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This article deals with stochastic processes endowed with the Markov(memoryless) property and evolving over general (uncountable) state spaces. Themodels further depend on a non-deterministic quantity in the form of a controlinput, which can be selected to affect the probabilistic dynamics. We addressthe computation of maximal reach-avoid specifications, together with thesynthesis of the corresponding optimal controllers. The reach-avoidspecification deals with assessing the likelihood that any finite-horizontrajectory of the model enters a given goal set, while avoiding a given set ofundesired states. This article newly provides an approximate computationalscheme for the reach-avoid specification based on the Fitted Value Iterationalgorithm, which hinges on random sample extractions, and gives a-prioricomputable formal probabilistic bounds on the error made by the approximationalgorithm: as such, the output of the numerical scheme is quantitativelyassessed and thus meaningful for safety-critical applications. Furthermore, weprovide tighter probabilistic error bounds that are sample-based. The overallcomputational scheme is put in relationship with alternative approximationalgorithms in the literature, and finally its performance is practicallyassessed over a benchmark case study.

Sampling-based Approximations with Quantitative Performance for the Probabilistic Reach-Avoid Problem over General Markov Processes