International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 2, Pages 347-366
doi:10.1155/S016117129700046X
New approach to asymptotic stability: time-varying nonlinear systems
Ecole Nationale d'Ingenieurs de Belfort, Espace Bartholdi, Belfort Technopole, B.P. 525, Belfort Cedex 90016, France
Received 24 February 1994; Revised 30 June 1995
Copyright © 1997 L. T. Grujić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The results of the paper concern a broad family of time-varying nonlinear systems with
differentiable motions. The solutions are established in a form of the necessary and sufficient conditions
for: 1) uniform asymptotic stability of the zero state, 2) for an exact single construction of a system
Lyapunov function and 3) for an accurate single determination of the (uniform) asymptotic stability
domain. They permit arbitrary selection of a function p(⋅) from a defined functional family to
determine a Lyapunov function v(⋅), [v(⋅)], by solving v′(⋅)=−p(⋅) {or equivalently,
v′(⋅)=−p(⋅)[1−v(⋅)]}, respectively. Illstrative examples are worked out.