March  2020, 13(3): 867-880. doi: 10.3934/dcdss.2020050

Mittag-Leffler input stability of fractional differential equations and its applications

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

* Corresponding author: Ndolane Sene

Received  August 2018 Revised  October 2018 Published  March 2019

This paper addresses the Mittag-Leffler input stability of the fractional differential equations with exogenous inputs. We continuous the first note. We discuss three properties of the Mittag-Leffler input stability: converging-input converging-state, bounded-input bounded-state, and Mittag-Leffler stability of the unforced fractional differential equation. We present the Lyapunov characterization of the Mittag-Leffler input stability, and conclude by introducing the fractional input stability for delay fractional differential equations, and we provide its Lyapunov-Krasovskii characterization. Several examples are treated to highlight the Mittag-Leffler input stability.

Citation: Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050
References:
[1]

T. Abdeljawad and V. Gejji, Lyapunov-Krasovskii stability theorem for fractional systems with delay, Rom. J. Phys., 56 (2011), 636-643.   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]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Scien., https://arXiv.org/abs/1602.03408 (2016). Google Scholar

[4]

D. BaleanuA. K. Golmankhaneh and A. K. Golmankhaneh, The dual action of the fractional multi time hamilton equations, Inter. J. of Theo. Phys., 48 (2009), 2558-2569.  doi: 10.1007/s10773-009-0042-x.  Google Scholar

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D. Baleanu, Z. B. Guvenc and J. A. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, 2009. doi: 10.1007/978-90-481-3293-5.  Google Scholar

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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

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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

[8]

M. Eslami, Exact traveling wave solutions to the fractional coupled nonlinear schrodinger equations, Appl. Math. Comput., 285 (2016), 141-148.  doi: 10.1016/j.amc.2016.03.032.  Google Scholar

[9]

E. F. D. Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos, 26 (2016), 084305, 10 pp. doi: 10.1063/1.4958921.  Google Scholar

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E. F. D. Goufo, An application of the Caputo-Fabrizio operator to replicator-mutator dynamics: Bifurcation, chaotic limit cycles and control, The Euro. Phys. J. Plus, 133 (2018), 80. doi: 10.1140/epjp/i2018-11933-0.  Google Scholar

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E. F. D. Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The Euro. Phys. J. Plus, 131 (2016), 269. Google Scholar

[12]

E. F. D. Goufo and T. Toudjeu, Around chaotic disturbance and irregularity for higher order traveling waves, J. of Math., 2018 (2018), Art. ID 2391697, 11 pp. doi: 10.1155/2018/2391697.  Google Scholar

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E. F. D. Goufo and J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, J. of Comp. and Appl. Math., 339 (2018), 329-342.  doi: 10.1016/j.cam.2017.08.026.  Google Scholar

[14]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their caputo modification, J. Nonlinear Sci. Appl, 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.  Google Scholar

[15]

F. Jarad, E. Ugurlu T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adva. in Diff. Equa., 2017 (2017), 247. doi: 10.1186/s13662-017-1306-z.  Google Scholar

[16]

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

[17]

N. Laskin, Fractional schrodinger equation, Phys. Review E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[18]

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

[19]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[20]

K. Oldham and J. Spanier, The Fractional Calculus Theory and Application Of Differentiation and Integration to Arbitrary Order, New York-London, 1974.  Google Scholar

[21]

P. Pepe and Z. P. Jiang, A lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Syst. Contr. Lett., 55 (2006), 1006-1014.  doi: 10.1016/j.sysconle.2006.06.013.  Google Scholar

[22]

I. Petras, Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Science and Business Media, 2011. Google Scholar

[23] 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
[24]

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

[25]

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

[26]

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

[27]

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

[28]

N. Sene, Fractional input stability and its application to neural network, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020). Google Scholar

[29]

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

[30]

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

[31]

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

[32]

A. R. Teel, Connections between razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE trans. on Auto. Control., 43 (1998), 960-964.  doi: 10.1109/9.701099.  Google Scholar

[33]

N. Yeganefar, P. Pepe and M. Dambrine, Input-to-state stability and exponential stability for time-delay systems: Further results, In Deci. and Contr., (2007), 2059–2064. Google Scholar

[34]

T. ZouJ. QuL. ChenYi 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 and V. Gejji, Lyapunov-Krasovskii stability theorem for fractional systems with delay, Rom. J. Phys., 56 (2011), 636-643.   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]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Scien., https://arXiv.org/abs/1602.03408 (2016). Google Scholar

[4]

D. BaleanuA. K. Golmankhaneh and A. K. Golmankhaneh, The dual action of the fractional multi time hamilton equations, Inter. J. of Theo. Phys., 48 (2009), 2558-2569.  doi: 10.1007/s10773-009-0042-x.  Google Scholar

[5]

D. Baleanu, Z. B. Guvenc and J. A. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, 2009. doi: 10.1007/978-90-481-3293-5.  Google Scholar

[6]

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

[7]

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

[8]

M. Eslami, Exact traveling wave solutions to the fractional coupled nonlinear schrodinger equations, Appl. Math. Comput., 285 (2016), 141-148.  doi: 10.1016/j.amc.2016.03.032.  Google Scholar

[9]

E. F. D. Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos, 26 (2016), 084305, 10 pp. doi: 10.1063/1.4958921.  Google Scholar

[10]

E. F. D. Goufo, An application of the Caputo-Fabrizio operator to replicator-mutator dynamics: Bifurcation, chaotic limit cycles and control, The Euro. Phys. J. Plus, 133 (2018), 80. doi: 10.1140/epjp/i2018-11933-0.  Google Scholar

[11]

E. F. D. Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The Euro. Phys. J. Plus, 131 (2016), 269. Google Scholar

[12]

E. F. D. Goufo and T. Toudjeu, Around chaotic disturbance and irregularity for higher order traveling waves, J. of Math., 2018 (2018), Art. ID 2391697, 11 pp. doi: 10.1155/2018/2391697.  Google Scholar

[13]

E. F. D. Goufo and J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, J. of Comp. and Appl. Math., 339 (2018), 329-342.  doi: 10.1016/j.cam.2017.08.026.  Google Scholar

[14]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their caputo modification, J. Nonlinear Sci. Appl, 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.  Google Scholar

[15]

F. Jarad, E. Ugurlu T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adva. in Diff. Equa., 2017 (2017), 247. doi: 10.1186/s13662-017-1306-z.  Google Scholar

[16]

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

[17]

N. Laskin, Fractional schrodinger equation, Phys. Review E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[18]

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

[19]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[20]

K. Oldham and J. Spanier, The Fractional Calculus Theory and Application Of Differentiation and Integration to Arbitrary Order, New York-London, 1974.  Google Scholar

[21]

P. Pepe and Z. P. Jiang, A lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Syst. Contr. Lett., 55 (2006), 1006-1014.  doi: 10.1016/j.sysconle.2006.06.013.  Google Scholar

[22]

I. Petras, Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Science and Business Media, 2011. Google Scholar

[23] 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
[24]

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

[25]

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

[26]

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

[27]

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

[28]

N. Sene, Fractional input stability and its application to neural network, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020). Google Scholar

[29]

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

[30]

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

[31]

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

[32]

A. R. Teel, Connections between razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE trans. on Auto. Control., 43 (1998), 960-964.  doi: 10.1109/9.701099.  Google Scholar

[33]

N. Yeganefar, P. Pepe and M. Dambrine, Input-to-state stability and exponential stability for time-delay systems: Further results, In Deci. and Contr., (2007), 2059–2064. Google Scholar

[34]

T. ZouJ. QuL. ChenYi 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

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