Abstract
The main object of this paper is to discuss the problem of the uniform exponential stability and uniform observability of time-varying linear stochastic equations in Hilbert spaces. We give a representation of the covariance operator associated to the mild solutions of these equations which allow us to obtain a characterization of the uniform exponential stability of uniformly observable systems in terms of Lyapunov equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. Arveson, An Invitation to C*-Algebra. Springer Verlag, New York, 1976.
I. Gelfand, H. Vilenkin, Funcţii generalizate. Aplicaţii ale analizei armonice. Editura Ştiinţifică şi Enciclopedică, (romanian translation) Bucureşti, 1985.
A. Germani, L. Jetto, M. Piccioni, Galerkin Approximations for Optimal Linear Filtering of Infinite-Dimensional Linear Systems. SIAM J. Control and Optim. 26 (1988), 1287–2305.
I. Gohberg, S. Goldberg, Basic Operator Theory. Birkhäuser, 1981
W. Grecksch, C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach Math. Res. vol. 75, Acad. Verlag, 1995.
C.S. Kubrusly, Mean Square Stability for Discrete Bounded Linear Systems in Hilbert Space, SIAM J. Control and Optimization 23 (1985), 19–29.
T. Morozan, Stochastic Uniform Observability and Riccati Equations of Stochastic Control, Rev. Roumaine Math. Pures Appl. 38 (1993), 771–481.
T. Morozan, On the Riccati Equation of Stochastic Control, International Series on Numerical Mathematics, vol. 107, Birkhäuser Verlag Basel, 1992, pp. 231–236.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44, Springer-Verlag, Berlin-New York, 1983.
G. Da Prato, A. Ichikawa, Quadratic Control for Linear Time-Varying Systems. SIAM. J. Control and Optimization 28 (1990), 359–381.
G. Da Prato, A. Ichikawa, Lyapunov Equations for Time-Varying Linear Systems. Systems and Control Letters 9 (1987), 165–172.
G. Da Prato, J. Zabczyc, Stochastic Equations in Infinite Dimensions. University Press Cambridge, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Ungureanu, V.M. (2004). Uniform Exponential Stability and Uniform Observability of Time-Varying Linear Stochastic Systems in Hilbert Spaces. In: Gaşpar, D., Timotin, D., Zsidó, L., Gohberg, I., Vasilescu, FH. (eds) Recent Advances in Operator Theory, Operator Algebras, and their Applications. Operator Theory: Advances and Applications, vol 153. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7314-8_19
Download citation
DOI: https://doi.org/10.1007/3-7643-7314-8_19
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7127-2
Online ISBN: 978-3-7643-7314-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)