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September  2020, 9(3): 795-816. doi: 10.3934/eect.2020034

## Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise

 Alexandru Ioan Cuza University, Department of Mathematics and Octav Mayer Institute of Mathematics (Romanian Academy), 700506 Iaşi, România

Received  August 2019 Revised  September 2019 Published  September 2020 Early access  December 2019

Fund Project: This work was supported by a grant of the "Alexandru Ioan Cuza" University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2018-03

Here we study the problem of boundary feedback stabilization to unbounded trajectories for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. The simple-form feedback allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument its stability is proved. The approach requires the initial data to be a random variable implying the fact that the solution of the random equation is not adapted. Thus, one cannot recover the solution of the initial stochastic equation from the random one. Hence, the designed feedback controller stabilizes the associated random equation and not the original stochastic equation. Anyway, it stabilizes its random version.

Citation: Ionuţ Munteanu. Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise. Evolution Equations & Control Theory, 2020, 9 (3) : 795-816. doi: 10.3934/eect.2020034
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