×

Stochastic uniform observability of general linear differential equations. (English) Zbl 1155.93017

Author’s resume: The aim of this article is to discuss the uniform observability property of general linear differential equations with multiplicative white noise in Hilbert spaces. Based on perturbation theory for evolution operators on Hilbert Schmidt spaces and the space of nuclear operators, new representation of the covariance operator associated to the mild solutions of the investigated stochastic differential equations are given. Using this result we obtain deterministic characterization of the stochastic uniform observability property. We also identify an entire class of stochastic differential equations which are never stochastic uniformly observable. Some examples will illustrate the theory. The author developed here the ideas by J. Zabczyk [SIAM J. Control 13, 1217–1234 (1975; Zbl 0313.93067)] and by T. Morozan [Rev. Roum. Math. Pures Appl. 38, No. 9, 771–781 (1993; Zbl 0810.93069)].

MSC:

93B07 Observability
34F05 Ordinary differential equations and systems with randomness
34G20 Nonlinear differential equations in abstract spaces
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Kalman RE, Bull. Soc. Math. Mexicana 5 pp 102– (1960)
[2] DOI: 10.1137/0313076 · Zbl 0313.93067
[3] Morozan T, Rev. Roumaine Math. Pures Appl. 38 pp 771– (1993)
[4] DOI: 10.1007/978-3-0348-8625-3_21
[5] Morozan T, Rev. Roumaine Math. Pures Appl. 46 pp 783– (2001)
[6] DOI: 10.1007/978-0-387-35690-7_42
[7] DOI: 10.1093/imamci/21.3.323 · Zbl 1060.93019
[8] Ungureanu VM, Studia Universitatis Babes-Bolyai Mathematica (4) pp 73– (2005)
[9] DOI: 10.1137/0328019 · Zbl 0692.49006
[10] DOI: 10.1007/BF01443614 · Zbl 0647.93057
[11] Gelfand I, Generalized Functions, Part 4 (1964)
[12] Grecksch W, A Hilbert Space Approach Mathematical Research 75 (1995)
[13] DOI: 10.1017/CBO9780511666223
[14] DOI: 10.1137/0326072 · Zbl 0662.93073
[15] DOI: 10.1007/978-1-4612-5561-1 · Zbl 0516.47023
[16] DOI: 10.1007/BFb0006761
[17] DOI: 10.14232/ejqtde.2004.1.4 · Zbl 1072.60047
[18] DOI: 10.1016/0167-6911(87)90023-5 · Zbl 0678.93051
[19] DOI: 10.1137/0328014 · Zbl 0693.93086
[20] Prato GDa, J. Math. Pures Appl. 52 pp 353– (1973)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.