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doi: 10.3934/dcdsb.2021197
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## Practical partial stability of time-varying systems

 Faculty of Sciences of Sfax, Department of Mathematics

* Corresponding author: Nizar Hadj Taieb

Received  January 2021 Revised  April 2021 Early access August 2021

Fund Project: The authors wish to thank the editor and the anonymous reviewers for their valuable and careful comments

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.

Citation: Abdelfettah Hamzaoui, Nizar Hadj Taieb, Mohamed Ali Hammami. Practical partial stability of time-varying systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021197
##### 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.

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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.
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