Averaging Results and the Study of Uniform Asymptotic Stability of Homogeneous Differential Equations That Are Not Fast Time-Varying

Within the Liapunov framework, a sufficient condition for uniform asymptotic stability of ordinary differential equations is proposed. Unlike with classical Liapunov theory, the time derivative of the $V$-function, taken along solutions of the system, may have positive and negative values. It is shown that the proposed condition is useful for the study of uniform asymptotic stability of homogeneous systems with order $\tau 0$. In particular, it is established that asymptotic stability of the averaged homogeneous system implies local uniform asymptotic stability of the original time-varying homogeneous system. This shows that averaging techniques play a prominent role in the study of homogeneous---not necessarily fast time-varying---systems.