Practical partial stability of time-varying systems

Abdelfettah Hamzaoui; Nizar Hadj Taieb; Mohamed Ali Hammami

doi:10.3934/dcdsb.2021197

Article Contents

2022, Volume 27, Issue 7: 3585-3603. Doi: 10.3934/dcdsb.2021197

This issue Previous Article Existence and continuity of global attractors for ternary mixtures of solids Next Article Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

Practical partial stability of time-varying systems

Faculty of Sciences of Sfax, Department of Mathematics

^* Corresponding author: Nizar Hadj Taieb
^* Corresponding author: Nizar Hadj Taieb

Received: January 2021

Revised: April 2021

Early access: August 9, 2021

Published online: August 9, 2021

The authors wish to thank the editor and the anonymous reviewers for their valuable and careful comments.

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Abstract

In this paper we investigate the practical asymptotic and exponential partial stability of time-varying nonlinear systems. We derive some sufficient conditions that guarantee practical partial stability of perturbed systems using Lyapunov's theory where a converse theorem is presented. Therefore, we generalize some works which are already made in the literature. Furthermore, we present some illustrative examples to verify the effectiveness of the proposed methods.

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Mathematics Subject Classification: Primary: 34D20, 37B25; Secondary: 37B55.

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References

[1]	A. Benabdallah, M. Dlala and M. A. Hammami, A new Lyapunov function for stability of time-varying nonlinear perturbed systems, Systems and Control Letters, 56 (2007), 179-187. doi: 10.1016/j.sysconle.2006.08.009.
[2]	A. Benabdallah, I. Ellouze and M. A. Hammami, Practical stability of nonlinear time-varying cascade systems, Journal of Dynamical and Control Systems, 15 (2009), 45-62. doi: 10.1007/s10883-008-9057-5.
[3]	A. Ben Makhlouf, Partial practical stability for fractional-order nonlinear systems, Mathematical Methods in the Applied Sciences, (2020).
[4]	T. Caraballo, F. Ezzine, M. A. Hammami and L. Mchiri, Practical stability with respect to a part of variables of stochastic differential equations, Stochastics, 93 (2021), 647-664. doi: 10.1080/17442508.2020.1773826.
[5]	M. Yu. Filimonov, Global asymptotic stability with respect to part of the variable for solutions of systems of ordinary differential equations, Differential Equations, 56 (2020), 710-720. doi: 10.1134/S001226612006004X.
[6]	B. Ghanmi, N. Hadj Taieb and M. A. Hammami, Growth conditions for exponential stability of time-varying perturbed systems, International Journal of Contol, 86 (2013), 1086-1097. doi: 10.1080/00207179.2013.774464.
[7]	N. Hadj Taieb and M. A. Hammami, Some new results on the global uniform asymptotic stability of time-varying dynamical systems, IMA Journal of Mathematical Control and Information, 35 (2018), 901-922. doi: 10.1093/imamci/dnx006.
[8]	N. Hadj Taieb, Stability analysis for time-varying nonlinear systems, International Journal of control, (2020), https://doi.org/10.1080/00207179.2020.1861332.
[9]	N. Hadj Taieb, Indefinite derivative for stability of time-varying nonlinear systems, IMA Journal of Mathematical Control and Information, 38 (2021), 534-551. doi: 10.1093/imamci/dnaa040.
[10]	J. W. Hagood and S. T. Brian, Recovering a function from a Dini derivative, Amer. Math. Monthly, 113 (2006), 34-46. doi: 10.1080/00029890.2006.11920276.
[11]	H. Khalil, Nonlinear Systems, Third ed. Prentice-Hall Englewood Cliffs, NJ, 2002.
[12]	V. Lakshmikantham, S. Leela and A. A. Martynyuk, Practical Stability of Nonlinear Systems, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. doi: 10.1142/1192.
[13]	J. Lasalle and S. Lefschetz, Stability by lyapunov direct method, with application, Mathematics in Science and Engineering, Academic Press, New York-London, 4 (1961).
[14]	Q. G. Linda, M. L. Jaime and W. H. Herbert, Four SEI endemic models with periodicity and separatrices, Math. Biosci., 128 (1995), 157-184. doi: 10.1016/0025-5564(94)00071-7.
[15]	A. Martynyuk and Z. Sun, Practical stability and its applications, Beijing: Science Press, (2003).
[16]	S. Ruiqing, J. Xiaowu and C. Lansun, The effect of impulsive vaccination on an SIR epidemic model, Applied Mathematics and Computation, 212 (2009), 305-311. doi: 10.1016/j.amc.2009.02.017.
[17]	E. D. Sontag, Smoimoth stabilization implies copre factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018.
[18]	V. V. Rumyantsev, Partial stability of motion, Mosk. Gos. Univ., Mat. Mekh. Fiz. Astronom. Khim., 4 (1957), 9-16.
[19]	V. V. Rumyantsev, Stability of equilibrium of a body with a liquid-filled hollow, Dokl. Akad. Nauk SSSR, 124 (1959), 291-294.
[20]	V. V. Rumyantsev, Stability of rotational motion of a liquid-filled solid, Prikl. Mat. Mekh., 23 (1959), 1057-1065.
[21]	V. V. Rumyantsev, Stability of motion of a gyrostat, Prikl. Mat. Mekh., 25 (1961), 9-19. doi: 10.1016/0021-8928(61)90094-6.
[22]	V. V. Rumyantsev, Stability of motion of solids with liquid-filled hollows, Tr. II Vses. sâezda Po Teor. Prikl. Mekh. Proc. 2 All-Union Congress: Theoretical and Applied Mechanics, Moscow: Nauka, 1 (1965), 57–71.
[23]	B. Zhou, On asymtotic stability of linear time-varying systems, Automatica, 68 (2016), 266-276. doi: 10.1016/j.automatica.2015.12.030.
[24]	B. Zhou, Stability analysis of nonlinear time-varying systems by lyapunov functions with indefinite derivatives, IET Control Theory and Applications, 11 (2017), 1434-1442. doi: 10.1049/iet-cta.2016.1538.