{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T08:54:14Z","timestamp":1787302454329,"version":"build-2736575974"},"reference-count":37,"publisher":"SAGE Publications","issue":"7","license":[{"start":{"date-parts":[[2012,5,30]],"date-time":"2012-05-30T00:00:00Z","timestamp":1338336000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["The International Journal of Robotics Research"],"published-print":{"date-parts":[[2012,6]]},"abstract":"<jats:p>Recent trends pushing robots into unstructured environments with limited sensors have motivated considerable work on planning under uncertainty and stochastic optimal control, but these methods typically do not provide guaranteed performance. Here we consider the problem of bounding the probability of failure (defined as leaving a finite region of state space) over a finite time for stochastic non-linear systems with continuous state. Our approach searches for exponential barrier functions that provide bounds using a variant of the classical supermartingale result. We provide a relaxation of this search to a semidefinite program, yielding an efficient algorithm that provides rigorous upper bounds on the probability of failure for the original non-linear system. We give a number of numerical examples in both discrete and continuous time that demonstrate the effectiveness of the approach.<\/jats:p>","DOI":"10.1177\/0278364912444146","type":"journal-article","created":{"date-parts":[[2012,5,30]],"date-time":"2012-05-30T04:43:54Z","timestamp":1338353034000},"page":"901-923","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":77,"title":["Finite-time regional verification of stochastic non-linear systems"],"prefix":"10.1177","volume":"31","author":[{"given":"Jacob","family":"Steinhardt","sequence":"first","affiliation":[{"name":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Russ","family":"Tedrake","sequence":"additional","affiliation":[{"name":"Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, Cambridge, MA, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2012,5,30]]},"reference":[{"key":"bibr1-0278364912444146","doi-asserted-by":"publisher","DOI":"10.3166\/ejc.16.624-641"},{"key":"bibr2-0278364912444146","volume":"19","author":"Abbeel P","year":"2006","journal-title":"Proceedings of the Neural Information Processing Systems (NIPS \u201907)"},{"key":"bibr3-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-16684-6"},{"key":"bibr4-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(73)90071-1"},{"key":"bibr5-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4606-6"},{"key":"bibr6-0278364912444146","first-page":"12","volume":"4","author":"Bouadi H","year":"2008","journal-title":"International Journal of Applied Mathematics and Computer Sciences"},{"key":"bibr7-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441"},{"key":"bibr8-0278364912444146","doi-asserted-by":"publisher","DOI":"10.15607\/RSS.2008.IV.030"},{"key":"bibr9-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1177\/0278364909340446"},{"key":"bibr10-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.6.1.73"},{"key":"bibr11-0278364912444146","doi-asserted-by":"publisher","DOI":"10.2514\/6.2008-7256"},{"key":"bibr12-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-00031-1"},{"key":"bibr13-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84800-155-8_7"},{"key":"bibr14-0278364912444146","unstructured":"Grant M, Boyd S (2011) CVX: Matlab software for disciplined convex programming - version 1.1beta. http:\/\/cvxr.com\/cvx."},{"key":"bibr15-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1007\/BF02134013"},{"key":"bibr16-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.53.1.8"},{"key":"bibr17-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1966.1098315"},{"key":"bibr18-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2009.2017144"},{"key":"bibr19-0278364912444146","unstructured":"Lofberg J (2011) Strictly feasible sum-of-squares solutions. http:\/\/users.isy.liu.se\/johanl\/yalmip\/pmwiki.php?n=Blog.Strictly-feasible-SOS-solutions."},{"key":"bibr20-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1177\/027836499000900206"},{"key":"bibr21-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2003.1271743"},{"key":"bibr22-0278364912444146","volume-title":"Proceedings of the 12th International Symposium on Experimental Robotics (ISER 2010)","author":"Mellinger D","year":"2010"},{"key":"bibr23-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1007\/10997703_2"},{"key":"bibr24-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2008.2009619"},{"key":"bibr25-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2004.1428804"},{"key":"bibr26-0278364912444146","volume-title":"SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB User\u2019s guide","author":"Prajna S","year":"2004","edition":"2"},{"key":"bibr27-0278364912444146","author":"Prajna S.","year":"2004","journal-title":"Proceedings of the ASCC 2004"},{"key":"bibr28-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2008.2010886"},{"key":"bibr29-0278364912444146","doi-asserted-by":"publisher","DOI":"10.3182\/20080706-5-KR-1001.01833"},{"key":"bibr30-0278364912444146","volume-title":"Model Predictive Control: Theory and Design","author":"Rawlings JB","year":"2009"},{"key":"bibr31-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1177\/0278364910388315"},{"key":"bibr32-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1080\/10556789908805766"},{"key":"bibr33-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2007.914221"},{"key":"bibr34-0278364912444146","doi-asserted-by":"publisher","DOI":"10.3182\/20110828-6-IT-1002.03098"},{"key":"bibr35-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1201\/9781420053739"},{"key":"bibr36-0278364912444146","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2009.02.046"},{"key":"bibr37-0278364912444146","unstructured":"Yeager JC (1998) Implementation and testing of turbulence models for the F18-HARV simulation. Technical report."}],"container-title":["The International Journal of Robotics Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/0278364912444146","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/0278364912444146","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T10:17:47Z","timestamp":1777457867000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/0278364912444146"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,5,30]]},"references-count":37,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2012,6]]}},"alternative-id":["10.1177\/0278364912444146"],"URL":"https:\/\/doi.org\/10.1177\/0278364912444146","relation":{},"ISSN":["0278-3649","1741-3176"],"issn-type":[{"value":"0278-3649","type":"print"},{"value":"1741-3176","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,5,30]]}}}