  This paper investigates uniform almost sure stability of randomly switched
time-varying systems. To assess stability properties of diverse time-varying
subsystems, mode-dependent indefinite multiple Lyapunov functions are
introduced. We present a novel condition so-called mean uniformly stable
function focusing on the time-varying functions in derivatives' parameters of
indefinite multiple Lyapunov functions. These conditions do not need
pre-determined switching moments and provide enhanced flexibility to
accommodate unstable subsystems and stable but no-exponentially decay
subystems. A numerical example is provided to demonstrate the effectiveness and
advantages of our approach.
