ℋ− index for discrete-time stochastic systems with Markovian jump and multiplicative noise

In this paper, we discuss the ℋ− index problem for stochastic linear discrete-time systems subject to Markovian jump and multiplicative noise, for which, a necessary and sufficient condition for an ℋ− index larger than γ>0 is given in finite time horizon. It is shown that the ℋ− index larger than a given value is equivalent to the solvability of a certain generalized difference Riccati equation (GDRE). What we have obtained generalizes the results of deterministic systems to stochastic models. Moreover, the ℋ− index problem for square systems in infinite horizon is also studied. Finally, some examples are presented to illustrate the effectiveness of the proposed theoretical results.