{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:49:21Z","timestamp":1776847761145,"version":"3.51.2"},"reference-count":39,"publisher":"American Mathematical Society (AMS)","issue":"301","license":[{"start":{"date-parts":[[2017,2,11]],"date-time":"2017-02-11T00:00:00Z","timestamp":1486771200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100004616","name":"Consiliul National al Cercetarii Stiintifice","doi-asserted-by":"publisher","award":["Grant PN-II-ID-PCE-2011-3-0257"],"award-info":[{"award-number":["Grant PN-II-ID-PCE-2011-3-0257"]}],"id":[{"id":"10.13039\/501100004616","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004616","name":"Consiliul National al Cercetarii Stiintifice","doi-asserted-by":"publisher","award":["PN-II-ID-PCE-2011-3-0257"],"award-info":[{"award-number":["PN-II-ID-PCE-2011-3-0257"]}],"id":[{"id":"10.13039\/501100004616","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004616","name":"Consiliul National al Cercetarii Stiintifice","doi-asserted-by":"publisher","award":["Grant PN-II-ID-PCE-2011-3-0257"],"award-info":[{"award-number":["Grant PN-II-ID-PCE-2011-3-0257"]}],"id":[{"id":"10.13039\/501100004616","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004616","name":"Consiliul National al Cercetarii Stiintifice","doi-asserted-by":"publisher","award":["PN-II-ID-PCE-2011-3-0257"],"award-info":[{"award-number":["PN-II-ID-PCE-2011-3-0257"]}],"id":[{"id":"10.13039\/501100004616","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate boundary controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.<\/p>","DOI":"10.1090\/mcom\/3064","type":"journal-article","created":{"date-parts":[[2015,4,15]],"date-time":"2015-04-15T09:12:18Z","timestamp":1429089138000},"page":"2259-2303","source":"Crossref","is-referenced-by-count":27,"title":["Approximation of the controls for the beam equation with vanishing viscosity"],"prefix":"10.1090","volume":"85","author":[{"given":"Ioan","family":"Bugariu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sorin","family":"Micu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ionel","family":"Roven\u0163a","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,2,11]]},"reference":[{"key":"1","isbn-type":"print","volume-title":"Families of exponentials","author":"Avdonin, Sergei A.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521452430"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1051\/mmnp\/20149405","article-title":"A numerical method for the controls of the heat equation","volume":"9","author":"Bugariu, I. F.","year":"2014","journal-title":"Math. Model. Nat. Phenom.","ISSN":"https:\/\/id.crossref.org\/issn\/0973-5348","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1007\/BF02192213","article-title":"On exact and approximate boundary controllabilities for the heat equation: a numerical approach","volume":"82","author":"Carthel, C.","year":"1994","journal-title":"J. Optim. Theory Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-3239","issn-type":"print"},{"key":"4","series-title":"Mathematical Surveys and Monographs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/136","volume-title":"Control and nonlinearity","volume":"136","author":"Coron, Jean-Michel","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821836682"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/s00211-009-0235-5","article-title":"Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes","volume":"113","author":"Ervedoza, Sylvain","year":"2009","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1-2","key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3233\/asy-2010-1028","article-title":"Spectral conditions for admissibility and observability of Schr\u00f6dinger systems: applications to finite element discretizations","volume":"71","author":"Ervedoza, Sylvain","year":"2011","journal-title":"Asymptot. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0921-7134","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.matpur.2008.09.002","article-title":"Uniformly exponentially stable approximations for a class of damped systems","volume":"91","author":"Ervedoza, Sylvain","year":"2009","journal-title":"J. Math. Pures Appl. (9)","ISSN":"https:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"1375","DOI":"10.3934\/dcdsb.2010.14.1375","article-title":"A systematic method for building smooth controls for smooth data","volume":"14","author":"Ervedoza, Sylvain","year":"2010","journal-title":"Discrete Contin. Dyn. Syst. Ser. B","ISSN":"https:\/\/id.crossref.org\/issn\/1531-3492","issn-type":"print"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/978-3-642-27893-8_5","article-title":"The wave equation: control and numerics","author":"Ervedoza, Sylvain","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642278921"},{"key":"10","doi-asserted-by":"publisher","first-page":"272","DOI":"10.1007\/BF00250466","article-title":"Exact controllability theorems for linear parabolic equations in one space dimension","volume":"43","author":"Fattorini, H. O.","year":"1971","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1090\/qam\/510972","article-title":"Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations","volume":"32","author":"Fattorini, H. O.","year":"1974","journal-title":"Quart. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-569X","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/0021-9991(92)90396-G","article-title":"Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation","volume":"103","author":"Glowinski, R.","year":"1992","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF03167891","article-title":"A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods","volume":"7","author":"Glowinski, Roland","year":"1990","journal-title":"Japan J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0910-2043","issn-type":"print"},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1017\/s0962492900002543","article-title":"Exact and approximate controllability for distributed parameter systems","author":"Glowinski, R.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521482550"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/0022-247X(91)90252-U","article-title":"Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems","volume":"158","author":"Hansen, Scott W.","year":"1991","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"4","key":"16","first-page":"457","article-title":"S\u00e9ries lacunaires et contr\u00f4le semi-interne des vibrations d\u2019une plaque rectangulaire","volume":"68","author":"Haraux, A.","year":"1989","journal-title":"J. Math. Pures Appl. (9)","ISSN":"https:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"key":"17","isbn-type":"print","volume-title":"The finite element method","author":"Hughes, Thomas J. R.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/013317025X"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"1366","DOI":"10.1137\/070683787","article-title":"Numerical dispersive schemes for the nonlinear Schr\u00f6dinger equation","volume":"47","author":"Ignat, Liviu I.","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1112\/jlms\/s1-9.1.29","article-title":"A Note on Fourier Transforms","volume":"9","author":"Ingham, A. E.","year":"1934","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/BF01180426","article-title":"Some trigonometrical inequalities with applications to the theory of series","volume":"41","author":"Ingham, A. E.","year":"1936","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"21","unstructured":"E. Isaakson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, 1996."},{"issue":"4","key":"22","first-page":"423","article-title":"Contr\u00f4le interne exact des vibrations d\u2019une plaque rectangulaire","volume":"47","author":"Jaffard, S.","year":"1990","journal-title":"Portugal. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0032-5155","issn-type":"print"},{"key":"23","series-title":"Springer Monographs in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/b139040","volume-title":"Fourier series in control theory","author":"Komornik, Vilmos","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/0387223835"},{"key":"24","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1051\/cocv:2002025","article-title":"Boundary controllability of the finite-difference space semi-discretizations of the beam equation","volume":"8","author":"Le\u00f3n, Liliana","year":"2002","journal-title":"ESAIM Control Optim. Calc. Var.","ISSN":"https:\/\/id.crossref.org\/issn\/1292-8119","issn-type":"print"},{"key":"25","series-title":"Recherches en Math\\'{e}matiques Appliqu\\'{e}es [Research in Applied Mathematics]","isbn-type":"print","volume-title":"Contr\\^{o}labilit\\'{e} exacte, perturbations et stabilisation de syst\\`emes distribu\\'{e}s. Tome 2","volume":"9","author":"Lions, J.-L.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/2225814740"},{"issue":"4","key":"26","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1007\/s002110100338","article-title":"Uniform boundary controllability of a semi-discrete 1-D wave equation","volume":"91","author":"Micu, Sorin","year":"2002","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"6","key":"27","doi-asserted-by":"publisher","first-page":"2857","DOI":"10.1137\/070696933","article-title":"Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity","volume":"47","author":"Micu, Sorin","year":"2008","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4171\/JST\/20","article-title":"Resolvent conditions for the control of unitary groups and their approximations","volume":"2","author":"Miller, Luc","year":"2012","journal-title":"J. Spectr. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1664-039X","issn-type":"print"},{"issue":"2","key":"29","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1051\/cocv:2007009","article-title":"Uniform stabilization of a viscous numerical approximation for a locally damped wave equation","volume":"13","author":"M\u00fcnch, Arnaud","year":"2007","journal-title":"ESAIM Control Optim. Calc. Var.","ISSN":"https:\/\/id.crossref.org\/issn\/1292-8119","issn-type":"print"},{"key":"30","series-title":"American Mathematical Society Colloquium Publications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/coll\/019","volume-title":"Fourier transforms in the complex domain","volume":"19","author":"Paley, Raymond E. A. C.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0821810197"},{"issue":"3","key":"31","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1051\/cocv:2007020","article-title":"Uniformly exponentially stable approximations for a class of second order evolution equations\u2014application to LQR problems","volume":"13","author":"Ramdani, Karim","year":"2007","journal-title":"ESAIM Control Optim. Calc. Var.","ISSN":"https:\/\/id.crossref.org\/issn\/1292-8119","issn-type":"print"},{"issue":"1","key":"32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0001-8708(77)80002-9","article-title":"Completeness of sets of complex exponentials","volume":"24","author":"Redheffer, Raymond M.","year":"1977","journal-title":"Advances in Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"issue":"4","key":"33","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1137\/1020095","article-title":"Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions","volume":"20","author":"Russell, David L.","year":"1978","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"3","key":"34","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/BF02511154","article-title":"The \u201cwindow problem\u201d for series of complex exponentials","volume":"6","author":"Seidman, T. I.","year":"2000","journal-title":"J. Fourier Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1069-5869","issn-type":"print"},{"issue":"3","key":"35","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1007\/s00211-002-0442-9","article-title":"Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity","volume":"95","author":"Tcheugou\u00e9 T\u00e9bou, Louis Roder","year":"2003","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1-3","key":"36","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s10444-004-7629-9","article-title":"Uniform boundary stabilization of the finite difference space discretization of the 1-\ud835\udc51 wave equation","volume":"26","author":"Tebou, Louis T.","year":"2007","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"37","series-title":"Birkh\\\"{a}user Advanced Texts: Basler Lehrb\\\"{u}cher. [Birkh\\\"{a}user Advanced Texts: Basel Textbooks]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7643-8994-9","volume-title":"Observation and control for operator semigroups","author":"Tucsnak, Marius","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9783764389932"},{"key":"38","series-title":"Pure and Applied Mathematics","isbn-type":"print","volume-title":"An introduction to nonharmonic Fourier series","volume":"93","author":"Young, Robert M.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0127728503"},{"issue":"2","key":"39","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1137\/S0036144503432862","article-title":"Propagation, observation, and control of waves approximated by finite difference methods","volume":"47","author":"Zuazua, Enrique","year":"2005","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-301\/S0025-5718-2016-03064-1\/S0025-5718-2016-03064-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-301\/S0025-5718-2016-03064-1\/S0025-5718-2016-03064-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:53:20Z","timestamp":1776797600000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-301\/S0025-5718-2016-03064-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,11]]},"references-count":39,"journal-issue":{"issue":"301","published-print":{"date-parts":[[2016,9]]}},"alternative-id":["S0025-5718-2016-03064-1"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3064","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2016,2,11]]}}}