{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T13:33:35Z","timestamp":1761226415491,"version":"build-2065373602"},"reference-count":65,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,9,25]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In this paper, we address the problem of non-linear boundary control design for the one-dimensional viscous Burgers equation. Using the modal decomposition approach, we design a finite-dimensional homogeneous controller for the controlled subsystem containing unstable modes. We prove that the closed-loop system with the homogeneous state-feedback control is locally asymptotically stable by constructing an appropriate Lyapunov functional depending on the canonical homogeneous norm and the degree of homogeneity. Next, we construct a linear infinite-dimensional observer and design the homogeneous output-feedback controller to achieve the local stability of the closed-loop system. We analyse the well-posedness of the closed-loop system with the finite-dimensional homogeneous state\/output feedback control by using the Arzela\u2013Ascoli theorem. Finally, we present numerical simulations to support the analytical results and compare our proposed controller with a linear one regarding closed-loop performance and the peaking effect.<\/jats:p>","DOI":"10.1093\/imamci\/dnaf033","type":"journal-article","created":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T11:52:27Z","timestamp":1758973947000},"source":"Crossref","is-referenced-by-count":0,"title":["Homogeneous boundary control design for a one-dimensional viscous Burgers equation: modal decomposition approach"],"prefix":"10.1093","volume":"42","author":[{"given":"Mericel","family":"Ayamou","sequence":"first","affiliation":[{"name":"Univ. Lille, CNRS , Inria, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille ,","place":["France"]}]},{"given":"Andrey","family":"Polyakov","sequence":"additional","affiliation":[{"name":"Univ. Lille , CNRS, Inria, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille ,","place":["France"]}]},{"given":"Nicolas","family":"Espitia","sequence":"additional","affiliation":[{"name":"Univ. Lille , CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille ,","place":["France"]}]},{"given":"Emilia","family":"Fridman","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Tel Aviv University , Tel Aviv 69978 ,","place":["Israel"]}]}],"member":"286","published-online":{"date-parts":[[2025,10,23]]},"reference":[{"key":"2025102309102407000_ref1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4471-3805-1","volume-title":"Flow Control by Feedback: stabilization and Mixing","author":"Aamo","year":"2003"},{"key":"2025102309102407000_ref2","doi-asserted-by":"crossref","first-page":"1814","DOI":"10.1137\/060675861","article-title":"Homogeneous approximation, recursive observer design, and output feedback","volume":"47","author":"Andrieu","year":"2008","journal-title":"SIAM J. 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