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Deterministic and stochastic systems are considered, as well as finite- and infinite-horizon problems. We give conditions under which these systems have <jats:italic>degenerate<\/jats:italic> feedback optimal controls so that the optimal control actions <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$a(t,x) \\equiv a(t)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u2261<\/mml:mo>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> are independent of the state variable <jats:italic>x<\/jats:italic>. As a consequence, open-loop and feedback (or Markov) optimal controls coincide, the value (or optimal objective) function is linear in the state <jats:italic>x<\/jats:italic>, and the <jats:italic>certainty equivalence<\/jats:italic> principle is satisfied.\n<\/jats:p>","DOI":"10.1007\/s10957-025-02657-w","type":"journal-article","created":{"date-parts":[[2025,4,4]],"date-time":"2025-04-04T11:11:10Z","timestamp":1743765070000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linear\u2013State Control Problems and Differential Games: Deterministic and Stochastic Systems"],"prefix":"10.1007","volume":"205","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-3081-4723","authenticated-orcid":false,"given":"Jos\u00e9","family":"E. 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