Control of continuous-time Markov chains with safety upper bounds

This work introduces the notion of safety for the controlled Markov chains in the continuous-time horizon. The concept is a non-trivial extension of safety control for stochastic systems modeled as discrete-time Markov decision processes, where the safety means that the probability distributions of the system states will not visit the given forbidden set at any time. In this paper an unit-interval valued vector that serves as an upper bound on the state probability distribution vector characterizes the forbidden set. A probability distribution is then called safe if it does not exceed the upper bound. Under mild conditions the author derives two results: 1) the necessary and sufficient conditions that guarantee the all-time safety of the probability distributions if the starting distribution is safe, and 2) the characterization of the supreme set of safe initial probability vectors that remain safe as time passes. In particular the paper identifies an upper bound on time and shows that if a distribution is always safe before that time, the distribution is safe at all times. Numerical examples are provided to illustrate the two results.