{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:27:42Z","timestamp":1787322462286,"version":"build-2736575974"},"reference-count":43,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as the state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We discretize the proposed truncated state equation using first-degree tensor product finite elements on anisotropic meshes. For the control problem we analyze two approaches: one that is semidiscrete based on the so-called variational approach, where the control is not discretized, and the other one that is fully discrete via the discretization of the control by piecewise constant functions. For both approaches, we derive a priori error estimates with respect to degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement.<\/jats:p>","DOI":"10.1137\/140975061","type":"journal-article","created":{"date-parts":[[2015,12,2]],"date-time":"2015-12-02T14:16:20Z","timestamp":1449065780000},"page":"3432-3456","source":"Crossref","is-referenced-by-count":64,"title":["A FEM for an Optimal Control Problem of Fractional Powers of Elliptic Operators"],"prefix":"10.1137","volume":"53","author":[{"given":"Harbir","family":"Antil","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enrique","family":"Ot\u00e1rola","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,12,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"M. Abramowitz and I.A. Stegun,\n                      Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables\n                      , Nat. Bureau Standards Appl. Math. 55, U.S. Government Printing Office, Washington, D.C., 1964.","DOI":"10.1115\/1.3625776"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/140958360"},{"key":"atypb3","unstructured":"H. Antil, E. Ota\u0301rola, and A. J Salgado,\n                      A Space-Time Fractional Optimal Control Problem: Analysis and Discretization\n                      , preprint, \\burlalthttp:\/\/arxiv.org\/abs\/1504.00063arXiv:1504.00063, 2015."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-011-9427-x"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/080731724"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-007-0133-0"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1023\/A:1020576801966"},{"key":"atypb8","unstructured":"T.M. Atanackovic, S. Pilipovic, B. Stankovic, and D. Zorica,\n                      Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes\n                      , John Wiley & Sons, Hoboken, NJ, 2014."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/0726068"},{"key":"atypb10","unstructured":"M Bonforte, Y Sire, and J.L. Va\u0301zquez,\n                      Existence, Uniqueness and Asymptotic Behaviour for Fractional Porous Medium Equations on Bounded Domains\n                      , preprint, http:\/\/arxiv.org\/abs\/1404.6195arXiv:1404.6195, 2014."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1098\/rsif.2014.0352"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2010.01.025"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1080\/03605300600987306"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1080\/03605302.2011.562954"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-005-2180-2"},{"key":"atypb16","unstructured":"L Chen,\n                      iFEM: An Integrated Finite Element Method Package in MATLAB\n                      , Technical report, University of California at Irvine, Irvine, CA, 2009."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.01.001"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1063\/1.2208452"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"P.G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , Classics Appl. Math. 40, SIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898719208"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-05-01732-1"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"A. Ern and J.L. Guermond,\n                      Theory and Practice of Finite Elements\n                      , Appl. Math. Sci. 159, Springer-Verlag, New York, 2004.","DOI":"10.1007\/978-1-4757-4355-5"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-014-9959-1"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-09-04615-7"},{"key":"atypb25","unstructured":"P. Grisvard,\n                      Elliptic Problems in Nonsmooth Domains\n                      , Monogr. Stud. 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