Stability analysis for stochastic hybrid systems: a survey. (English) Zbl 1301.93168
Summary: This survey addresses stability analysis for Stochastic Hybrid Systems (SHS), which are dynamical systems that combine continuous change and instantaneous change and that also include random effects. We re-emphasize the common features found in most of the models that have appeared in the literature, which include stochastic switched systems, Markov jump systems, impulsive stochastic systems, switching diffusions, stochastic impulsive systems driven by renewal processes, diffusions driven by Lévy processes, piecewise-deterministic Markov processes, general stochastic hybrid systems, and stochastic hybrid inclusions. Then, we review many of the stability concepts that have been studied, including Lyapunov stability, Lagrange stability, asymptotic stability, and recurrence. Next, we detail Lyapunov-based sufficient conditions for these properties, and additional relaxations of Lyapunov conditions. Many other aspects of stability theory for SHS, like converse Lyapunov theorems and robustness theory, are not fully developed; hence, we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces.
MSC:
| 93E15 | Stochastic stability in control theory |
| 93E03 | Stochastic systems in control theory (general) |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
Keywords:
stochastic hybrid systems; stochastic switched systems; stochastic differential equations; switching diffusions; Markov chains; stochastic stability; Lyapunov functions; converse theorems; robustnessReferences:
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