TIME-VARYING CONTINUOUS NONLINEAR-SYSTEMS - UNIFORM ASYMPTOTIC STABILITY

The framework of the presented research is a large class of time-varying nonlinear systems with continuous motions. The study of the uniform asymptotic stability of the zero equilibrium state developed here goes back to, and relies on, the very foundations of the Lyapunov stability concept and the (second) Lyapunov method. Stability domains are first defined and examined. Then, their qualitative features are used to establish complete solutions to the problem of uniform asymptotic stability of the equilibrium for various subclasses of the systems. The solutions present conditions that are both necessary and sufficient for: (1) the uniform asymptotic stability, (2) an exact direct one-shot construction of a system Lyapunov function and (3) for a direct accurate one-shot determination of the asymptotic stability domain. In addition, the solutions establish a novel Lyapunov-method based approach to the asymptotic stability analysis. This enables an arbitrary selection of a function p(.) from a defined functional family to determine a Lyapunov function upsilon(.), [nu(.)], by solving D(+)upsilon(.)= -p(.) or, equivalently, D(+)v(.) = -p(.)[1 - nu(.)], respectively. Illustrative examples are presented.