{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T03:41:28Z","timestamp":1775014888338,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,6,19]],"date-time":"2018-06-19T00:00:00Z","timestamp":1529366400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2018,8]]},"DOI":"10.1007\/s10957-018-1311-8","type":"journal-article","created":{"date-parts":[[2018,6,19]],"date-time":"2018-06-19T14:15:44Z","timestamp":1529417744000},"page":"317-348","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems"],"prefix":"10.1007","volume":"178","author":[{"given":"Francesco","family":"Ludovici","sequence":"first","affiliation":[]},{"given":"Ira","family":"Neitzel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6571-8043","authenticated-orcid":false,"given":"Winnifried","family":"Wollner","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,6,19]]},"reference":[{"issue":"1","key":"1311_CR1","first-page":"5","volume":"37","author":"JC Los Reyes de","year":"2008","unstructured":"de Los Reyes, J.C., Merino, P., Rehberg, J., Tr\u00f6ltzsch, F.: Optimality conditions for state-constrained PDE control problems with time-dependent controls. Control Cybern. 37(1), 5\u201338 (2008)","journal-title":"Control Cybern."},{"key":"1311_CR2","volume-title":"Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications","author":"M Hinze","year":"2010","unstructured":"Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications. Springer, Netherlands (2010)"},{"issue":"2","key":"1311_CR3","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1137\/07068240X","volume":"19","author":"E Casas","year":"2008","unstructured":"Casas, E., de los Reyes, J.C., Tr\u00f6ltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19(2), 616\u2013643 (2008)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1311_CR4","doi-asserted-by":"publisher","first-page":"874","DOI":"10.1137\/140960645","volume":"53","author":"I Neitzel","year":"2015","unstructured":"Neitzel, I., Pfefferer, J., R\u00f6sch, A.: Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation. SIAM J. Control Optim. 53(2), 874\u2013904 (2015)","journal-title":"SIAM J. Control Optim."},{"issue":"3","key":"1311_CR5","doi-asserted-by":"publisher","first-page":"1150","DOI":"10.1137\/070694016","volume":"47","author":"D Meidner","year":"2008","unstructured":"Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints. SIAM J. Control Optim. 47(3), 1150\u20131177 (2008)","journal-title":"SIAM J. Control Optim."},{"issue":"3","key":"1311_CR6","doi-asserted-by":"publisher","first-page":"1301","DOI":"10.1137\/070694028","volume":"47","author":"D Meidner","year":"2008","unstructured":"Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. II. Problems with control constraints. SIAM J. Control Optim. 47(3), 1301\u20131329 (2008)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"1311_CR7","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1137\/140961778","volume":"53","author":"F Ludovici","year":"2015","unstructured":"Ludovici, F., Wollner, W.: A priori error estimates for a finite element discretization of parabolic optimization problems with pointwise constraints in time on mean values of the gradient of the state. SIAM J. Control Optim. 53(2), 745\u2013770 (2015)","journal-title":"SIAM J. Control Optim."},{"issue":"5","key":"1311_CR8","doi-asserted-by":"publisher","first-page":"2797","DOI":"10.1137\/120885772","volume":"51","author":"D Leykekhman","year":"2013","unstructured":"Leykekhman, D., Vexler, B.: Optimal a priori error estimates of parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 51(5), 2797\u20132821 (2013)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"1311_CR9","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/s00211-011-0409-9","volume":"120","author":"I Neitzel","year":"2012","unstructured":"Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120(2), 345\u2013386 (2012)","journal-title":"Numer. Math."},{"issue":"5","key":"1311_CR10","doi-asserted-by":"publisher","first-page":"1961","DOI":"10.1137\/100793888","volume":"49","author":"D Meidner","year":"2011","unstructured":"Meidner, D., Rannacher, R., Vexler, B.: A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time. SIAM J. Control Optim. 49(5), 1961\u20131997 (2011)","journal-title":"SIAM J. Control Optim."},{"issue":"3","key":"1311_CR11","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1007\/s10589-013-9537-8","volume":"55","author":"D Leykekhman","year":"2013","unstructured":"Leykekhman, D., Meidner, D., Vexler, B.: Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints. Comput. Optim. Appl. 55(3), 769\u2013802 (2013)","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"1311_CR12","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s00211-017-0906-6","volume":"138","author":"I Neitzel","year":"2018","unstructured":"Neitzel, I., Wollner, W.: A priori \n                    \n                      \n                    \n                    $$L^2$$\n                    \n                      \n                        \n                          L\n                          2\n                        \n                      \n                    \n                  -discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints. Numer. Math. 138(2), 273\u2013299 (2018). \n                    https:\/\/doi.org\/10.1007\/s00211-017-0906-6","journal-title":"Numer. Math."},{"issue":"5","key":"1311_CR13","doi-asserted-by":"publisher","first-page":"1473","DOI":"10.3934\/dcdsb.2012.17.1473","volume":"17","author":"K Chrysafinos","year":"2012","unstructured":"Chrysafinos, K., Karatzas, E.N.: Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE\u2019s. Discrete Contin. Dyn. Syst. Ser. B 17(5), 1473\u20131506 (2012)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"issue":"1","key":"1311_CR14","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1051\/m2an\/2009046","volume":"44","author":"K Chrysafinos","year":"2010","unstructured":"Chrysafinos, K.: Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE\u2019s. M2AN Math. Model. Numer. Anal. 44(1), 189\u2013206 (2010)","journal-title":"M2AN Math. Model. Numer. Anal."},{"issue":"1","key":"1311_CR15","first-page":"67","volume":"21","author":"E Casas","year":"2002","unstructured":"Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21(1), 67\u2013100 (2002)","journal-title":"Comput. Appl. Math."},{"issue":"1","key":"1311_CR16","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s10589-011-9410-6","volume":"52","author":"M Hinze","year":"2012","unstructured":"Hinze, M., Meyer, C.: Stability of semilinear elliptic optimal control problems with pointwise state constraints. Comput. Optim. Appl. 52(1), 87\u2013114 (2012)","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"1311_CR17","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.1137\/15M103412X","volume":"54","author":"D Leykekhman","year":"2016","unstructured":"Leykekhman, D., Vexler, B.: Pointwise best approximation results for Galerkin finite element solutions of parabolic problems. SIAM J. Numer. Anal. 54(3), 1365\u20131384 (2016)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1311_CR18","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s10589-013-9541-z","volume":"56","author":"W Gong","year":"2013","unstructured":"Gong, W., Hinze, M.: Error estimates for parabolic optimal control problems with control and state constraints. Comput. Optim. Appl. 56(1), 131\u2013151 (2013)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"1311_CR19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4208\/jcm.1006-m3213","volume":"29","author":"K Deckelnick","year":"2011","unstructured":"Deckelnick, K., Hinze, M.: Variational discretization of parabolic control problems in the presence of pointwise state constraints. J. Comput. Math. 29(1), 1\u201315 (2011)","journal-title":"J. Comput. Math."},{"issue":"1","key":"1311_CR20","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1365\/s13291-014-0109-3","volume":"117","author":"E Casas","year":"2015","unstructured":"Casas, E., Tr\u00f6ltzsch, F.: Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math. Ver. 117(1), 3\u201344 (2015)","journal-title":"Jahresber. Dtsch. Math. Ver."},{"key":"1311_CR21","volume-title":"Perturbation Analysis of Optimization Problems, Springer Series Operations Research Financial Engineering","author":"J Bonnans","year":"2010","unstructured":"Bonnans, J., Shapiro, A.: Perturbation Analysis of Optimization Problems, Springer Series Operations Research Financial Engineering, 1st edn. Springer, New York (2010)","edition":"1"},{"issue":"4","key":"1311_CR22","doi-asserted-by":"publisher","first-page":"1007","DOI":"10.1137\/0331045","volume":"31","author":"H Goldberg","year":"1993","unstructured":"Goldberg, H., Tr\u00f6ltzsch, F.: Second-order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J. Control Optim. 31(4), 1007\u20131025 (1993)","journal-title":"SIAM J. Control Optim."},{"issue":"4","key":"1311_CR23","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1137\/S0363012995283637","volume":"35","author":"E Casas","year":"1997","unstructured":"Casas, E.: Pontryagin\u2019s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35(4), 1297\u20131327 (1997)","journal-title":"SIAM J. Control Optim."},{"issue":"1","key":"1311_CR24","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1137\/120871687","volume":"51","author":"K Krumbiegel","year":"2013","unstructured":"Krumbiegel, K., Rehberg, J.: Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints. SIAM J. Control Optim. 51(1), 304\u2013331 (2013)","journal-title":"SIAM J. Control Optim."},{"issue":"7","key":"1311_CR25","doi-asserted-by":"publisher","first-page":"4550","DOI":"10.1137\/080744608","volume":"48","author":"JF Bonnans","year":"2010","unstructured":"Bonnans, J.F., Jaisson, P.: Optimal control of a parabolic equation with time-dependent state constraints. SIAM J. Control Optim. 48(7), 4550\u20134571 (2010)","journal-title":"SIAM J. Control Optim."},{"issue":"4","key":"1311_CR26","doi-asserted-by":"publisher","first-page":"1182","DOI":"10.1137\/S0363012998345627","volume":"39","author":"E Casas","year":"2000","unstructured":"Casas, E., Raymond, J.P., Zidani, H.: Pontryagin\u2019s principle for local solutions of control problems with mixed control-state constraints. SIAM J. Control Optim. 39(4), 1182\u20131203 (2000)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"1311_CR27","doi-asserted-by":"publisher","first-page":"431","DOI":"10.3934\/dcds.2000.6.431","volume":"6","author":"JP Raymond","year":"2000","unstructured":"Raymond, J.P., Tr\u00f6ltzsch, F.: Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst. 6(2), 431\u2013450 (2000)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"1311_CR28","unstructured":"Ludovici, F.: Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative. Ph.D. thesis, Technische Universit\u00e4t Darmstadt (2017). \n                    http:\/\/tuprints.ulb.tu-darmstadt.de\/6781\/"},{"key":"1311_CR29","volume-title":"Optimal Control of Partial Differential Equations. Theory, Methods and Applications., Graduate Studies in Mathematics","author":"F Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications., Graduate Studies in Mathematics, vol. 112. AMS, Providence (2010)"},{"issue":"5","key":"1311_CR30","doi-asserted-by":"publisher","first-page":"1431","DOI":"10.1137\/S0363012900382011","volume":"40","author":"E Casas","year":"2002","unstructured":"Casas, E., Mateos, M.: Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40(5), 1431\u20131454 (2002). (electronic)","journal-title":"SIAM J. Control Optim."},{"key":"1311_CR31","volume-title":"Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math.","author":"V Thom\u00e8e","year":"2006","unstructured":"Thom\u00e8e, V.: Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math., 2nd edn. Springer, Berlin (2006)","edition":"2"},{"issue":"1","key":"1311_CR32","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s10589-005-4559-5","volume":"30","author":"M Hinze","year":"2005","unstructured":"Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45\u201361 (2005)","journal-title":"Comput. Optim. Appl."},{"key":"1311_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics","author":"SC Brenner","year":"2008","unstructured":"Brenner, S.C., Scott, R.L.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)","edition":"3"},{"issue":"2","key":"1311_CR34","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1007\/s00211-013-0562-4","volume":"126","author":"RE Bank","year":"2014","unstructured":"Bank, R.E., Yserentant, H.: On the \n                    \n                      \n                    \n                    $$H^1$$\n                    \n                      \n                        \n                          H\n                          1\n                        \n                      \n                    \n                  -stability of the \n                    \n                      \n                    \n                    $$L_2$$\n                    \n                      \n                        \n                          L\n                          2\n                        \n                      \n                    \n                  -projection onto finite element spaces. Numer. Math. 126(2), 361\u2013381 (2014)","journal-title":"Numer. Math."},{"issue":"3","key":"1311_CR35","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/BF01396760","volume":"54","author":"RH Nochetto","year":"1988","unstructured":"Nochetto, R.H.: Sharp \n                    \n                      \n                    \n                    $$L^\\infty $$\n                    \n                      \n                        \n                          L\n                          \u221e\n                        \n                      \n                    \n                  -error estimates for semilinear elliptic problems with free boundaries. Numer. Math. 54(3), 243\u2013255 (1988)","journal-title":"Numer. Math."},{"key":"1311_CR36","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/0022-247X(73)90022-X","volume":"44","author":"RS Falk","year":"1973","unstructured":"Falk, R.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28\u201347 (1973)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1311_CR37","first-page":"51","volume":"37","author":"C Meyer","year":"2008","unstructured":"Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37(1), 51\u201383 (2008)","journal-title":"Control Cybern."},{"issue":"3","key":"1311_CR38","first-page":"695","volume":"31","author":"E Casas","year":"2002","unstructured":"Casas, E., Tr\u00f6ltzsch, F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybern. 31(3), 695\u2013712 (2002)","journal-title":"Control Cybern."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-018-1311-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-018-1311-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-018-1311-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,18]],"date-time":"2019-06-18T19:17:37Z","timestamp":1560885457000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-018-1311-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,19]]},"references-count":38,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,8]]}},"alternative-id":["1311"],"URL":"https:\/\/doi.org\/10.1007\/s10957-018-1311-8","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,19]]},"assertion":[{"value":"13 November 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 June 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}