# The steady-state control problem for Markov decision processes

2 SUMO - SUpervision of large MOdular and distributed systems
Inria Rennes – Bretagne Atlantique , IRISA-D4 - LANGAGE ET GÉNIE LOGICIEL
5 MEXICO - Modeling and Exploitation of Interaction and Concurrency
LSV - Laboratoire Spécification et Vérification [Cachan], Inria Saclay - Ile de France
Abstract : 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.
Document type :
Conference papers

Cited literature [11 references]

https://hal.inria.fr/hal-00879355
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Submitted on : Saturday, November 2, 2013 - 10:21:32 PM
Last modification on : Thursday, January 20, 2022 - 5:33:32 PM
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• HAL Id : hal-00879355, version 1

### Citation

Sundararaman Akshay, Nathalie Bertrand, Serge Haddad, Loïc Hélouët. The steady-state control problem for Markov decision processes. Qest 2013, Sep 2013, Buenos Aires, Argentina. pp.290-304. ⟨hal-00879355⟩

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