We address stability of a class of Markovian discrete-time stochastic hybrid
systems. This class of systems is characterized by the state-space of the
system being partitioned into a safe or target set and its exterior, and the
dynamics of the system being different in each domain. We give conditions for
$L_1$-boundedness of Lyapunov functions based on certain negative drift
conditions outside the target set, together with some more minor assumptions.
We then apply our results to a wide class of randomly switched systems (or
iterated function systems), for which we give conditions for global asymptotic
stability almost surely and in $L_1$. The systems need not be time-homogeneous,
and our results apply to certain systems for which functional-analytic or
martingale-based estimates are difficult or impossible to get.