This paper investigates the problem of ℋ∞ static output feedback control design for discrete-time Markov jump linear systems (MJLS), assuming that the transition probability matrix is not precisely known, but affected by different classes of uncertainties: polytopic, bounded or completely unknown elements. All types of uncertainties are modeled through one single representation, expressed in terms of the Cartesian product of simplexes, called multi-simplex. The main novelty of the proposed design procedure is that, differently from previous approaches in the literature, parameter-dependent Lyapunov matrices are used to certify the closed-loop stability with an ℋ∞ bound for the discrete-time MJLS. The proposed conditions are based on linear matrix inequality relaxations performed in two steps: the first step generates a parameter-dependent state feedback controller that is employed as an input for the second stage, which synthesizes a robust static output feedback gain assuring an ℋ∞ guaranteed cost. The proposed strategy can also cope with ℋ ∞ state feedback control for discrete-time MJLS. Numerical examples illustrate the advantages of the proposed methodology when compared to other methods from the literature. © 2014 American Automatic Control Council. 489494Boeing,et al.,GE Global Research,Honeywell,MathWorks,MitsubishiSyrmos, V.L., Abdallah, C.T., Dorato, P., Grigoriadis, K., Static output feedback - A survey (1997) Automatica, 33 (2), pp. 125-137. , FebruaryDe Oliveira, M.C., Geromel, J.C., Bernussou, J., Extended H2 and H∞ characterization and controller parametrizations for discrete-time systems (2002) Int. J. 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