A note on input-to-state stability and averaging of systems with inputs

Two different definitions of an average for time-varying systems with inputs and a small parameter that were recently introduced in the literature are considered: "strong" and "weak" averages. It is shown that if the strong average is input-to-state stable (ISS), then the solutions of the actual system satisfy an integral bound in a semiglobal practical sense. The integral bound that we prove can be viewed as a generalization of the notion of finite-gain L₂ stability, that was recently introduced in the literature. A similar result is proved for weak averages but the class of inputs for which the integral bound holds is smaller (Lipschitz inputs) than in the case of strong averages (measurable inputs)