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

Arash Bahari Kordabad; Rupak Majumdar; Sadegh Soudjani

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arXiv · 2510.25513

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

Arash Bahari Kordabad·Rupak Majumdar·Sadegh Soudjani

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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.

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Arash Bahari Kordabad, Rupak Majumdar, Sadegh Soudjani. 2025-10-29. Sum-of-Squares Certificates for Almost-Sure Reachability of Stochastic Polynomial Systems. https://arxiv.org/abs/2510.25513

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