Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T23:26:34.305Z Has data issue: false hasContentIssue false
  • English
  • Français

On stabilization of partial differential equations by noise

Published online by Cambridge University Press:  22 January 2016

Tomás Caraballo ,
Kai Liu  and
Xuerong Mao
Tomás Caraballo
Affiliation:
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apartado de Correos 1160, 41080-Sevilla, Spain, [email protected]
Kai Liu*
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom, [email protected]
Xuerong Mao
Affiliation:
Department of Statistics and Modelling Science, University of Strathclyde, Glasglow G1 1XH, Scotland, United Kingdom, [email protected]
*
Department of Probability and Statistics, University of Sheffield, The Hicks Building, Sheffield S3 7RH, United Kingdom, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Some results on stabilization of (deterministic and stochastic) partial differential equations are established. In particular, some stability criteria from Chow [4] and Haussmann [6] are improved and subsequently applied to certain situations, on which the original criteria commonly do not work, to ensure almost sure exponential stability. This paper also extends to infinite dimension some results due to Mao [9] on stabilization of differential equations in finite dimension.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 161 , March 2001 , pp. 155 - 170
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

References

[1] Arnold, L., Crauel, H. and Wihstutz, V., Stabilization of linear systems by noise, SIAM J. Control Optim., 21 (1983), 451–461.Google Scholar
[2] Caraballo, T., Existence and uniqueness of solutions for non-linear stochastic partial differential equations, Collect. Math., 42 (1) (1991), 51–74.Google Scholar
[3] Caraballo, T. and Liu, K., On exponential stability criteria of stochastic partial differential equations, Stoch. Proc. Appl., 83 (1999), 289–301.CrossRefGoogle Scholar
[4] Chow, P. L., Stability of nonlinear stochastic evolution equation, J. Math. Anal. Appl., 89 (2) (1982), 400–419.Google Scholar
[5] Has’minskii, R.Z., Stochastic Stability of Differential Equations, Sijthoff and Noord-hoff, Alphen aan den Rijn, The Netherlands, 1980.Google Scholar
[6] Haussmann, U. G., Asymptotic Stability of the Linear Ito Equation in Infinite Dimensions, J. Math. Anal. Appl. (1978), 219–235.Google Scholar
[7] Lions, J.L., Quelque méthodes de résolution des problemes aux limites non lineaires, Dunod Gauthier–Villars, Paris, 1969.Google Scholar
[8] Liu, K., On stability for a class of semilinear stochastic evolution equations, Stoch. Proc. Appl., 70 (1997), 219–241.CrossRefGoogle Scholar
[9] Mao, X. R., Stochastic stabilization and destabilization, Systems and Control Letts., 23 (1) (1994), 279–290.CrossRefGoogle Scholar
[10] Pardoux, E., Equations aux Dérivées Partielles Stochastiques Nonlinéaires Monotones, Thesis, Université Paris Sud, 1975, pp. 279–290.Google Scholar
[11] Scheutzow, M., Stabilization and destabilization by noise in the plane, Stoch. Anal. Appl., 11 (1) (1993), 97–113.CrossRefGoogle Scholar