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

Ungureanu, Viorica Mariela

Dyn. Syst. 23, No. 3, 333-350 (2008).

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)].

Reviewer: Jozef Myjak (L’Aquila)

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)

Keywords:

stochastic differential equations; stochastic uniform observability; covariance operators; Hilbert Schmidt spaces; evolution operators; nuclear operators; mild solutions

Citations:

Zbl 0313.93067; Zbl 0810.93069

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