Sum-of-Squares Certificates for Almost-Sure Reachability of Stochastic Polynomial Systems

TL;DR

This paper introduces a sum-of-squares based computational method to certify almost-sure reachability in discrete-time polynomial stochastic systems, using drift and variant certificates solved via semidefinite programming.

Contribution

It develops a novel SOS framework combining drift and variant certificates, with an alternating optimization scheme to handle bilinearities, enabling certification of almost-sure reachability.

Findings

Successfully applied to two case studies demonstrating effectiveness.

Provides a systematic SOS-based certification approach.

Handles bilinearities with an alternating optimization scheme.

Abstract

In this paper, we present a computational approach to certify almost sure reachability for discrete-time polynomial stochastic systems by turning drift--variant criteria into sum-of-squares (SOS) programs solved with standard semidefinite solvers. Specifically, we provide an SOS method based on two complementary certificates: (i) a drift certificate that enforces a radially unbounded function to be non-increasing in expectation outside a compact set of states; and (ii) a variant certificate that guarantees a one-step decrease with positive probability and ensures the target contains its nonpositive sublevel set. We transform these conditions to SOS constraints. For the variant condition, we enforce a robust decrease over a parameterized disturbance ball with nonzero probability and encode the constraints via an S-procedure with polynomial multipliers. The resulting bilinearities are handled by an alternating scheme that alternates between optimizing multipliers and updating the variant and radius until a positive slack is obtained. Two case studies illustrate the workflow and certifies almost-sure reachability.