{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,13]],"date-time":"2025-12-13T07:15:07Z","timestamp":1765610107565,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,10,25]],"date-time":"2020-10-25T00:00:00Z","timestamp":1603584000000},"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)","PhD fellowship PD\/BD\/150273\/2019"],"award-info":[{"award-number":["UIDB\/04106\/2020 (CIDMA)","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":["Axioms"],"abstract":"<jats:p>Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives\/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin\u2019s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.<\/jats:p>","DOI":"10.3390\/axioms9040124","type":"journal-article","created":{"date-parts":[[2020,10,26]],"date-time":"2020-10-26T02:34:54Z","timestamp":1603679694000},"page":"124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Distributed-Order Non-Local Optimal Control"],"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":[[2020,10,25]]},"reference":[{"key":"ref_1","unstructured":"Caputo, M. (1969). Elasticit\u00e0 e Dissipazione, Zanichelli."},{"key":"ref_2","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_3","first-page":"865","article-title":"On the existence of the order domain and the solution of distributed order equations. I","volume":"2","author":"Bagley","year":"2000","journal-title":"Int. J. Appl. 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