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.

本文在时间尺度参数\（\ epsilon \）的情况下，解决了由两个时标马尔可夫链驱动的连续时间马尔可夫跳跃线性系统（MJLSs）的平均稳定性分析和\（L_1 \）性能。趋于零。这里考虑的跳跃过程是双变量的，具有慢速和快速分量。我们的方法依赖于涉及半群的收敛性分析，当马尔可夫链的快速部分的开关频率趋于无穷大时，该半群会生成MJLS的第一矩动力学。在此设置中，我们引入了极限情况下稳定性的新定义，并将其与平均MJLS的平均稳定性联系起来。在平均MJLS为正的特殊情况下，我们还得出评估平均稳定性和\（L_1 \）的合适标准性能。这些标准用矩阵的Hurwitz稳定性表示（矩阵的维数与快速切换组件的状态空间的基数无关），线性规划以及某个传递矩阵的1范数表示，这使得它们适合于计算目的。我们还建立了我们的（两次尺度）方法与文献中现有的一次尺度方法之间的比较，并表明我们的标准基于相对较小维度的矩阵，而不依赖于尺度参数\（ \ epsilon \）。通过流行病学和分类模型的数值示例讨论了主要结果的有效性。