A note on the Liapunov approach to the study of asymptotic stability for differential equations

A new sufficient condition for asymptotic stability of ordinary differential equations is proposed. Unlike classical Liapunov theory, the time derivative along solutions of the Liapunov function may have positive and negative values. The classical Liapunov approach may be regarded as an infinitesimal version of the present theorem. Verification in practical problems is harder than in the classical case; an example is included in order to indicate how the present theorem may be applied.