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Stochastic stability properties of jump linear systems. (English) Zbl 0747.93079
Summary: Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. In this paper, we study stochastic stability properties of jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented. Finally, for one-dimensional jump linear systems, we prove that the region for $\delta$-moment stability is monotonically converging to the region for almost sure stability as $\delta ↓{0}^{+}$.
##### MSC:
 93E15 Stochastic stability 93C05 Linear control systems