{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:35:57Z","timestamp":1787322957332,"version":"build-2736575974"},"reference-count":13,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely characterized by a linear system of first order partial differential equations for one set of variables ($\\,\\lambda$) followed by a linear system of first order partial differential equations for the second set of variables ($\\,{\\widehat g}$, $\\,{\\widehat V}$). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the $\\,\\lambda$-equations. We give four examples illustrating different features of matching control laws. The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case.<\/jats:p>","DOI":"10.1137\/s0363012901393304","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1372-1388","source":"Crossref","is-referenced-by-count":60,"title":["On the $\\lambda$-Equations for Matching Control Laws"],"prefix":"10.1137","volume":"41","author":[{"given":"David","family":"Auckly","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lev","family":"Kapitanski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"F. Andreev, D. Auckly, S. Gosavi, L. Kapitanski, A. Kelkar, and W. White,\n                      Matching linear system, and ball and beam\n                      , Automatica J. IFAC, 2002, to appear; archived online at http:\/\/xxx.lanl.gov\/abs\/math.OC\/0003177.","DOI":"10.1016\/S0005-1098(02)00145-0"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"F. Andreev, D. Auckly, L. Kapitanski, A. Kelkar, and W. White,\n                      Matching control laws for a ball and beam system\n                      , in Proceedings of the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton, NJ, 2000, pp. 161\u2013162.","DOI":"10.1016\/S1474-6670(17)35563-5"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"F. Andreev, D. Auckly, L. Kapitanski, A. Kelkar, and W. 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Hamberg,\n                      Controlled Lagrangians, symmetries and conditions for strong matching\n                      , in Proceedings of the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton, NJ, 2000, pp. 62\u201367.","DOI":"10.1016\/S1474-6670(17)35547-7"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"J. Hamberg,\n                      Simplified conditions for matching and for generalized matching in the theory of controlled Lagrangians\n                      , in Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 3918\u20133923.","DOI":"10.1109\/ACC.2000.876957"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"D. V. Zenkov, A. M. Bloch, N. E. Leonard, and J. E. Marsden,\n                      Matching and stabilization of low\u2010dimensional nonholonomic systems\n                      , in Proceedings of the 39th IEEE Conference on Decision and Control, IEEE Control Systems Society, Piscataway, NJ, 2000, pp. 1289\u20131295.","DOI":"10.1109\/CDC.2000.912033"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0363012901393304","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:35:36Z","timestamp":1787319336000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0363012901393304"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":13,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["10.1137\/S0363012901393304"],"URL":"https:\/\/doi.org\/10.1137\/s0363012901393304","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}