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We consider a Markov control model in discrete time with countable both statespace and action space. Using the value function of a suitable long-run averagereward problem, we study various reachability/controllability problems. First,we characterize the domain of attraction and escape set of the system, and ageneralization called $p$-domain of attraction, using the aforementioned valuefunction. Next, we solve the problem of maximizing the probability of reachinga set $A$ while avoiding a set $B$. Finally, we consider a constrained versionof the previous problem where we ask for the probability of reaching the set$B$ to be bounded. In the finite case, we use linear programming formulationsto solve these problems. Finally, we apply our results to a example of anobject that navigates under stochastic influence.

On reachability of Markov chains: A long-run average approach