A Lyapunov Criterion for the Existence of Stationary Probability Distributions for Systems Perturbed by Noise

Previous article Next article A Lyapunov Criterion for the Existence of Stationary Probability Distributions for Systems Perturbed by NoiseMoshe ZakaiMoshe Zakaihttps://doi.org/10.1137/0307028PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. Z. Khasminskii, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Theor. Probability Appl., 5 (1960), 179–196 10.1137/1105016 0093.14902 LinkGoogle Scholar[2] W. M. Wonham, Liapunov criteria for weak stochastic stability, J. Differential Equations, 2 (1966), 195–207 10.1016/0022-0396(66)90043-X MR0194699 0133.40601 CrossrefISIGoogle Scholar[3] W. M. Wonham, A Liapunov method for the estimation of statistical averages, J. Differential Equations, 2 (1966), 365–377 10.1016/0022-0396(66)90047-7 MR0206438 0168.15903 CrossrefISIGoogle Scholar[4] V. E. Beneš, Finite regular invariant measures for Feller processes, J. Appl. Probability, 5 (1968), 203–209 MR0225382 CrossrefISIGoogle Scholar[5] J. C. Oxtoby and , S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2), 40 (1939), 560–566 MR0000097 0021.41202 CrossrefGoogle Scholar[6] S. R. Foguel, Existence of invariant measures for Markov processes. II, Proc. Amer. Math. Soc., 17 (1966), 387–389 MR0193673 0168.16405 CrossrefISIGoogle Scholar[7] E. B. Dynkin, Markov processes. Vols. I, II, Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wi ssenschaften, Bände 121, Vol. 122, Academic Press Inc., Publishers, New York, 1965Vol. I: xii+365 pp.; Vol. II: viii+274 MR0193671 0132.37901 CrossrefGoogle Scholar[8] Moshe Zakai, Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations, Israel J. Math., 5 (1967), 170–176 MR0225397 0166.13803 CrossrefISIGoogle Scholar[9] Moshe Zakai, On the ultimate boundedness of moments associated with solutions of stochastic differential equations, SIAM J. Control, 5 (1967), 588–593 10.1137/0305037 MR0225398 0267.60066 LinkGoogle Scholar[10] Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965xi+458 MR0180824 0126.11504 CrossrefGoogle Scholar[11] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[12] J. L. Doob, Asymptotic properties of Markoff transition prababilities, Trans. Amer. Math. Soc., 63 (1948), 393–421 MR0025097 0041.45406 ISIGoogle Scholar[13] A. V. Skorokhod, Studies in the theory of random processes, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965viii+199, New York MR0185620 0146.37701 Google Scholar[14] L. A. Zadeh and , C. A. Desoer, Linear System Theory, McGraw-Hill, New York, 1963 Google Scholar[15] H. J. Kushner, The Cauchy problem for a class of degenerate parabolic equations and asymptotic properties of the related diffusion processes, Tech. Rep., 68-3, Center for Dynamical Systems, Division of Applied Mathematics, Brown University, 1968 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functionsDebasish Chatterjee and Daniel Liberzon26 July 2006 | SIAM Journal on Control and Optimization, Vol. 45, No. 1AbstractPDF (308 KB)Minimax LQG Control of Stochastic Partially Observed Uncertain SystemsValery A. Ugrinovskii and Ian R. Petersen26 July 2006 | SIAM Journal on Control and Optimization, Vol. 40, No. 4AbstractPDF (324 KB)Stability and Control of Stochastic Systems with Wide-band Noise Disturbances. IG. Blankenship and G. C. Papanicolaou12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 34, No. 3AbstractPDF (2926 KB)Stationary Probability Distributions for Linear Time-Invariant SystemsJakov Snyders18 July 2006 | SIAM Journal on Control and Optimization, Vol. 15, No. 3AbstractPDF (914 KB)Stability of Linear Systems with Control Dependent NoiseU. G. Haussmann18 July 2006 | SIAM Journal on Control, Vol. 11, No. 2AbstractPDF (1026 KB)Weak Solutions of a Partial Differential Equation of Dynamic ProgrammingRaymond W. Rishel18 July 2006 | SIAM Journal on Control, Vol. 9, No. 4AbstractPDF (809 KB)Optimal Stationary Control with State Control Dependent NoiseU. G. Haussmann18 July 2006 | SIAM Journal on Control, Vol. 9, No. 2AbstractPDF (1050 KB) Volume 7, Issue 3| 1969SIAM Journal on Control History Submitted:01 October 1968Published online:18 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0307028Article page range:pp. 390-397ISSN (print):0036-1402Publisher:Society for Industrial and Applied Mathematics