{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T10:26:26Z","timestamp":1762079186070,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,10]],"date-time":"2022-12-10T00:00:00Z","timestamp":1670630400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CIDMA\u2014Center for Research and Development in Mathematics and Applications","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}]},{"DOI":"10.13039\/501100001871","name":"Portuguese Foundation for Science and Technology (FCT-Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examples.<\/jats:p>","DOI":"10.3390\/fractalfract6120731","type":"journal-article","created":{"date-parts":[[2022,12,12]],"date-time":"2022-12-12T04:34:20Z","timestamp":1670819660000},"page":"731","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels"],"prefix":"10.3390","volume":"6","author":[{"given":"F\u00e1tima","family":"Cruz","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1305-2411","authenticated-orcid":false,"given":"Ricardo","family":"Almeida","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3535-3909","authenticated-orcid":false,"given":"Nat\u00e1lia","family":"Martins","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s11071-004-3764-6","article-title":"A general formulation and solution scheme for fractional optimal control problems","volume":"38","author":"Agrawal","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1051\/cocv\/2019021","article-title":"Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints","volume":"26","author":"Bergounioux","year":"2020","journal-title":"ESAIM Control Optim. 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