{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:52:32Z","timestamp":1787320352409,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>We study the local exponential stabilization, near a given steady-state flow, of solutions of the Navier\u2013Stokes equations in a bounded domain. The control is performed through a Dirichlet boundary condition. We apply a linear feedback controller, provided by a well-posed infinite-dimensional Riccati equation. We give a characterization of the domain of the closed-loop operator which is obtained from the closed-loop linearized Navier\u2013Stokes system. We give a class of initial data for which a Lyapunov function is obtained. For all $s\\in[0,1\/2[$, the stabilization of the two-dimensional Navier\u2013Stokes equations is proved for initial data in $\\mathbf{H}^s(\\Omega)\\cap V_n^0(\\Omega)$, where $V_n^0(\\Omega)$ is the space in which the Stokes operator is defined. We also obtain a three-dimensional stabilization result but only for a very specific set of initial data.<\/jats:p>","DOI":"10.1137\/070682630","type":"journal-article","created":{"date-parts":[[2009,5,22]],"date-time":"2009-05-22T18:14:57Z","timestamp":1243016097000},"page":"1797-1830","source":"Crossref","is-referenced-by-count":25,"title":["Lyapunov Function and Local Feedback Boundary Stabilization of the Navier\u2013Stokes Equations"],"prefix":"10.1137","volume":"48","author":[{"given":"Mehdi","family":"Badra","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,5,22]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1080\/01630560701348434"},{"key":"R2","unstructured":"M. Badra,\n                      Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system\n                      , ESAIM Control Optim. Calc. Var., to appear."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2003009"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2005.09.012"},{"key":"R5","doi-asserted-by":"crossref","first-page":"x","DOI":"10.1090\/memo\/0852","volume":"181","author":"Barbu V.","year":"2006","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"V. Barbu, I. Lasiecka, and R. Triggiani,\n                      Local exponential stabilization strategies of the Navier-Stokes equations, $d=2, 3$, via feedback stabilization of its linearization\n                      , in Control of Coupled Partial Differential Equations, Internat. Ser. Numer. Math. 155, Birkh\u00e4user, Basel, 2007, pp. 13\u201346.","DOI":"10.1007\/978-3-7643-7721-2_2"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2004.53.2445"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"A. Bensoussan, G. Da Prato, M. C. Delfour, and S. Mitter,\n                      Representation and Control of Infinite-Dimensional Systems. Vol. 1, Systems & Control: Foundations & Applications\n                      , Birkh\u00e4user Boston, Boston, MA, 1992.","DOI":"10.1007\/978-1-4612-2750-2"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"P. Constantin and C. Foias,\n                      Navier-Stokes Equations\n                      , Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1988.","DOI":"10.7208\/chicago\/9780226764320.001.0001"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0520006"},{"key":"R11","doi-asserted-by":"crossref","first-page":"viii","DOI":"10.1090\/memo\/0788","volume":"166","author":"Denk R.","year":"2003","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2004.02.010"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.3792\/pja\/1195526510"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"G. P. Galdi,\n                      An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol\n                      . 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