{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:41:02Z","timestamp":1787323262130,"version":"build-2736575974"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"8","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>In arbitrary dimension, we consider the semidiscrete elliptic operator $-\\partial_t^2+\\mathcal{A}^{\\scriptscriptstyle\\mathfrak{M}}$, where $\\mathcal{A}^{\\scriptscriptstyle\\mathfrak{M}}$ is a finite-difference approximation of the operator $-\\nabla_x(\\Gamma(x)\\nabla_x)$. For this operator we derive a global Carleman estimate, in which the usual large parameter is connected to the discretization step-size. We address discretizations on some families of smoothly varying meshes. We present consequences of this estimate, such as a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano for $\\mathcal{A}^{\\scriptscriptstyle\\mathfrak{M}}$ and a null-controllability result for the parabolic operator $\\partial_t+\\mathcal{A}^{\\scriptscriptstyle\\mathfrak{M}}$ for the lower part of the spectrum of $\\mathcal{A}^{\\scriptscriptstyle\\mathfrak{M}}$. With the control function that we construct (whose norm is uniformly bounded) we prove that the $L^2$-norm of the final state converges to zero exponentially, as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced.<\/jats:p>","DOI":"10.1137\/100784278","type":"journal-article","created":{"date-parts":[[2010,11,9]],"date-time":"2010-11-09T18:51:20Z","timestamp":1289328680000},"page":"5357-5397","source":"Crossref","is-referenced-by-count":39,"title":["Discrete Carleman Estimates for Elliptic Operators in Arbitrary Dimension and Applications"],"prefix":"10.1137","volume":"48","author":[{"given":"Franck","family":"Boyer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Florence","family":"Hubert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J\u00e9r\u00f4me Le","family":"Rousseau","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,11,9]]},"reference":[{"key":"R1","unstructured":"J.P. Aubin and I. Ekeland,\n                      Applied Nonlinear Analysis\n                      , John Wiley & Sons, New York, 1984."},{"key":"R2","unstructured":"F. Boyer, F. Hubert, and J. Le Rousseau,\n                      Uniform Null-Controllability Properties for Space\/Time-Discretized Parabolic Equations\n                      , preprint, 2009; available online from http:\/\/hal.archives-ouvertes.fr\/hal-00429197\/fr\/."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"F. Boyer, F. Hubert, and J. Le Rousseau,\n                      Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations\n                      , in preparation, 2010.","DOI":"10.1016\/j.matpur.2009.11.003"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2009.11.003"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012904439696"},{"key":"R6","unstructured":"A. Fursikov and O. Yu. Imanuvilov,\n                      Controllability of Evolution Equations\n                      , Lecture Notes 34, Seoul National University, Korea, 1996."},{"key":"R7","doi-asserted-by":"crossref","unstructured":"L. H\u00f6rmander,\n                      Linear Partial Differential Operators\n                      , Springer-Verlag, Berlin, 1963.","DOI":"10.1007\/978-3-642-46175-0"},{"key":"R8","unstructured":"L. H\u00f6rmander,\n                      The Analysis of Linear Partial Differential Operators\n                      , Vol. IV, Springer-Verlag, Berlin, 1985."},{"key":"R9","unstructured":"D. Jerison and G. Lebeau,\n                      Nodal sets of sums of eigenfunctions\n                      , in Harmonic Analysis and Partial Differential Equations (Chicago, IL, 1996), Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, 1999, pp. 223\u2013239."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0151085"},{"key":"R11","unstructured":"J. Le Rousseau,\n                      Repr\u00e9sentation Microlocale de Solutions de Syst\u00e8mes Hyperboliques, Application \u00e0 l'Imagerie, et Contributions au Contr\u00f4le et aux Probl\u00e8mes Inverses pour des \u00e9quations Paraboliques\n                      , M\u00e9moire d'habilitation \u00e0 diriger des recherches, Universit\u00e9s d'Aix-Marseille, Universit\u00e9 de Provence, 2007; available online from http:\/\/tel.archives-ouvertes.fr\/tel-00201887\/fr\/."},{"key":"R12","unstructured":"J. Le Rousseau and G. Lebeau,\n                      On Carleman Estimates for Elliptic and Parabolic Operators. Applications to Unique Continuation and Control of Parabolic Equations\n                      , preprint, 2009."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1080\/03605309508821097"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-97-08614-2"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2006.01.004"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/s002050050078"},{"key":"R17","unstructured":"A. Lopez and E. Zuazua,\n                      Some new results to the null controllability of the\n                      1\n                      -d heat equation\n                      , in S\u00e9minaire sur les \u00c9quations aux D\u00e9riv\u00e9es Partielles, 1997\u20131998, Exp. VIII, \u00c9cole Polytech., Palaiseau, 1998."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"E. Zuazua,\n                      Control and numerical approximation of the wave and heat equations\n                      , in International Congress of Mathematicians (Madrid, Spain), Vol. III, 2006, pp. 1389\u20131417.","DOI":"10.4171\/022-3\/67"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"C. Zuily,\n                      Uniqueness and Nonuniqueness in the Cauchy Problem\n                      , Progr. Math. 33, Birkh\u00e4user, Boston, 1983.","DOI":"10.1007\/978-1-4899-6656-8"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100784278","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:51:40Z","timestamp":1787320300000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100784278"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":19,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/100784278"],"URL":"https:\/\/doi.org\/10.1137\/100784278","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}