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.

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In this paper we formulate, within the Liapunov framework, a su½-cient condition for exponential stability of a di¨erential equation. This condition gives rise to a new averaging result referred to as``partial averaging'': exponential stability of a system xt f xY tY t, with su½ciently large, is implied by exponential stability of a time-varying system xt f xY t.