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When the intersection of these two sets is empty, typically because the bounds on the control variables are too tight, the problem becomes infeasible. In this paper, we prove that, under a controllability assumption, the \u201cbest approximation\u201d optimal control minimizing the distance (and thus finding the \u201cgap\u201d) between the two sets is of bang\u2013bang type, with the \u201cgap function\u201d playing the role of a switching function. The critically feasible control solution (the case when one has the smallest control bound for which the problem is feasible) is also shown to be of bang\u2013bang type. We present the full analytical solution for the critically feasible problem involving the (simple but rich enough) double integrator. We illustrate the overall results numerically on various challenging example problems.<\/jats:p>","DOI":"10.1007\/s10957-024-02419-0","type":"journal-article","created":{"date-parts":[[2024,4,10]],"date-time":"2024-04-10T11:01:46Z","timestamp":1712746906000},"page":"1219-1245","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Infeasible and Critically Feasible Optimal Control"],"prefix":"10.1007","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1332-6213","authenticated-orcid":false,"given":"Regina S.","family":"Burachik","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7962-7153","authenticated-orcid":false,"given":"C. 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