New results on exponential stability of time-varying systems using logarithmic norm

References

[1] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (1943), 643–647. 10.1215/S0012-7094-43-01059-2Search in Google Scholar

[2] A. Benabdallah, I. Ellouze and M. A. Hammami, Practical exponential stability of perturbed triangular systems and a separation principle, Asian J. Control 13 (2011), no. 3, 445–448. 10.1002/asjc.325Search in Google Scholar

[3] B. Ben Hamed, I. Ellouze and M. A. Hammami, Practical uniform stability of nonlinear differential delay equations, Mediterr. J. Math. 8 (2011), no. 4, 603–616. 10.1007/s00009-010-0083-7Search in Google Scholar

[4] A. Ben Makhlouf, M. A. Hammami and M. Hammi, A new approach for stabilization of control-affine systems via integral inequalities, IMA J. Math. Control Inform. 39 (2022), no. 3, 837–860. 10.1093/imamci/dnac007Search in Google Scholar

[5] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath, Boston, 1965. Search in Google Scholar

[6] W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math. 629, Springer, Berlin, 1978. 10.1007/BFb0067780Search in Google Scholar

[7] M. Corless, Guaranteed rates of exponential convergence for uncertain systems, J. Optim. Theory Appl. 64 (1990), no. 3, 481–494. 10.1007/BF00939420Search in Google Scholar

[8] G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Almqvist & Wiksells Boktryckeri AB, Uppsala, 1958. Search in Google Scholar

[9] C. A. Desoer, Slowly varying system x ˙ = A ⁢ ( t ) ⁢ x , IEEE Trans. Automatic Control AC-14 (1969), 780–781. 10.1109/TAC.1969.1099336Search in Google Scholar

[10] A. Dorgham, M. Hammi and M. A. Hammami, Asymptotic behavior of a class of perturbed differential equations, Ukrainian Math. J. 73 (2021), no. 5, 731–745. 10.1007/s11253-021-01956-5Search in Google Scholar

[11] S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science, Hauppauge, 2003. Search in Google Scholar

[12] Š. G. Gamidov, Certain integral inequalities for boundary value problems of differential equations, Differ. Uravn. 5 (1969), 463–472. Search in Google Scholar

[13] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. (2) 20 (1919), no. 4, 292–296. 10.2307/1967124Search in Google Scholar

[14] N. Hadj Taieb, M. A. Hammami and M. Hammi, On the global uniform stability analysis of non-autonomous dynamical systems: A survey, Math. Morav. 26 (2022), no. 2, 1–48. 10.5937/MatMor2202001TSearch in Google Scholar

[15] W. Hahn, Stability of Motion, Grundlehren Math. Wiss. 138, Springer, New York, 1967. 10.1007/978-3-642-50085-5Search in Google Scholar

[16] M. Hammi and M. A. Hammami, Gronwall–Bellman type integral inequalities and applications to global uniform asymptotic stability, Cubo 17 (2015), no. 3, 53–70. 10.4067/S0719-06462015000300004Search in Google Scholar

[17] M. Hammi and M. A. Hammami, Non-linear integral inequalities and applications to asymptotic stability, IMA J. Math. Control Inform. 32 (2015), no. 4, 717–735. Search in Google Scholar

[18] H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice-Hall, New York, 2002. Search in Google Scholar

[19] S. M. Lozinskiĭ, Error estimate for numerical integration of ordinary differential equations. I (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1958 (1958), no. 5(6), 52–90; erratum, Izv. Vyssh. Uchebn. Zaved. Mat. 1959 (1959), no. 5(12), 222. Search in Google Scholar

[20] A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control 55 (1992), no. 3, 531–534. 10.1080/00207179208934253Search in Google Scholar

[21] A. I. Perov and I. D. Kostrub, On the spectral abscissa and the logarithmic norm (in Russian), Mat. Zametki 101 (2017), no. 4, 562–575; transaltion in Math. Notes 101 (2017), no. 3–4, 677–687. Search in Google Scholar

[22] H. H. Rosenbrock, The stability of linear time-dependent control systems, J. Electronics Control (1) 15 (1963), 73–80. 10.1080/00207216308937556Search in Google Scholar

[23] W. J. Rugh, Linear System Theory, Prentice Hall Inform. Syst. Sci. Ser., Prentice Hall, Englewood Cliffs, 1993. Search in Google Scholar

[24] V. Solo, On the stability of slowly time-varying linear systems, Math. Control Signal Systems 7 (1994), 331–350. 10.1007/BF01211523Search in Google Scholar

[25] R. Vrabel, Criterion for robustness of global asymptotic stability to external perturbations of linear time-varying systems, Int. J. Gen. Syst. 50 (2021), no. 2, 211–222. 10.1080/03081079.2020.1870223Search in Google Scholar

[26] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Publ. Math. Soc. Japan 9, Mathematical Society of Japan, Tokyo, 1966. Search in Google Scholar