The ϕ-null controllability for semi-discrete stochastic semilinear parabolic equations

¹ School of Mathematics, Southwest Jiaotong University, Chengdu, PR China
² School of Mathematics Sciences, Sichuan Normal University, Chengdu, PR China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 8 March 2025
Accepted: 28 September 2025

Abstract

The global null controllability of stochastic semilinear parabolic equations with globally Lipschitz nonlinearities has been addressed in recent literature. However, there are no results concerning their numerical approximation and the behavior of discrete controls when the discretization parameter goes to zero. This paper is intended to studying the ϕ-null controllability for semi-discrete stochastic semilinear parabolic equations, where the spatial variable is discretized using a finite difference scheme while the temporal variable remains continuous. The ϕ-null controllability means that terminal state can be dominated by a function that tends to zero as the spatial discretization parameter h approaches zero. The proof is based on a new refined semi-discrete Carleman estimate and Banach fixed point method. The main novelty here is that the Carleman parameters and discretization parameter are made explicit and are then used in a Banach fixed point method.