On exponential stability of nonlinear time-varying differential equations

Abstract

Within the Liapunov framework, a sufficient condition for exponential stability of ordinary differential equations is proposed. Unlike with classical Liapunov theory, the time derivative of the Liapunov function, taken along solutions of the system, may have positive and negative values. Verification of the conditions of the main theorem may be harder than in the classical case. It is shown that the proposed conditions are useful for the investigation of the exponential stability of fast time-varying systems. This sets the stability study by means of averaging in a Liapunov context. In particular, it is established that exponential stability of the averaged system implies exponential stability of the original fast time-varying system. A comparison of our work with results taken from the literature is included.