• Previous Article
    Attractors for a class of perturbed nonclassical diffusion equations with memory
  • DCDS-B Home
  • This Issue
  • Next Article
    Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate
doi: 10.3934/dcdsb.2021197
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021197
References:
[1]

A. BenabdallahM. 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.  Google Scholar

[2]

A. BenabdallahI. 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.  Google Scholar

[3]

A. Ben Makhlouf, Partial practical stability for fractional-order nonlinear systems, Mathematical Methods in the Applied Sciences, (2020). Google Scholar

[4]

T. CaraballoF. EzzineM. 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.  Google Scholar

[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.  Google Scholar

[6]

B. GhanmiN. 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.  Google Scholar

[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.  Google Scholar

[8]

N. Hadj Taieb, Stability analysis for time-varying nonlinear systems, International Journal of control, (2020), https://doi.org/10.1080/00207179.2020.1861332. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Khalil, Nonlinear Systems, Third ed. Prentice-Hall Englewood Cliffs, NJ, 2002. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[14]

Q. G. LindaM. 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.  Google Scholar

[15]

A. Martynyuk and Z. Sun, Practical stability and its applications, Beijing: Science Press, (2003). Google Scholar

[16]

S. RuiqingJ. 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.  Google Scholar

[17]

E. D. Sontag, Smoimoth stabilization implies copre factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[18]

V. V. Rumyantsev, Partial stability of motion, Mosk. Gos. Univ., Mat. Mekh. Fiz. Astronom. Khim., 4 (1957), 9-16.   Google Scholar

[19]

V. V. Rumyantsev, Stability of equilibrium of a body with a liquid-filled hollow, Dokl. Akad. Nauk SSSR, 124 (1959), 291-294.   Google Scholar

[20]

V. V. Rumyantsev, Stability of rotational motion of a liquid-filled solid, Prikl. Mat. Mekh., 23 (1959), 1057-1065.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

B. Zhou, On asymtotic stability of linear time-varying systems, Automatica, 68 (2016), 266-276.  doi: 10.1016/j.automatica.2015.12.030.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. BenabdallahM. 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.  Google Scholar

[2]

A. BenabdallahI. 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.  Google Scholar

[3]

A. Ben Makhlouf, Partial practical stability for fractional-order nonlinear systems, Mathematical Methods in the Applied Sciences, (2020). Google Scholar

[4]

T. CaraballoF. EzzineM. 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.  Google Scholar

[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.  Google Scholar

[6]

B. GhanmiN. 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.  Google Scholar

[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.  Google Scholar

[8]

N. Hadj Taieb, Stability analysis for time-varying nonlinear systems, International Journal of control, (2020), https://doi.org/10.1080/00207179.2020.1861332. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Khalil, Nonlinear Systems, Third ed. Prentice-Hall Englewood Cliffs, NJ, 2002. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[14]

Q. G. LindaM. 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.  Google Scholar

[15]

A. Martynyuk and Z. Sun, Practical stability and its applications, Beijing: Science Press, (2003). Google Scholar

[16]

S. RuiqingJ. 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.  Google Scholar

[17]

E. D. Sontag, Smoimoth stabilization implies copre factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[18]

V. V. Rumyantsev, Partial stability of motion, Mosk. Gos. Univ., Mat. Mekh. Fiz. Astronom. Khim., 4 (1957), 9-16.   Google Scholar

[19]

V. V. Rumyantsev, Stability of equilibrium of a body with a liquid-filled hollow, Dokl. Akad. Nauk SSSR, 124 (1959), 291-294.   Google Scholar

[20]

V. V. Rumyantsev, Stability of rotational motion of a liquid-filled solid, Prikl. Mat. Mekh., 23 (1959), 1057-1065.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

B. Zhou, On asymtotic stability of linear time-varying systems, Automatica, 68 (2016), 266-276.  doi: 10.1016/j.automatica.2015.12.030.  Google Scholar

[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.  Google Scholar

[1]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116

[2]

Jifeng Chu, Jinzhi Lei, Meirong Zhang. Lyapunov stability for conservative systems with lower degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 423-443. doi: 10.3934/dcdsb.2011.16.423

[3]

Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks & Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787

[4]

Sibel Senan, Eylem Yucel, Zeynep Orman, Ruya Samli, Sabri Arik. A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1415-1428. doi: 10.3934/dcdss.2020358

[5]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172

[6]

Rafael Ortega. Variations on Lyapunov's stability criterion and periodic prey-predator systems. Electronic Research Archive, 2021, 29 (6) : 3995-4008. doi: 10.3934/era.2021069

[7]

Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761

[8]

Giovanni Russo, Fabian Wirth. Matrix measures, stability and contraction theory for dynamical systems on time scales. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021188

[9]

Guoshan Zhang, Peizhao Yu. Lyapunov method for stability of descriptor second-order and high-order systems. Journal of Industrial & Management Optimization, 2018, 14 (2) : 673-686. doi: 10.3934/jimo.2017068

[10]

Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192

[11]

Sergey Zelik. On the Lyapunov dimension of cascade systems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 971-985. doi: 10.3934/cpaa.2008.7.971

[12]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[13]

Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319

[14]

Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2006, 5 (3) : 515-528. doi: 10.3934/cpaa.2006.5.515

[15]

PaweŁ Hitczenko, Georgi S. Medvedev. Stability of equilibria of randomly perturbed maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 369-381. doi: 10.3934/dcdsb.2017017

[16]

Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2007, 6 (1) : 69-82. doi: 10.3934/cpaa.2007.6.69

[17]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

[18]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

[19]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[20]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527

2020 Impact Factor: 1.327

Article outline

[Back to Top]