Mean stability and $${\varvec{L}}_\mathbf{1 }$$ performance of a class of two-time-scale Markov jump linear systems

This paper addresses the mean stability analysis and $$L_1$$ performance of continuous-time Markov jump linear systems (MJLSs) driven by a two time-scale Markov chain, in the scenario in which the temporal scale parameter $$\epsilon $$ tends to zero. The jump process considered here is bivariate, with slow and fast components. Our approach relies on a convergence analysis involving the semigroup that generates the first-moment dynamics of the MJLS when the switching frequency of the fast part of the Markov chain tends to infinity. In this setup, we introduce a new definition of stability in a limit case, and connect it with the mean stability of an averaged MJLS. In the particular case where the averaged MJLS is positive, we also derive suitable criteria for assessing mean stability and $$L_1$$ performance. These criteria are expressed in terms of the Hurwitz stability of a matrix (whose dimension is independent of the cardinality of the state space of the fast switching component), of linear programming, and of the 1-norm of a certain transfer matrix, which makes them suitable for computational purposes. We also establish comparisons between our (two-time-scale) approach and existing one-time-scale approaches from the literature, and show that our criteria are based on matrices of relatively smaller dimensions, which do not depend on the scale parameter $$\epsilon $$ . The effectiveness of the main results is discussed through numerical examples of epidemiological and compartmental models.