{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T18:09:50Z","timestamp":1774375790387,"version":"3.50.1"},"reference-count":37,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1418784"],"award-info":[{"award-number":["DMS-1418784"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009068","name":"Consejo Nacional de Innovaci\u00f3n, Ciencia y Tecnolog\u00eda","doi-asserted-by":"publisher","award":["3160201"],"award-info":[{"award-number":["3160201"]}],"id":[{"id":"10.13039\/501100009068","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity results together with first-order optimality conditions. In order to propose a solution technique, we realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an equivalent optimal control problem with a nonuniformly elliptic equation as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme: piecewise constant functions for the control variable and first-degree tensor product finite elements for the state variable. We derive a priori error estimates for the control and state variables.<\/jats:p>","DOI":"10.1515\/cmam-2017-0030","type":"journal-article","created":{"date-parts":[[2017,9,2]],"date-time":"2017-09-02T10:02:50Z","timestamp":1504346570000},"page":"95-110","source":"Crossref","is-referenced-by-count":15,"title":["Sparse Optimal Control for Fractional Diffusion"],"prefix":"10.1515","volume":"18","author":[{"given":"Enrique","family":"Ot\u00e1rola","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica , Universidad T\u00e9cnica Federico Santa Mar\u00eda , Valpara\u00edso , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abner J.","family":"Salgado","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Tennessee , Knoxville , TN 37996 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,9,2]]},"reference":[{"key":"2023033115122833204_j_cmam-2017-0030_ref_001_w2aab3b7e1865b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"H.  Antil and E.  Ot\u00e1rola,\nA FEM for an optimal control problem of fractional powers of elliptic operators,\nSIAM J. Control Optim. 53 (2015), no. 6, 3432\u20133456.","DOI":"10.1137\/140975061"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_002_w2aab3b7e1865b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"H.  Antil, E.  Ot\u00e1rola and A. J.  Salgado,\nA space-time fractional optimal control problem: Analysis and discretization,\nSIAM J. Control Optim. 54 (2016), no. 3, 1295\u20131328.","DOI":"10.1137\/15M1014991"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_003_w2aab3b7e1865b1b6b1ab2ab3Aa","unstructured":"T.  Atanackovic, S.  Pilipovic, B.  Stankovic and D.  Zorica,\nFractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes,\nJohn Wiley & Sons, Hoboken, 2014."},{"key":"2023033115122833204_j_cmam-2017-0030_ref_004_w2aab3b7e1865b1b6b1ab2ab4Aa","unstructured":"L.  Banjai, J.  Melenk, R. H.  Nochetto, E.  Ot\u00e1rola, A. J.  Salgado and C.  Schwab,\nTensor FEM for spectral fractional diffusion,\npreprint (2017), https:\/\/arxiv.org\/abs\/1707.07367."},{"key":"2023033115122833204_j_cmam-2017-0030_ref_005_w2aab3b7e1865b1b6b1ab2ab5Aa","unstructured":"A.  Bonito, J. P.  Borthagaray, R. H.  Nochetto, E.  Ot\u00e1rola and A. J.  Salgado,\nNumerical methods for fractional diffusion,\npreprint (2017), https:\/\/arxiv.org\/abs\/1707.01566."},{"key":"2023033115122833204_j_cmam-2017-0030_ref_006_w2aab3b7e1865b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"A.  Bueno-Orovio, D.  Kay, V.  Grau, B.  Rodriguez and K.  Burrage,\nFractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization,\nJ. R. Soc. Interface 11 (2014), no. 97.","DOI":"10.1098\/rsif.2014.0352"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_007_w2aab3b7e1865b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"X.  Cabr\u00e9 and J.  Tan,\nPositive solutions of nonlinear problems involving the square root of the Laplacian,\nAdv. Math. 224 (2010), no. 5, 2052\u20132093.","DOI":"10.1016\/j.aim.2010.01.025"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_008_w2aab3b7e1865b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"L.  Caffarelli and L.  Silvestre,\nAn extension problem related to the fractional Laplacian,\nComm. Partial Differential Equations 32 (2007), no. 7\u20139, 1245\u20131260.","DOI":"10.1080\/03605300600987306"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_009_w2aab3b7e1865b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"A.  Capella, J.  D\u00e1vila, L.  Dupaigne and Y.  Sire,\nRegularity of radial extremal solutions for some non-local semilinear equations,\nComm. Partial Differential Equations 36 (2011), no. 8, 1353\u20131384.","DOI":"10.1080\/03605302.2011.562954"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_010_w2aab3b7e1865b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"E.  Casas, R.  Herzog and G.  Wachsmuth,\nApproximation of sparse controls in semilinear equations by piecewise linear functions,\nNumer. Math. 122 (2012), no. 4, 645\u2013669.","DOI":"10.1007\/s00211-012-0475-7"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_011_w2aab3b7e1865b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"E.  Casas, R.  Herzog and G.  Wachsmuth,\nOptimality conditions and error analysis of semilinear elliptic control problems with L1L^{1} cost functional,\nSIAM J. Optim. 22 (2012), no. 3, 795\u2013820.","DOI":"10.1137\/110834366"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_012_w2aab3b7e1865b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"W.  Chen,\nA speculative study of 2\/32\/3-order fractional laplacian modeling of turbulence: Some thoughts and conjectures,\nChaos 16 (2006), no. 2, 1\u201311.","DOI":"10.1063\/1.2208452"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_013_w2aab3b7e1865b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"P.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nSIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898719208"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_014_w2aab3b7e1865b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"F. H.  Clarke,\nOptimization and Nonsmooth Analysis, 2nd ed.,\nClass. Appl. Math. 5,\nSIAM, Philadelphia, 1990.","DOI":"10.1137\/1.9781611971309"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_015_w2aab3b7e1865b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"R.  Dur\u00e1n and A.  Lombardi,\nError estimates on anisotropic Q1Q_{1} elements for functions in weighted Sobolev spaces,\nMath. Comp. 74 (2005), no. 252, 1679\u20131706.","DOI":"10.1090\/S0025-5718-05-01732-1"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_016_w2aab3b7e1865b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nTheory and Practice of Finite Elements,\nAppl. Math. Sci. 159,\nSpringer, New York, 2004.","DOI":"10.1007\/978-1-4757-4355-5"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_017_w2aab3b7e1865b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"D.  Fujiwara,\nConcrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,\nProc. Japan Acad. 43 (1967), 82\u201386.","DOI":"10.3792\/pja\/1195521686"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_018_w2aab3b7e1865b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"P.  Gatto and J.  Hesthaven,\nNumerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising,\nJ. Sci. Comput. 65 (2015), no. 1, 249\u2013270.","DOI":"10.1007\/s10915-014-9959-1"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_019_w2aab3b7e1865b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"V.  Gol\u2019dshtein and A.  Ukhlov,\nWeighted Sobolev spaces and embedding theorems,\nTrans. Amer. Math. Soc. 361 (2009), no. 7, 3829\u20133850.","DOI":"10.1090\/S0002-9947-09-04615-7"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_020_w2aab3b7e1865b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"R.  Ishizuka, S.-H.  Chong and F.  Hirata,\nAn integral equation theory for inhomogeneous molecular fluids: The reference interaction site model approach,\nJ. Chem. Phys. 128 (2008), no. 3.","DOI":"10.1063\/1.2819487"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_021_w2aab3b7e1865b1b6b1ab2ac21Aa","unstructured":"D.  Kinderlehrer and G.  Stampacchia,\nAn Introduction to Variational Inequalities and Their Applications,\nPure Appl. Math. 88,\nAcademic Press, New York, 1980."},{"key":"2023033115122833204_j_cmam-2017-0030_ref_022_w2aab3b7e1865b1b6b1ab2ac22Aa","doi-asserted-by":"crossref","unstructured":"N.  Landkof,\nFoundations of Modern Potential Theory,\nGrundlehren Math. Wiss. 180,\nSpringer, Berlin, 1972.","DOI":"10.1007\/978-3-642-65183-0"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_023_w2aab3b7e1865b1b6b1ab2ac23Aa","doi-asserted-by":"crossref","unstructured":"S.  Levendorski\u012d,\nPricing of the American put under L\u00e9vy processes,\nInt. J. Theor. Appl. Finance 7 (2004), no. 3, 303\u2013335.","DOI":"10.1142\/S0219024904002463"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_024_w2aab3b7e1865b1b6b1ab2ac24Aa","doi-asserted-by":"crossref","unstructured":"J.-L.  Lions and E.  Magenes,\nNon-homogeneous Boundary Value Problems and Applications. Vol. I,\nSpringer, New York, 1972.","DOI":"10.1007\/978-3-642-65217-2"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_025_w2aab3b7e1865b1b6b1ab2ac25Aa","doi-asserted-by":"crossref","unstructured":"B.  Muckenhoupt,\nWeighted norm inequalities for the Hardy maximal function,\nTrans. Amer. Math. Soc. 165 (1972), 207\u2013226.","DOI":"10.1090\/S0002-9947-1972-0293384-6"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_026_w2aab3b7e1865b1b6b1ab2ac26Aa","doi-asserted-by":"crossref","unstructured":"R.  Musina and A. I.  Nazarov,\nOn fractional Laplacians,\nComm. Partial Differential Equations 39 (2014), no. 9, 1780\u20131790.","DOI":"10.1080\/03605302.2013.864304"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_027_w2aab3b7e1865b1b6b1ab2ac27Aa","doi-asserted-by":"crossref","unstructured":"R. H.  Nochetto, E.  Ot\u00e1rola and A. J.  Salgado,\nA PDE approach to fractional diffusion in general domains: A priori error analysis,\nFound. Comput. Math. 15 (2015), no. 3, 733\u2013791.","DOI":"10.1007\/s10208-014-9208-x"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_028_w2aab3b7e1865b1b6b1ab2ac28Aa","doi-asserted-by":"crossref","unstructured":"R. H.  Nochetto, E.  Ot\u00e1rola and A. J.  Salgado,\nA PDE approach to space-time fractional parabolic problems,\nSIAM J. Numer. Anal. 54 (2016), no. 2, 848\u2013873.","DOI":"10.1137\/14096308X"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_029_w2aab3b7e1865b1b6b1ab2ac29Aa","doi-asserted-by":"crossref","unstructured":"E.  Ot\u00e1rola,\nA piecewise linear FEM for an optimal control problem of fractional operators: Error analysis on curved domains,\nESAIM Math. Model. Numer. Anal. 51 (2017), no. 4, 1473\u20131500.","DOI":"10.1051\/m2an\/2016065"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_030_w2aab3b7e1865b1b6b1ab2ac30Aa","doi-asserted-by":"crossref","unstructured":"W.  Schirotzek,\nNonsmooth Analysis,\nUniversitext,\nSpringer, Berlin, 2007.","DOI":"10.1007\/978-3-540-71333-3"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_031_w2aab3b7e1865b1b6b1ab2ac31Aa","doi-asserted-by":"crossref","unstructured":"G.  Stadler,\nElliptic optimal control problems with L1L^{1}-control cost and applications for the placement of control devices,\nComput. Optim. Appl. 44 (2009), no. 2, 159\u2013181.","DOI":"10.1007\/s10589-007-9150-9"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_032_w2aab3b7e1865b1b6b1ab2ac32Aa","doi-asserted-by":"crossref","unstructured":"P. R.  Stinga and J. L.  Torrea,\nExtension problem and Harnack\u2019s inequality for some fractional operators,\nComm. Partial Differential Equations 35 (2010), no. 11, 2092\u20132122.","DOI":"10.1080\/03605301003735680"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_033_w2aab3b7e1865b1b6b1ab2ac33Aa","unstructured":"L.  Tartar,\nAn Introduction to Sobolev Spaces and Interpolation Spaces,\nLect. Notes Unione Mat. Ital. 3,\nSpringer, Berlin, 2007."},{"key":"2023033115122833204_j_cmam-2017-0030_ref_034_w2aab3b7e1865b1b6b1ab2ac34Aa","doi-asserted-by":"crossref","unstructured":"F.  Tr\u00f6ltzsch,\nOptimal Control of Partial Differential Equations,\nGrad. Stud. Math. 112,\nAmerican Mathematical Society, Providence, 2010.","DOI":"10.1090\/gsm\/112\/07"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_035_w2aab3b7e1865b1b6b1ab2ac35Aa","doi-asserted-by":"crossref","unstructured":"B. O.  Turesson,\nNonlinear Potential Theory and Weighted Sobolev Spaces,\nSpringer, Berlin, 2000.","DOI":"10.1007\/BFb0103908"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_036_w2aab3b7e1865b1b6b1ab2ac36Aa","doi-asserted-by":"crossref","unstructured":"G.  Vossen and H.  Maurer,\nOn L1L^{1}-minimization in optimal control and applications to robotics,\nOptimal Control Appl. Methods 27 (2006), no. 6, 301\u2013321.","DOI":"10.1002\/oca.781"},{"key":"2023033115122833204_j_cmam-2017-0030_ref_037_w2aab3b7e1865b1b6b1ab2ac37Aa","doi-asserted-by":"crossref","unstructured":"G.  Wachsmuth and D.  Wachsmuth,\nConvergence and regularization results for optimal control problems with sparsity functional,\nESAIM Control Optim. Calc. Var. 17 (2011), no. 3, 858\u2013886.","DOI":"10.1051\/cocv\/2010027"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/1\/article-p95.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0030\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0030\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:48:02Z","timestamp":1680299282000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0030\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,2]]},"references-count":37,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,6,17]]},"published-print":{"date-parts":[[2018,1,1]]}},"alternative-id":["10.1515\/cmam-2017-0030"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0030","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,9,2]]}}}