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, 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.

[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]

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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

[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.
[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[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.

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.

[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]

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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

[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.
[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[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.

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