{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:40:55Z","timestamp":1787323255344,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"8","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by output injection. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a left factorization over H-infinity. Another equivalent condition is that the filter Riccati equation (of an arbitrary realization) has a solution (in general, unbounded and even nondensely defined). We further show that after renorming the state space in terms of the inverse of the smallest solution of the filter Riccati equation, the closed-loop system is not only input-output stable but also strongly internally $*$-stable.<\/jats:p>","DOI":"10.1137\/090762233","type":"journal-article","created":{"date-parts":[[2010,10,19]],"date-time":"2010-10-19T18:09:09Z","timestamp":1287511749000},"page":"5084-5107","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems by Output Injection"],"prefix":"10.1137","volume":"48","author":[{"given":"Mark R.","family":"Opmeer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Olof J.","family":"Staffans","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,10,19]]},"reference":[{"key":"R1","first-page":"393","volume":"55","author":"Arov D. Z.","year":"2006","journal-title":"J. Operator Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0379-4024","issn-type":"print"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"D. Z. Arov and O. J. Staffans,\n                      State\/signal linear time-invariant systems theory,\n                      I\n                      . Discrete time systems\n                      , in The State Space Method Generalizations and Applications, Oper. Theory Adv. Appl. 161, Birkh\u00e4user, Basel, 2006, pp. 115\u2013177.","DOI":"10.1007\/3-7643-7431-4_3"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1002\/rnc.1089"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2009.05.029"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0330003"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"G. Da Prato and M. Delfour,\n                      Linear quadratic control problem without stabilizability\n                      , in Stabilization of Flexible Structures, Lecture Notes in Control and Inform. Sci. 147, Springer, Berlin, 1990, pp. 126\u2013147.","DOI":"10.1007\/BFb0005151"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/9.29425"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1109\/9.29424"},{"key":"R9","unstructured":"M. Green and D. J. N. Limebeer,\n                      Linear Robust Control\n                      , Prentice Hall, Englewood Cliffs, NJ, 1995."},{"key":"R10","doi-asserted-by":"crossref","unstructured":"T. Kailath,\n                      Lectures on Wiener and Kalman Filtering\n                      , Springer-Verlag, Wien, 1981.","DOI":"10.1007\/978-3-7091-2804-6"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1115\/1.3662552"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1115\/1.3658902"},{"key":"R13","unstructured":"E. Kreyszig,\n                      Introductory Functional Analysis with Applications\n                      , John Wiley & Sons, New York, London, Sydney, 1978."},{"key":"R14","unstructured":"H. Kwakernaak and R. Sivan,\n                      Linear Optimal Control Systems\n                      , John Wiley & Sons, New York, 1972."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s00498-008-0034-z"},{"key":"R16","unstructured":"M. R. Opmeer and O. J. Staffans,\n                      Coprime Factorization and Optimal Control on the Whole Discrete Time Axis\n                      , manuscript, 2010."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s11785-007-0035-9"},{"key":"R18","unstructured":"W. Rudin,\n                      Functional Analysis\n                      , McGraw-Hill, New York, 1973."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S036301299833519X"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4076(03)00146-5"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090762233","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:51:14Z","timestamp":1787320274000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090762233"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":20,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090762233"],"URL":"https:\/\/doi.org\/10.1137\/090762233","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}