First-Moment Stability of Continuous-Time Markov Jump Linear Systems with Stationary, Time-Varying, and Polytopic Transition Rates

This work studies 1-moment stability for continuous-time Markov Jump Linear Systems under both stationary and time-varying transition rates. For stationary transition rates, we propose novel sufficient stability conditions that are stated in terms of linear programs. These conditions offer a less restrictive and computationally simpler stability characterization compared to the commonly used mean square stability analysis. Moreover, they offer a simple approach to also investigate the weaker notion of almost sure stability, which is implied by 1-moment stability. For the more general case of time-varying transition rates, we also propose 1-moment stability conditions expressed as linear inequalities. These conditions are derived by appropriately extending recent results on positive time-varying systems to general (i.e., not necessarily positive) systems. However, in the general case of time-varying transition rates without any particular structure, the conditions require infinitely many tests. Therefore, we explore finitely testable scenarios in which the transition rate matrix takes values within a polytope, as well as other relaxations.