Two characterizations of switched nonlinear systems with average dwell time

This note aims to establish the fast switching condition with average dwell time satisfying an upper bound. Important results are obtained on the behaviour of switched nonlinear dynamical systems. In specific, this note contributes in the following three aspects: (1) establish the condition of fast switching of switched nonlinear systems; (2) obtain the condition of arbitrary switching stability of switched nonlinear dynamical systems using a weak Lyapunov functions approach; and (3) prove the necessity of the average dwell time condition associated with the conventional multiple Lyapunov functions’ framework.