Null controllability for stochastic fourth order semi-discrete parabolic equations

TL;DR

This paper investigates the null controllability of stochastic fourth order semi-discrete parabolic equations, establishing new Carleman estimates and observability inequalities that enable control results for discretized stochastic PDEs.

Contribution

It introduces a novel global Carleman estimate for stochastic semi-discrete fourth order parabolic operators, leading to new controllability results.

Findings

Established a new Carleman estimate for backward stochastic semi-discrete operators.

Derived an explicit observability constant depending on discretization parameters.

Proved $oldsymbol{ extphi}$-null controllability using duality techniques.

Abstract

This paper is devoted to studying null controllability for a class of stochastic fourth order semi-discrete parabolic equations, where the spatial variable is discretized with finite difference scheme and the time is kept as a continuous variable. For this purpose, we establish a new global Carleman estimate for a backward stochastic fourth order semi-discrete parabolic operators, in which the large parameter is connected to the mesh size. A relaxed observability estimate is established for backward stochastic fourth order semi-discrete parabolic equations by this new Carleman estimate, with an explicit observability constant that depends on the discretization parameter and coefficients of lower order terms. Then, the $\phi$-null controllability of the stochastic fourth order semi-discrete parabolic equations is proved using the standard duality technique.