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In a pair of papers, we construct a first order optimization algorithm to address this problem. Our approach, which we prove in this paper converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical chattering lemma that we formalize. In the second part of this pair of papers, we describe how this conceptual algorithm can be recast in order to devise an implementable algorithm that constructs a sequence of points by recursive application that converge to local minimizers of the optimal control problem for switched systems.<\/jats:p>","DOI":"10.1137\/120901490","type":"journal-article","created":{"date-parts":[[2013,12,11]],"date-time":"2013-12-11T13:35:30Z","timestamp":1386768930000},"page":"4463-4483","source":"Crossref","is-referenced-by-count":49,"title":["Consistent Approximations for the Optimal Control of Constrained Switched Systems---Part 1: A Conceptual Algorithm"],"prefix":"10.1137","volume":"51","author":[{"given":"Ramanarayan","family":"Vasudevan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Humberto","family":"Gonzalez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ruzena","family":"Bajcsy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S. 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