Null controllability for semi-discrete stochastic semilinear parabolic equations
Yu Wang, Qingmei Zhao
TL;DR
This work studies null controllability for semi-discrete stochastic semilinear parabolic equations, discretizing space with a finite difference scheme and keeping time continuous. It develops a refined global Carleman estimate for backward stochastic semi-discrete parabolic equations with explicit dependence on the discretization parameter $h$ and Carleman parameters, enabling control-theoretic analysis in a non-compact stochastic setting. The linear problem is shown to be $\phi$-null controllable via Carleman estimates and the penalized Hilbert Uniqueness Method, with explicit $h$-dependent bounds on the final state and controls. The nonlinear result follows by a Banach fixed point argument around the linear theory, establishing $\phi$-null controllability for the semi-discrete nonlinear system under assumptions (A1)-(A5) with an exponential decay in $h$ for the final state, and clarifying the discretization-induced behavior of controls.
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 null controllability for semi-discrete stochastic semilinear parabolic equations, where the spatial variable is discretized with finite difference scheme and the time is kept as a continuous variable. 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.