Lyapunov stability theorem about fractional system without and with delay. (English) Zbl 1311.34125

Summary: The difficulty of fractional direct Lyapunov stable theorem lies in that how to design a positive definite function \(V\) and easily ascertain whether fractional derivative of the function \(V\) is less than zero. In view of this difficulty, we propose a Lyapunov stability theorem for fractional system without delay and extend the newly proposed theorem to fractional system with delay. The obvious difference of the proposed theory with the fractional Lyapunov direct theory is taking the integer derivative instead of the fractional derivative of the positive definite function \(V\). Four examples are provided to illustrate the proposed theorem. The studying results in this paper show that the proposed theorem is not only applicable to the fractional autonomous system with and without delay, but also applicable to the fractional non-autonomous system with and without delay.


34D20 Stability of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
34K37 Functional-differential equations with fractional derivatives
34K20 Stability theory of functional-differential equations
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[1] Nagih, A.; Plateau, G., Fractional problems: overview of applications and solutions, Rairo-Recherche Operationnelle-Oper Res, 33, 383-419 (1999) · Zbl 1016.90065
[2] Caponetto, R.; Fazzino, S., A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems, Commun Nonlinear Sci Numer Simul, 18, 22-27 (2013) · Zbl 1253.35199
[3] Ferraris, L.; Ferraris, P.; Poskovic, E.; Tenconi, A., Theoretic and experimental approach to the adoption of bonded magnets in fractional machines for automotive applications, IEEE Trans Ind Electron, 59, 2309-2318 (2012)
[4] He, J., An approximation to solution of space and time fractional telegraph equations by the variational iteration method, Math Probl Eng, 394212 (2012) · Zbl 1264.65172
[5] Isfer, L.; Lenzi, E.; Teixeira, G.; Lenzi, M., Fractional control of an industrial furnace, Acta Sci-Technol, 32, 279-285 (2010)
[6] Kilbas, A.; Marzan, S., Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ Equ, 41, 84-89 (2005) · Zbl 1160.34301
[7] Chen, W.; Ye, L.; Sun, H., Fractional diffusion equations by the Kansa method, Comput Math Appl, 59, 1614-1620 (2010) · Zbl 1189.35356
[8] Szabo, T.; Wu, J., A model for longitudinal and shear wave propagation in viscoelastic media, J Acoust Soc Am, 107, 2437-2446 (2000)
[9] Lu, R.; Sheng, G.; Hu, Y.; Zheng, P.; Jiang, H.; Tang, Y., Fractional characterization of a bio-oil derived from rice husk, Biomass Bioenergy, 35, 671-678 (2011)
[10] Rossikhin, Y.; Shitikova, M., Application of fractional operators to the analysis of damped vibrations of viscoelastic single-mass systems, J Sound Vib, 199, 567-586 (1997) · Zbl 0901.73030
[11] Saito, J.; Sato, S.; Fukuhara, A.; Sato, Y.; Nikaido, T.; Inokoshi, Y., Association of asthma education with asthma control evaluated by asthma control test, fev1, and fractional exhaled nitric oxide, J Asthma, 50, 97-102 (2013)
[12] Sejdic, E.; Djurovic, I.; Stankovic, L., Fractional fourier transform as a signal processing tool: an overview of recent developments, Signal Process, 91, 1351-1369 (2011) · Zbl 1220.94024
[13] de Oliveira, P.; Oliveira, R.; Leite, V.; Montagner, V.; Peres, P., H(infinity) guaranteed cost computation by means of parameter-dependent Lyapunov functions, Automatica, 40, 1053-1061 (2004) · Zbl 1110.93021
[14] Li, X., Numerical solution of fractional differential equations using cubic b-spline wavelet collocation method, Commun Nonlinear Sci Numer Simul, 17, 3934-3946 (2012) · Zbl 1250.65094
[15] Pan, L.; Zhou, W.; Fang, J.; Li, D., Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control, Commun Nonlinear Sci Numer Simul, 15, 3754-3762 (2010) · Zbl 1222.34063
[16] Devi, J.; Mc Rae, F.; Drici, Z., Variational Lyapunov method for fractional differential equations, Comput Math Appl, 64, 2982-2989 (2012) · Zbl 1268.34031
[17] Giovanni, A.; Ouaknine, M.; Triglia, J., Determination of largest Lyapunov exponents of vocal signal: application to unilateral laryngeal paralysis, J Voice, 13, 341-354 (1999)
[18] Li, Y.; Chen, Y.; Podlubny, I., Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput Math Appl, 59, 1810-1821 (2010) · Zbl 1189.34015
[19] Baleanu, D.; Ranjbar, A.; Sadati, S.; Delavari, R.; Abdeljawad, T.; Gejji, V., Lyapunov-Krasovskii stability theorem for fractional systems with delay, Rom J Phys, 56, 636-643 (2011) · Zbl 1231.34005
[20] Tang, Y.; Wang, Z.; Fang, J., Pinning control of fractional-order weighted complex networks, Chaos, 19, 013112 (2009) · Zbl 1311.34018
[21] Anastassiou, G., Univariate mixed Caputo fractional Ostrowski inequalities, J Comput Anal Appl, 14, 706-713 (2012) · Zbl 1260.26006
[22] Hu, J.; Zhao, L., Stability theorem and control of fractional systems, Acta Phys Sin, 62, 240504 (2013)
[23] Sheu, L.; Chen, H.; Chen, J.; Tam, L.; Chen, W.; Lin, K., Chaos in the Newton-Leipnik system with fractional order, Chaos Solitons Fractals, 36, 98-103 (2008) · Zbl 1152.37319
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