New criteria for exponential and uniform asymptotic stability of nonlinear time-variant differential equations

Within the Lyapunov framework, sufficient conditions for exponential and uniform asymptotic stability of ordinary differential equations are proposed. Unlike with classical Lyapunov theory, the time derivative of the Lyapunov function, taken along solutions of the system, may have positive and negative values. Verification of the conditions of the main theorems 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 Lyapunov context. In particular, it is established that exponential stability of the averaged system implies exponential stability of the original fast time-varying system