International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 2, Pages 347-366

New approach to asymptotic stability: time-varying nonlinear systems

L. T. Grujić

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.


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()[1v()]}, respectively. Illstrative examples are worked out.