Sum-of-Squares Certificates for Almost-Sure Reachability of Stochastic Polynomial Systems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909876286914560 |
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| author | Kordabad, Arash Bahari Majumdar, Rupak Soudjani, Sadegh |
| author_facet | Kordabad, Arash Bahari Majumdar, Rupak Soudjani, Sadegh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sum-of-Squares Certificates for Almost-Sure Reachability of Stochastic Polynomial Systems Kordabad, Arash Bahari Majumdar, Rupak Soudjani, Sadegh Optimization and Control Systems and Control 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. |
| title | Sum-of-Squares Certificates for Almost-Sure Reachability of Stochastic Polynomial Systems |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2510.25513 |