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

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.