Controllability for semi-discrete semilinear stochastic parabolic operators

TL;DR

This paper extends controllability results to semi-discrete semilinear stochastic parabolic operators in any dimension, using a new Carleman estimate and fixed-point methods.

Contribution

It introduces a novel Carleman estimate for the adjoint operator and proves $$-null controllability for semi-discrete semilinear stochastic systems with gradient-dependent nonlinearities.

Findings

Established $$-null controllability for linear systems in arbitrary dimensions.

Extended controllability results to semilinear operators with gradient dependence.

Unified previous one-dimensional and multidimensional results under a common framework.

Abstract

In \cite{LPP:2025}, it was shown that, in arbitrary dimension, the spatial semi-discretization of a controlled stochastic parabolic operator is generically not null-controllable. Nevertheless, $\phi$-null controllability results remain attainable. The present paper extends those results to semi-discrete semilinear stochastic operators in arbitrary dimension, whose nonlinearities may also depend on the first-order spatial derivatives. The approach relies on establishing a new Carleman estimate for the adjoint backward stochastic parabolic operator, which yields $\phi$-null controllability for the associated linear system via a duality argument. The semilinear case is handeld by means of a fixed-point argument. As particular cases, our results recover the one-dimensional linear results of \cite{zhao:2024}, the multidimensional linear results of \cite{LPP:2025}, and the semilinear one-dimensional framework of \cite{WZ:2025} in the absence of gradient dependence.