Fractional input stability and its application to neural network

Ndolane Sene

doi:10.3934/dcdss.2020049

Article Contents

2020, Volume 13, Issue 3: 853-865. Doi: 10.3934/dcdss.2020049

This issue Previous Article Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations Next Article Mittag-Leffler input stability of fractional differential equations and its applications

Fractional input stability and its application to neural network

Ndolane Sene^,

Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Laboratoire Lmdan, BP 5683 Dakar Fann, Sénégal

^* Corresponding author: Ndolane Sene
^* Corresponding author: Ndolane Sene

Received: June 2018

Revised: August 2018

Published: December 2019

Abstract Full Text(HTML) Figure(3) Related Papers Cited by

Abstract

This paper deals with fractional input stability, and contributes to introducing a new stability notion in the stability analysis of fractional differential equations (FDEs) with exogenous inputs using the Caputo fractional derivative. In particular, we study the fractional input stability of FDEs with exogenous inputs. A Lyapunov characterization of this notion is proposed and several examples are provided to explain the fractional input stability of FDEs with exogenous inputs. The applicability and simulation of this method are illustrated by studying the particular class of fractional neutral networks.

Keywords:

Mathematics Subject Classification: Primary: 26A33, 93D05; Secondary: 93D25.

Citation:

Full Text(HTML)

Figure 1. Not CICS and not BIBS

Download: Full-size image PowerPoint slide

Figure 2. FIS of fractional neural network

Download: Full-size image PowerPoint slide

Figure 3. Asymptotic stability of trivial $ x = 0 $ solution of FDE with a converging input

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 570-66. doi: 10.1016/j.cam.2014.10.016.
[2]	Y. Adjabi, F. Jarad and T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filo., 31 (2017), 5457-5473. doi: 10.2298/FIL1717457A.
[3]	R. Almeida, D. Tavares and D. F. M. Torres, The Variable-Order Fractional Calculus of Variations, SpringerBriefs in Applied Sciences and Technology. Springer, Cham, 2019, arXiv: 1805.00720. doi: 10.1007/978-3-319-94006-9.
[4]	D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability, IEEE Trans. Auto. Contr., 45 (2000), 1082-1097. doi: 10.1109/9.863594.
[5]	A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with markovian and non-markovian properties, Phys. A: Stat. Mech. Appl., 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056.
[6]	A. Atangana and I. Koca, New direction in fractional differentiation, Math. Nat. Sci., 1 (2017), 18-25. doi: 10.22436/mns.01.01.02.
[7]	N. A. Camacho, M. A. Duarte-Mermoud and J. A. Gallegos, Lyapunov functions for fractional order systems, Comm. Nonl. Sci. Num. Simul., 19 (2014), 2951-2957. doi: 10.1016/j.cnsns.2014.01.022.
[8]	A. Chaillet, D. Angeli and H. Ito, Combining iISS and ISS with respect to small inputs: The Strong iISS property, IEEE Trans. Auto. Contr., 59 (2014), 2518-2524. doi: 10.1109/TAC.2014.2304375.
[9]	H. Chen and and U. N. Katugampola, Hermite adamard and hermite-hadamard-fej r type inequalities for generalized fractional integrals, J. Math.l Anal. and Appl., 446 (2017), 1274-1291. doi: 10.1016/j.jmaa.2016.09.018.
[10]	S. K. Choi, B. Kang and N. Koo, Stability for caputo fractional differential systems, Abstract and Applied Analysis, 2014 (2014), Art. ID 631419, 6 pp. doi: 10.1155/2014/631419.
[11]	W. S. Chung, Fractional newton mechanics with conformable fractional derivative, J. Comput. Appl. Math., 290 (2015), 150-158. doi: 10.1016/j.cam.2015.04.049.
[12]	S. Dashkovskiy and P. Feketa, Input-to-state stability of impulsive systems and their networks, Nonl. Anal.: Hybr. Syst., 26 (2017), 190-200. doi: 10.1016/j.nahs.2017.06.004.
[13]	H. Delavari, D. Baleanu and J. Sadati, Stability analysis of caputo fractional-order nonlinear systems revisited, Nonl. Dyn., 67 (2012), 2433-2439. doi: 10.1007/s11071-011-0157-5.
[14]	H. Haimovich and J. L. Mancilla-Aguilar, A characterization of integral iss for switched and time-varying systems, IEEE Contr. Syst. Soc., 63 (2018), 578-585. doi: 10.1109/TAC.2017.2729284.
[15]	U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.
[16]	U. N. Katugampola, Correction to "What is a fractional derivative?" by Ortigueira and Machado [journal of computational physics, volume 293, 15 july 2015, pages 4-13. special issue on fractional pdes], J. Comput. Phys., 321 (2016), 1255-1257. doi: 10.1016/j.jcp.2016.05.052.
[17]	H. K. Khalil and J. W. Grizzle, Nonlinear Systems, volume 3. Prentice hall New Jersey, 1996.
[18]	R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. doi: 10.1016/j.cam.2014.01.002.
[19]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, elsevier, 2006.
[20]	Y. Li, Y. Q. Chen and I. Podlubny, Mittag-leffler stability of fractional order nonlinear dynamic systems, Auto., 45 (2009), 1965-1969. doi: 10.1016/j.automatica.2009.04.003.
[21]	S. Liu, W. Jiang, X. Li and X. F. Zhou, Lyapunov stability analysis of fractional nonlinear systems, Appl. Math. Lett., 51 (2016), 13-19. doi: 10.1016/j.aml.2015.06.018.
[22]	K. S. Miller and S. G. Samk, A note on the complete monotonicity of the generalized mittag-leffler function, Real Anal. Exch., 23 (1997), 753–755, https://www.jstor.org/stable/44153996.
[23]	M. Ortigueira and J. Machado, Which derivative?, Fractal and Fract., 1 (2017), 3. doi: 10.3390/fractalfract1010003.
[24]	S. Priyadharsini, Stability Of Fractional Neutral and Integrodifferential Systems, J. Fract. Calc. Appl., 7 (2016), 87-102.
[25]	I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.
[26]	I. Podlubny, Matrix approach to discrete fractional calculus ii: Partial fractional differential equations, J. Comput. Phys., 228 (2009), 3137-3153. doi: 10.1016/j.jcp.2009.01.014.
[27]	A. Pratap, R. Raja, C. Sowmiya, O. Bagdasar, J. Cao and G. Rajchakit, Robust generalized mittag-leffler synchronization of fractional order neural networks with discontinuous activation and impulses, Neur. Net., 103 (2018), 128-141. doi: 10.1016/j.neunet.2018.03.012.
[28]	D. Qian, C. Li, R. P. Agarwal and P. J. Y. Wong, Stability analysis of fractional differential system with riemann-liouville derivative, Math. Comput. Model., 52 (2010), 862-874. doi: 10.1016/j.mcm.2010.05.016.
[29]	N. Sene, Lyapunov characterization of the fractional nonlinear systems with exogenous input, Fractal Fract., 2 (2018), 17. doi: 10.3390/fractalfract2020017.
[30]	N. Sene, On stability analysis of the fractional nonlinear systems with hurwitz state matrix, J. Fract. Calc. Appl., 10 (2019), 1-9.
[31]	N. Sene, A. Chaillet and M. Balde, Relaxed conditions for the stability of switched nonlinear triangular systems under arbitrary switching, Syst. Contr. Let., 84 (2015), 52-56. doi: 10.1016/j.sysconle.2015.06.004.
[32]	N. Sene, Exponential form for Lyapunov function and stability analysis of the fractional differential equations, J. Math. Comp. Scien., 18 (2018), 388-397. doi: 10.22436/jmcs.018.04.01.
[33]	E. D. Sontag, Smooth stabilization implies coprime factorization, Syst. Contr. Let., 34 (1989), 435-443. doi: 10.1109/9.28018.
[34]	E. D. Sontag, On the input-to-state stability property, Euro. J. Contr., 1 (1995), 24-36. doi: 10.1016/S0947-3580(95)70005-X.
[35]	E. D. Sontag, Input to state stability: Basic concepts and results, Springer, 1932 (2008), 163-220. doi: 10.1007/978-3-540-77653-6_3.
[36]	E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Syst. Contr. Let., 24 (1995), 351-359. doi: 10.1016/0167-6911(94)00050-6.
[37]	A. Souahi, A. B. Makhlouf and M. A. Hammami, Stability analysis of conformable fractional-order nonlinear system, Inda. Math., 28 (2017), 1265-1274. doi: 10.1016/j.indag.2017.09.009.
[38]	T. Zou, J. Qu, L. Chen, Y. Chai and Z. Yang, Stability analysis of a class of fractional-order neural networks, Indo. J. Elect. Engi. Comput. Sci., 12 (2014), 1086-1093.