We develop a novel framework for formulating a class of stochastic reachability problems with state constraints as a stochastic optimal control problem. Previous approaches to solving these problems are either confined to the deterministic setting or address almostsure stochastic notions. In contrast, we propose a new methodology to tackle probabilistic specifications that are less specific than almost sure requirements. To this end, we first establish a connection between two stochastic reach-avoid problems and classes of different stochastic optimal control problems for diffusions with discontinuous payoff functions. In the sequel, we shall focus on one of the class of stochastic optimal control problem, exit-time problem, which indeed addresses both reachability type questions. We derive a weak version of dynamic programming principle (DPP) for the value functions. Moreover, based on our DPP, we give an alternative characterization of the value function as the solution to a partial differential equation in the sense of discontinuous viscosity solutions along with Dirichlet type boundary conditions. Finally we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.
A partial differential equation characterization of continuous-time stochastic reach-avoid problems is studied in REF based on the theory of discontinuous viscosity solutions.
The Stochastic Reach-Avoid Problem and Set Characterization for Diffusions
Automatica, Vol. 70, pp. 43-56, 2016
Mathematics Computer Science