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Stability of random-switching systems of differential equations. (English) Zbl 1163.93036

Summary: This work is devoted to the stability of random-switching systems of differential equations. After presenting the formulation of random-switching systems, the notion of stability is recalled, and sufficient conditions in terms of the Lyapunov function are presented. Then easily verifiable conditions for stability and instability of systems arising in approximation are established. Using a logarithm transformation, necessary and sufficient conditions are derived for systems that are linear in the continuous state component. Several examples are provided as demonstrations. Among other things, a somewhat different behavior from the well-known Hartman-Grobman theorem is observed.

MSC:

93E15 Stochastic stability in control theory
60J27 Continuous-time Markov processes on discrete state spaces
60J75 Jump processes (MSC2010)
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D20 Asymptotic stability in control theory
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References:

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