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NECESSARY AND SUFFICIENT CONDITIONS FOR EXPONENTIAL STABILITY AND ULTIMATE BOUNDEDNESS OF SYSTEMS GOVERNED BY STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  30 October 2000

KAI LIU
KAI LIU
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP; k.liu@swansea.ac.uk
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Abstract

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

formula here

It is proved that under certain mild assumptions, the strong solution Xt(x0)∈V[rarrhk ]H[rarrhk ]V*, t [ges ] 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·)[ratio ]H×R+→R1 which satisfies the following conditions:

formula here

formula here

where [Lscr ] 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.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 311 - 320
Copyright
The London Mathematical Society 2000

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