{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T00:22:43Z","timestamp":1787358163867,"version":"build-2736575974"},"reference-count":48,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>We consider the development of a general nonlinear input-output theory which encompasses systems with initial conditions. Systems are defined in a set theoretic manner from input-output pairs on a doubly infinite time axis, and a general construction of the initial conditions is given in terms of an equivalence class of trajectories on the negative time axis. Input-output operators are then defined for given initial conditions, and a suitable notion of input-output stability on the positive time axis with initial conditions is given. This notion of stability is closely related to the ISS\/IOS concepts of Sontag. A fundamental robust stability theorem is derived which represents a generalization of the input-output operator robust stability theorem of Georgiou and Smith, to include the case of initial conditions. This includes a suitable generalization of the nonlinear gap metric. Some applications are given to show the utility of the robust stability theorem.<\/jats:p>","DOI":"10.1137\/120903373","type":"journal-article","created":{"date-parts":[[2015,6,25]],"date-time":"2015-06-25T11:08:04Z","timestamp":1435230484000},"page":"1625-1653","source":"Crossref","is-referenced-by-count":7,"title":["Robust Stability of Input-Output Systems with Initial Conditions"],"prefix":"10.1137","volume":"53","author":[{"given":"Jing","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"French","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,6,25]]},"reference":[{"key":"atypb1","unstructured":"B. D. O. Anderson and J. B. Moore,\n                      Linear Optimal Control\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1971."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-6911(99)00055-9"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/0005-1098(69)90026-0"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1967.1098635"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"C. A. Desoer and M. Vidyasagar,\n                      Feedback Systems: Input-Output Properties\n                      , Academic Press, New York, 1975.","DOI":"10.1016\/B978-0-12-212050-3.50011-6"},{"key":"atypb6","unstructured":"B. A. 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Liu,\n                      Robust Stability for Nonlinear Control: State-Space and Input-Output Synthesis\n                      , Ph.D. thesis, University of Southampton, Southampton, UK, 2014."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1958-0135681-9"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"J. W. Polderman and J. C. Willems,\n                      Introduction to Mathematical Systems Theory: A Behavioral Approach\n                      , Springer, New York, 1998.","DOI":"10.1007\/978-1-4757-2953-5"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"J. Sakarovitch,\n                      Elements of Automata Theory\n                      , Cambridge University Press, Cambridge, 2009.","DOI":"10.1017\/CBO9781139195218"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1964.tb04098.x"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1965.tb01671.x"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1109\/9.28018"},{"key":"atypb28","first-page":"163","author":"Sontag E. D.","year":"2008","journal-title":"New York"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(94)00050-6"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1109\/9.536498"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-6911(97)90013-X"},{"key":"atypb32","unstructured":"E. D. Sontag,\n                      Mathematical Control Theory: Deterministic Finite Dimensional Systems\n                      , Springer, New York, 1998."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-6911(99)00070-5"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1016\/S0016-0032(02)00022-4"},{"key":"atypb35","unstructured":"M. Vidyasagar,\n                      Nonlinear Systems Analysis\n                      , 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1993."},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1994-1179402-6"},{"key":"atypb37","doi-asserted-by":"crossref","unstructured":"J. C. Willems,\n                      The Analysis of Feedback Systems\n                      , MIT Press, Cambridge, MA, 1971.","DOI":"10.7551\/mitpress\/1258.001.0001"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1971.1099831"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/BF00276493"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1007\/BF00276494"},{"key":"atypb41","doi-asserted-by":"crossref","unstructured":"J. C. Willems,\n                      Models for Dynamics\n                      , Dynamics Reported 2, Springer, New York, 1989, pp. 171-269.","DOI":"10.1007\/978-3-322-96657-5_5"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1109\/9.73561"},{"key":"atypb43","unstructured":"L. A. Zadeh and C. A. Desoer,\n                      Linear System Theory: The State Space Approach\n                      , McGraw-Hill, New York, 1963."},{"key":"atypb44","unstructured":"G. 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