{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:45:23Z","timestamp":1760233523763,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Universidad Nacional de San Agust\u00edn de Arequipa, UNSA, Arequipa, Per\u00fa","award":["IBA-IB-04-2020-UNSA"],"award-info":[{"award-number":["IBA-IB-04-2020-UNSA"]}]},{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["Fondecyt 1190142"],"award-info":[{"award-number":["Fondecyt 1190142"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many significant real world challenges arise as optimization problems on different classes of control systems. In particular, ordinary differential equations with symmetries. The purpose of this review article is twofold. First, we give the information we have about the class of Linear Control Systems \u03a3G on a low dimension matrix Lie group G. Second, we invite the Mathematical community to consider possible applications through the Pontryagin Maximum Principle for \u03a3G. In addition, we challenge some theoretical open problems. The class \u03a3G is a perfect generalization of the classical Linear Control System on the Abelian group Rn. Let G be a Lie group of dimension two or three. Related to \u03a3G, this review describes the actual results about controllability, the time-optimal Hamiltonian equations and, the Pontryagin Maximum Principle. We show how to build \u03a3G, through several examples on low dimensional matrix groups.<\/jats:p>","DOI":"10.3390\/sym13020167","type":"journal-article","created":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T12:28:31Z","timestamp":1611577711000},"page":"167","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Toward Applications of Linear Control Systems on the Real World and Theoretical Challenges"],"prefix":"10.3390","volume":"13","author":[{"given":"V\u00edctor","family":"Ayala","sequence":"first","affiliation":[{"name":"Instituto de Alta Investigaci\u00f3n, Universidad de Tarapac\u00e1, Casilla 7D, Arica 1000000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mar\u00eda","family":"Torreblanca","sequence":"additional","affiliation":[{"name":"Departamento Acad\u00e9mico de Matem\u00e1ticas, Universidad Nacional de San Agust\u00edn de Arequipa, Calle Santa Catalina, Nro. 117, Arequipa 04001, Peru"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"William","family":"Valdivia","sequence":"additional","affiliation":[{"name":"Instituto de Alta Investigaci\u00f3n, Universidad de Tarapac\u00e1, Casilla 7D, Arica 1000000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,22]]},"reference":[{"key":"ref_1","unstructured":"Brockett, R.W. 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