{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,12]],"date-time":"2025-07-12T01:09:39Z","timestamp":1752282579202},"reference-count":40,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T00:00:00Z","timestamp":1685404800000},"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":[[2023,9,20]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2\\pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2\\pi )$-norm by the use of an appropriate Lyapunov\u2013Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.<\/jats:p>","DOI":"10.1093\/imamci\/dnad016","type":"journal-article","created":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T20:11:47Z","timestamp":1684959107000},"page":"516-533","source":"Crossref","is-referenced-by-count":2,"title":["Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay"],"prefix":"10.1093","volume":"40","author":[{"given":"Habib","family":"Ayadi","sequence":"first","affiliation":[{"name":"University of Kairouan, Higher Institute of Applied Mathematics and Computer Sciences Mathematics, , Avenue Ibn Fourat, Kairouan Sfax 3100, Tunisia"}]},{"given":"Mariem","family":"Jlassi","sequence":"additional","affiliation":[{"name":"University of Sfax, Faculty of Sciences Mathematics, , Soukra Road, Km 4, Kairouan Sfax 3000, Tunisia"}]}],"member":"286","published-online":{"date-parts":[[2023,5,30]]},"reference":[{"key":"2023092216581118500_ref1","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1109\/TAC.1982.1103023","article-title":"Linear systems with delayed controls: a reduction","volume":"27","author":"Artstein","year":"2013","journal-title":"IEEE Trans. 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