{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T22:14:38Z","timestamp":1787955278943,"version":"build-2784847793"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS; the cases of summation, maximization, and separation with respect to external gains are obtained as corollaries.<\/jats:p>","DOI":"10.1137\/090746483","type":"journal-article","created":{"date-parts":[[2010,5,12]],"date-time":"2010-05-12T19:34:17Z","timestamp":1273692857000},"page":"4089-4118","source":"Crossref","is-referenced-by-count":336,"title":["Small Gain Theorems for Large Scale Systems and Construction of ISS Lyapunov Functions"],"prefix":"10.1137","volume":"48","author":[{"given":"Sergey N.","family":"Dashkovskiy","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bj\u00f6rn S.","family":"R\u00fcffer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fabian R.","family":"Wirth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,5,12]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"V. 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Wirth,\n                      A small-gain type stability criterion for large scale networks of ISS systems\n                      , in 44th IEEE Conference on Decision and Control and European Control Conference, CDC\/ECC 2005, Seville, Spain, 2005, pp. 5633\u20135638.","DOI":"10.1109\/CDC.2005.1583060"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s00498-007-0014-8"},{"key":"R7","unstructured":"S. Dashkovskiy, B. R\u00fcffer, and F. Wirth,\n                      A Lyapunov ISS small gain theorem for strongly connected networks\n                      , in Proceedings of the 7th IFAC Symposium on Nonlinear Control Systems, NOLCOS2007, Pretoria, South Africa, 2007, pp. 283\u2013288."},{"key":"R8","unstructured":"S. Dashkovskiy, B. S. R\u00fcffer, and F. R. 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Pritchard,\n                      Mathematical Systems Theory\n                      I\n                      : Modelling, State Space Analysis, Stability and Robustness\n                      , Springer, New York, 2005."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/070707919"},{"key":"R17","doi-asserted-by":"crossref","unstructured":".28em plus .1em minus .1em Z.P. Jiang, I. M. Y. Mareels, and Y. Wang,\n                      A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems\n                      , Automatica J. IFAC, 32 (1996), pp. 1211\u20131215.","DOI":"10.1016\/0005-1098(96)00051-9"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01211469"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"Z.P. Jiang and Y. 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