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In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.<\/jats:p>","DOI":"10.3390\/sym13112092","type":"journal-article","created":{"date-parts":[[2021,11,4]],"date-time":"2021-11-04T22:25:54Z","timestamp":1636064754000},"page":"2092","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Manifold Calculus in System Theory and Control\u2014Fundamentals and First-Order Systems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5964-7464","authenticated-orcid":false,"given":"Simone","family":"Fiori","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria dell\u2019Informazione, Universit\u00e0 Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Abraham, R., Marsden, J., and Ratiu, T. 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