Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain

Abstract

A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future, and the minimum among all such compact limit-sets is called a forward controller. The existence of a forward controller closely relates to the forward compactness of the attractor, which is further argued by the forward-pullback asymptotic compactness of the system. The abstract results are applied to the non-autonomous stochastic sine-Gordon equation on an unbounded domain. The existence of a forward compact attractor is proved, which leads to the existence of a forward controller. The measurability of the attractor is proved by considering two different universes.

Regular dynamics of stochastic nondissipative retarded Kuramoto–Sivashinsky equations

“…Hence, a natural idea is to consider the time-dependent property of nonautonomous attractors. Recently, this subject for stochastic delay-free PDEs has been studied, see, for example, [6,9,20,24,25]. Caraballo et al [6] discussed the backward asymptotic autonomy of pullback random attractors (PRAs)  = {(𝜏, 𝜔) ∶ 𝜏 ∈ ℝ, 𝜔 ∈ Ω}: lim 𝜏→−∞ 𝑑𝑖𝑠𝑡 𝑋 ((𝜏, 𝜔),  ∞ (𝜔)) = 0, ∀ 𝜔 ∈ Ω, where (𝑋, ‖ ⋅ ‖ 𝑋 ) is a Banach space, 𝑑𝑖𝑠𝑡 𝑋 (⋅, ⋅) denotes the Hausdorff semidistance, and  ∞ = { ∞ (𝜔) ∶ 𝜔 ∈ Ω} is a random attractor associated with .…”

“…where 𝐴(𝜔) is a nonempty smallest compact set such that (2) holds. In addition, 𝐴(⋅) is named that forward controller of  in [25]. Yang and Li [24] established the forward asymptotic autonomy of PRAs: lim 𝜏→+∞ 𝑑𝑖𝑠𝑡 𝑋 ((𝜏, 𝜔),  ∞ (𝜔)) = 0, ∀ 𝜔 ∈ Ω.…”

“…Hence, a natural idea is to consider the time‐dependent property of nonautonomous attractors. Recently, this subject for stochastic delay‐free PDEs has been studied, see, for example, [6, 9, 20, 24, 25]. Caraballo et al.…”

“…In [9], the authors studied the finite time stability of PRAs: $$$$\begin{align*} \lim _{\tau \rightarrow \tau _{0}}dist_{X}(\mathcal {A}(\tau , \omega ), \mathcal {A}(\tau _{0}, \omega ))=0, \ \ \forall \ \omega \in \Omega . \end{align*}$$$$In [20, 25], the authors investigated the long‐time stability of PRAs: $$$$\begin{align} \lim _{\tau \rightarrow -(+)\infty }dist_{X}(\mathcal {A}(\tau , \omega ), A(\omega ))=0, \ \ \forall \ \omega \in \Omega , \end{align}$$$$where $$A(\omega )$$ is a nonempty smallest compact set such that (2) holds. In addition, $$A(\cdot )$$ is named that forward controller of $$\mathcal {A}$$ in [25].…”

Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations

“…In the deterministic case, the asymptotic autonomy had been studied by Kloeden et al [16,17,18], where the criterion of the uniform compactness of the pullback attractor was reduced in [22] and [10], see also [26,30] for the random case.…”