{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T11:10:45Z","timestamp":1764587445228,"version":"3.38.0"},"reference-count":15,"publisher":"SAGE Publications","issue":"6","license":[{"start":{"date-parts":[[2009,3,31]],"date-time":"2009-03-31T00:00:00Z","timestamp":1238457600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Vibration and Control"],"published-print":{"date-parts":[[2009,6]]},"abstract":"<jats:p> We prove a necessary optimality condition for isoperimetric problems under nabla-differentiable curves. As a consequence, the recent results of Caputo (2008), that put together seemingly dissimilar optimal control problems in economics and physics, are extended to a generic time scale. We end with an illustrative example of the application of our main result to a dynamic optimization problem from economics. <\/jats:p>","DOI":"10.1177\/1077546309103268","type":"journal-article","created":{"date-parts":[[2009,4,1]],"date-time":"2009-04-01T00:45:05Z","timestamp":1238546705000},"page":"951-958","source":"Crossref","is-referenced-by-count":54,"title":["Isoperimetric Problems on Time Scales with Nabla Derivatives"],"prefix":"10.1177","volume":"15","author":[{"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"given":"Delfim F.M.","family":"Torres","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal,"}]}],"member":"179","published-online":{"date-parts":[[2009,3,31]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2005.08.014"},{"volume-title":"Proceedings of the Conference on Qualitative Theory of Differential Equations","author":"Aulbach, B.","key":"atypb2"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.3176\/phys.math.2007.3.03"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2007.914243"},{"issue":"3","key":"atypb5","first-page":"339","volume":"13","author":"Bohner, M.","year":"2004","journal-title":"Dynamic Systems and Applications"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0201-1"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8230-9"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2008.04.004"},{"issue":"07","key":"atypb9","first-page":"65","volume":"9","author":"Ferreira, R.A.C.","year":"2007","journal-title":"International Journal of Ecological Economics & Statistics"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-69532-5_9"},{"volume-title":"Proceedings of the Nonlinear Analysis and Optimization Conference","author":"Ferreira, R.A.C.","key":"atypb11"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s10958-009-9601-1"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2008.11.035"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.3176\/phys.math.2007.4.06"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/b97436"}],"container-title":["Journal of Vibration and Control"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1077546309103268","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1077546309103268","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T03:24:09Z","timestamp":1741058649000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/1077546309103268"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,3,31]]},"references-count":15,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2009,6]]}},"alternative-id":["10.1177\/1077546309103268"],"URL":"https:\/\/doi.org\/10.1177\/1077546309103268","relation":{},"ISSN":["1077-5463","1741-2986"],"issn-type":[{"type":"print","value":"1077-5463"},{"type":"electronic","value":"1741-2986"}],"subject":[],"published":{"date-parts":[[2009,3,31]]}}}