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A Discussion on the Existence of Smooth Lyapunov Functions for Continuous Stable Systems | ||
| International Journal of Industrial Electronics Control and Optimization | ||
| مقاله 7، دوره 4، شماره 3، آبان 2021، صفحه 333-340 اصل مقاله (774.92 K) | ||
| نوع مقاله: Research Articles | ||
| شناسه دیجیتال (DOI): 10.22111/ieco.2021.35409.1300 | ||
| نویسندگان | ||
| Majid Akbarian1؛ Naser Pariz* 2 | ||
| 1Department of Electrical Engineering, Ferdowsi University of Mashhad, 91775-1111, Iran | ||
| 2Ferdowsi University of Mashhad | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Stability؛ Smooth Lyapunov functions؛ Stable equilibrium points؛ Limit cycles | ||
| مراجع | ||
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