Document Type : Research Articles

Authors

1 Department of Electrical Engineering, Ferdowsi University of Mashhad, 91775-1111, Iran

2 Ferdowsi University of Mashhad

Abstract

Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability.

Keywords

Main Subjects

[1] H. K. Khalil and J. Grizzle, "Nonlinear systems, vol. 3,"
Prentice hall Upper Saddle River, 2002.

[2] S. Jafari Fesharaki, F. Sheikholeslam, M. Kamali, and A.
Talebi, "Tractable robust model predictive control with
adaptive sliding mode for uncertain nonlinear systems,"
International Journal of Systems Science, vol. 51, no. 12,
pp. 2204-2216, 2020.

[3] T. S. Gardner, C. R. Cantor, and J. J. Collins,
"Construction of a genetic toggle switch in Escherichia
coli," Nature, vol. 403, no. 6767, pp. 339-342, 2000.

[4] L. Li, H. Zhang, and Y. Wang, "Stabilization and optimal
control of discrete-time systems with multiplicative noise
and multiple input delays," Systems & Control Letters,
vol. 147, p. 104833, 2021.

[5] Z. Shen, C. Li, H. Li, and Z. Cao, "Estimation of the
domain of attraction for discrete-time linear impulsive
control systems with input saturation," Applied
Mathematics and Computation, vol. 362, p. 124502, 2019.

[6] X.-F. Wang, A. R. Teel, K.-Z. Liu, and X.-M. Sun,
"Stability analysis of distributed convex optimization
under persistent attacks: A hybrid systems approach,"
Automatica, vol. 111, p. 108607, 2020.

[7] J.J. E. Slotine and W. Li, Applied nonlinear control (no.
1). Prentice hall Englewood Cliffs, NJ, 1991.

[8] A. I. Doban and M. Lazar, "Computation of Lyapunov
functions for nonlinear differential equations via a
Yoshizawatype construction," IFAC-PapersOnLine,
vol. 49, no. 18, pp. 29-34, 2016.

[9] A. I. Doban and M. Lazar, "Computation of Lyapunov
functions for nonlinear differential equations via a
Massera-type construction," IEEE Transactions on
Automatic Control, vol. 63, no. 5, pp. 1259-1272, 2017.

[10] P. Giesl and S. Hafstein, "Review on computational
methods for Lyapunov functions," Discrete and
Continuous Dynamical Systems-Series B, vol. 20, no. 8,
pp. 2291-2331, 2015.

[11] G. Bidari, N. Pariz, and A. Karimpour, "Sufficient
conditions for stabilization of interval uncertain LTI
switched systems with unstable subsystems,"
International Journal of Industrial Electronics, Control
and Optimization, vol. 2, no. 1, pp. 1-6, 2019.

[12] M. Akbarian, N. Eghbal, and N. Pariz, "A Novel Method
for Optimal Control of Piecewise Affine Systems Using
Semi-Definite Programming," International Journal of
Industrial Electronics, Control and Optimization, vol. 3,
no. 1, pp. 59-68, 2020.

[13] S. S. S. Farahani and S. Fakhimi Derakhshan, "LMI-based
congestion control algorithms for a delayed network," vol.
2, no. 2, pp. 91-98, 2019.

[14] K. Persidskii, "On a theorem of Liapunov," in CR (Dokl.)
Acad. Sci. URSS, 1937, vol. 14, pp. 541-543.

[15] C. M. Kellett, "Classical converse theorems in Lyapunov's
second method," arXiv preprint arXiv:1502.04809, 2015.

[16] A. Bacciotti, L. Rosier, and Z. Lin, "Liapunov Functions
and Stability in Control Theory. Lecture Notes in Control
and Information Sciences 267," Appl. Mech. Rev., vol. 55,
no. 5, pp. B88-B89, 2002.

[17] J. Auslander and P. Seibert, "Prolongations and stability in
dynamical systems," in Annales de l'institut Fourier,
1964, vol. 14, no. 2, pp. 237-267.

[18] A. A. Ahmadi, M. Krstic, and P. A. Parrilo, "A globally
asymptotically stable polynomial vector field with no
polynomial Lyapunov function," in 2011 50th IEEE
Conference on Decision and Control and European
Control Conference, 2011, pp. 7579-7580: IEEE.

[19] A. A. Ahmadi and B. El Khadir, "A globally asymptotically
stable polynomial vector field with rational coefficients
and no local polynomial Lyapunov function," Systems &
Control Letters, vol. 121, pp. 50-53, 2018.

[20] A. A. Ahmadi and B. El Khadir, "On algebraic proofs of
stability for homogeneous vector fields," IEEE
Transactions on Automatic Control, 2019.

[21] W. Hahn, Stability of motion. Springer, 1967.

[22] F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R.
Wolenski, Nonsmooth analysis and control theory.
Springer Science & Business Media, 2008.

[23] A. Bacciotti and L. Rosier, Liapunov functions and stability
in control theory. Springer Science & Business Media,
2006.

[24] A. Bacciotti and L. Rosier, "Regularity of Liapunov
functions for stable systems," Systems & control letters,
vol. 41, no. 4, pp. 265-270, 2000.

[25] J. LaSalle, "Some extensions of Liapunov's second
method," IRE Transactions on circuit theory, vol. 7, no. 4,
pp. 520-527, 1960.