{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:35:10Z","timestamp":1787322910784,"version":"build-2736575974"},"reference-count":18,"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 control systems of the type $\\dot{x}=Ax+\\alpha(t)bu$, where $u\\in\\mathbf{R}$, $(A,b)$ is a controllable pair, and $\\alpha$ is an unknown measurable time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition of the type $\\int_t^{t+T}\\alpha(s)ds\\geq\\mu$ for every $t\\geq0$, with $0&lt;\\mu\\leq T$ independent of t. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,\\mu)$ if the eigenvalues of A have a nonpositive real part. We also show that stabilizability does not hold for an arbitrary matrix A. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon depending on the parameter $\\mu\/T$.<\/jats:p>","DOI":"10.1137\/080737812","type":"journal-article","created":{"date-parts":[[2010,5,5]],"date-time":"2010-05-05T18:08:16Z","timestamp":1273082896000},"page":"4032-4055","source":"Crossref","is-referenced-by-count":23,"title":["On the Stabilization of Persistently Excited Linear Systems"],"prefix":"10.1137","volume":"48","author":[{"given":"Yacine","family":"Chitour","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mario","family":"Sigalotti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,5,5]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1109\/9.701102"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(99)00012-6"},{"key":"R3","unstructured":"B. Anderson, R. Bitmead, C. Johnson, P. Kokotovic, R. Kosut, I. Mareels, L. Praly, and B. Riedle,\n                      Stability of Adaptive Systems: Passivity and Averaging Analysis\n                      , MIT Press, Cambridge, MA, 1986."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2008.7.1"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012900382837"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2006.15.415"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/9.664150"},{"key":"R8","unstructured":"H. Brezis,\n                      Analyse Fonctionnelle, Th\u00e9orie et applications\n                      , Masson, Paris, 1983."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/s00498-008-0024-1"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2007.07.001"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"F. Colonius and W. Kliemann,\n                      The dynamics of control\n                      , Systems & Control: Foundations & Applications, Birkh\u00e4user Boston, Boston, MA, 2000, with an appendix by Lars Gr\u00fcne.","DOI":"10.1007\/978-1-4612-1350-5"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012991221791"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"D. Liberzon,\n                      Switching in systems and control\n                      , Systems & Control: Foundations & Applications, Birkh\u00e4user Boston, Boston, MA, 2003.","DOI":"10.1007\/978-1-4612-0017-8"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0315013"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s004980200008"},{"key":"R16","unstructured":"W. Rudin,\n                      Real and complex analysis\n                      , 3rd edition, McGraw-Hill, New York, 1987."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/05063516X"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"E. D. Sontag,\n                      Mathematical control theory\n                      , Texts in Applied Mathematics 6, 2nd edition, Springer-Verlag, New York, 1998.","DOI":"10.1007\/978-1-4612-0577-7"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080737812","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:33:01Z","timestamp":1787319181000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080737812"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":18,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080737812"],"URL":"https:\/\/doi.org\/10.1137\/080737812","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}