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New approach to asymptotic stability: Time-varying nonlinear systems. (English) Zbl 0876.34060
Summary: The results of the paper concern a broad family of time-varying nonlinear systems with differentiable motions. The solutions are established in form of 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 $$p(\cdot)$$ from a defined functional family to determine a Lyapunov function $$\text{v} (\cdot)$$, $$[v(\cdot)]$$, by solving $$\text{v}'(\cdot)= -p(\cdot)$$ (or equivalently $$v'(\cdot)= -p(\cdot) [1-v(\cdot)])$$, respectively. Illustrative examples are worked out.

##### MSC:
 34D20 Stability of solutions to ordinary differential equations 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93D20 Asymptotic stability in control theory
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