Probabilistic Invariance of Mixed Deterministic-Stochastic Dynamical Systems

This work is concerned with the computation of probabilistic invariance (or safety) over a finite horizon for mixed deterministic-stochastic, discrete-time processes over a continuous state space. The models of interest are made up of two sets of (possibly coupled) variables: the first set of variables has associated dynamics that are described by deterministic maps (vector fields), whereas the complement has dynamics that are characterized by a stochastic kernel. The contribution shows that the probabilistic invariance problem can be separated into two parts: a deterministic reachability analysis, and a probabilistic invariance problem that depends on the outcome of the first. This technique shows advantages over a fully probabilistic approach, and allows putting forward an approximation algorithm with explicit error bounds. The technique is tested on a case study modeling a chemical reaction network.