Input-to-state stability for nonlinear time-varying systems via averaging

We introduce two definitions of an averaged system for a time-varying ordin ary differential equation with exogenous disturbances ("strong average" and "weak average"). The class of systems for which the strong average exists is shown to be strictly smaller than the class of systems for which the wea k average exists. It is shown that input-to-state stability (ISS) of the st rong average of a system implies uniform semi-global practical ISS of the a ctual system, This result generalizes the result of [TPA] which states that global asymptotic stability of the averaged system implies uniform semi-gl obal practical stability of the actual system. On the other hand, we illust rate by an example that ISS of the weak average of a system does not necess arily imply uniform semi-global practical ISS of the actual system. However , ISS of the weak average of a system does imply a weaker semi-global pract ical "ISS-like" property for the actual system when the disturbances w are absolutely continuous and w, w is an element of L-x, ISS of the weak averag e of a system is shown to be useful in a stability analysis of time-varying cascaded systems.