Almost Sure Reachability in Continuous-time Stochastic Systems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866909014446571520 |
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| author | Kordabad, Arash Bahari Majumdar, Rupak Soudjani, Sadegh |
| author_facet | Kordabad, Arash Bahari Majumdar, Rupak Soudjani, Sadegh |
| contents | We provide certificates for almost sure reachability of continuous-time stochastic systems governed by stochastic differential equations (SDEs). We first show that a standard Euler-Maruyama discretization may fail to preserve almost sure reachability property of the system using a double-well Langevin system. This observation motivates us to develop certificates for almost sure reachability directly on the continuous-time system. We introduce a pair of certificates, a drift function and a variant function, and prove necessity and sufficiency for almost sure reachability of an open bounded target set. Using these certificates, for linear SDEs, we give a characterization of almost sure reachability in terms of the spectral structure of the system matrices. For polynomial SDEs, we fix a polynomial template for the drift function and choose the variant function template as an exponential function composed with a polynomial. This allows us to translate the conditions in the certificates into sum-of-squares (SOS) constraints. We then propose an alternating scheme to resolve bilinearities. We illustrate the approach on the double-well Langevin example, showing that continuous-time SOS certificates recover almost sure reachability that is lost under time discretization. Moreover, we verify the SOS approach on a polynomial system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03595 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Almost Sure Reachability in Continuous-time Stochastic Systems Kordabad, Arash Bahari Majumdar, Rupak Soudjani, Sadegh Systems and Control We provide certificates for almost sure reachability of continuous-time stochastic systems governed by stochastic differential equations (SDEs). We first show that a standard Euler-Maruyama discretization may fail to preserve almost sure reachability property of the system using a double-well Langevin system. This observation motivates us to develop certificates for almost sure reachability directly on the continuous-time system. We introduce a pair of certificates, a drift function and a variant function, and prove necessity and sufficiency for almost sure reachability of an open bounded target set. Using these certificates, for linear SDEs, we give a characterization of almost sure reachability in terms of the spectral structure of the system matrices. For polynomial SDEs, we fix a polynomial template for the drift function and choose the variant function template as an exponential function composed with a polynomial. This allows us to translate the conditions in the certificates into sum-of-squares (SOS) constraints. We then propose an alternating scheme to resolve bilinearities. We illustrate the approach on the double-well Langevin example, showing that continuous-time SOS certificates recover almost sure reachability that is lost under time discretization. Moreover, we verify the SOS approach on a polynomial system. |
| title | Almost Sure Reachability in Continuous-time Stochastic Systems |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2605.03595 |