  We study the synthesis of mode switching protocols for a class of
discrete-time switched linear systems in which the mode jumps are governed by
Markov decision processes (MDPs). We call such systems MDP-JLS for brevity.
Each state of the MDP corresponds to a mode in the switched system. The
probabilistic state transitions in the MDP represent the mode transitions. We
focus on finding a policy that selects the switching actions at each mode such
that the switched system that follows these actions is guaranteed to be stable.
Given a policy in the MDP, the considered MDP-JLS reduces to a Markov jump
linear system (MJLS). {We consider both mean-square stability and stability
with probability one. For mean-square stability, we leverage existing stability
conditions for MJLSs and propose efficient semidefinite programming
formulations to find a stabilizing policy in the MDP. For stability with
probability one, we derive new sufficient conditions and compute a stabilizing
policy using linear programming. We also extend the policy synthesis results to
MDP-JLS with uncertain mode transition probabilities.
