Skip Navigation

Journal of the London Mathematical Society 2000 62(1):311-320; doi:10.1112/S0024610700001162
© 2000 by London Mathematical Society
This Article
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Liu, K.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The London Mathematical Society

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Kai Liu

Department of Mathematics, University of Wales Swansea Singleton Park, Swansea SA2 8PP, k.liu{at}swansea.ac.uk

Received 4 March 1999. Revision received 29 April 1999.

Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm |·|:


Formula

It is proved that under certain mild assumptions, the strong solution Xt(x0)isinV{rightarrowhook}H{rightarrowhook}V*, t ≥ 0, is mean square exponentially stable if and only if there exists a Lyapunov functional {Lambda}(·, ·):HxR+->R1 which satisfies the following conditions:

(i)c1|x|2–k1e–µ1t≤{Lambda}(x,t)≤c2|x|2+k2+k2e–µ2t;

(ii) L{Lambda}(x,t)≤–c3{Lambda}(x,t)+k3e–µ3t, {forall} xisinV, t≥0;

where L is the infinitesimal generator of the Markov process Xt and ci, ki, µi, i = 1, 2, 3, are positive constants. As a by-product, the characterization of exponential ultimate boundedness of the strong solution is established as the null decay rates (that is, µi = 0) are considered.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.