Variational inequalities for the stochastic reachability problem

Unlike the pure discrete or continuous cases, for stochastic hybrid systems the reachability analysis is still a difficult problem. The early mathematical approaches to the stochastic reachability turned out into constructive verification methods with tool support from dynamic programming. This line of research suggests the need for a deeper mathematical investigation of connections between reachability based verification and optimal control for stochastic hybrid systems. In this paper, we prove a new characterization of the stochastic reachability problem in terms of variational inequalities on Hilbert spaces. These inequalities are well studied in stochastic optimal control, for which theoretical and numerical methods exist.