Reach-Avoid Analysis for Polynomial Stochastic Differential Equations

In this article, we propose a novel semidefinite programming approach that solves reach-avoid problems over open (i.e., not bounded a priori) time horizons for dynamical systems modeled by polynomial stochastic differential equations. The reach-avoid problem in this article is a probabilistic guarantee: we approximate from the inner a
$p$
-reach-avoid set, i.e., the set of initial states guaranteeing with probability larger than
$p$
that the system eventually enters a given target set while remaining inside a specified safe set till the target hit. Our approach begins with the construction of a bounded value function, whose strict
$p$
super-level set is equal to the
$p$
-reach-avoid set. This value function is then reduced to a twice continuously differentiable solution to a system of equations. The system of equations facilitates the construction of a semidefinite program using sum-of-squares decomposition for multivariate polynomials and thus, the transformation of nonconvex reach-avoid problems into a convex optimization problem. We would like to point out that our approach can straightforwardly be specialized to address classical safety verification by, a.o., stochastic barrier certificate methods and reach-avoid analysis for ordinary differential equations. In addition, several examples are provided to demonstrate theoretical and algorithmic developments of the proposed method.