March  2020, 13(3): 853-865. doi: 10.3934/dcdss.2020049

Fractional input stability and its application to neural network

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

* Corresponding author: Ndolane Sene

Received  June 2018 Revised  August 2018 Published  March 2019

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.

Citation: Ndolane Sene. Fractional input stability and its application to neural network. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 853-865. doi: 10.3934/dcdss.2020049
References:
[1]

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 570-66.  doi: 10.1016/j.cam.2014.10.016.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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H. DelavariD. 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.  Google Scholar

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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.  Google Scholar

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U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.   Google Scholar

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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.  Google Scholar

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R. KhalilM. A. HoraniA. 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.  Google Scholar

[19]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, elsevier, 2006.  Google Scholar

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Y. LiY. 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.  Google Scholar

[21]

S. LiuW. JiangX. 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.  Google Scholar

[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.  Google Scholar

[23]

M. Ortigueira and J. Machado, Which derivative?, Fractal and Fract., 1 (2017), 3. doi: 10.3390/fractalfract1010003.  Google Scholar

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S. Priyadharsini, Stability Of Fractional Neutral and Integrodifferential Systems, J. Fract. Calc. Appl., 7 (2016), 87-102.   Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[27]

A. PratapR. RajaC. SowmiyaO. BagdasarJ. 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.  Google Scholar

[28]

D. QianC. LiR. 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.  Google Scholar

[29]

N. Sene, Lyapunov characterization of the fractional nonlinear systems with exogenous input, Fractal Fract., 2 (2018), 17. doi: 10.3390/fractalfract2020017.  Google Scholar

[30]

N. Sene, On stability analysis of the fractional nonlinear systems with hurwitz state matrix, J. Fract. Calc. Appl., 10 (2019), 1-9.   Google Scholar

[31]

N. SeneA. 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.  Google Scholar

[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.  Google Scholar

[33]

E. D. Sontag, Smooth stabilization implies coprime factorization, Syst. Contr. Let., 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

A. SouahiA. 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.  Google Scholar

[38]

T. ZouJ. QuL. ChenY. Chai and Z. Yang, Stability analysis of a class of fractional-order neural networks, Indo. J. Elect. Engi. Comput. Sci., 12 (2014), 1086-1093.   Google Scholar

show all references

References:
[1]

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 570-66.  doi: 10.1016/j.cam.2014.10.016.  Google Scholar

[2]

Y. AdjabiF. Jarad and T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filo., 31 (2017), 5457-5473.  doi: 10.2298/FIL1717457A.  Google Scholar

[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.  Google Scholar

[4]

D. AngeliE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. A. CamachoM. 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.  Google Scholar

[8]

A. ChailletD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. DelavariD. 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.  Google Scholar

[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.  Google Scholar

[15]

U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.   Google Scholar

[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.  Google Scholar

[17]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems, volume 3. Prentice hall New Jersey, 1996. Google Scholar

[18]

R. KhalilM. A. HoraniA. 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.  Google Scholar

[19]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, elsevier, 2006.  Google Scholar

[20]

Y. LiY. 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.  Google Scholar

[21]

S. LiuW. JiangX. 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.  Google Scholar

[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.  Google Scholar

[23]

M. Ortigueira and J. Machado, Which derivative?, Fractal and Fract., 1 (2017), 3. doi: 10.3390/fractalfract1010003.  Google Scholar

[24]

S. Priyadharsini, Stability Of Fractional Neutral and Integrodifferential Systems, J. Fract. Calc. Appl., 7 (2016), 87-102.   Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[27]

A. PratapR. RajaC. SowmiyaO. BagdasarJ. 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.  Google Scholar

[28]

D. QianC. LiR. 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.  Google Scholar

[29]

N. Sene, Lyapunov characterization of the fractional nonlinear systems with exogenous input, Fractal Fract., 2 (2018), 17. doi: 10.3390/fractalfract2020017.  Google Scholar

[30]

N. Sene, On stability analysis of the fractional nonlinear systems with hurwitz state matrix, J. Fract. Calc. Appl., 10 (2019), 1-9.   Google Scholar

[31]

N. SeneA. 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.  Google Scholar

[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.  Google Scholar

[33]

E. D. Sontag, Smooth stabilization implies coprime factorization, Syst. Contr. Let., 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

A. SouahiA. 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.  Google Scholar

[38]

T. ZouJ. QuL. ChenY. Chai and Z. Yang, Stability analysis of a class of fractional-order neural networks, Indo. J. Elect. Engi. Comput. Sci., 12 (2014), 1086-1093.   Google Scholar

Figure 1.  Not CICS and not BIBS
Figure 2.  FIS of fractional neural network
Figure 3.  Asymptotic stability of trivial $ x = 0 $ solution of FDE with a converging input
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