Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations |
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Authors: | Liu Kai |
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Institution: | Department of Mathematics, University of Wales Swansea Singleton Park, Swansea SA2 8PP, k.liu{at}swansea.ac.uk |
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Abstract: | Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|:
It is proved that under certain mild assumptions, the strongsolution Xt(x0)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR+R1 which satisfies the following conditions: (i)c1|x|2k1eµ1t(x,t)c2|x|2+k2+k2eµ2t; (ii) L(x,t)c3(x,t)+k3eµ3t, xV, t0; where L is the infinitesimal generator of the Markov processXt and ci, ki, µi, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µi = 0) are considered. |
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