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We prove necessary optimality conditions of Pontryagin maximum principle type. Here, a special control perturbation is used that respects the <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> constraint. First, the maximum principle is obtained in integral form, which is then turned into a pointwise form. In addition, an optimization algorithm of proximal gradient type is analyzed. Under some assumptions, the sequence of iterates contains strongly converging subsequences, whose limits are feasible and satisfy a subset of the necessary optimality conditions.<\/jats:p>","DOI":"10.1007\/s10589-023-00456-5","type":"journal-article","created":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T21:57:02Z","timestamp":1675807022000},"page":"811-833","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Optimal control problems with $$L^0(\\Omega )$$ constraints: maximum principle and proximal gradient method"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7828-5614","authenticated-orcid":false,"given":"Daniel","family":"Wachsmuth","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,7]]},"reference":[{"key":"456_CR1","doi-asserted-by":"publisher","unstructured":"Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. 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