Abstract
This chapter presents techniques for solving stochastic reachability, based strictly on the probabilistic structure of stochastic hybrid systems. These techniques are based on the use of operator semigroup, infinitesimal generator and associated quadratic form, reduced function and balayage operator, Newtonian and Martin kernel, and properties of martingale associated to the stochastic hybrid process. All these techniques use the above concepts and provide schemes to extend results available Markov chains to stochastic hybrid processes. These give rise to approximation results or upper/lower bounds for the stochastic reachability probabilities. It seems that good and satisfying results in this research direction are based on finding the suitable methods that provide the best Markov chain abstractions for stochastic hybrid systems. As soon as a Markov chain preserves more desired features of a stochastic hybrid system and abstracts away its continuous dynamics, these probabilistic methods can provide us quite accurate results for stochastic reachability.
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Bujorianu, L.M. (2012). Probabilistic Methods for Stochastic Reachability. In: Stochastic Reachability Analysis of Hybrid Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-2795-6_6
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