Practical semiglobal uniform exponential stability of nonlinear nonautonomous systems

A.  Kicha; M. A.  Hammami; I.-E.  Abbes

doi:10.37863/umzh.v75i5.7071

A. Kicha University of Jijel, Faculty of Exact Sciences and Computer Sciences, Department of Mathematics, LAOTI Laboratory, Algeria
M. A. Hammami University of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Stability and Control of Systems and PDEs Laboratory, Tunisia
I.-E. Abbes University of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Stability and Control of Systems and PDEs Laboratory, Tunisia

DOI: https://doi.org/10.37863/umzh.v75i5.7071

Keywords: Lyaponov theory, Parametrized systems, Practical semi-global uniform exponential stability

Abstract

UDC 517.9

We solve the following twofold problem: In the first part, we deduce Lyapunov sufficient conditions for practical uniform exponential stability of nonlinear perturbed systems under different conditions for the perturbed term. The second part presents a converse Lyapunov theorem for the notion of semiglobal uniform exponential stability for parametrized nonlinear time-varying systems. We establish the possibility of application of a perturbed parametrized system, by using Lyapunov theory, to the investigation of the robustness properties that may provide practical semiglobal uniform exponential stability with respect to perturbations.

References

A. Ben Abdallaha, M. Dlalaa, M. A. Hammami, A new Lyapunov function for stability of time-varying nonlinear perturbed systems, Systems Control Lett., 56, № 3, 179–187 (2007). DOI: https://doi.org/10.1016/j.sysconle.2006.08.009

A. Ben Abdallah, M. Dlala, M. A. Hammami, Exponential stability of perturbed nonlinear systems, Nonlinear Dyn. Syst. Theory, 5, № 4, 357–367 (2005).

A. Ben Abdallah, I. Ellouze, M. A. Hammami, Practical stability of nonlinear time-varying cascade systems, J. Dyn. Control Syst., 15, № 1, 45–62 (2009). DOI: https://doi.org/10.1007/s10883-008-9057-5

B. Ben Hamed, I. Ellouze, M. A. Hammami, Practical uniform stability of nonlinear differential delay equations, Mediterr. J. Math., 8, № 4, 603–616 (2011). DOI: https://doi.org/10.1007/s00009-010-0083-7

A. Ben Makhlouf, M. A. Hammami, A nonlinear inequality and application to global asymptotic stability of perturbed systems, Math. Methods Appl. Sci., 38, № 12, 2496–2505 (2015). DOI: https://doi.org/10.1002/mma.3236

A. Chaillet, A. Loría, Uniform semiglobal practical asymptotic stability for nonautonomous cascaded systems and applications, Automatica, 44, № 2, 337–347 (2008). DOI: https://doi.org/10.1016/j.automatica.2007.05.019

M. Corless, L. Glielmo, On the exponential stability of singularly perturbed systems, SIAM J. Control and Optim., 30, № 6, 1338–1360 (1992). DOI: https://doi.org/10.1137/0330071

A. Dorgham, M. Hammi, M. A. Hammami, Asymptotic behavior of a class of perturbed differential equations, Ukr. Math. J., 73, № 5, 731–745 (2021). DOI: https://doi.org/10.1007/s11253-021-01956-5

I. Ellouze, On the practical separation principle of time-varying perturbed systems, IMA J. Math. Control and Inform., 37, № 1, 260–275 (2020). DOI: https://doi.org/10.1093/imamci/dny049

T. I. Fossen, K. Y. Pettersen, On uniform semiglobal exponential stability (USGES) of proportional line-of-sight guidance laws, Automatica, 50, № 11, 2912–2917 (2014). DOI: https://doi.org/10.1016/j.automatica.2014.10.018

B. Ghanmi, N. Hadj Taieb, M. A. Hammami, Growth conditions for exponential stability of time-varying perturbed systems, Int. J. Control, 86, № 6, 1086–1097 (2013). DOI: https://doi.org/10.1080/00207179.2013.774464

E. I. Grǿtli, A. Chaillet, J. T. Gravdahl, Output control of spacecraft in leader follower formation, Proc. 47th IEEE Conf. Decision and Control, Cancun, Mexico (2008), p.~ 1030–1035.

Z. HajSalem, M. A. Hammami, M. Mabrouk, On the global uniform asymptotic stability of time-varying dynamical systems, Stud. Univ. Babeş-Bolyai Math., 59, № 1, 57–67 (2014).

M. Hammi, M. A. Hammami, Non-linear integral inequalities and applications to asymptotic stability, IMA J. Math. Control and Inform., 32, № 4, 717–735 (2015).

M. Hammi, M. A. Hammami, Gronwall–Bellman type integral inequalities and applications to global uniform asymptotic stability, Cubo, 17, № 3, 53–70 (2015). DOI: https://doi.org/10.4067/S0719-06462015000300004

M. A. Hammami, On the stability of nonlinear control systems with uncertainty, J. Dyn. Control Syst., 7, № 2, 171–179 (2011).

H. Khalil, Nonlinear systems, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ (2002).

N. N. Krasovsky, Some problems of the stability theory of motion, Fizmatlit, Moscow (1959).

A. Loria, E. Panteley, Cascaded nonlinear time-varying systems: analysis and design, Advanced Topics in Control Systems Theory, Springer-Verlag, London (2004), p. 23–64. DOI: https://doi.org/10.1007/11334774_2

G. A. Los', Stability, asymptotic and exponential stability of a linear differential system, Ukr. Math. J., 30, № 1, 80–83 (1978). DOI: https://doi.org/10.1007/BF01130637

M. F. M. Naser, F. Ikhouane, Stability of time-varying systems in the absence of strict Lyapunov functions, IMA J. Math. Control and Inform., 36, 461–483 (2019). DOI: https://doi.org/10.1093/imamci/dnx056

S. K. Persidskii, On the exponential stability of some nonlinear systems, Ukr. Math. J., 57, № 2, 157–164 (2005). DOI: https://doi.org/10.1007/s11253-005-0178-7

K. Y. Pettersen, Lyapunov sufficient conditions for uniform semiglobal exponential stability, Automatica, 78, 97–102 (2017). DOI: https://doi.org/10.1016/j.automatica.2016.12.004

B. Zhou, Stability analysis of nonlinear time-varying systems by Lyaponov function with indefinite derivative, IET Control. Theory Appl., 11, 1434–1442 (2017). DOI: https://doi.org/10.1049/iet-cta.2016.1538