{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:18:17Z","timestamp":1787321897185,"version":"build-2736575974"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>We derive simplified formulas for analyzing the moment stability of stochastic parametrically forced linear systems. This analysis extends the results in [T. Blass and L. A. Romero, SIAM J. Control Optim., 51 (2013), pp. 1099--1127], where, under the assumption that the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this sum to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this approach leads to an enormous simplification. We give two examples of systems with rank one forcing, including the problem of stochastically forced Faraday waves.<\/jats:p>","DOI":"10.1137\/140965375","type":"journal-article","created":{"date-parts":[[2015,7,8]],"date-time":"2015-07-08T11:30:47Z","timestamp":1436355047000},"page":"1842-1859","source":"Crossref","is-referenced-by-count":1,"title":["On the Moment Stability of Stochastic Parametrically Forced Equations with Rank One Forcing"],"prefix":"10.1137","volume":"53","author":[{"given":"T. J.","family":"Blass","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L. A.","family":"Romero","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. R.","family":"Torczynski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,7,8]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1954.0218"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(02)00684-X"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/110855302"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.78.859"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112098001578"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02450690"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1098\/rstl.1831.0018"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"M. I. Freidlin and A. D. Wentzell,\n                      Random perturbations of dynamical systems\n                      , 3rd ed., Grundlehren Math. Wiss. 260, Springer, Heidelberg, 2012 (translated from the 1979 Russian original by J. Szu\u0308cs).","DOI":"10.1007\/978-3-642-25847-3_8"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"C. W. Gardiner,\n                      Handbook of Stochastic Methods, For Physics, Chemistry and the Natural Sciences\n                      , 2nd ed., Springer Ser. Synergetics 13, Springer-Verlag, Berlin, 1985.","DOI":"10.1007\/978-3-662-02452-2"},{"key":"atypb10","unstructured":"G. H. Golub and C. Greif,\n                      Techniques for Solving General KKT Systems\n                      , Technical report, SCCM, Stanford, Palo Alto, CA, 2000."},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"R. Khasminskii,\n                      Stochastic Stability of Differential Equations\n                      , 2nd ed., Stoch. Model. Appl. Probab. 66, Springer, Heidelberg, 2012.","DOI":"10.1007\/978-3-642-23280-0"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1996.0056"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.62.1416"},{"key":"atypb14","unstructured":"L. D. Landau and E. M. Lifshitz,\n                      Course of Theoretical Physics. Vol.6, Fluid Mechanics\n                      , 2nd ed., Pergamon Press, Oxford, 1987 (translated from the 3rd Russian edition by J. B. Sykes and W. H. Reid)."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160270503"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1063\/1.1518690"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7462(86)90025-9"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/0266-8920(88)90028-8"},{"key":"atypb19","unstructured":"J. W. Strutt, Lord Rayleigh,\n                      The Theory of Sound, Vol.\n                      II, Dover Publications, New York, 1945."},{"key":"atypb20","unstructured":"N. G. van Kampen,\n                      Stochastic Processes in Physics and Chemistry\n                      , Lecture Notes in Math. 888, North-Holland, Amsterdam, 1981."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1063\/1.858723"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/140965375","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:16:01Z","timestamp":1787318161000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/140965375"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/140965375"],"URL":"https:\/\/doi.org\/10.1137\/140965375","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}