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

Abstract

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 surely It is proved that under certain mild assumptions, the strong solution Xt(x0)∈V↪H↪V*, t ⩾ 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·):H×R+→R1 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 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.