{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:44:21Z","timestamp":1753886661041,"version":"3.41.2"},"reference-count":9,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,10,4]],"date-time":"2012-10-04T00:00:00Z","timestamp":1349308800000},"content-version":"vor","delay-in-days":277,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71073072","11171135","09KJB110003","BK2010329"],"award-info":[{"award-number":["71073072","11171135","09KJB110003","BK2010329"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71073072","11171135","09KJB110003","BK2010329"],"award-info":[{"award-number":["71073072","11171135","09KJB110003","BK2010329"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Jiangsu Higher Education Institutions of China","award":["71073072","11171135","09KJB110003","BK2010329"],"award-info":[{"award-number":["71073072","11171135","09KJB110003","BK2010329"]}]},{"name":"Nature Science Foundation of Jiangsu","award":["71073072","11171135","09KJB110003","BK2010329"],"award-info":[{"award-number":["71073072","11171135","09KJB110003","BK2010329"]}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>This paper studies the adaptive control problem of the Korteweg\u2010de Vries\u2010Burgers equation. Using the Lyapunov function method, we prove that the closed\u2010loop system including the parameter estimator as a dynamic component is globally <jats:italic>L<\/jats:italic><jats:sup>2<\/jats:sup> stable. Furthermore, we show that the state of the system is regulated to zero by developing an alternative to Barbalat\u2032s lemma which cannot be used in the present situation. The closed\u2010loop system is shown to be well posed.<\/jats:p>","DOI":"10.1155\/2012\/389450","type":"journal-article","created":{"date-parts":[[2012,10,4]],"date-time":"2012-10-04T21:06:21Z","timestamp":1349384781000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adaptive Stabilization of the Korteweg\u2010de Vries\u2010Burgers Equation with Unknown Dispersion"],"prefix":"10.1155","volume":"2012","author":[{"given":"Xiaoyan","family":"Deng","sequence":"first","affiliation":[]},{"given":"Lixin","family":"Tian","sequence":"additional","affiliation":[]},{"given":"Wenxia","family":"Chen","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,10,4]]},"reference":[{"key":"e_1_2_8_1_2","doi-asserted-by":"publisher","DOI":"10.1155\/S1024123X00001320"},{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/acs.699"},{"key":"e_1_2_8_3_2","first-page":"23","article-title":"Neumann boundary control of MKdV-Burgers equation","volume":"24","author":"Tian L.","year":"2003","journal-title":"Journal of Jiangsu University"},{"key":"e_1_2_8_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10483-006-0114-z"},{"key":"e_1_2_8_5_2","doi-asserted-by":"publisher","DOI":"10.1023\/B:NODY.0000040023.92220.09"},{"key":"e_1_2_8_6_2","first-page":"319","article-title":"Adaptive control of generalized viscous Burgers\u2032 equation","volume":"7","author":"Deng X. Y.","year":"2009","journal-title":"International Journal of Nonlinear Science"},{"key":"e_1_2_8_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2010.05.001"},{"key":"e_1_2_8_8_2","first-page":"51","article-title":"Barbalat Lemma and its application in analysis of system stability","volume":"37","author":"Min Y. Y.","year":"2007","journal-title":"Journal of Shandong University"},{"volume-title":"Sobolev Spaces","year":"1975","author":"Adams R.","key":"e_1_2_8_9_2"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/389450.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/389450.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/389450","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T06:39:35Z","timestamp":1718087975000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/389450"}},"subtitle":[],"editor":[{"given":"Junjie","family":"Wei","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/389450"],"URL":"https:\/\/doi.org\/10.1155\/2012\/389450","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2012-07-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-09-10","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-10-04","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"389450"}}