A Hamilton-Jacobi-Bellman Approach for the Numerical Computation of Probabilistic State Constrained Reachable Sets

Aim of this work is to characterise and compute the set of initial conditions for a system of controlled diffusion processes which allow to reach a terminal target satisfying pointwise state constraints with a given probability of success. Defining a suitable auxiliary optimal control problem, the characterization of this set is related to the solution of a particular Hamilton-Jacobi-Bellman equation. A semi-Lagrangian numerical scheme is defined and its convergence to the unique viscosity solution of the equation is proved. The validity of the proposed approach is then tested on some numerical examples.