Stability for the stochastic heat equation with multiplicative noise via finite-dimensional feedback

Abstract

In this paper, we study the long-time behavior of a stochastic heat equation with a class of infinite-dimensional multiplicative noises and localized control. We first illustrate through simple examples of the uncontrolled dynamics that mean-square and almost sure exponential stability may hold under different conditions on the parameters, reflecting the interplay between the drift and the multiplicative noise. We then construct a finite-dimensional feedback control acting on a measurable subset of positive measure, built from finitely many Fourier modes of the solution. In particular, we show that the number of controlled modes determines the decay rate and allows for arbitrarily fast stabilization in the mean-square sense. As a consequence, almost sure exponential stability is recovered via a probabilistic argument, so that both notions of stability are achieved within the same framework and with the same guaranteed decay rate. We also establish null-controllability at any positive time through an iterative construction of adapted controls from finite-dimensional feedback laws. The construction relies on the quantitative stabilization estimates and requires no observability inequality for the adjoint equation.

Improved version. New assumptions on the noise term