TY - JOUR
ID - 6267
TI - A Discussion on the Existence of Smooth Lyapunov Functions for Continuous Stable Systems
JO - International Journal of Industrial Electronics Control and Optimization
JA - IECO
LA - en
SN - 2645-3517
AU - Akbarian, Majid
AU - Pariz, Naser
AD - Department of Electrical Engineering, Ferdowsi University of Mashhad, 91775-1111, Iran
AD - Ferdowsi University of Mashhad
Y1 - 2021
PY - 2021
VL - 4
IS - 3
SP - 333
EP - 340
KW - stability
KW - Smooth Lyapunov functions
KW - Stable equilibrium points
KW - Limit cycles
DO - 10.22111/ieco.2021.35409.1300
N2 - 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 donâ€™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.
UR - https://ieco.usb.ac.ir/article_6267.html
L1 - https://ieco.usb.ac.ir/article_6267_ea8e2dd3e5b26ec18fe0a2fdc0292592.pdf
ER -