{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T05:48:14Z","timestamp":1782366494636,"version":"3.54.5"},"reference-count":25,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"name":"Horizon Europe MSCA project Modconflex","award":["101073558"],"award-info":[{"award-number":["101073558"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,3,24]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In this paper, we study the well-posedness and the input-to-state type stability of a one-dimensional fluid\u2013particle interaction system. A distinctive feature, not yet considered in the input-to-state stability literature, is that our system involves a free boundary. More precisely, the fluid is described by the viscous Burgers equation and the motion of particle obeys Newton\u2019s second law. The point mass is subject to both a feedback control and an open-loop control. We first establish the well-posedness of the system for any open-loop input in $ L^{2}(0,\\infty ) $. Then, assuming the input also belongs to $ L^{1}(0,\\infty ) $, we prove that the particle\u2019s position remains uniformly bounded and that the system is input-to-state type stable. The proof is based on the construction of a Lyapunov functional derived from a special test function.<\/jats:p>","DOI":"10.1093\/imamci\/dnag004","type":"journal-article","created":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T06:18:17Z","timestamp":1774505897000},"source":"Crossref","is-referenced-by-count":0,"title":["Input-to-state type stability for simplified fluid\u2013particle interaction system"],"prefix":"10.1093","volume":"43","author":[{"given":"Zhuo","family":"Xu","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques de Bordeaux UMR 5251, Universit\u00e9 de Bordeaux\/Bordeaux INP\/CNRS , 351 Cours de la Lib\u00e9ration, 33 405 Talence, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,3,25]]},"reference":[{"key":"2026062501163094200_ref1","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/j.matpur.2015.02.009","article-title":"Particle supported control of a fluid\u2013particle system","volume":"104","author":"C\u00eendea","year":"2015","journal-title":"J. Math. Pures Appl."},{"key":"2026062501163094200_ref2","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1142\/S0218202505000522","article-title":"Some control results for simplified one-dimensional models of fluid-solid interaction","volume":"15","author":"Doubova","year":"2005","journal-title":"Math. Models Methods Appl. Sci."},{"key":"2026062501163094200_ref3","volume-title":"Collisions of Smooth Bodies in Viscous Fluids: A Mathematical Investigation","author":"Hesla","year":"2004"},{"key":"2026062501163094200_ref4","doi-asserted-by":"crossref","first-page":"1345","DOI":"10.1080\/03605300601088740","article-title":"Lack of collision between solid bodies in a 2d incompressible viscous flow","volume":"32","author":"Hillairet","year":"2007","journal-title":"Comm. Partial Differ Equ"},{"key":"2026062501163094200_ref5","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.anihpc.2009.09.007","article-title":"Blow up and grazing collision in viscous fluid solid interaction systems","volume":"27","author":"Hillairet","year":"2010","journal-title":"Annales de l\u2019IHP Analyse non lin\u00e9aire"},{"key":"2026062501163094200_ref6","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1137\/23M155788X","article-title":"Input-to-state stability for bilinear feedback systems","volume":"62","author":"Hosfeld","year":"2024","journal-title":"SIAM J. Control Optim."},{"key":"2026062501163094200_ref7","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1109\/CDC.2016.7798600","article-title":"On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems","volume-title":"2016 IEEE 55th Conference on Decision and Control (CDC)","author":"Jacob","year":"2016"},{"key":"2026062501163094200_ref8","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1137\/16M1099467","article-title":"Infinite-dimensional input-to-state stability and orlicz spaces","volume":"56","author":"Jacob","year":"2018","journal-title":"SIAM J. Control Optim."},{"key":"2026062501163094200_ref9","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s00498-021-00302-1","article-title":"Input-to-state stability for infinite-dimensional systems","volume":"34","author":"Jacob","year":"2022","journal-title":"Math. Control Signals Systems"},{"key":"2026062501163094200_ref10","doi-asserted-by":"crossref","first-page":"2247","DOI":"10.1109\/CDC.2016.7798597","article-title":"Input-to-state stability with respect to boundary disturbances for the 1-d heat equation","volume-title":"2016 IEEE 55th Conference on Decision and Control (CDC)","author":"Karafyllis","year":"2016"},{"key":"2026062501163094200_ref11","doi-asserted-by":"crossref","first-page":"3712","DOI":"10.1109\/TAC.2016.2519762","article-title":"ISS with respect to boundary disturbances for 1-d parabolic PDEs","volume":"61","author":"Karafyllis","year":"2016","journal-title":"IEEE Trans. Autom. Control"},{"key":"2026062501163094200_ref12","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1137\/16M1073753","article-title":"ISS in different norms for 1-d parabolic PDEs with boundary disturbances","volume":"55","author":"Karafyllis","year":"2017","journal-title":"SIAM J. Control Optim."},{"key":"2026062501163094200_ref13","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1051\/cocv\/2021053","article-title":"ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs","volume":"27","author":"Karafyllis","year":"2021","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"2026062501163094200_ref14","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1051\/cocv\/2011196","article-title":"Single input controllability of a simplified fluid-structure interaction model","volume":"19","author":"Liu","year":"2013","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"2026062501163094200_ref15","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1137\/19M1291248","article-title":"Input-to-state stability of infinite-dimensional systems: Recent results and open questions","volume":"62","author":"Mironchenko","year":"2020","journal-title":"SIAM Rev."},{"key":"2026062501163094200_ref16","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1137\/17M1161877","article-title":"Monotonicity methods for input-to-state stability of nonlinear parabolic pdes with boundary disturbances","volume":"57","author":"Mironchenko","year":"2019","journal-title":"SIAM J. Control Optim."},{"key":"2026062501163094200_ref17","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1007\/BF00252910","article-title":"An optimal poincar\u00e9 inequality for convex domains","volume":"5","author":"Payne","year":"1960","journal-title":"Arch. Rational Mech. Anal."},{"key":"2026062501163094200_ref18","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0167-6911(94)00050-6","article-title":"On characterizations of the input-to-state stability property","volume":"24","author":"Sontag","year":"1995","journal-title":"Systems Control Lett."},{"key":"2026062501163094200_ref19","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1109\/9.28018","article-title":"Smooth stabilization implies coprime factorization","volume":"34","author":"Sontag","year":"1989","journal-title":"IEEE Trans. Autom. Control"},{"key":"2026062501163094200_ref20","first-page":"1499","article-title":"Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain","volume":"8","author":"Takahashi","year":"2003","journal-title":"Adv. Differ. Equ."},{"key":"2026062501163094200_ref21","doi-asserted-by":"crossref","first-page":"1757","DOI":"10.1016\/j.automatica.2014.04.016","article-title":"Well-posed systems\u2014The LTI case and beyond","volume":"50","author":"Tucsnak","year":"2014","journal-title":"Automatica J. IFAC"},{"key":"2026062501163094200_ref22","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.jde.2025.113490","article-title":"Global exponential stabilization for a simplified fluid-particle interaction system","volume":"442","author":"Tucsnak","year":"2025","journal-title":"J. Differ. Equ."},{"key":"2026062501163094200_ref23","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1081\/PDE-120024530","article-title":"Large time behavior for a simplified 1D model of fluid\u2013solid interaction\u2020","volume":"28","author":"V\u00e1zquez","year":"2003","journal-title":"Commun. Partial Differ. Equ."},{"key":"2026062501163094200_ref24","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1142\/S0218202506001303","article-title":"Lack of collision in a simplified 1d model for fluid\u2013Solid interaction","volume":"16","author":"V\u00e1zquez","year":"2006","journal-title":"Math. Models Methods Appl. Sci."},{"key":"2026062501163094200_ref25","volume-title":"Nonlinear Functional Analysis and its Applications: II\/B: Nonlinear Monotone Operators","author":"Zeidler","year":"2013"}],"container-title":["IMA Journal of Mathematical Control and Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/43\/2\/dnag004\/67508963\/dnag004.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/43\/2\/dnag004\/67508963\/dnag004.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T05:16:40Z","timestamp":1782364600000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imamci\/article\/doi\/10.1093\/imamci\/dnag004\/8540367"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,24]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3,24]]}},"URL":"https:\/\/doi.org\/10.1093\/imamci\/dnag004","relation":{},"ISSN":["0265-0754","1471-6887"],"issn-type":[{"value":"0265-0754","type":"print"},{"value":"1471-6887","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,6]]},"published":{"date-parts":[[2026,3,24]]},"article-number":"dnag004"}}