Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity

TL;DR

This paper establishes null controllability and solves an inverse source problem for a stochastic Grushin equation with boundary degeneracy and singularity, using Carleman estimates and weighted identities.

Contribution

It introduces new Carleman estimates tailored for stochastic Grushin equations with boundary degeneracy and singularity, enabling controllability and inverse source problem solutions.

Findings

Proved null controllability for any time T and degeneracy level γ>0.

Established uniqueness for the inverse source problem.

Developed novel Carleman estimates for stochastic Grushin equations.

Abstract

In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any $T$ and any degeneracy $\gamma>0$, when our control domain touches the degeneracy line $\{x=0\}$. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem.