  We investigate absorption, i.e., almost sure convergence to an absorbing
state, in time-varying (non-homogeneous) discrete-time Markov chains with
finite state space. We consider systems that can switch among a finite set of
transition matrices, which we call the modes. Our analysis is focused on two
properties: 1) almost sure convergence to an absorbing state under any
switching, and 2) almost sure convergence to a desired set of absorbing states
via a proper switching policy. We derive necessary and sufficient conditions
based on the structures of the transition graphs of modes. More specifically,
we show that a switching policy that ensures almost sure convergence to a
desired set of absorbing states from any initial state exists if and only if
those absorbing states are reachable from any state on the union of simplified
transition graphs. We then show three sufficient conditions for absorption
under arbitrary switching. While the first two conditions depend on the
acyclicity (weak acyclicity) of the union (intersection) of simplified
transition graphs, the third condition is based on the distances of each state
to the absorbing states in all the modes. These graph theoretic conditions can
verify the stability and stabilizability of absorbing states based only on the
feasibility of transitions in each mode.
