{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T22:01:11Z","timestamp":1768514471564,"version":"3.49.0"},"reference-count":30,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2020-114837GB-I00"],"award-info":[{"award-number":["PID2020-114837GB-I00"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We study the numerical approximation of a control problem governed by a semilinear parabolic problem, where the usual Tikhonov regularization term in the cost functional is replaced by a non-differentiable sparsity-promoting term.<\/jats:p>","DOI":"10.1515\/cmam-2022-0130","type":"journal-article","created":{"date-parts":[[2022,11,7]],"date-time":"2022-11-07T21:21:42Z","timestamp":1667856102000},"page":"877-898","source":"Crossref","is-referenced-by-count":1,"title":["Error Estimates for the Numerical Approximation of Unregularized Sparse Parabolic Control Problems"],"prefix":"10.1515","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8364-9416","authenticated-orcid":false,"given":"Eduardo","family":"Casas","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada y Ciencias de la Computaci\u00f3n , Universidad de Cantabria , Santander , Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3100-412X","authenticated-orcid":false,"given":"Mariano","family":"Mateos","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas , Universidad de Oviedo , Gij\u00f3n , Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,11,8]]},"reference":[{"key":"2023100411150512037_j_cmam-2022-0130_ref_001","doi-asserted-by":"crossref","unstructured":"J. Bergh and J. L\u00f6fstr\u00f6m,\nInterpolation Spaces. An Introduction,\nGrundlehren Math. Wiss. 223,\nSpringer, Berlin, 1976.","DOI":"10.1007\/978-3-642-66451-9"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_002","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_003","doi-asserted-by":"crossref","unstructured":"E. Casas,\nPontryagin\u2019s principle for state-constrained boundary control problems of semilinear parabolic equations,\nSIAM J. Control Optim. 35 (1997), no. 4, 1297\u20131327.","DOI":"10.1137\/S0363012995283637"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_004","doi-asserted-by":"crossref","unstructured":"E. Casas,\nSecond order analysis for bang-bang control problems of PDEs,\nSIAM J. Control Optim. 50 (2012), no. 4, 2355\u20132372.","DOI":"10.1137\/120862892"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_005","doi-asserted-by":"crossref","unstructured":"E. Casas, R. Herzog and G. Wachsmuth,\nOptimality conditions and error analysis of semilinear elliptic control problems with \n                  \n                     \n                        \n                           L\n                           1\n                        \n                     \n                     \n                     L^{1}\n                  \n                cost functional,\nSIAM J. Optim. 22 (2012), no. 3, 795\u2013820.","DOI":"10.1137\/110834366"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_006","doi-asserted-by":"crossref","unstructured":"E. Casas, R. Herzog and G. Wachsmuth,\nAnalysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations,\nESAIM Control Optim. Calc. Var. 23 (2017), no. 1, 263\u2013295.","DOI":"10.1051\/cocv\/2015048"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_007","doi-asserted-by":"crossref","unstructured":"E. Casas and K. Kunisch,\nParabolic control problems in space-time measure spaces,\nESAIM Control Optim. Calc. Var. 22 (2016), no. 2, 355\u2013370.","DOI":"10.1051\/cocv\/2015008"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_008","doi-asserted-by":"crossref","unstructured":"E. Casas and K. Kunisch,\nOptimal control of semilinear parabolic equations with non-smooth pointwise-integral control constraints in time-space,\nAppl. Math. Optim. 85 (2022), no. 1, Paper No. 12.","DOI":"10.1007\/s00245-022-09850-7"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_009","doi-asserted-by":"crossref","unstructured":"E. Casas, K. Kunish and M. Mateos,\nError estimates for the numerical approximation of optimal control problems with non-smooth pointwise-integral control constraints,\nIMA J. Numer. Anal. (2022), 10.1093\/imanum\/drac027","DOI":"10.1093\/imanum\/drac027"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_010","doi-asserted-by":"crossref","unstructured":"E. Casas and M. Mateos,\nOptimal control of partial differential equations,\nComputational Mathematics, Numerical Analysis and Applications,\nSEMA SIMAI Springer Ser. 13,\nSpringer, Cham (2017), 3\u201359.","DOI":"10.1007\/978-3-319-49631-3_1"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_011","doi-asserted-by":"crossref","unstructured":"E. Casas and M. Mateos,\nCritical cones for sufficient second order conditions in PDE constrained optimization,\nSIAM J. Optim. 30 (2020), no. 1, 585\u2013603.","DOI":"10.1137\/19M1258244"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_012","doi-asserted-by":"crossref","unstructured":"E. Casas and M. Mateos,\nState error estimates for the numerical approximation of sparse distributed control problems in the absence of Tikhonov regularization,\nVietnam J. Math. 49 (2021), no. 3, 713\u2013738.","DOI":"10.1007\/s10013-021-00491-x"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_013","doi-asserted-by":"crossref","unstructured":"E. Casas and M. Mateos,\nCorrigendum: Critical cones for sufficient second order conditions in PDE constrained optimization,\nSIAM J. Optim. 32 (2022), no. 1, 319\u2013320.","DOI":"10.1137\/21M1466839"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_014","doi-asserted-by":"crossref","unstructured":"E. Casas, M. Mateos and A. R\u00f6sch,\nFinite element approximation of sparse parabolic control problems,\nMath. Control Relat. Fields 7 (2017), no. 3, 393\u2013417.","DOI":"10.3934\/mcrf.2017014"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_015","doi-asserted-by":"crossref","unstructured":"E. Casas, M. Mateos and A. R\u00f6sch,\nImproved approximation rates for a parabolic control problem with an objective promoting directional sparsity,\nComput. Optim. Appl. 70 (2018), no. 1, 239\u2013266.","DOI":"10.1007\/s10589-018-9979-0"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_016","doi-asserted-by":"crossref","unstructured":"E. Casas, M. Mateos and A. R\u00f6sch,\nError estimates for semilinear parabolic control problems in the absence of Tikhonov term,\nSIAM J. Control Optim. 57 (2019), no. 4, 2515\u20132540.","DOI":"10.1137\/18M117220X"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_017","doi-asserted-by":"crossref","unstructured":"E. Casas, C. Ryll and F. Tr\u00f6ltzsch,\nSecond order and stability analysis for optimal sparse control of the FitzHugh\u2013Nagumo equation,\nSIAM J. Control Optim. 53 (2015), no. 4, 2168\u20132202.","DOI":"10.1137\/140978855"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_018","doi-asserted-by":"crossref","unstructured":"E. Casas and F. Tr\u00f6ltzsch,\nSecond order analysis for optimal control problems: Improving results expected from abstract theory,\nSIAM J. 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Ural\u2019tseva,\nLinear and Quasilinear Elliptic Equations,\nAmerican Mathematical Society, Providence, 1988."},{"key":"2023100411150512037_j_cmam-2022-0130_ref_028","doi-asserted-by":"crossref","unstructured":"M. Mateos,\nSparse Dirichlet optimal control problems,\nComput. Optim. Appl. 80 (2021), no. 1, 271\u2013300.","DOI":"10.1007\/s10589-021-00290-7"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_029","doi-asserted-by":"crossref","unstructured":"D. Meidner and B. Vexler,\nOptimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations,\nESAIM Math. Model. Numer. Anal. 52 (2018), no. 6, 2307\u20132325.","DOI":"10.1051\/m2an\/2018040"},{"key":"2023100411150512037_j_cmam-2022-0130_ref_030","doi-asserted-by":"crossref","unstructured":"F. P\u00f6rner and D. Wachsmuth,\nTikhonov regularization of optimal control problems governed by semi-linear partial differential equations,\nMath. Control Relat. Fields 8 (2018), no. 1, 315\u2013335.","DOI":"10.3934\/mcrf.2018013"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0130\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0130\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T11:16:41Z","timestamp":1696418201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0130\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,8]]},"references-count":30,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4,1]]},"published-print":{"date-parts":[[2023,10,1]]}},"alternative-id":["10.1515\/cmam-2022-0130"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0130","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,8]]}}}