Controlled Markov chains with safety upper bound

In this note, we introduce and study the notion of safety control of stochastic discrete-event systems (DESs), modeled as controlled Markov chains. For nonstochastic DESs modeled by state machines or automata, safety is specified as a set of forbidden states, or equivalently by a binary valued vector that imposes an upper bound on the set of states permitted to be visited. We generalize this notion of safety to the setting of stochastic DESs by specifying it as an unit-interval valued vector that imposes an upper bound on the state probability distribution vector. Under the assumption of complete state observation, we identify: 1) the set of all state feedback controllers that satisfy the safety requirement for any given safe initial state probability distribution, and 2) the set of all safe initial state probability distributions for a given state feedback controller.