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Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations
Author(s) -
Liu Kai
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001162
Subject(s) - mathematics , exponential stability , hilbert space , exponential function , exponential decay , combinatorics , product (mathematics) , mathematical analysis , pure mathematics , mathematical physics , physics , quantum mechanics , geometry , nonlinear system
Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm ∣·∣:X t = x 0 + ∫ 0 t f ( X s , s ) d s + ∫ 0 t g ( X s , s ) d W s , t ⩾ 0 , P almost surelyIt is proved that under certain mild assumptions, the strong solution X t ( x 0 )∈ V ↪ H ↪ V *, t ⩾ 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·): H × R + → R 1 which satisfies the following conditions: (i)c 1 | x | 2 − k 1 e − μ 1 t ⩽ Λ ( x , t ) ⩽ c 2 | x | 2 + k 2 e − μ 2 t ;(ii) ℒ Λ ( x , t ) ⩽ − c 3 Λ ( x , t ) + k 3 e − μ 3 t , ∀ x ∈ V , t ⩾ 0 ;where L is the infinitesimal generator of the Markov process X t and c i , k i , μ i , 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, μ i = 0) are considered.