{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:11:28Z","timestamp":1757311888881,"version":"3.37.3"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,3,22]],"date-time":"2021-03-22T00:00:00Z","timestamp":1616371200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,3,22]],"date-time":"2021-03-22T00:00:00Z","timestamp":1616371200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,4]]},"DOI":"10.1007\/s40314-021-01488-5","type":"journal-article","created":{"date-parts":[[2021,3,22]],"date-time":"2021-03-22T17:03:31Z","timestamp":1616432611000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Some numerical results for control of 3D heat equations using Nash equilibrium"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0949-2921","authenticated-orcid":false,"given":"Pit\u00e1goras Pinheiro","family":"de Carvalho","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,22]]},"reference":[{"key":"1488_CR1","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1051\/cocv\/2014052","volume":"21","author":"FD Araruna","year":"2015","unstructured":"Araruna FD, Fern\u00e1ndez-Cara E, Santos MC (2015) Stackelberg-Nash exact controllability for linear and semilinear parabolic equations. ESAIM Control Optim Calc Var 21:835\u2013856","journal-title":"ESAIM Control Optim Calc Var"},{"key":"1488_CR2","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.sysconle.2017.03.009","volume":"104","author":"FD Araruna","year":"2017","unstructured":"Araruna FD, Fern\u00e1ndez-Cara E, Guerrero S, Santos MC (2017) New results on the Stackelberg-Nash exact control of linear parabolic equations. Syst Control Lett 104:78\u201385","journal-title":"Syst Control Lett"},{"key":"1488_CR3","first-page":"41","volume":"1950034","author":"FD Araruna","year":"2019","unstructured":"Araruna FD, Fern\u00e1ndez-Cara E, da Silva LC (2019) Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls. Commun Contemp Math 1950034:41","journal-title":"Commun Contemp Math"},{"key":"1488_CR4","unstructured":"Araruna FD, Fern\u00e1ndez-Cara E, da Silva LC (2021) Hierarchic control for the wave equation. J Optim Theory Appl"},{"key":"1488_CR5","unstructured":"Bauso D (2016) Game theory with engineering applications, Advances in Design and Control, 30. In: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA"},{"key":"1488_CR6","doi-asserted-by":"crossref","unstructured":"Bruce LM (2013) Game theory applied to big data analytics in geosciences and remote sensing. IEEE International Geoscience and Remote Sensing Symposium - IGARSS. Melbourne, VIC, pp 4094\u20134097","DOI":"10.1109\/IGARSS.2013.6723733"},{"key":"1488_CR7","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1007\/s10915-018-0764-0","volume":"78","author":"PP de Carvalho","year":"2019","unstructured":"de Carvalho PP, Enrique F-C (2019) Carvalho, Enrique Fern\u00e1ndez-Cara on the computation of nash and pareto equilibria for some bi-objective control problems. J Sci Comput 78:246\u2013273","journal-title":"J Sci Comput"},{"issue":"5","key":"1488_CR8","doi-asserted-by":"publisher","first-page":"A2678","DOI":"10.1137\/19M1253320","volume":"42","author":"PP de Carvalho","year":"2021","unstructured":"de Carvalho PP, Fern\u00e1ndez-Cara E (2021) Numerical Stackelberg-Nash control for the heat equation. SIAM J Sci Comput 42(5):A2678\u2013A2700","journal-title":"SIAM J Sci Comput"},{"key":"1488_CR9","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s10444-020-09812-z","volume":"46","author":"PP de Carvalho","year":"2020","unstructured":"de Carvalho PP, Enrique F-C, Ferrel JL (2020) On the computation of nash and pareto equilibria for some bi-objective wave equations. Adv Comput Math 46:73","journal-title":"Adv Comput Math"},{"key":"1488_CR10","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1007\/s10957-016-0984-0","volume":"171","author":"P de Isa\u00edas","year":"2016","unstructured":"de Isa\u00edas P (2016) Jesus, hierarchical control for the wave equation with a moving boundary. J Optim Theory Appl 171:336\u2013350","journal-title":"J Optim Theory Appl"},{"issue":"4","key":"1488_CR11","doi-asserted-by":"publisher","first-page":"780","DOI":"10.1080\/00036811.2014.901503","volume":"94","author":"IP de Jesus","year":"2015","unstructured":"de Jesus IP, de Menezes SB (2015) On the approximate controllability of Stackelberg-Nash strategies for linear heat equations in $${\\mathbb{R}}^N$$ with potentials. Appl Anal 94(4):780\u2013799","journal-title":"Appl Anal"},{"issue":"4","key":"1488_CR12","doi-asserted-by":"publisher","first-page":"2101","DOI":"10.3934\/cpaa.2020093","volume":"19","author":"E Fern\u00e1ndez-Cara","year":"2019","unstructured":"Fern\u00e1ndez-Cara E, Gayte IM (2019) Theoretical and numerical results for some bi-objective optimal control problems. Commun Pure Applied Anal 19(4):2101\u20132126","journal-title":"Commun Pure Applied Anal"},{"key":"1488_CR13","doi-asserted-by":"crossref","unstructured":"Guillen-Gonzalez F, Marques-Lopes F, Rojas MMA (2013) On the approximate controllability of Stackelberg-Nash strategies for Stokes equations. In: Proceedings of the American Mathematical Society","DOI":"10.1090\/S0002-9939-2012-11459-5"},{"issue":"3\u20134","key":"1488_CR14","first-page":"251","volume":"20","author":"F Hecht","year":"2012","unstructured":"Hecht F (2012) New development in freefem++. J Numer Math 20(3\u20134):251\u2013265","journal-title":"J Numer Math"},{"key":"1488_CR15","unstructured":"Heliou A (2017) Molecular conformations and game theory. Bioinformatics [q-bio.QM]. Universit\u00e9 Paris-Saclay. English. ffNNT : 2017SACLX033ff"},{"key":"1488_CR16","doi-asserted-by":"crossref","unstructured":"Hern\u00e1ndez-Santamar\u00eda V, de Teresa L (2018) Robust stackelberg controllability for linear and semilinear heat equations. Evolut Equ Control Theory 7(2):247","DOI":"10.3934\/eect.2018012"},{"key":"1488_CR17","doi-asserted-by":"crossref","unstructured":"Hern\u00e1ndez-Santamar\u00eda V, de Teresa L (2018) Some remarks on the hierarchic control for coupled parabolic PDEs. Recent advances in PDEs: analysis. Numerics and Control. Springer, Cham, pp 117\u2013137","DOI":"10.1007\/978-3-319-97613-6_7"},{"key":"1488_CR18","unstructured":"Hern\u00e1ndez-Santamar\u00eda V, de Teresa L, Poznyak A (2015) Hierarchic control for a coupled parabolic system. Port Math 73(2):115\u2013137"},{"key":"1488_CR19","unstructured":"Kassay G, R\u01cedulescu V (2019) Equilibrium problems and applications (mathematics in science and engineering) (English Edition), 1 Edition"},{"key":"1488_CR20","doi-asserted-by":"publisher","first-page":"286","DOI":"10.2307\/1969529","volume":"54","author":"JF Nash","year":"1951","unstructured":"Nash JF (1951) Noncooperative games. Ann. Math 54:286\u2013295","journal-title":"Ann. Math"},{"key":"1488_CR21","unstructured":"Ramos AM (2021) Nash equilibria strategies and equivalent single-objective optimization problems. The case of linear partial differential equations. arXiv:1908.11858"},{"key":"1488_CR22","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1023\/A:1017907930931","volume":"112","author":"AM Ramos","year":"2001","unstructured":"Ramos AM, Glowinski R, Periaux J (2001) Pointwise control of the Burgers equation and related Nash equilibria problems: a computational approach. J Optim Theory Appl 112:499\u2013516","journal-title":"J Optim Theory Appl"},{"key":"1488_CR23","doi-asserted-by":"crossref","unstructured":"Ramos AM, Glowinski R, Periaux J (2002) Nash equilibria for the multi-objective control of linear partial partial differential equations. J Optim Theory Appl: 457\u2013498","DOI":"10.1023\/A:1017981514093"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01488-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-021-01488-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01488-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,8]],"date-time":"2021-04-08T22:06:19Z","timestamp":1617919579000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-021-01488-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,22]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["1488"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01488-5","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2021,3,22]]},"assertion":[{"value":"14 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 January 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 March 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"92"}}