{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T01:18:29Z","timestamp":1778548709915,"version":"3.51.4"},"reference-count":47,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T00:00:00Z","timestamp":1646265600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"German Federal Ministry of Education and Research","award":["01LY1706B"],"award-info":[{"award-number":["01LY1706B"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The power output of Stirling engines can be optimized by several means. In this study, the focus is on potential performance improvements that can be achieved by optimizing the piston motion of an alpha-Stirling engine in the presence of dissipative processes, in particular mechanical friction. We use a low-effort endoreversible Stirling engine model, which allows for the incorporation of finite heat and mass transfer as well as the friction caused by the piston motion. Instead of performing a parameterization of the piston motion and optimizing these parameters, we here use an indirect iterative gradient method that is based on Pontryagin\u2019s maximum principle. For the varying friction coefficient, the optimization results are compared to both, a harmonic piston motion and optimization results found in a previous study, where a parameterized piston motion had been used. Thus we show how much performance can be improved by using the more sophisticated and numerically more expensive iterative gradient method.<\/jats:p>","DOI":"10.3390\/e24030362","type":"journal-article","created":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T09:24:53Z","timestamp":1646299493000},"page":"362","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Power-Optimal Control of a Stirling Engine\u2019s Frictional Piston Motion"],"prefix":"10.3390","volume":"24","author":[{"given":"Raphael","family":"Paul","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdellah","family":"Khodja","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas","family":"Fischer","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robin","family":"Masser","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karl Heinz","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,3]]},"reference":[{"key":"ref_1","unstructured":"Stirling, R. 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