L₂-gain analysis via time-delay approach to periodic averaging with stochastic extension

We consider perturbed linear systems with fast-varying coefficients that are piecewise-continuous in time. Recently, a constructive time-delay approach to periodic averaging of such systems was introduced that provided an upper bound on the small parameter preserving their input-to-state stability (ISS). In the present paper, we present an improved time-delay approach and extend it to L₂-gain analysis. By the backward averaging, we transform the system to a modified time-delay system that leads to fewer terms to be compensated in the Lyapunov–Krasovskii (L–K) analysis. As a result we derive less conservative and simpler conditions for ISS and L₂-gain analysis in the form of linear matrix inequalities (LMIs). We further extend our results to stochastic systems where we employ a stochastic extension of Lyapunov functionals that we use for the deterministic case. Two numerical examples (stabilization by vibrational control and by time-dependent switching) illustrate the efficiency of the method.