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 and 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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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. 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.

[19]

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

[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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 

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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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. 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.

[19]

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

[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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 

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
[1]

Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050

[2]

Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 1017-1029. doi: 10.3934/dcdss.2020060

[3]

Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030

[4]

Platon Surkov. Dynamical estimation of a noisy input in a system with a Caputo fractional derivative. The case of continuous measurements of a part of phase coordinates. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022020

[5]

Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025

[6]

Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. doi: 10.3934/dcdss.2020040

[7]

Piotr Grabowski. On analytic semigroup generators involving Caputo fractional derivative. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022014

[8]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3659-3683. doi: 10.3934/dcdss.2021023

[9]

Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 975-993. doi: 10.3934/dcdss.2020057

[10]

Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031

[11]

Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021026

[12]

Iman Malmir. Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 395-426. doi: 10.3934/naco.2021013

[13]

Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021021

[14]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282

[15]

Pierre Aime Feulefack, Jean Daniel Djida, Atangana Abdon. A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317

[16]

Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control and Related Fields, 2020, 10 (1) : 141-156. doi: 10.3934/mcrf.2019033

[17]

Kolade M. Owolabi, Abdon Atangana, Jose Francisco Gómez-Aguilar. Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2455-2469. doi: 10.3934/dcdss.2021060

[18]

Abdon Atangana, Ali Akgül. On solutions of fractal fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3441-3457. doi: 10.3934/dcdss.2020421

[19]

Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037

[20]

Tran Bao Ngoc, Nguyen Huy Tuan, R. Sakthivel, Donal O'Regan. Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative. Evolution Equations and Control Theory, 2022, 11 (2) : 439-455. doi: 10.3934/eect.2021007

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (500)
  • HTML views (640)
  • Cited by (5)

Other articles
by authors

[Back to Top]