{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T12:30:30Z","timestamp":1771677030232,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2021,8,8]],"date-time":"2021-08-08T00:00:00Z","timestamp":1628380800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020 (CIDMA)"],"award-info":[{"award-number":["UIDB\/04106\/2020 (CIDMA)"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PhD fellowship PD\/BD\/150273\/2019"],"award-info":[{"award-number":["PhD fellowship PD\/BD\/150273\/2019"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.<\/jats:p>","DOI":"10.3390\/math9161883","type":"journal-article","created":{"date-parts":[[2021,8,8]],"date-time":"2021-08-08T21:49:41Z","timestamp":1628459381000},"page":"1883","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Pontryagin Maximum Principle for Distributed-Order Fractional Systems"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0119-6178","authenticated-orcid":false,"given":"Fa\u00ef\u00e7al","family":"Nda\u00efrou","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF02826009","article-title":"Mean fractional-order-derivatives differential equations and filters","volume":"41","author":"Caputo","year":"1995","journal-title":"Ann. Univ. Ferrara Sez. VII"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"046129","DOI":"10.1103\/PhysRevE.66.046129","article-title":"Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations","volume":"66","author":"Chechkin","year":"2002","journal-title":"Phys. Rev. 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