Stochastic stability for Markovian jump linear systems subject to a crucial failure event

This paper is concerned with the stability of discrete-time linear systems subject to random jumps in the parameters, described by an underlying finite-state Markov chain. In the model studied, a stopping time τ_Δ is associated with the occurrence of a "crucial failure" after which the system is brought to a halt for maintenance. The usual stochastic stability concepts and associated results are not indicated, since they are tailored to pure infinite horizon problems. Using the concept named stochastic τ-stability, equivalent conditions to ensure the stochastic stability of the system until the occurrence of τ_Δ is obtained. In addition, an intermediary and mixed case for which τ represents the minimum between the occurrence of a fix number N of failures and the occurrence of a "crucial failure" τ_Δ is also considered. Necessary and sufficient conditions to ensure the stochastic τ-stability are provided in this setting that are auxiliary to the main result.