This paper addresses a control problem for probabilistic models in the setting of Markov decision processes (\MDP). We are interested in the \emph{steady-state control problem} which asks, given an ergodic \MDP\ $\mathcal M$ and a distribution $\delta_{goal}$, whether there exists a (history-dependent randomized) policy $\pi$ ensuring that the steady-state distribution of $\mathcal M$ under $\pi$ is exactly $\delta_{goal}$. We first show that stationary randomized policies suffice to achieve a given steady-state distribution. Then we infer that the steady-state control problem is decidable for \MDP, and can be represented as a linear program which is solvable in PTIME. This decidability result extends to labeled \MDP\ (\LMDP) where the objective is a steady-state distribution on labels carried by the states, and we provide a PSPACE algorithm. We also show that a related \emph{steady-state language inclusion problem} is decidable in EXPTIME for \LMDP. Finally, we prove that if we consider \MDP\ under partial observation (\POMDP), the steady-state control problem becomes undecidable.