. (1999), Fractional Differential Equation, Academic Press: New York.
28.  [28] Qian, D., Li, C., Agarwal, R.P., and Wongd, J.Y. (2010), Stability analysis of fractional differential system with Riemann-Liouville derivative, Mathematical and Computer Modelling, 52, 862-874.
29.  [29] Corduneanu, C. (1971), Principles of Differential and Integral Equations, Allyn and Bacon: Boston.
30.  [30] Odibat, Z. and Momani, S. (2007), Numerical approach to differential equations of fractional order, Computers and Mathematics with Applications, 207, 96-110. %
31.  [31] Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., and Castro-Linares, R. (2015), Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Communications in Nonlinear Science Numerical Simulation, 22, 650-659. % % %
32.  [32] Gao, J. and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis Applications, 328, 1075-1081. % %
33.  [33] Govindaraj, V. and Balachandran, K. (2014), Stability of Basset equation, %Journal of Fractional Calculus and Applications, 20, 1-15. % % % %
34.  [34] Guo, P., Zeng, C., Li, C., and Chen, Y. (2013), Numerics for the fractional Langevin equation driven by the fractional Brownian motion, Fractional Calculus and Applied Analysis, 16, 123-141. % % % %
35.  [35] Krol, K. (2011), Asymptotic properties of fractional delay differential equations, Applied Mathematical Computation, 218, 1515-1532. % % %
36.  [36] Liu, S., Li, X., Jiang, W., and Zhou, X.F. (2012), Mittag-Leffler stability of nonlinear fractional neutral singular systems, Communications in Nonlinear Sciences Numerical Simulation, 17, 3961-3966. % % % %
37.  [37] Liu, S., Li, X., Zhou, X.F., and Jiang, W. (2017), Asymptotical stability of Riemann-Liouville fractional singular systems with multiple time - varying delays, Applied Mathematics Letters, 65, 32-39. % % %
38.  [38] Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley: New York. % %
39.  [39] Momani, S. and Hadid, S. (2004), Lyapunov stability solutions of fractional integrodifferential equations, International Journal of Mathematical Sciences, %47, 2503-2507. % %
40.  [40] Oldham, K.B. and Spanier, J. (1974), The Fractional Calculus, Academic Press: California. % %
41.  [41] Odibat, Z. (2006), Approximations of fractional integral and Caputo fractional derivatives, Computers and Mathematics with Applications, 178, 527-533. % % % %
42.  [42] Odibat, M. and Momani, S. (2008), An algorithm for the numerical solutions of differential equations of fractional order, Applied Mathematics and Informatics, 26, 15-27. % % %
43.  [43] Porra, J.M., Wang, K.G., and Masoliver, J. (1996), Generalized Langevin equations: Anomalous diffusion and probability distributions, Physics Reviews, 53, 58-72. %% %
44.  [44] Priyadharsini, S. (2016), Stability of fractional neutral and integrodifferential systems, Journal of Fractional Calculus and Applications, 7, 87-102. % %
45.  [45] Priyadharsini, S. (2016), Stability analysis of fractional differential systems with constant delay, Journal of Indian Mathematical Society, 83, 337-350. % % % %
46.  [46] Wu, R.C., Hei, X.D., and Chen, L.P. (2013), Finite-time stability of fractional-order neural networks with delay, Communications in Theoretical Physics, 60, 189-193.