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

## TL;DR

This paper develops explicit boundary feedback controllers for stabilizing non-autonomous cubic stochastic heat equations using eigenfunction-based feedback, transforming the problem into a deterministic form for analysis.

## Contribution

It introduces a simple, explicit linear boundary feedback stabilizer for a class of stochastic heat equations with cubic non-linearity, supported on boundary subsets.

## Key findings

- Successfully stabilizes unbounded trajectories.
- Transforms stochastic equations into deterministic forms for analysis.
- Provides explicit feedback control formulas.

## Abstract

Here we design boundary feedback stabilizers 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. Then, the simple form of the feedback, we propose here, allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument the existence \& stabilization result is proved.