{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T19:42:31Z","timestamp":1775072551990,"version":"3.50.1"},"reference-count":34,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T00:00:00Z","timestamp":1701388800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100004410","name":"Scientific and Technological Research Council of Turkey","doi-asserted-by":"publisher","award":["T\u00dcB\u0130TAK 1001"],"award-info":[{"award-number":["T\u00dcB\u0130TAK 1001"]}],"id":[{"id":"10.13039\/501100004410","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004410","name":"Scientific and Technological Research Council of Turkey","doi-asserted-by":"publisher","award":["# 117F449"],"award-info":[{"award-number":["# 117F449"]}],"id":[{"id":"10.13039\/501100004410","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Research Committee of the Academy of Athens","award":["# 200\/984"],"award-info":[{"award-number":["# 200\/984"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,12,24]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We prove that the solutions of the Schr\u00f6dinger and biharmonic Schr\u00f6dinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of $L^2$ boundary controls with data spaces $H^{-1}(\\mathbb{R}_+)$ and $H^{-2}(\\mathbb{R}_+)$ for the classical and biharmonic Schr\u00f6dinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces.<\/jats:p>","DOI":"10.1093\/imamci\/dnad032","type":"journal-article","created":{"date-parts":[[2023,12,2]],"date-time":"2023-12-02T00:23:37Z","timestamp":1701476617000},"page":"789-803","source":"Crossref","is-referenced-by-count":8,"title":["Existence of unattainable states for Schr\u00f6dinger type flows on the half-line"],"prefix":"10.1093","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4240-5252","authenticated-orcid":false,"given":"T\u00fcrker","family":"\u00d6zsar\u0131","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bilkent University , \u00c7ankaya, Ankara 06800 , Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7337-6231","authenticated-orcid":false,"given":"Konstantinos","family":"Kalimeris","sequence":"additional","affiliation":[{"name":"Mathematics Research Center, Academy of Athens , Soranou Efesiou 4, Athens 115 27 , Greece"}]}],"member":"286","published-online":{"date-parts":[[2023,12,1]]},"reference":[{"key":"2023122615292165900_ref1","first-page":"15","article-title":"Fokas method for linear boundary value problems involving mixed spatial derivatives","volume":"476","author":"Batal","year":"2020","journal-title":"Proc. A."},{"key":"2023122615292165900_ref2","first-page":"20","article-title":"Nonlinear Schr\u00f6dinger equations on the half-line with nonlinear boundary conditions","volume":"No. 222","author":"Batal","year":"2016","journal-title":"Electron. J. Diff. Equ."},{"key":"2023122615292165900_ref3","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1016\/j.matpur.2005.02.005","article-title":"Local controllability of a 1-D Schr\u00f6dinger equation","volume":"84","author":"Beauchard","year":"2005","journal-title":"J. Math. Pures Appl."},{"key":"2023122615292165900_ref4","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1016\/j.jfa.2005.03.021","article-title":"Controllability of a quantum particle in a moving potential well","volume":"232","author":"Beauchard","year":"2006","journal-title":"J. Funct. Anal."},{"key":"2023122615292165900_ref5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.matpur.2017.11.001","article-title":"Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schr\u00f6dinger equations","volume":"9","author":"Bona","year":"2018","journal-title":"J. Math. Pures Appl."},{"key":"2023122615292165900_ref6","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s00245-019-09640-8","article-title":"Stabilization and control for the biharmonic Schr\u00f6dinger equation","volume":"84","author":"Capistrano-Filho","year":"2021","journal-title":"Appl. Math. Optim."},{"key":"2023122615292165900_ref7","doi-asserted-by":"crossref","DOI":"10.1142\/q0348","volume-title":"Modern Mathematical Methods for Scientists and Engineers: a street-smart introduction","author":"Fokas","year":"2023"},{"key":"2023122615292165900_ref8","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1098\/rspa.1997.0077","article-title":"A unified transform method for solving linear and certain nonlinear PDEs","volume":"453","author":"Fokas","year":"1997","journal-title":"Proc. Roy. Soc. London Ser. A"},{"key":"2023122615292165900_ref9","volume-title":"A unified approach to boundary value problems, volume 78 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM)","author":"Fokas","year":"2008"},{"key":"2023122615292165900_ref10","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1090\/tran\/6734","article-title":"The nonlinear Schr\u00f6dinger equation on the half-line","volume":"369","author":"Fokas","year":"2017","journal-title":"Trans. Am. Math. Soc."},{"key":"2023122615292165900_ref11","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1051\/cocv:2006014","article-title":"Limitations on the control of Schr\u00f6dinger equations","volume":"12","author":"Illner","year":"2006","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"2023122615292165900_ref12","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.aml.2020.106241","article-title":"An elementary proof of the lack of null controllability for the heat equation on the half line","volume":"104","author":"Kalimeris","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"2023122615292165900_ref13","first-page":"1","article-title":"Numerical computation of Neumann controls for the heat equation on a finite interval","author":"Kalimeris","year":"2023","journal-title":"IEEE Trans. Autom. Control, Early Access"},{"key":"2023122615292165900_ref14","doi-asserted-by":"crossref","first-page":"461","DOI":"10.3934\/eect.2020075","article-title":"A remark on the attainable set of the Schr\u00f6dinger equation","volume":"10","author":"Lampart","year":"2021","journal-title":"Evol. Equ. Control Theory"},{"key":"2023122615292165900_ref15","first-page":"521","article-title":"Optimal regularity, exact controllability and uniform stabilization of Schr\u00f6dinger equations with Dirichlet control","volume":"5","author":"Lasiecka","year":"1992","journal-title":"Differ. Integral Equ."},{"key":"2023122615292165900_ref16","volume-title":"Control theory for partial differential equations: continuous and approximation theories. II, vol. 75 of Encyclopedia of Mathematics and its Applications","author":"Lasiecka","year":"2000"},{"key":"2023122615292165900_ref17","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1051\/cocv\/2009001","article-title":"Global controllability and stabilization for the nonlinear Schr\u00f6dinger equation on an interval","volume":"16","author":"Laurent","year":"2010","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"2023122615292165900_ref18","first-page":"267","article-title":"Contr\u00f4le de l\u2019\u00e9quation de Schr\u00f6dinger","volume":"71","author":"Lebeau","year":"1992","journal-title":"J. Math. Pures Appl."},{"key":"2023122615292165900_ref19","volume-title":"Probl\u00e8mes aux limites non homog\u00e8nes et applications","author":"Lions","year":"1968"},{"key":"2023122615292165900_ref20","volume-title":"Non-homogeneous boundary value problems and applications. Vol. I, volume Band 181 of Die Grundlehren der mathematischen Wissenschaften","author":"Lions","year":"1972"},{"key":"2023122615292165900_ref21","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1137\/S0363012991223145","article-title":"Exact controllability for the Schr\u00f6dinger equation","volume":"32","author":"Machtyngier","year":"1994","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292165900_ref22","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1090\/S0002-9947-00-02665-9","article-title":"On the lack of null-controllability of the heat equation on the half-line","volume":"353","author":"Micu","year":"2001","journal-title":"Trans. Am. Math. Soc."},{"key":"2023122615292165900_ref23","doi-asserted-by":"crossref","first-page":"3285","DOI":"10.3934\/cpaa.2019148","article-title":"The initial-boundary value problem for the biharmonic Schr\u00f6dinger equation on the half-line","volume":"18","author":"\u00d6zsar\u0131","year":"2019","journal-title":"Commun. Pure Appl. Anal."},{"key":"2023122615292165900_ref24","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1137\/S0363012900368405","article-title":"Observability and control of Schr\u00f6dinger equations","volume":"40","author":"Phung","year":"2001","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292165900_ref25","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1137\/S0363012999353229","article-title":"Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line","volume":"39","author":"Rosier","year":"2000","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292165900_ref26","doi-asserted-by":"crossref","first-page":"4129","DOI":"10.1016\/j.jde.2008.11.004","article-title":"Exact boundary controllability of the nonlinear Schr\u00f6dinger equation","volume":"246","author":"Rosier","year":"2009","journal-title":"J. Differ. Equ."},{"key":"2023122615292165900_ref27","doi-asserted-by":"crossref","first-page":"972","DOI":"10.1137\/070709578","article-title":"Local exact controllability and stabilizability of the nonlinear Schr\u00f6dinger equation on a bounded interval","volume":"48","author":"Rosier","year":"2009","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292165900_ref28","first-page":"367","article-title":"Carleman estimates and unique continuation for solutions to boundary value problems","volume":"75","author":"Tataru","year":"1996","journal-title":"J. Math. Pures Appl."},{"key":"2023122615292165900_ref29","first-page":"109","article-title":"The role of an L$_2\\,(\\Omega)$ energy estimate in the theories of uniform stabilization and exact controllability for Schr\u00f6dinger equations with Neumann boundary control","author":"Triggiani","year":"2007"},{"key":"2023122615292165900_ref30","first-page":"613","article-title":"Exact controllability in L$_2\\,(\\Omega)$ of the Schr\u00f6dinger equation in a Riemannian manifold with L$_2(\\varSigma _1$) Neumann boundary control","volume-title":"Functional analysis and evolution equations","author":"Triggiani","year":"2008"},{"key":"2023122615292165900_ref31","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/978-3-642-57237-1_4","article-title":"On the controllability of bilinear quantum systems","volume-title":"Mathematical models and methods for ab initio quantum chemistry, volume 74 of Lecture Notes in Chem","author":"Turinici","year":"2000"},{"key":"2023122615292165900_ref32","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1137\/120902744","article-title":"Well-posedness and exact controllability of fourth order Schr\u00f6dinger equation with boundary control and collocated observation","volume":"52","author":"Wen","year":"2014","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292165900_ref33","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-44778-6","volume-title":"Mathematical control theory\u2014an introduction. Systems & Control: Foundations & Applications","author":"Zabczyk","year":"2020"},{"key":"2023122615292165900_ref34","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1090\/crmp\/033\/12","article-title":"Remarks on the controllability of the Schr\u00f6dinger equation","volume-title":"Quantum control: mathematical and numerical challenges, volume 33 of CRM Proc. Lecture Notes","author":"Zuazua","year":"2003"}],"container-title":["IMA Journal of Mathematical Control and Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/40\/4\/789\/54804595\/dnad032.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/40\/4\/789\/54804595\/dnad032.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,26]],"date-time":"2023-12-26T15:29:50Z","timestamp":1703604590000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imamci\/article\/40\/4\/789\/7457345"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,1]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,12,1]]},"published-print":{"date-parts":[[2023,12,24]]}},"URL":"https:\/\/doi.org\/10.1093\/imamci\/dnad032","relation":{},"ISSN":["0265-0754","1471-6887"],"issn-type":[{"value":"0265-0754","type":"print"},{"value":"1471-6887","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,12]]},"published":{"date-parts":[[2023,12,1]]}}}