## Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain.(English)Zbl 1456.35043

Summary: 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.

### MSC:

 35B41 Attractors 35L71 Second-order semilinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35R60 PDEs with randomness, stochastic partial differential equations 37L55 Infinite-dimensional random dynamical systems; stochastic equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 93B05 Controllability

### Keywords:

forward compactness; forward controller; pullback attractor
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### References:

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