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Here, we focus on control cost functionals that promote sparsity, which includes functionals of <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$<\/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:mi>p<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-type for <jats:inline-formula><jats:alternatives><jats:tex-math>$$p\\in [0,1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mo>[<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin\u2019s maximum principle and weaker than <jats:italic>L<\/jats:italic>-stationarity.<\/jats:p>","DOI":"10.1007\/s10589-021-00308-0","type":"journal-article","created":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T19:18:29Z","timestamp":1630696709000},"page":"639-677","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A proximal gradient method for control problems with non-smooth and non-convex control cost"],"prefix":"10.1007","volume":"80","author":[{"given":"Carolin","family":"Natemeyer","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Wachsmuth","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,9,3]]},"reference":[{"key":"308_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511897450","volume-title":"Nonlinear Superposition Operators, Volume 95 of Cambridge Tracts in Mathematics","author":"J Appell","year":"1990","unstructured":"Appell, J., Zabrejko, P.P.: Nonlinear Superposition Operators, Volume 95 of Cambridge Tracts in Mathematics. 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